RISK MANAGEMENT IN BANKING
RISK MANAGEMENT IN BANKING ¨ Bessis Joel
Copyright 2002 by
John Wiley & Sons Ltd, Baf...

Author:
Joël Bessis

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RISK MANAGEMENT IN BANKING

RISK MANAGEMENT IN BANKING ¨ Bessis Joel

Copyright 2002 by

John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex, PO19 1UD, England National 01243 779777 International (+44) 1243 779777 e-mail (for orders and customer service enquiries): [email protected] Visit our Home Page on http://www.wileyeurope.com or http://www.wiley.com

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London, UK W1P 9HE, without the permission in writing of the publisher. Jo¨el Bessis has asserted his right under the Copyright, Designs and Patents Act 1988, to be identified as the author of this work. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA WILEY-VCH Verlag GmbH, Pappelallee 3, D-69469 Weinheim, Germany John Wiley & Sons (Australia) Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W 1L1, Canada Library of Congress Cataloguing-in-Publication Data Bessis, Jo¨el. [Gestion des risques et gestion actif-passif des banques. English] Risk management in banking/Jo¨el Bessis.—2nd ed. p. cm. Includes bibliographical references and index. ISBN 0-471-49977-3 (cloth)—ISBN 0-471-89336-6 (pbk.) 1. Bank management. 2. Risk management. 3. Asset-liability management. I. Title. HG1615 .B45713 2001 332.1′ 068′ 1—dc21 2001045562 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-471-49977-3 (cloth) ISBN 0-471-89336-6 (paper) Typeset in 10/12pt Times Roman by Laserwords Private Limited, Chennai, India. Printed and bound in Great Britain by TJ International Ltd, Padstow, England. This book is printed on acid-free paper responsibly manufactured from sustainable forestation, for which at least two trees are planted for each one used for paper production.

Contents

Introduction

ix

SECTION 1 Banking Risks

1

1 2

Banking Business Lines Banking Risks

SECTION 2 Risk Regulations 3

Banking Regulations

SECTION 3 Risk Management Processes 4 5

Risk Management Processes Risk Management Organization

SECTION 4 Risk Models 6 7 8 9

Risk Measures VaR and Capital Valuation Risk Model Building Blocks

3 11 23 25 51 53 67 75 77 87 98 113

SECTION 5 Asset–Liability Management

129

10 11 12 13 14

131 136 151 164 180

ALM Overview Liquidity Gaps The Term Structure of Interest Rates Interest Rate Gaps Hedging and Derivatives

vi

CONTENTS

SECTION 6 Asset–Liability Management Models

191

15 16 17 18 19

193 201 210 224 233

Overview of ALM Models Hedging Issues ALM Simulations ALM and Business Risk ALM ‘Risk and Return’ Reporting and Policy

SECTION 7 Options and Convexity Risk in Banking

245

20 Implicit Options Risk 21 The Value of Implicit Options

247 254

SECTION 8 Mark-to-Market Management in Banking

269

22 23 24 25

271 280 289 300

Market Value and NPV of the Balance Sheet NPV and Interest Rate Risk NPV and Convexity Risks NPV Distribution and VaR

SECTION 9 Funds Transfer Pricing

309

26 FTP Systems 27 Economic Transfer Prices

311 325

SECTION 10 Portfolio Analysis: Correlations

337

28 Correlations and Portfolio Effects

339

SECTION 11 Market Risk

357

29 30 31 32

359 363 384 396

Market Risk Building Blocks Standalone Market Risk Modelling Correlations and Multi-factor Models for Market Risk Portfolio Market Risk

SECTION 12 Credit Risk Models

417

33 Overview of Credit Risk Models

419

SECTION 13 Credit Risk: ‘Standalone Risk’

433

34 35 36 37 38

435 443 451 459 479

Credit Risk Drivers Rating Systems Credit Risk: Historical Data Statistical and Econometric Models of Credit Risk The Option Approach to Defaults and Migrations

CONTENTS

39 40 41 42 43

Credit Risk Exposure From Guarantees to Structures Modelling Recoveries Credit Risk Valuation and Credit Spreads Standalone Credit Risk Distributions

vii

495 508 521 538 554

SECTION 14 Credit Risk: ‘Portfolio Risk’

563

44 45 46 47 48 49 50

565 580 586 595 608 622 627

Modelling Credit Risk Correlations Generating Loss Distributions: Overview Portfolio Loss Distributions: Example Analytical Loss Distributions Loss Distributions: Monte Carlo Simulations Loss Distribution and Transition Matrices Capital and Credit Risk VaR

SECTION 15 Capital Allocation

637

51 52

639 655

Capital Allocation and Risk Contributions Marginal Risk Contributions

SECTION 16 Risk-adjusted Performance

667

53 54

669 679

Risk-adjusted Performance Risk-adjusted Performance Implementation

SECTION 17 Portfolio and Capital Management (Credit Risk)

689

55 56 57 58 59 60

691 701 714 721 733 744

Portfolio Reporting (1) Portfolio Reporting (2) Portfolio Applications Credit Derivatives: Definitions Applications of Credit Derivatives Securitization and Capital Management

Bibliography

762

Index

781

Introduction

Risk management in banking designates the entire set of risk management processes and models allowing banks to implement risk-based policies and practices. They cover all techniques and management tools required for measuring, monitoring and controlling risks. The spectrum of models and processes extends to all risks: credit risk, market risk, interest rate risk, liquidity risk and operational risk, to mention only major areas. Broadly speaking, risk designates any uncertainty that might trigger losses. Risk-based policies and practices have a common goal: enhancing the risk–return profile of the bank portfolio. The innovation in this area is the gradual extension of new quantified risk measures to all categories of risks, providing new views on risks, in addition to qualitative indicators of risks. Current risks are tomorrow’s potential losses. Still, they are not as visible as tangible revenues and costs are. Risk measurement is a conceptual and a practical challenge, which probably explains why risk management suffered from a lack of credible measures. The recent period has seen the emergence of a number of models and of ‘risk management tools’ for quantifying and monitoring risks. Such tools enhance considerably the views on risks and provide the ability to control them. This book essentially presents the risk management ‘toolbox’, focusing on the underlying concepts and models, plus their practical implementation. The move towards risk-based practices accelerated in recent years and now extends to the entire banking industry. The basic underlying reasons are: banks have major incentives to move rapidly in that direction; regulations developed guidelines for risk measurement and for defining risk-based capital (equity); the risk management ‘toolbox’ of models enriched considerably, for all types of risks, providing tools making risk measures instrumental and their integration into bank processes feasible.

THE RATIONALE FOR RISK-BASED PRACTICES Why are visibility and sensitivity to risks so important for bank management? Certainly because banks are ‘risk machines’: they take risks, they transform them, and they embed

x

INTRODUCTION

them in banking products and services. Risk-based practices designate those practices using quantified risk measures. Their scope evidently extends to risk-taking decisions, under an ‘ex ante’ perspective, and risk monitoring, under an ‘ex post’ perspective, once risk decisions are made. There are powerful motives to implement risk-based practices: to provide a balanced view of risk and return from a management point of view; to develop competitive advantages, to comply with increasingly stringent regulations. A representative example of ‘new’ risk-based practices is the implementation of riskadjusted performance measures. In the financial universe, risk and return are two sides of the same coin. It is easy to lend and to obtain attractive revenues from risky borrowers. The price to pay is a risk that is higher than the prudent bank’s risk. The prudent bank limits risks and, therefore, both future losses and expected revenues, by restricting business volume and screening out risky borrowers. The prudent bank avoids losses but it might suffer from lower market share and lower revenues. However, after a while, the risk-taker might find out that higher losses materialize, and obtain an ex post performance lower than the prudent lender performance. Who performs best? Unless assigning some measure of risk to income, it is impossible to compare policies driven by different risk appetites. Comparing performances without risk adjustment is like comparing apples and oranges. The rationale of risk adjustment is in making comparable different performances attached to different risk levels, and in general making comparable the risk–return profiles of transactions and portfolios. Under a competitive perspective, screening borrowers and differentiating the prices accordingly, given the borrowers’ standing and their contributions to the bank’s portfolio risk–return profile, are key issues. Not doing so results in adverse economics for banks. Banks who do not differentiate risks lend to borrowers rejected by banks who better screen and differentiate risks. By overpricing good risks, they discourage good borrowers. By underpricing risks to risky customers, they attract them. By discouraging the relatively good ones and attracting the relatively bad ones, the less advanced banks face the risk of becoming riskier and poorer than banks adopting sound risk-based practices at an earlier stage. Those banking institutions that actively manage their risks have a competitive advantage. They take risks more consciously, they anticipate adverse changes, they protect themselves from unexpected events and they gain the expertise to price risks. The competitors who lack such abilities may gain business in the short-term. Nevertheless, they will lose ground with time, when those risks materialize into losses. Under a management perspective, without a balanced view of expected return and risk, banks have a ‘myopic’ view of the consequences of their business policies in terms of future losses, because it is easier to measure income than to capture the underlying risks. Even though risks remain a critical factor to all banks, they suffer from the limitations of traditional risk indicators. The underlying major issue is to assign a value to risks in order to make them commensurable with income and fully address the risk–return trade-off. Regulation guidelines and requirements have become more stringent on the development of risk measures. This single motive suffices for developing quantified risk-based practices. However, it is not the only incentive for structuring the risk management tools and processes. The above motivations inspired some banks who became pioneers in this field many years before the regulations set up guidelines that led the entire industry towards more ‘risk-sensitive’ practices. Both motivations and regulations make risk measurement a core building block of valuable risk-based practices. However, both face the same highly challenging risk measuring issue.

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xi

RISK QUANTIFICATION IS A MAJOR CHALLENGE Since risks are so important in banking, it is surprising that risk quantification remained limited until recently. Quantitative finance addresses extensively risk in the capital markets. However, the extension to the various risks of financial institutions remained a challenge for multiple reasons. Risks are less tangible and visible than income. Academic models provided foundations for risk modelling, but did not provide instrumental tools helping decision-makers. Indeed, a large fraction of this book addresses the gap between conceptual models and banking risk management issues. Moreover, the regulators’ focus on risks is still relatively recent. It dates from the early stages of the reregulation phase, when the Cooke ratio imposed a charge in terms of capital for any credit risk exposure. Risk-based practices suffered from real challenges: simple solutions do not help; risk measures require models; models not instrumental; quantitative finance aimed at financial markets more than at financial institutions. For such reasons, the prerequisites for making instrumental risk quantifications remained out of reach.

Visibility on Losses is Not Visibility on Risks Risks remain intangible and invisible until they materialize into losses. Simple solutions do not really help to capture risks. For instance, a credit risk exposure from a loan is not the risk. The risk depends on the likelihood of losses and the magnitude of recoveries in addition to the size of the amount at risk. Observing and recording losses and their frequencies could help. Unfortunately, loss histories are insufficient. It is not simple to link observable losses and earning declines with specific sources of risks. Tracking credit losses does not tell whether they result from inadequate limits, underestimating credit risk, inadequate guarantees, or excessive risk concentration. Recording the fluctuations of the interest income is easy, but tracing back such changes to interest rates is less obvious. Without links to instrumental risk controls, earning and loss histories are of limited interest because they do not help in taking forward looking corrective actions. Visibility on losses is not visibility on risks.

Tracking Risks for Management Purposes Requires Models Tracking risks for management purposes requires models for better capturing risks and relating them to instrumental controls. Intuitively, the only way to quantify invisible risks is to model them. Moreover, multiple risk indicators are not substitutes for quantified measures. Surveillance of risk typically includes such various items as exposure size, watch lists for credit risk, or value changes triggered by market movements for market instruments. These indicators capture the multiple dimensions of risk, but they do not add them up into a quantified measure. Finally, missing links between future losses from current risks and risk drivers, which are instrumental for controlling risk, make it unfeasible to timely monitor risks. The contribution of models addresses such issues. They provide quantified measures of risk or, in other words, they value the risk of banks. Moreover, they do so in a way that allows tracing back risks to management controls over risk exposures of financial institutions. Without such links, risk measures would ‘float in the air’, without providing management tools.

xii

INTRODUCTION

Financial Markets versus Financial Institutions The abundance of models in quantitative finance did not address the issues that financial institutions face until recently, except in certain specific areas such as asset portfolio management. They undermined the foundations of risk management, without bridging the gap between models and the needs of financial institutions. Quantitative finance became a huge field that took off long ago, with plenty of pioneering contributions, many of them making their authors Nobel Prize winners. In the market place, quantification is ‘natural’ because of the continuous observation of prices and market parameters (interest rates, equity indexes, etc.). For interest rate risk, modelling the term structure of interest rates is a classical field in market finance. The pioneering work of Sharpe linked stock prices to equity risk in the stock market. The Black–Scholes option model is the foundation for pricing derivative instruments, options and futures, which today are standard instruments for managing risks. The scientific literature also addressed credit risk a long time ago. The major contribution of Robert Merton on modelling default within an option framework, a pillar of current credit risk modelling, dates from 1974. These contributions fostered major innovations, from pricing market instruments and derivatives (options) that serve for investing and hedging risks, to defining benchmarks and guidelines for the portfolios management of market instruments (stocks and bonds). They also helped financial institutions to develop their business through ever-changing product innovations. Innovation made it feasible to customize products for matching investors’ needs with specific risk–return bundles. It also allowed both financial and corporate entities to hedge their risks with derivatives. The need for investors to take exposures and, for those taking exposures, to hedge them provided business for both risk-takers and risk-hedgers. However, these developments fell short of directly addressing the basic prerequisites of a risk management system in financial institutions.

Prerequisites for Risk Management in Financial Institutions The basic prerequisites for deploying risk management in banks are: • Risks measuring and valuation. • Tracing risks back to risk drivers under the management control. Jumping to market instruments for managing risks without prior knowledge of exposures to the various risks is evidently meaningless unless we know the magnitude of the risks to keep under control, and what they actually mean in terms of potential value lost. The risk valuation issue is not simple. It is much easier to address in the market universe. However, interest rate risk requires other management models and tools. All banking business lines generate exposures to interest rate risks. However, linking interest income and rates requires modelling the balance sheet behaviour. Since the balance sheet generates both interest revenues and interest costs, they offset each other to a certain extent, depending on matches and mismatches between sizes of assets and liabilities and interest rate references. Capturing the extent of offsetting effects between assets and liabilities also requires dedicated models.

INTRODUCTION

xiii

Credit risk remained a challenge until recently, even though it is the oldest of all banking risks. Bank practices rely on traditional indicators, such as credit risk exposures measured by outstanding balances of loans at risk with borrowers, or amounts at risk, and internal ratings measuring the ‘quality’ of risk. Banking institutions have always monitored credit risk actively, through a number of systems such as limits, delegations, internal ratings and watch lists. Ratings agencies monitor credit risk of public debt issues. However, credit risk assessment remained judgmental, a characteristic of the ‘credit culture’, focusing on ‘fundamentals’: all qualitative variables that drive the credit worthiness of a borrower. The ‘fundamental’ view on credit risk still prevails, and it will obviously remain relevant. Credit risk benefits from diversification effects that limit the size of credit losses of a portfolio. Credit risk focus is more on transactions. When moving to the global portfolio view, we know that a large fraction of the risk of individual transactions is diversified away. A very simple question is: By how much? This question remained unanswered until portfolio models, specifically designed for that purpose, emerged in the nineties. It is easy to understand why. Credit risk is largely invisible. The simultaneous default of two large corporate firms, for whom the likelihood of default is small, is probably an unobservable event. Still, this is the issue underlying credit risk diversification. Because of the scarcity of data available, the diversification issue for credit risk remained beyond reach until new modelling techniques appeared. Portfolio models, which appeared only in the nineties, turned around the difficulty by modelling the likelihood of modelled defaults, rather than actual defaults. This is where modelling risks contributes. It pushes further away the frontier between measurable risks and invisible–intangible risks and, moreover, it links risks to the sources of uncertainty that generate them.

PHASES OF DEVELOPMENT OF THE REGULATORY GUIDELINES Banks have plenty of motives for developing risk-based practices and risk models. In addition, regulators made this development a major priority for the banking industry, because they focus on ‘systemic risk’, the risk of the entire banking industry made up of financial institutions whose fates are intertwined by the density of relationships within the financial system. The risk environment has changed drastically. Banking failures have been numerous in the past. In recent periods their number has tended to decrease in most, although not all, of the Organization for Economic Coordination and Development (OECD) countries, but they became spectacular. Banking failures make risks material and convey the impression that the banking industry is never far away from major problems. Mutual lending–borrowing and trading create strong interdependencies between banks. An individual failure of a large bank might trigger the ‘contagion’ effect, through which other banks suffer unsustainable losses and eventually fail. From an industry perspective, ‘systemic risk’, the risk of a collapse of the entire industry because of dense mutual relations, is always in the background. Regulators have been very active in promoting pre-emptive policies for avoiding individual bank failures and for helping the industry absorb the shock of failures when they happen. To achieve these results, regulators have totally renovated the regulatory framework. They promoted and enforced new guidelines for measuring and controlling the risks of individual players.

xiv

INTRODUCTION

Originally, regulations were traditional conservative rules, requiring ‘prudence’ from each player. The regulatory scheme was passive and tended to differentiate prudent rules for each major banking business line. Differentiated regulations segmented the market and limited competition because some players could do what others could not. Obvious examples of segmentation of the banking industry were commercial versus investment banking, or commercial banks versus savings institutions. Innovation made rules obsolete, because players found ways to bypass them and to compete directly with other segments of the banking industry. Obsolete barriers between the business lines of banks, plus failures, triggered a gradual deregulation wave, allowing players to move from their original business field to the entire spectrum of business lines of the financial industry. The corollary of deregulation is an increased competition between unequally experienced players, and the implication is increased risks. Failures followed, making the need for reregulation obvious. Reregulation gave birth to the current regulatory scheme, still evolving with new guidelines, the latest being the New Basel Accord of January 2001. Under the new regulatory scheme, initiated with the Cooke ratio in 1988, ‘risk-based capital’ or, equivalently, ‘capital adequacy’ is a central concept. The philosophy of ‘capital adequacy’ is that capital should be capable of sustaining the future losses arising from current risks. Such a sound and simple principle is hardly debatable. The philosophy provides an elegant and simple solution to the difficult issue of setting up a ‘pre-emptive’, ‘ex ante’ regulatory policy. By contrast, older regulatory policies focused more on corrective actions, or ‘after-the-fact’ actions, once banks failed. Such corrective actions remain necessary. They were prompt when spectacular failures took place in the financial industry (LTCM, Baring Brothers). Nevertheless, avoiding ‘contagion’ when bank failures occur is not a substitute for pre-emptive actions aimed at avoiding them. The practicality of doing so remains subject to adequate modelling. The trend towards more internal and external assessment on risks and returns emerged and took momentum in several areas. Through successive accords, regulators promoted the building up of information on all inputs necessary for risk quantification. Accounting standards evolved as well. The ‘fair value’ concept gained ground, raising hot debates on what is the ‘right’ value of bank assets and how to accrue earnings in traditional commercial banking activities. It implies that a loan providing a return not in line with its risk and cost of funding should appear at lower than face value. The last New Basel Accord promotes the ‘three pillars’ foundation of supervision: new capital requirements for credit risk and operational risks; supervisory processes; disclosure of risk information by banks. Together, the three pillars allow external supervisors to audit the quality of the information, a basic condition for assessing the quality and reliability of risk measures in order to gain more autonomy in the assessment of capital requirements. Regulatory requirements for market, credit and operational risk, plus the closer supervision of interest rate risk, pave the way for a comprehensive modelling of banking risks, and a tight integration with risk management processes, leading to bank-wide risk management across all business lines and all major risks.

FROM RISK MODELS TO RISK MANAGEMENT Risk models have two major contributions: measuring risks and relating these measures to management controls over risks. Banking risk models address both issues by embedding the specifics of each major risk. As a direct consequence, there is a wide spectrum of

INTRODUCTION

xv

modelling building blocks, differing across and within risks. They share the risk-based capital and the ‘Value at Risk’ (VaR) concepts that are the basic foundations of the new views on risk modelling, risk controlling and risk regulations. Risk management requires an entire set of models and tools for linking risk management (business) issues with financial views on risks and profitability. Together, they make up the risk management toolbox, which provides the necessary inputs that feed and enrich the risk process, to finally close the gap between models and management processes.

Risk Models and Risks Managing the banking exposure to interest rate risk and trading interest rate risk are different businesses. Both commercial activities and trading activities use up liquidity that financial institutions need to fund in the market. Risk management, in this case, relates to the structural posture that banks take because of asset and liability mismatches of volumes, maturity and interest rate references. Asset–Liability Management (ALM) is in charge of managing this exposure. ALM models developed gradually until they became standard references for managing the liquidity and interest rate risk of the banking portfolio. For market risk, there is a large overlap between modelling market prices and measuring market risk exposures of financial institutions. This overlap covers most of the needs, except one: modelling the potential losses from trading activities. Market risk models appeared soon after the Basel guidelines started to address the issues of market risk. They appeared sufficiently reliable to allow internal usage by banks, under supervision of regulators, for defining their capital requirements. For credit risk, the foundations exist for deploying instrumental tools fitting banks’ requirements and, potentially, regulators’ requirements. Scarce information on credit events remains a major obstacle. Nevertheless, the need for quantification increased over time, necessitating measuring the size of risk, the likelihood of losses, the magnitude of losses under default and the magnitude of diversification within banks’ portfolios. Modelling the qualitative assessment of risk based on the fundamentals of borrowers has a long track record of statistical research, which rebounds today because of the regulators’ emphasis on extending the credit risk data. Since the early nineties, portfolio models proposed measures of credit risk diversification within portfolios, offering new paths for quantifying risks and defining the capital capable of sustaining the various levels of portfolio losses. Whether the banks should go along this path, however, is no longer a question since the New Basel Accord of January 2001 set up guidelines for credit risk-sensitive measures, therefore preparing the foundations for the full-blown modelling of the credit risk of banks’ portfolios. Other major risks appeared when progressing in the knowledge of risks. Operational risk became a major priority, since January 2001, when the regulatory authorities formally announced the need to charge bank capital against this risk.

Capital and VaR It has become impossible to discuss risk models without referring to economic capital and VaR. The ‘capital adequacy’ principle states that the bank’s capital should match risks. Since capital is the most scarce and costly resource, the focus of risk monitoring and

xvi

INTRODUCTION

risk measurement follows. The central role of risk-based capital in regulations is a major incentive to the development of new tools and management techniques. Undoubtedly a most important innovation of recent years in terms of the modelling ‘toolbox’ is the VaR concept for assessing capital requirements. The VaR concept is a foundation of risk-based capital or, equivalently, ‘economic capital’. The VaR methodology aims at valuing potential losses resulting from current risks and relies on simple facts and principles. VaR recognizes that the loss over a portfolio of transactions could extend to the entire portfolio, but this is an event that has a zero probability given the effective portfolio diversification of banks. Therefore, measuring potential losses requires some rule for defining their magnitude for a diversified portfolio. VaR is the upper bound of losses that should not be exceeded in more than a small fraction of all future outcomes. Management and regulators define benchmarks for this small preset fraction, called the ‘confidence level’, measuring the appetite for risk of banks. Economic capital is VaRbased and crystallizes the quantified present value of potential future losses for making sure that banks have enough capital to sustain worst-case losses. Such risk valuation potentially extends to all main risks. Regulators made the concept instrumental for VaR-based market risk models in 1996. Moreover, even though the New Accord of 2001 falls short of allowing usage of credit models for measuring credit risk capital, it ensures the development of reliable inputs for such models.

The Risk Management Toolbox Risk-based practices require the deployment of multiple tools, or models, to meet the specifications of risk management within financial institutions. Risk models value risks and link them to their drivers and to the business universe. By performing these tasks, risk models contribute directly to risk processes. The goal of risk management is to enhance the risk–return profiles of transactions, of business lines’ portfolios of transactions and of the entire bank’s portfolio. Risk models provide these risk–return profiles. The risk management toolbox also addresses other major specifications. Since two risks of 1 add up to less than 2, unlike income and costs, we do not know how to divide a global risk into risk allocations for individual transactions, product families, market segments and business lines, unless we have some dedicated tools for performing this function. The Funds Transfer Pricing (FTP) system allocates income and the capital allocation system allocates risks. These tools provide a double link: • The top-down/bottom-up link for risks and income. • The transversal business-to-financial sphere linkage. Without such links, between the financial and the business spheres and between global risks and individual transaction profiles, there would be no way to move back and forth from a business perspective to a financial perspective and along the chain from individual transactions to the entire bank’s global portfolio.

Risk Management Processes Risk management processes are evolving with the gradual emergence of new risk measures. Innovations relate to:

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xvii

• The recognition of the need for quantification to develop risk-based practices and meet risk-based capital requirements. • The willingness of bankers to adopt a more proactive view on risks. • The gradual development of regulator guidelines for imposing risk-based techniques, enhanced disclosures on risks and ensuring a sounder and safer level playing field for the financial system. • The emergence of new techniques of managing risks (credit derivatives, new securitizations that off-load credit risk from the banks’ balance sheets) serving to reshape the risk–return profile of banks. • The emergence of new organizational processes for better integrating these advances, such as loan portfolio management. Without risk models, such innovations would remain limited. By valuing risks, models contribute to a more balanced view of income and risks and to a better control of risk drivers, upstream, before they materialize into losses. By linking the business and the risk views, the risk management ‘toolbox’ makes models instrumental for management. By feeding risk processes with adequate risk–return measures, they contribute to enriching them and leveraging them to new levels. Figure 1 shows how models contribute to the ‘vertical’ top-down and bottom-up processes, and how they contribute as well to the ‘horizontal’ links between the risk and return views of the business dimensions (transactions, markets and products, business lines). Risk Models & Tools

Credit Risk

FTP Capital Allocation

Market Risk ALM Others ...

Risk Processes Global Policy

Business Policy

Risk−Return Policy

Reporting Risk & Capital

Earnings

Business Lines

FIGURE 1

Comprehensive and consistent set of models for bank-wide risk management

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INTRODUCTION

THE STRUCTURE OF THE BOOK The structure of the book divides each topic into single modular pieces addressing the various issues above. The first section develops general issues, focusing on risks, risk measuring and risk management processes. The next major section addresses sequentially ALM, market risk and credit risk.

Book Outline The structure of the book is in 17 sections, each divided into several chapters. This structure provides a very distinct division across topics, each chapter dedicated to a major single topic. The benefit is that it is possible to move across chapters without necessarily following a sequential process throughout the book. The drawback is that only a few chapters provide an overview of interrelated topics. These chapters provide a synthesis of subsequent specific topics, allowing the reader to get both a summary and an overview of a series of interrelated topics. Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Banking Risks Risk Regulations Risk Management Processes Risk Models Asset–Liability Management Asset–Liability Management Models Options and Convexity Risk in Banking Mark-to-Market Management in Banking Funds Transfer Pricing Portfolio Analysis: Correlations Market Risk Credit Risk Models Credit Risk: ‘Standalone Risk’ Credit Risk: ‘Portfolio Risk’ Capital Allocation Risk-adjusted Performance Portfolio and Capital Management (Credit Risk)

Building Block Structure The structure of the book follows from several choices. The starting point is a wide array of risk models and tools that complement each other and use sometimes similar, sometimes different techniques for achieving the same goals. This raises major structuring issues for ensuring a consistent coverage of the risk management toolbox. The book relies on a building block structure shared by models; some of them extending across many blocks, some belonging to one major block. The structuring by building blocks of models and tools remedies the drawbacks of a sequential presentation of industry models; these bundle modelling techniques within each building block according to the model designers’ assembling choices. By contrast,

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a building block structure lists issues separately, and allows us to discuss explicitly the various modelling options. To facilitate the understanding of vendors’ models, some chapters provide an overview of all existing models, while detailed presentations provide an overview of all techniques applying to a single basic building block. Moreover, since model differentiation across risks is strong, there is a need to organize the structure by nature of risk. The sections of the book dealing directly with risk modelling cross-tabulate the main risks with the main building blocks of models. The main risks are interest rate risk, market risk and credit risk. The basic structure, within each risk, addresses four major modules as shown in Figure 2.

FIGURE 2

I

Risk drivers and transaction risk (standalone)

II

Portfolio risk

III

Top-down & bottom-up tools

IV

Risk & return measures

The building block structure of risk models

The basic blocks, I and II, are dedicated by source of risk. The two other blocks, III and IV, are transversal to all risks. The structure of the book follows from these principles.

Focus The focus is on risk management issues for financial institutions rather than risk modelling applied to financial markets. There is an abundant literature on financial markets and financial derivatives. A basic understanding of what derivatives achieve in terms of hedging or structuring transactions is a prerequisite. However, sections on instruments are limited to the essentials of what hedging instruments are and their applications. Readers can obtain details on derivatives and pricing from other sources in the abundant literature. We found that textbooks rarely address risk management in banking and in financial institutions in a comprehensive manner. Some focus on the technical aspects. Others focus on pure implementation issues to the detriment of technical substance. In other cases, the scope is unbalanced, with plenty of details on some risks (market risk notably) and fewer on others. We have tried to maintain a balance across the main risks without sacrificing scope. The text focuses on the essential concepts underlying the risk analytics of existing models. It does detail the analytics without attempting to provide a comprehensive coverage of each existing model. This results in a more balanced view of all techniques for modelling banking risk. In addition, model vendors’ documentation is available directly from sites dedicated to risk management modelling. There is simply no need to replicate such documents. When developing the analytics, we considered that providing a

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universal framework, allowing the contrast of various techniques based on their essential characteristics, was of greater help than replicating public documentation. Accordingly, we skipped some details available elsewhere and located some technicalities in appendices. However, readers will find the essentials, the basics for understanding the technicalities, and the examples for grasping their practical value added. In addition, the text develops many numerical examples, while restricting the analytics to the essential ingredients. A first motive is that simple examples help illustrate the essentials better than a detailed description. Of course, simple examples are no substitute for full-blown models. The text is almost self-contained. It details the prerequisites for understanding fullblown models. It summarizes the technicalities of full-blown models and it substitutes examples and applications for further details. Still, it gathers enough substance to provide an analytical framework, developed sufficiently to make it easy to grasp details not expanded here and map them to the major building blocks of risk modelling. Finally, there is a balance to strike between technicalities and applications. The goal of risk management is to use risk models and tools for instrumental purposes, for developing risk-based practices and enriching risk processes. Such an instrumental orientation strongly inspired this text. *

*

*

The first edition of this book presented details on ALM and introduced major advances in market risk and credit risk modelling. This second edition expands considerably on credit risk and market risk models. In addition, it does so within a unified framework for capturing all major risks and deploying bank-wide risk management tools and processes. Accordingly, the volume has roughly doubled in size. This is illustrative of the fast and continuous development of the field of risk management in financial institutions.

SECTION 1 Banking Risks

1 Banking Business Lines

The banking industry has a wide array of business lines. Risk management practices and techniques vary significantly between the main poles, such as retail banking, investment banking and trading, and within the main poles, between business lines. The differences across business lines appear so important, say between retail banking and trading for example, that considering using the same concepts and techniques for risk management purposes could appear hopeless. There is, indeed, a differentiation, but risk management tools, borrowing from the same core techniques, apply across the entire spectrum of banking activities generating financial risks. However, risks and risk management differ across business lines. This first chapter provides an overview of banking activities. It describes the main business poles, and within each pole the business lines. Regulations make a clear distinction between commercial banking and trading activities, with the common segmentation between ‘banking book’ and ‘trading book’. In fact, there are major distinctions within business lines of lending activities, which extend from retail banking to specialized finance. This chapter provides an overview of the banking business lines and of the essentials of financial statements.

BUSINESS POLES IN THE BANKING INDUSTRY The banking industry has a wide array of business lines. Figure 1.1 maps these activities, grouping them into main poles: traditional commercial banking; investment banking, with specialized transactions; trading. Poles subdivide into various business lines. Management practices are very different across and within the main poles. Retail banking tends to be mass oriented and ‘industrial’, because of the large number of transactions. Lending to individuals relies more on statistical techniques. Management reporting on such large numbers of transactions focuses on large subsets of transactions.

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Business Poles

Business Lines

Retail Financial Services Commercial Banking

Corporate−Middle Market Large Corporate

Lending and collecting deposits Individuals and small businesses Identified borrowers and relationship banking

Advisory Services Mergers & Acquisitions LBO ... Investment Banking

Banks & Financial Firms Assets Financing (Aircrafts ...)

'Structured Finance'

Commodities Securitization

Derivatives Trading Trading

Equity

Traded Instruments

Fixed Income

FIGURE 1.1

Private Private Banking Banking

Assets Management

Others ... Others..

Custody ...

Breaking down the bank portfolio along organizational dimensions

Criteria for grouping the transactions include date of origination, type of customer, product family (consumer loans, credit cards, leasing). For medium and large corporate borrowers, individual decisions require more judgment because mechanical rules are not sufficient to assess the actual credit standing of a corporation. For the middle market segment to large corporate businesses, ‘relationship banking’ prevails. The relation is stable, based on mutual confidence, and generates multiple services. Risk decisions necessitate individual evaluation of transactions. Obligors’ reviews are periodical. Investment banking is the domain of large transactions customized to the needs of big corporates or financial institutions. ‘Specialized finance’ extends from specific fields with standard practices, such as export and commodities financing, to ‘structured financing’, implying specific structuring and customization for making large and risky transactions feasible, such as project finance or corporate acquisitions. ‘Structuring’ designates the assembling of financial products and derivatives, plus contractual clauses for monitoring risk (‘covenants’). Without such risk mitigants, transactions would not be feasible. This domain overlaps with traditional ‘merchant’ banking and market business lines. Trading involves traditional proprietary trading and trading for third parties. In the second case,

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5

Market Transactions

Specialized Finance

Off-balance Sheet

Markets

Corporate Lending

RFS

Product Groups

traders interact with customers and other banking business units to bundle customized products for large borrowers, including ‘professionals’ of the finance industry, banks, insurance, brokers and funds. Other activities do not generate directly traditional banking risks, such as private banking, or asset management and advisory services. However, they generate other risks, such as operational risk, similarly to other business lines, as defined in the next chapter. The view prevailing in this book is that all main business lines share the common goals of risk–expected return enhancement, which also drives the management of global bank portfolios. Therefore, it is preferable to differentiate business lines beyond the traditional distinctions between the banking portfolio and the trading portfolio. The matrix shown in Figure 1.2 is a convenient representation of all major lines across which practices differ. Subsequent chapters differentiate, whenever necessary, risk and profitability across the cells of the matrix, cross-tabulating main product lines and main market segments. Banking business lines differ depending on the specific organizations of banks and on their core businesses. Depending on the degree of specialization, along with geographical subdivisions, they may or may not combine one or several market segments and product families. Nevertheless, these two dimensions remain the basic foundations for differentiating risk management practices and designing ‘risk models’.

Consumers Corporate−Middle Market Large Corporate Firms Financial Institutions Specialized Finance

FIGURE 1.2 Main product–market segments RFS refers to ‘Retail Financial Services’. LBO refers to ‘Leveraged Buy-Out’, a transaction allowing a major fraction of the equity of a company to be acquired using a significant debt (leverage). Specialized finance refers to structured finance, project finance, LBO or assets financing.

Product lines vary within the above broad groups. For instance, standard lending transactions include overnight loans, short-term loans (less than 1 year), revolving facilities, term loans, committed lines of credit, or large corporate general loans. Retail financial services cover all lending activities, from credit card and consumer loans to mortgage loans. Off-balance sheet transactions are guarantees and backup lines of credit

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providing significant revenues to banks. Specialized finance includes project finance, commodities financing, asset financing (from real estate to aircraft) and trade financing. Market transactions cover all basic compartments, fixed income, equity and foreign exchange trading, including derivatives from standard swaps and options to exotic and customized products. Major market segments appear explicitly in the matrix. They also subdivide. Financial institutions include banks as well as insurance or brokers. Specialized finance includes various fields, including structured finance. The greyed cell represents a basic market–product couple. Risk management involves risk and expected return measuring, reporting and management for such transactions, for the bank portfolio as a whole and for such basic couples. The next section reverts to the basic distinction between banking book and trading book, which is essentially product-driven.

THE BANKING AND THE TRADING BOOKS The ‘banking book’ groups and records all commercial banking activities. It includes all lending and borrowing, usually both for traditional commercial activities, and overlaps with investment banking operations. The ‘trading book’ groups all market transactions tradable in the market. The major difference between these two segments is that the ‘buy and hold’ philosophy prevails for the banking book, contrasting with the trading philosophy of capital markets. Accounting rules differ for the banking portfolio and the trading portfolio. Accounting rules use accrual accounting of revenues and costs, and rely on book values for assets and liabilities. Trading relies on market values (mark-to-market) of transactions and Profit and Loss (P&L), which are variations of the mark-to-market value of transactions between two dates. The rationale for separating these ‘portfolios’ results from such major characteristics.

The Banking Book The banking portfolio follows traditional accounting rules of accrued interest income and costs. Customers are mainly non-financial corporations or individuals, although interbanking transactions occur between professional financial institutions. The banking portfolio generates liquidity and interest rate risks. All assets and liabilities generate accrued revenues and costs, of which a large fraction is interest rate-driven. Any maturity mismatch between assets and liabilities results in excesses or deficits of funds. Mismatch also exists between interest references, ‘fixed’ or ‘variable, and results from customers’ demand and the bank’s business policy. In general, both mismatches exist in the ‘banking book’ balance sheet. For instance, there are excess funds when collection of deposits and savings is important, or a deficit of funds whenever the lending activity uses up more resources than the deposits from customers. Financial transactions (on the capital markets) serve to manage such mismatches between commercial assets and liabilities through either investment of excess funds or long-term debt by banks. Asset–Liability Management (ALM) applies to the banking portfolio and focuses on interest rate and liquidity risks. The asset side of the banking portfolio also generates credit risk. The liability side contributes to interest rate risk, but does not generate credit risk, since the lenders or depositors are at risk with the bank. There is no market risk for the banking book.

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7

The Trading Portfolio The market transactions are not subject to the same management rules. The turnover of tradable positions is faster than that of the banking portfolio. Earnings are P&L equal to changes of the mark-to-market values of traded instruments. Customers include corporations (corporate counterparties) or other financial players belonging to the banking industry (professional counterparties). The market portfolio generates market risk, defined broadly as the risk of adverse changes in market values over a liquidation period. It is also subject to market liquidity risk, the risk that the volume of transactions narrows so much that trades trigger price movements. The trading portfolio extends across geographical borders, just as capital markets do, whereas traditional commercial banking is more ‘local’. Many market transactions use non-tradable instruments, or derivatives such as swaps and options traded over-the-counter. Such transactions might have a very long maturity. They trigger credit risk, the risk of a loss if the counterparty fails.

Off-balance Sheet Transactions Off-balance sheet transactions are contingencies given and received. For banking transactions, contingencies include guarantees given to customers or to third parties, committed credit lines not yet drawn by customers, or backup lines of credit. Those are contractual commitments, which customers use at their initiative. A guarantee is the commitment of the bank to fulfil the obligations of the customer, contingent on some event such as failure to face payment obligations. For received contingencies, the beneficiary is the bank. ‘Given contingencies’ generate revenues, as either upfront and/or periodic fees, or interest spreads calculated as percentages of outstanding balances. They do not generate ‘immediate’ exposures since there is no outflow of funds at origination, but they do trigger credit risk because of the possible future usage of contingencies given. The outflows occur conditionally on what happens to the counterparty. If a borrower draws on a credit line previously unused, the resulting loan moves up on the balance sheet. ‘Off-balance sheet’ lines turn into ‘on-balance sheet’ exposures when exercised. Derivatives are ‘off-balance sheet’ market transactions. They include swaps, futures contracts, foreign exchange contracts and options. As other contingencies, they are obligations to make contractual payments conditional upon occurrence of a specified event. Received contingencies create symmetrical obligations for counterparties who sold them to the bank.

BANKS’ FINANCIAL STATEMENTS There are several ways of grouping transactions. The balance sheet provides a snapshot view of all assets and liabilities at a given date. The income statement summarizes all revenues and costs to determine the income of a period.

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Balance Sheet In a simplified view (Table 1.1), the balance sheet includes four basic levels, in addition to the off-balance sheet, which divide it horizontally: • • • •

Treasury and banking transactions. Intermediation (lending and collecting deposits). Financial assets (trading portfolio). Long-term assets and liabilities: fixed assets, investments in subsidiaries and equity plus long-term debt.

TABLE 1.1

Simplified balance sheet

Assets

Equity and liabilities

Cash Lending Financial assets Fixed assets

Short-term debt Deposits Financial assets Long-term debt Equity Off-balance sheet (contingencies given)

Off-balance sheet (contingencies received)

The relative weights of the major compartments vary from one institution to another, depending on their core businesses. Equity is typically low in all banks’ balance sheets. Lending and deposits are traditionally large in retail and commercial banking. Investment banking, including both specialized finance and trading, typically funds operations in the market. In European banks, ‘universal banking’ allows banking institutions to operate over the entire spectrum of business lines, contrasting with the separation between investment banking and commercial banking, which still prevails in the United States.

Income Statement and Valuation The current national accounting standards use accrual measures of revenues and costs to determine the net income of the banking book. Under such standards, net income ignores any change of ‘mark-to-market’ value, except for securities traded in the market or considered as short-term holdings. International accounting standards progress towards ‘fair value’ accounting for the banking book, and notably for transactions hedging risk. Fair value is similar to mark-to-market1 , except that it extends to non-tradable assets such as loans. There is a strong tendency towards generalizing the ‘fair value’ view of the balance sheet for all activities, which is subject to hot debates for the ‘banking book’. Accounting standards are progressively evolving in that direction. The implications are major in terms of profitability, since gains and losses of the balance sheet ‘value’ between two 1 See

Chapter 8 for details on calculating ‘mark-to-market’ values on non-tradable transactions.

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9

dates would count as profit and losses. A major driving force for looking at ‘fair values’ is the need to use risk-adjusted values in modelling risks. In what follows, because of the existing variations around the general concept of ‘fair value’, we designate such values equivalently as ‘economic values’, ‘mark-to-market’ or ‘mark-to-model’ values depending on the type of valuation technique used. Fair values have become so important, if only because of risk modelling, that we discuss them in detail in Chapter 8. For the banking portfolio, the traditional accounting measures of earnings are contribution margins calculated at various levels of the income statement. They move from the ‘interest income’ of the bank, which is the difference between all interest revenues plus fees and all interest costs, down to net income. The total revenue cumulates the interest margin with all fees for the period. The interest income of commercial banking commonly serves as the main target for management policies of interest rate risk because it is entirely interest-driven. Another alternative target variable is the Net Present Value (NPV) of the balance sheet, measured as a Mark-to-Market (MTM) of assets minus that of liabilities. Commercial banks try to increase the fraction of revenues made up of fees for making the net income less interest rate-sensitive. Table 1.2 summarizes the main revenues and costs of the income statement. TABLE 1.2

Income statement and earnings

Interest margin plus fees Capital gains and losses −Operating costs =Operating income (EBTD)a −Depreciation −Provisions −Tax =Net income a EBTD

= Earnings Before Tax and Depreciation.

Provisions for loan losses deserve special attention. The provision policy should ideally be an indicator of the current credit risk of banking loans. However, provisions have to comply with accounting and fiscal rules and differ from economic provisions. Economic provisions are ‘ex ante’ provisions, rather than provisions resulting from the materialization of credit risk. They should anticipate the effective credit risk without the distortions due to legal and tax constraints. Economic provisioning is a debated topic, because unless new standards and rules emerge for implementation, it remains an internal risk management tool without impact on the income statement bottom line.

Performance Measures Performance measures derive directly from the income statement. The ratio of net income to equity is the accounting Return On Equity (ROE). It often serves as a target profitability measure at the overall bank level. The accounting ROE ratio is not the market return on equity, which is a price return, or the ratio of the price variation between two dates of

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the bank’s stock (ignoring dividends). Under some specific conditions2 , it might serve as a profitability benchmark. Both the ROE and the market return on equity should be in line with shareholders’ expectations for a given level of risk of the bank’s stock. A current order of magnitude for the target ROE is 15% after tax, or about 25% before tax. When considering banking transactions, the Return On Assets (ROA) is another measure of profitability for banking transactions. The most common calculation of ROA is the ratio of the current periodical income, interest income and current fees, divided by asset balance. The current ROA applies both to single individual transactions and to the total balance sheet. The drawback of accounting ROE and ROA measures, and of the P&L of the trading portfolio, is that they do not include any risk adjustment. Hence, they are not comparable from one borrower to another, because their credit risk differs, from one trading transaction to another, and because the market risk varies across products. This drawback is the origin of the concept of risk-adjusted performance measures. This is an incentive for moving, at least in internal reports of risks and performances, to ‘economic values’, ‘mark-to-market’ or ‘mark-to-model’ values, because these are both risk- and revenue-adjusted3 .

2 It

can be shown that a target accounting ROE implies an identical value for the market return on the bank’s equity under the theoretical condition that the Price–Earnings Ratio (PER) remains constant. See Chapter 53. 3 See Chapter 8 to explain how mark-to-market values embed both risk and expected return.

2 Banking Risks

Risks are uncertainties resulting in adverse variations of profitability or in losses. In the banking universe, there are a large number of risks. Most are well known. However, there has been a significant extension of focus, from the traditional qualitative risk assessment towards the quantitative management of risks, due to both evolving risk practices and strong regulatory incentives. The different risks need careful definition to provide sound bases serving for quantitative measures of risk. As a result, risk definitions have gained precision over the years. The regulations, imposing capital charges against all risks, greatly helped the process. The underlying philosophy of capital requirement is to bring capital in line with risks. This philosophy implies modelling the value of risk. The foundation of such risk measures is in terms of potential losses. The capital charge is a quantitative value. Under regulatory treatment, it follows regulatory rules applying to all players. Under an economic view, it implies modelling potential losses from each source of risk, which turns out to be the ‘economic’ capital ‘adequate’ to risk. Most of the book explains how to assign economic values to risks. Therefore, the universal need to value risks, which are intangible and invisible, requires that risks be well-defined. Risk definitions serve as the starting point for both regulatory and economic treatments of risks. This book focuses on three main risks: interest rate risk for the banking book; market risk for the trading book; credit risk. However, this chapter does provide a comprehensive overview of banking risks.

BANKING RISKS Banking risks are defined as adverse impacts on profitability of several distinct sources of uncertainty (Figure 2.1). Risk measurement requires capturing the source of the uncertainty and the magnitude of its potential adverse effect on profitability. Profitability refers to both accounting and mark-to-market measures.

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Credit Interest rate Market Banking Risks Liquidity Operational Foreign exchange

Other risks: country risk, settlement risk, performance risk ...

FIGURE 2.1

Main bank risks

This book focuses on financial risks, or risks related to the market movements or the economic changes of the environment. Market risk is relatively easy to quantify thanks to the large volume of price observations. Credit risk ‘looked like’ a ‘commercial’ risk because it is business-driven. Innovation changed this view. Since credit risk is a major risk, the regulators insisted on continuously improving its measurement in order to quantify the amount of capital that banks should hold. Credit risk and the principle of diversification are as old as banks are. It sounds like a paradox that major recent innovations focus on this old and well-known risk. Operational risk also attracts attention1 . It covers all organizational malfunctioning, of which consequences can be highly important and, sometimes, fatal to an institution. Following the regulators’ focus on valuing risk as a capital charge, model designers developed risk models aimed at the quantification of potential losses arising from each source of risk. The central concept of such models is the well-known ‘Value at Risk’ (VaR). Briefly stated, a VaR is a potential loss due to a defined risk. The issue is how to define a potential loss, given that the loss can be as high as the current portfolio value. Of course, the probability of such an event is zero. In order to define the potential adverse deviation of value, or loss, a methodology is required to identify what could be a ‘maximum’ deviation. Under the VaR methodology, the worst-case loss is a ‘maximum’ bound not exceeded in more than a preset fraction (for instance 1%) of all possible states over a defined period. Models help to determine a market risk VaR and a credit risk VaR. The VaR concept also extends to other risks. Subsequent developments explain how to move from the following definitions of risk to VaR modelling and measuring. 1 The

New Basel Accord of January 2001 requires a capital charge against operational risk.

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13

CREDIT RISK Credit risk is the first of all risks in terms of importance. Default risk, a major source of loss, is the risk that customers default, meaning that they fail to comply with their obligations to service debt. Default triggers a total or partial loss of any amount lent to the counterparty. Credit risk is also the risk of a decline in the credit standing of an obligor of the issuer of a bond or stock. Such deterioration does not imply default, but it does imply that the probability of default increases. In the market universe, a deterioration of the credit standing of a borrower does materialize into a loss because it triggers an upward move of the required market yield to compensate the higher risk and triggers a value decline. ‘Issuer’ risk designates the obligors’ credit risk, to make it distinct from the specific risk of a particular issue, among several of the same issuer, depending on the nature of the instrument and its credit mitigants (seniority level and guarantees). The view of credit risk differs for the banking portfolio and the trading portfolio.

Banking Portfolio Credit risk is critical since the default of a small number of important customers can generate large losses, potentially leading to insolvency. There are various default events: delay in payment obligations; restructuring of debt obligations due to a major deterioration of the credit standing of the borrower; bankruptcies. Simple delinquencies, or payment delays, do not turn out as plain defaults, with a durable inability of lenders to face debt obligations. Many are resolved within a short period (say less than 3 months). Restructuring is very close to default because it results from the view that the borrower will not face payment obligations unless its funding structure changes. Plain defaults imply that the non-payment will be permanent. Bankruptcies, possibly liquidation of the firm or merging with an acquiring firm, are possible outcomes. They all trigger significant losses. Default means any situation other than a simple delinquency. Credit risk is difficult to quantify on an ‘ex ante’ basis, since we need an assessment of the likelihood of a default event and of the recoveries under default, which are contextdependent2 . In addition, banking portfolios benefit from diversification effects, which are much more difficult to capture because of the scarcity of data on interdependencies between default events of different borrowers are interdependent.

Trading Portfolio Capital markets value the credit risk of issuers and borrowers in prices. Unlike loans, the credit risk of traded debts is also indicated by the agencies’ ratings, assessing the quality of public debt issues, or through changes of the value of their stocks. Credit risk is also visible through credit spreads, the add-ons to the risk-free rate defining the required market risk yield on debts. The capability of trading market assets mitigates the credit risk since there is no need to hold these securities until the deterioration of credit risk materializes into effective losses. 2 Context

refers to all factors influencing loss under default, such as the outstanding balance of debt at default, the existence of guarantees, or the policy of all stakeholders with respect to existing debt.

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If the credit standing of the obligor declines, it is still possible to sell these instruments in the market at a lower value. The loss due to credit risk depends on the value of these instruments and their liquidity. If the default is unexpected, the loss is the difference between the pre- and post-default prices. The faculty of trading the assets limits the loss if sale occurs before default. The selling price depends on the market liquidity. Therefore, there is some interaction between credit risk and trading risk. For over-the-counter instruments, such as derivatives (swaps and options), whose development has been spectacular in the recent period, sale is not readily feasible. The bank faces the risk of losing the value of such instruments when it is positive. Since this value varies constantly with the market parameters, credit risk changes with market movements during the entire residual life of the instrument. Credit risk and market risk interact because these values depend on the market moves. Credit risk for traded instruments raises a number of conceptual and practical difficulties. What is the value subject to loss, or exposure, in future periods? Does the current price embed already the credit risk, since market prices normally anticipate future events, and to what extent? Will it be easy to sell these instruments when signs of deterioration get stronger, and at what discount from the current value since the market for such instruments might narrow when credit risk materializes? Will the bank hold these instruments longer than under normal conditions?

Measuring Credit Risk Even though procedures for dealing with credit risk have existed since banks started lending, credit risk measurement raises several issues. The major credit risk components are exposure, likelihood of default, or of a deterioration of the credit standing, and the recoveries under default. Scarcity of data makes the assessment of these components a challenge. Ratings are traditional measures of the credit quality of debts. Some major features of ratings systems are3 : • Ratings are ordinal or relative measures of risk rather than cardinal or absolute measures, such as default probability. • External ratings are those of rating agencies, Moody’s, Standard & Poor’s (S&P) and Fitch, to name the global ones. They apply to debt issues rather than issuers because various debt issues from the same issuer have different risks depending on seniority level and guarantees. Detailed rating scales of agencies have 20 levels, ignoring the near default rating levels. • By contrast, an issuer’s rating characterizes only the default probability of the issuer. • Banks use internal rating scales because most of their borrowers do not have publicly rated debt issues. Internal rating scales of banks are customized to banks’ requirements, and usually characterize both borrower’s risk and facility’s risk. • There are various types of ratings. Ratings characterize sovereign risk, the risk of country debt and the risk of the local currency. Ratings are also either short-term

3 Chapters

35 and 36 detail further the specifications of rating systems.

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15

or long-term. There are various types of country-related ratings: sovereign ratings of government sponsored borrowers; ratings of currencies; ratings of foreign currencies held locally; ratings of transfer risk, the risk of being unable to transfer cash out of the country. • Because ratings are ordinal measures of credit risk, they are not sufficient to value credit risk. Moreover, ratings apply only to individual debts of borrowers, and they do not address the bank’s portfolio risk, which benefits from diversification effects. Portfolio models show that portfolio risk varies across banks depending on the number of borrowers, the discrepancies in size between exposures and the extent of diversification among types of borrowers, industries and countries. The portfolio credit risk is critical in terms of potential losses and, therefore, for finding out how much capital is required to absorb such losses. Modelling default probability directly with credit risk models remained a major challenge, not addressed until recent years. A second challenge of credit risk measurement is capturing portfolio effects. Due to the scarcity of data in the case of credit risk, quantifying the diversification effect sounds like a formidable challenge. It requires assessing the joint likelihood of default for any pair of borrowers, which gets higher if their individual risks correlate. Given its importance for banks, it is not surprising that banks, regulators and model designers made a lot of effort to better identify the relevant inputs for valuing credit risk and model diversification effects with ‘portfolio models’. Accordingly, a large fraction of this book addresses credit risk modelling.

COUNTRY AND PERFORMANCE RISKS Credit risk is the risk of loss due to a deterioration of the credit standing of a borrower. Some risks are close to credit risk, but distinct, such as country risk and performance risk.

Country Risk Country risk is, loosely speaking, the risk of a ‘crisis’ in a country. There are many risks related to local crises, including: • Sovereign risk, which is the risk of default of sovereign issuers, such as central banks or government sponsored banks. The risk of default often refers to that of debt restructuring for countries. • A deterioration of the economic conditions. This might lead to a deterioration of the credit standing of local obligors, beyond what it should be under normal conditions. Indeed, firms’ default frequencies increase when economic conditions deteriorate. • A deterioration of the value of the local foreign currency in terms of the bank’s base currency.

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• The impossibility of transferring funds from the country, either because there are legal restrictions imposed locally or because the currency is not convertible any more. Convertibility or transfer risks are common and restrictive definitions of country risks. • A market crisis triggering large losses for those holding exposures in the local markets. A common practice stipulates that country risk is a floor for the risk of a local borrower, or equivalently, that the country rating caps local borrowers’ ratings. In general, country ratings serve as benchmarks for corporate and banking entities. The rationale is that, if transfers become impossible, the risk materializes for all corporates in the country. There are debates around such rules, since the intrinsic credit standing of a borrower is not necessarily lower than on that of the country.

Performance Risk Performance risk exists when the transaction risk depends more on how the borrower performs for specific projects or operations than on its overall credit standing. Performance risk appears notably when dealing with commodities. As long as delivery of commodities occurs, what the borrower does has little importance. Performance risk is ‘transactional’ because it relates to a specific transaction. Moreover, commodities shift from one owner to another during transportation. The lender is at risk with each one of them sequentially. Risk remains more transaction-related than related to the various owners because the commodity value backs the transaction. Sometimes, oil is a major export, which becomes even more strategic in the event of an economic crisis, making the financing of the commodity immune to country risk. In fact, a country risk increase has the paradoxical effect of decreasing the risk of the transaction because exports improve the country credit standing.

LIQUIDITY RISK Liquidity risk refers to multiple dimensions: inability to raise funds at normal cost; market liquidity risk; asset liquidity risk. Funding risk depends on how risky the market perceives the issuer and its funding policy to be. An institution coming to the market with unexpected and frequent needs for funds sends negative signals, which might restrict the willingness to lend to this institution. The cost of funds also depends on the bank’s credit standing. If the perception of the credit standing deteriorates, funding becomes more costly. The problem extends beyond pure liquidity issues. The cost of funding is a critical profitability driver. The credit standing of the bank influences this cost, making the rating a critical factor for a bank. In addition, the rating drives the ability to do business with other financial institutions and to attract investors because many follow some minimum rating guidelines to invest and lend. The liquidity of the market relates to liquidity crunches because of lack of volume. Prices become highly volatile, sometimes embedding high discounts from par, when counterparties are unwilling to trade. Funding risk materializes as a much higher cost

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17

of funds, although the cause lies more with the market than the specific bank. Market liquidity risk materializes as an impaired ability to raise money at a reasonable cost. Asset liquidity risk results from lack of liquidity related to the nature of assets rather than to the market liquidity. Holding a pool of liquid assets acts as a cushion against fluctuating market liquidity, because liquid assets allow meeting short-term obligations without recourse to external funding. This is the rationale for banks to hold a sufficient fraction of their balance sheet of liquid assets, which is a regulatory rule. The ‘liquidity ratio’ of banks makes it mandatory to hold more short-term assets than short-term liabilities, in order to meet short-run obligations. In order to fulfil this role, liquid assets should mature in the short-term because market prices of long-term assets are more volatile4 , possibly triggering substantial losses in the event of a sale. Moreover, some assets are less tradable than others, because their trading volume is narrow. Some stocks trade less than others do, and exotic products might not trade easily because of their high level of customization, possibly resulting in depressed prices. In such cases, any sale might trigger price declines, so that the proceeds from a one-shot or a progressive sale become uncertain and generate losses. To a certain extent, funding risk interacts with market liquidity and asset liquidity because the inability to face payment obligations triggers sales of assets, possibly at depressed prices. Liquidity risk might become a major risk for the banking portfolio. Extreme lack of liquidity results in bankruptcy, making liquidity risk a fatal risk. However, extreme conditions are often the outcome of other risks. Important unexpected losses raise doubts with respect to the future of the organization and liquidity issues. When a commercial bank gets into trouble, depositors ‘run’ to get their money back. Lenders refrain from further lending to the troubled institution. Massive withdrawals of funds or the closing of credit lines by other institutions are direct outcomes of such situations. A brutal liquidity crisis follows, which might end up in bankruptcy. In what follows, we adopt an Asset–Liability Management (ALM) view of the liquidity situation. This restricts liquidity risk to bank-specific factors other than the credit risk of the bank and the market liquidity. The time profiles of projected uses and sources of funds, and their ‘gaps’ or liquidity mismatches, capture the liquidity position of a bank. The purpose of debt management is to manage these future liquidity gaps within acceptable limits, given the market perception of the bank. This perspective does not fully address liquidity risk, and the market risk definitions below address this only partially. Liquidity risk, in terms of market liquidity or asset liquidity, remains a major issue that current techniques do not address fully. Practices rely on empirical and continuous observations of market liquidity, while liquidity risk models remain too theoretical to allow instrumental applications. This is presumably a field necessitating increased modelling research and improvement of practices.

INTEREST RATE RISK The interest rate risk is the risk of a decline in earnings due to the movements of interest rates. 4 Long-term

interest-bearing assets are more sensitive to interest rate movements. See the duration concept used for capturing the sensitivity of the mark-to-market value of the balance sheet in Chapter 22.

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Most of the items of banks’ balance sheets generate revenues and costs that are interest rate-driven. Since interest rates are unstable, so are earnings. Any one who lends or borrows is subject to interest rate risk. The lender earning a variable rate has the risk of seeing revenues reduced by a decline in interest rates. The borrower paying a variable rate bears higher costs when interest rates increase. Both positions are risky since they generate revenues or costs indexed to market rates. The other side of the coin is that interest rate exposure generates chances of gains as well. There are various and complex indexations on market rates. Variable rate loans have rates periodically reset using some market rate references. In addition, any transaction reaching maturity and renewed will stick to the future and uncertain market conditions. Hence, fixed rates become variable at maturity for loan renewals and variable rates remain fixed between two reset dates. In addition, the period between two rate resets is not necessarily constant. For instance, the prime rate of banks remains fixed between two resets, over periods of varying lengths, even though market rates constantly move. The same happens for the rates of special savings deposits, when they are subject to legal and fiscal rules. This variety makes the measure of interest rate sensitivity of assets and liabilities to market rates more complex. Implicit options in banking products are another source of interest rate risk. A wellknown case is that of the prepayment of loans that carry a fixed rate. A person who borrows can always repay the loan and borrow at a new rate, a right that he or she will exercise when interest rates decline substantially. Deposits carry options as well, since deposit holders transfer funds to term deposits earning interest revenues when interest rates increase. Optional risks are ‘indirect’ interest rate risks. They do not arise directly and only from a change in interest rate. They also result from the behaviour of customers, such as geographic mobility or the sale of their homes to get back cash. Economically, fixed rate borrowers compare the benefits and the costs of exercising options embedded in banking products, and make a choice depending on market conditions. Given the importance of those products in the balance sheets of banks, optional risk is far from negligible. Measuring the option risk is more difficult than measuring the usual risk which arises from simple indexation to market rates. Section 5 of this book details related techniques.

MARKET RISK Market risk is the risk of adverse deviations of the mark-to-market value of the trading portfolio, due to market movements, during the period required to liquidate the transactions. The period of liquidation is critical to assess such adverse deviations. If it gets longer, so do the deviations from the current market value. Earnings for the market portfolio are Profit and Loss (P&L) arising from transactions. The P&L between two dates is the variation of the market value. Any decline in value results in a market loss. The potential worst-case loss is higher when the holding period gets longer because market volatility tends to increase over longer horizons. However, it is possible to liquidate tradable instruments or to hedge their future changes of value at any time. This is the rationale for limiting market risk to the liquidation period. In general, the liquidation period varies with the type of instruments. It could be short (1 day) for foreign exchange and much longer for ‘exotic’ derivatives. The regulators

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provide rules to set the liquidation period. They use as reference a 10-day liquidation period and impose a multiple over banks’ internal measures of market value potential losses (see Chapter 3). Liquidation involves asset and market liquidity risks. Price volatility is not the same in high-liquidity and poor-liquidity situations. When liquidity is high, the adverse deviations of prices are much lower than in a poor-liquidity environment, within a given horizon. ‘Pure’ market risk, generated by changes of market parameters (interest rates, equity indexes, exchange rates), differs from market liquidity risk. This interaction raises important issues. What is the ‘normal’ volatility of market parameters under fair liquidity situations? What could it become under poorer liquidity situations? How sensitive are the prices to liquidity crises? The liquidity issue becomes critical in emerging markets. Prices in emerging markets often diverge considerably from a theoretical ‘fair value’. Market risk does not refer to market losses due to causes other than market movements, loosely defined as inclusive of liquidity risk. Any deficiency in the monitoring of the market portfolio might result in market values deviating by any magnitude until liquidation finally occurs. In the meantime, the potential deviations can exceed by far any deviation that could occur within a short liquidation period. This risk is an operational risk, not a market risk5 . In order to define the potential adverse deviation, a methodology is required to identify what could be a ‘maximum’ adverse deviation of the portfolio market value. This is the VaR methodology. The market risk VaR technique aims at capturing downside deviations of prices during a preset period for liquidating assets, considering the changes in the market parameters. Controlling market risk means keeping the variations of the value of a given portfolio within given boundary values through actions on limits, which are upper bounds imposed on risks, and hedging for isolating the portfolio from the uncontrollable market movements.

FOREIGN EXCHANGE RISK The currency risk is that of incurring losses due to changes in the exchange rates. Variations in earnings result from the indexation of revenues and charges to exchange rates, or of changes of the values of assets and liabilities denominated in foreign currencies. Foreign exchange risk is a classical field of international finance, so that we can rely on traditional techniques in this book, without expanding them. For the banking portfolio, foreign exchange risk relates to ALM. Multi-currency ALM uses similar techniques for each local currency. Classical hedging instruments accommodate both interest rate and exchange rate risk. For market transactions, foreign exchange rates are a subset of market parameters, so that techniques applying to other market parameters apply as well. The conversion risk resuls from the need to convert all foreign currency-denominated transactions into a base reference currency. This risk does exist, beyond accounting conversion in a single currency, if the capital base that protects the bank from losses 5 An

example is the failure of Baring Brothers, due to deficiencies in the control of the risk positions (see Leeson, 1996).

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is in local currency. A credit loss in a foreign country might result in magnified losses in local currency if the local currency depreciates relative to the currency of the foreign exposure.

SOLVENCY RISK Solvency risk is the risk of being unable to absorb losses, generated by all types of risks, with the available capital. It differs from bankruptcy risk resulting from defaulting on debt obligations and inability to raise funds for meeting such obligations. Solvency risk is equivalent to the default risk of the bank. Solvency is a joint outcome of available capital and of all risks. The basic principle of ‘capital adequacy’, promoted by regulators, is to define what level of capital allows a bank to sustain the potential losses arising from all current risks and complying with an acceptable solvency level. The capital adequacy principle follows the major orientations of risk management. The implementation of this principle requires: • Valuing all risks to make them comparable to the capital base of a bank. • Adjusting capital to a level matching the valuation of risks, which implies defining a ‘tolerance’ level for the risk that losses exceed this amount, a risk that should remain very low to be acceptable. Meeting these specifications drives the regulators’ philosophy and prudent rules. The VaR concept addresses these issues directly by providing potential loss values for various confidence levels (probability that actual losses exceed an upper bound).

OPERATIONAL RISK Operational risks are those of malfunctions of the information system, reporting systems, internal risk-monitoring rules and internal procedures designed to take timely corrective actions, or the compliance with internal risk policy rules. The New Basel Accord of January 2001 defines operational risk as ‘the risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events’. In the absence of efficient tracking and reporting of risks, some important risks remain ignored, do not trigger any corrective action and can result in disastrous consequences. In essence, operational risk is an ‘event risk’. There is a wide range of events potentially triggering losses6 . The very first step for addressing operational risk is to set up a common classification of events that should serve as a receptacle for data gathering processes on event frequencies and costs. Such taxonomy is still flexible and industry standards will emerge in the future. What follows is a tentative classification. Operational risks appear at different levels: 6 See

Marshall (2001) for a comprehensive view of operational risk and how to handle it for assessing potential losses.

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• • • •

21

People. Processes. Technical. Information technology.

‘People’ risk designates human errors, lack of expertise and fraud, including lack of compliance with existing procedures and policies. Process risk scope includes: • Inadequate procedures and controls for reporting, monitoring and decision-making. • Inadequate procedures on processing information, such as errors in booking transactions and failure to scrutinize legal documentation. • Organizational deficiencies. • Risk surveillance and excess limits: management deficiencies in risk monitoring, such as not providing the right incentives to report risks, or not abiding by the procedures and policies in force. • Errors in the recording process of transactions. • The technical deficiencies of the information system or the risk measures. Technical risks relate to model errors, implementation and the absence of adequate tools for measuring risks. Technology risks relate to deficiencies of the information system and system failure. For operational risks, there are sources of historical data on various incidents and their costs, which serve to measure the number of incidents and the direct losses attached to such incidents. Beyond external statistics, other proxy sources on operational events are expert judgments, questioning local managers on possible events and what would be their implications, pooling data from similar institutions and insurance costs that should relate to event frequencies and costs. The general principle for addressing operational risk measurement is to assess the likelihood and cost of adverse events. The practical difficulties lie in agreeing on a common classification of events and on the data gathering process, with several potential sources of event frequencies and costs. The data gathering phase is the first stage, followed by data analysis and statistical techniques. They help in finding correlations and drivers of risks. For example, business volume might make some events more frequent, while others depend on different factors. The process ends up with some estimate of worst-case losses due to event risks.

MODEL RISK Model risk is significant in the market universe, which traditionally makes relatively intensive usage of models for pricing purposes. Model risk is growing more important, with the extension of modelling techniques to other risks, notably credit risk, where scarcity of data remains a major obstacle for testing the reliability of inputs and models. This book details many modelling approaches, making model risk self-explanatory.

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Model risk materializes, for instance, as gaps between predicted values of variables, such as the VaR, and actual values observed from experience. Pricing models used for market instruments predict prices, which are readily comparable to observed prices. Often, modelled prices do not track actual prices accurately. When P&L calculations for the trading portfolio rely on ‘pricers’, there are ‘P&L leaks’ due to models. Such ‘P&L leaks’ represent the dollar value of model risk for ‘pricers’. Unfortunately, it is much easier to track gaps between predicted prices and actual prices than between modelled worst-case losses and actual large losses. The difficulty is even greater for credit risk, since major credit risk events remain too scarce. The sources of model risk, however, relate to common sense. Models are subject to misspecifications, because they ignore some parameters for practical reasons. Model implementation suffers from errors of statistical techniques, lack of observable data for obtaining reliable fits (credit risk) and judgmental choices on dealing with ‘outliers’, those observations that models fail to capture with reasonable accuracy. Credit risk models of interdependencies between individual defaults of borrowers rely on inferring such relations from ‘modelled’ events rather than observable events, because default data remains scarce. Consequently, measuring errors is not yet feasible until credit risk data becomes more reliable than it is currently. This is an objective of the New Basel Accord, from January 2001 (Chapter 3), making the data gathering process a key element in implementing the advanced techniques for measuring credit risk capital.

SECTION 2 Risk Regulations

3 Banking Regulations

Regulations have several goals: improving the safety of the banking industry, by imposing capital requirements in line with banks’ risks; levelling the competitive playing field of banks through setting common benchmarks for all players; promoting sound business and supervisory practices. Regulations have a decisive impact on risk management. The regulatory framework sets up the constraints and guidelines that inspire risk management practices, and stimulates the development and enhancement of the internal risk models and processes of banks. Regulations promote better definitions of risks, and create incentives for developing better methodologies for measuring risks. They impose recognition of the core concept of the capital adequacy principle and of ‘risk-based capital’, stating that banks’ capital should be in line with risks, and that defining capital requirements implies a quantitative assessment of risks. Regulations imposing capital charge against risks are a strong incentive to improve risk measures and controls. They set minimum standards for sound practices, while the banking industry works on improving the risk measures and internal models for meeting their own goals in terms of best practices of risk management. Starting from crude estimates of capital charges, with the initial so-called ‘Cooke ratio’, the regulators evolved towards increasingly ‘risk-sensitive’ capital requirements. The Existing Accord on credit risk dates from 1988. The Accord Amendment for market risk (1996) and the New Basel Accord (2001) provide very significant enhancements of risk measures. The existing accord imposed capital charge against credit risk. The amendment provided a standardized approach for dealing with market risk, and offered the opportunity to use internal models, subject to validation by the supervisory bodies, allowing banks to use their own models for assessing the market risk capital. The new accord imposes a higher differentiation of credit risk based on additional risk inputs characterizing banks’ facilities. By doing so, it paves the way towards internal credit risk modelling, already instrumental in major institutions.

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Regulations are inspired by economic principles, expanded on in subsequent chapters. However, regulations tend to strike a compromise between moving towards economic measures while being practical enough to be applicable by all players. This chapter describes the main features of regulations, and discusses the underlying economic rationale, as a transition towards subsequent developments. It does not provide a comprehensive and detailed presentation of regulations. Details are easily available from public documents on the Bank of International Settlements (BIS) site, and there is no need to replicate them. A number of dilemmas inspired the deregulation process in the eighties, followed by the reregulation wave. An ideal scheme would combine a minimal insurance mechanism without providing any incentive for taking more risk, and still foster competition among institutions. Old schemes failed to ensure system safety, as spectacular failures of the eighties demonstrate. New regulations make the ‘capital adequacy’ principle a central foundation, starting from the end of the eighties up to the current period. Regulatory requirements use minimum forfeit amounts of capital held against risks to ensure that capital is high enough to sustain future losses materializing from current risks. The main issue is to assess potential losses arising from current measures. Regulations provide sets of forfeits and rules defining the capital base as a function of banks’ exposures. The risk modelling approach addresses the same issue, but specifies all inputs and models necessary to obtain the best feasible economic estimates of the required capital. Regulations lag behind in sophistication because they need to set up rules that apply to all. As a result, regulatory capital does not coincide with modelled economic capital. The notable exception is market risk, since the 1996 Accord Amendment for market risk that allowed usage of internal models, subject to validation and supervision by the regulatory bodies. The New Accord of January 2001 progresses significantly towards more risksensitive measures of credit risk at the cost of an additional complexity. It proposes three approaches for defining capital requirements: the simplest is the ‘standardized’ approach, while the ‘foundation’ and the ‘advanced’ approach allow further refinements of capital requirements without, however, accepting internal models for credit risk. It has become obvious, when looking at regulatory accords, that the capital adequacy concept imposes continuous progress for quantifying potential losses, for measuring capital charges, making risk valuation a major goal and a major challenge for the industry. The regulatory system has evolved through the years. The 1996 Accord Amendment refined market risk measures, allowed internal models and is still in force. Capital requirements still do not apply to interest rate risk, although periodical enhanced reports are mandatory and the regulatory bodies can impose corrective actions. The last stage of the process is the New Basel Accord, published in early 2001, focusing on capital requirement definition (Pillar 1), with major enhancements to credit risk measures, a first coverage of operational risk, disclosure, supervisory review process (Pillar 2) and market discipline (Pillar 3). Simultaneously, it enhances some features of the 1988 Existing Accord, currently under implementation. The New Basel Accord target implementation date is early 2004. The first section discusses the basic issues and dilemmas that inspired the new regulation waves. The second section introduces the central concept of ‘capital adequacy’, which plays a pivotal role for regulations, quantitative risk analytics and risk management. The

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third section summarizes the rules of the Current Accord, which still applies, and the 1996 Amendment. Finally, the fourth section details the New Basel Accord in its January 2001 format, subject to revisions from banking industry feedback in mid-2001.

REGULATORY ISSUES The source of regulations lies in the differences between the objectives of banks and those of the regulatory authorities. Expected profitability is a major incentive for taking risks. Individual banks’ risks create ‘systemic risk’, the risk that the whole banking system fails. Systemic risk results from the high interrelations between banks through mutual lending and borrowing commitments. The failure of a single institution generates a risk of failure for all banks that have ongoing commitments with the defaulting bank. Systemic risk is a major challenge for the regulator. Individual institutions are more concerned with their own risk. The regulators tend to focus on major goals: • The risk of the whole system, or ‘systemic risk’, leading to pre-emptive actions against bank failures by the regulators, all inspired by the most important capital adequacy to risk principle. • Promoting a level playing field, without distortions in terms of unfair competitive advantages. • Promoting sound practices, contributing to the financial system safety. The regulators face several dilemmas when attempting to control risks. Regulation and competition conflict, since many regulations restrict the operations of banks. New rules may create unpredictable behaviour to turn around the associated constraints. For this reason, regulators avoid making brutal changes in the environment that would generate other uncertainties.

The Need for Regulation Risk taking is a normal behaviour of financial institutions, given that risk and expected return are so tightly interrelated. Because of the protection of bank depositors that exists in most countries, banks also benefit from ‘quasi-free’ insurance, since depositors cannot impose a real market discipline on banks. The banking system is subject to ‘moral hazard’: enjoying risk protection is an incentive for taking more risks because of the absence of penalties. Sometimes, adverse conditions are incentives to maximize risks. When banks face serious difficulties, the barriers that limit risks disappear. In such situations, and in the absence of aggressive behaviour, failure becomes almost unavoidable. By taking additional risks, banks maximize the chances of survival. The higher the risk, the wider the range of possible outcomes, including favourable ones. At the same time, the losses of shareholders and managers do not increase because of limited liability. In the absence of real downside risk, it becomes rational to increase risk. This is the classical attitude of a

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call option holder. He gets the upside of the bet with a limited downside. The potential for upside gain without the downside risk encourages risk taking because it maximizes the expected gains.

The Dilemmas of the Regulator A number of factors helped stabilize the banking environment in the seventies. Strong and constraining regulations weighed heavily on the banks’ management. Commercial banking meant essentially collecting resources and lending. Limited competition facilitated a fair and stable profitability. Concerns for the safety of the industry and the control of its money creation power were the main priorities for regulators. The rules limited the scope of the operations of the various credit institutions, and limited their risks as well. There were low incentives for change and competition. The regulators faced a safety–competition trade-off in addition to the negative side effects of risk insurance. In the period preceding deregulation, too many regulations segmented the banking industry, restraining competition. Deregulation increased competition between players unprepared by their past experiences, thereby resulting in increasing risks for the system. Reregulation aimed at setting up a regulatory framework reconciling risk control and fair competition. Regulation and Competition

Regulations limit the scope of operations of the various types of financial institutions, thereby interfering directly with free competition. Examples of inconsistency between competition and regulations have grown more and more numerous. A well-known example, in the United States in the early eighties, was the unfair competition between commercial banks, subject to regulation Q, imposing a ceiling on the interest paid on deposits, and investment bankers offering money market funds through interestearning products. Similar difficulties appear whenever rules enforce the segmentation of industry between those who operate in financial markets and others, savings and banking institutions, etc. The unfair competition stimulated the disappearance of such rules and the barriers to competition were progressively lifted. This old dilemma led to the deregulation of the banking industry. The seventies and the eighties were the periods of the first drastic waves of change in the industry. The disappearance of old rules created a vacancy. The regulators could not rely any longer on rules that segmented the industry as barriers to risk-taking behaviour. They started redefining new rules that could ensure the safety of the banking industry. The BIS (Bank for International Settlements) in Basel defined these new regulations, and national regulators relayed them for implementation within the national environments. Deregulation drastically widened the range of products and services offered by banks. Most credit institutions diversified their operations out of their original businesses. Moreover, the pace of creation of new products remained constantly high, especially for those acting in the financial markets, such as derivatives or futures. The active research for new

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market opportunities and products stimulated the growth of fields other than intermediation. Value-added services, such as advisory, structured transactions, asset acquisition, Leveraged Buy-Out (LBO), project finance, securitization of mortgages and credit card debts, derivatives and off-balance sheet operations, developed at a rapid pace. The banks entered new business fields and faced new risks. The market share of bank lending decreased with the development of capital markets and the competition within existing market shares rose abruptly. Those waves of change generated risks. Risks increased because of new competition, product innovations, a shift from commercial banking to capital markets, increased market volatility and the disappearance of old barriers. This was a radical change in the banking industry. Lifting existing constraints stimulates new competition and increases risks and failures for those players who are less ready to enter new businesses. Deregulation implies that players can freely enter new markets. The process necessitates an orderly and gradual progress to avoid bank management disruption. This is still an ongoing process, notably in the United States and Japan, where the Glass–Steagall Act and the Article 65 enforced a separation between commercial banking and the capital markets. For those countries where universal banking exists, the transition is less drastic, but leads to a significant restructuring of the industry. It is not surprising that risk management has emerged with such strong force at the time of these waves of transformations. Risk Control versus Risk Insurance

Risk control is pre-emptive, while insurance is passive, after the fact. The regulatory framework aims at making it an obligation to have a capital level matching risks. This is a pre-emptive policy. However, after the fact, there is still a need to limit the consequences of failures. Losses of depositors are the old consequences of bank failures. The consequences of bank failures are that they potentially trigger systemic risk, because of banks’ interdependencies, such as the recent LTCM collapse. Hence, there is a need for afterthe-fact regulatory action. If such actions rely on an insurance-like scheme, such as the insurance deposit scheme, it can foster risk-taking behaviour. Until reregulation through capital changes came into force, old regulations created insurance-based incentives for taking risks. Any rule that limits the adverse consequences of risk-taking behaviour is an incentive for risk taking rather than risk reduction. Regulators need to minimize adverse effects of failures for the whole system without at the same time encouraging the risk-taking behaviour. The ideal solution would be to control the risks without insuring them. In practice, there is no such ideal solution. The deposit insurance scheme protects depositors against bank failures. Nevertheless, it does generate the adverse side effect of encouraging risk taking. Theoretically, depositors should monitor the bank behaviour, just as any lender does. In practice, if depositors’ money were at risk, the functioning of the banking system would greatly deteriorate. Any sign of increased risks would trigger withdrawals of funds that would maximize the difficulties, possibly leading to failures. Such an adverse effect alone is a good reason to insure depositors, at least largely, against bank failure.

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The new regulation focuses on pre-emptive (ex ante) actions, while still enforcing after-the-fact (ex post) corrective actions.

CAPITAL ADEQUACY A number of rules, aimed at limiting risks in a simple manner, have been in force for a long time. For instance, several ratios are subject to minimum values. The liquidity ratio imposes that short-term assets be greater than short-term liabilities. Individual exposures to single borrowers are subject to caps. The main ‘pillar’ of the regulations is ‘capital adequacy’: by enforcing a capital level in line with risks, regulators focus on pre-emptive actions limiting the risk of failure. Guidelines are defined by a group of regulators meeting in Basel at the BIS, hence the name ‘Basel’ Accord. The process attempts to reach a consensus on the feasibility of implementing new complex guidelines by interacting with the industry. Basel guidelines are subject to some implementation variations from one country to another, according to the view of local supervisors. The first implemented accord focused on credit risk, with the famous ‘Cooke ratio’. The Cooke ratio sets up the minimum required capital as a fixed percentage of assets weighted according to their nature in 1988. The scope of regulations extended progressively later. The extension to market risk, with the 1996 Amendment, was a major step. The New Basel Accord of January 2001 considerably enhances the old credit risk regulations. The schedule of successive accords is as follows: • • • • • • •

1988 Current Accord published. 1996 Market Risk Amendment allowing usage of internal models. 1999 First Consultative Package on the New Accord. January 2001 Second Consultative Package. End-May 2001 Deadline for comments. End-2001 Publication of the New Accord. 2004 Implementation of the New Basel Capital Accord.

The next subsections refer to the 1988 ‘Current Accord’ plus the market risk capital regulations. The last section expands the ‘New Accord’ in its current ‘consultative’ form1 , which builds on the ‘Existing Accord’ on credit risk, while making capital requirements more risk-sensitive.

Risk-based Capital Regulations The capital base is not limited to equity plus retained earnings. It includes any debt subordinated to other commitments by the bank. Equity represents at least 50% of the total capital base for credit risk. Equity is also the ‘Tier 1’ of capital or the ‘core capital’. 1 The

consultation with industry expires by mid-2001, and a final version should follow.

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The Cooke ratio stipulates that the capital base should be at least 8% of weighted assets. The weights depend on the credit quality of the borrowers, as described below. For market risk, the rules are more complex because the regulations aim at a greater accuracy in capturing the economics of market risk, taking advantage of the widely available information on market parameters and prices. The 1996 Amendment for market risk opened the door to internal models that have the capability to value capital requirements with more accurate and sophisticated tools than simple forfeits. Regulations allow the usage of internal bank models for market risk only, subject to validation by supervisors. Market risk models provide ‘Value at Risk’ (VaR)-based measurements of economic capital, complying with specific guidelines for implementing models set by Basel, which substitute for regulatory forfeits. The New Accord of January 2001 paves the way for modelling credit risk, but recognizes that it is still of major importance to lay down more reliable foundations for inputs to credit risk models, rather than allowing banks to use model credit risk with unreliable inputs, mainly due to scarce information on credit risk.

The Implications of Capital Requirements Before detailing the regulations, the implications of risk-based capital deserve attention. Traditionally, capital represents a very small fraction of total assets of banks, especially when comparing the minimum requirements to similar ratios of non-financial institutions. A capital percentage of 8% of assets is equivalent to a leverage ratio (debt/equity ratio) of 92/8 = 11.5. This leverage ratio would be unsustainable with non-financial institutions, since borrowers would consider it as impairing too much the repayment ability and causing an increase in the bankruptcy risk beyond acceptable levels. The high leverage of banking institutions results from a number of factors. Economically, the discipline imposed by borrowers does not apply to depositors who benefit from deposit insurance programmes in all countries. Smooth operations of banks require easy and immediate access to financial markets, so that funding is not a problem, as long as the perceived risk by potential lenders remains acceptable. In addition, bank ratings by specialized agencies make risks visible and explicit.

The Function of Capital

The theoretical reason for holding capital is that it should provide protection against unexpected losses. Without capital, banks will fail at the first dollar of loss not covered by provisions. It is possible to compare regulatory forfeits to historical default rates for a first assessment of the magnitude of the safety cushion against losses embedded in regulatory capital. Default rates of corporate borrowers above 1% are speculative. Investment grade borrowers have yearly default rates between 0% and 0.1%. At the other extreme of the risk scale, corporate borrowers (small businesses) can have default rates ranging from 3% to 16% or more across the economic cycle. The 8% ratio of capital to weighted assets seems too high if compared to average observed default rates under good conditions, and perhaps too low with portfolios of high-risk borrowers when a downturn in economic conditions occurs. However, the 8%

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ratio does not aim at protecting banks against average losses. Provisions and reserves should, theoretically, take care of those. Rather, it aims at a protection against deviations from average. The ratio means that capital would cover deviations of losses, in excess of loan loss provisions, up to 8%, given the average portfolio structure of banks. However, without modelling potential portfolio losses, it is impossible to say whether this capital in excess of provisions is very or moderately conservative. Portfolio models provide ‘loss distributions’ for credit risk, the combinations of loss values plus the probability that losses hit each level, and allow us to see how likely various loss levels are, for each specific portfolio. In addition, loss provisioning might not be conservative enough, so that capital should also be sufficient to absorb a fraction of average losses in addition to loss in excess of average. Legal reserves restrain banks from economic provisioning, as well as window dressing or the need to maximize retained earnings that feed into capital. Economic provisioning designates a system by which provisions follow ex ante estimates of what the future average losses could be, while actual provisions are more ex post, after the fact. Still, regulatory capital is not a substitute for provisions. The Dilemma between Risk Controlling and Risk Taking

There is a trade-off between risk taking and risk controlling. Available capital puts a limit on risk taking which, in turn, limits the ability to develop business. This generates a conflict between business goals, in terms of volume and risks, and the limits resulting from available capital. The available capital might not suffice to sustain new business developments. Capital availability could result in tighter limits for account officers and traders, and for the volume of operations. On the other hand, flexible constraints allow banks to take more risks, which leads to an increase in risk for the whole system. Striking the right balance between too much risk control and not enough is not an easy task. Risk-based Capital and Growth

The capital ratio sets the minimum value of capital, given the volume of operations and their risks. The constraint might require raising new equity, liquidation of assets or risk reduction actions. Raising additional capital requires that the profitability of shareholders is in line with their expectations. When funding capital growth through retained earnings only, the profitability should be high enough for the capital to grow in line with the requirements. These are two distinct constraints. A common proxy for the minimum required return of shareholders is an accounting Return On Equity (ROE) of 15% after tax, or 25% before tax. When the target is funding capital, the minimum accounting ROE should be in line with this funding constraint. The higher the growth of the volume of operations, the higher the increase in required capital (if the ‘average’ risk level remains unchanged). Profitability limits the sustainable growth if outside sources of capital are not used. The ROE becomes equal to the growth rate of

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capital if no dividends are paid2 . Therefore, any growth above ROE is not sustainable when funding through retained earnings only, under constant average risk. The first implication of capital requirements is that the ROE caps the growth of banks when they rely on retained earnings only. The second implication is that capital should provide the required market return to stockholders given the risk of the bank’s stocks. Sometimes, the bank growth is too high to be sustainable given capital requirements. Sometimes, the required compensation to shareholders is the ‘effective’ constraint, while funding capital growth internally is not3 . The only options available to escape the capital constraints are to liquidate assets or to reduce risks. Direct sales of assets or securitizations serve this purpose, among others. Direct loan sales become feasible through the emerging loan trading market. Chapter 60 discusses structured transactions as a tool for capital management.

THE ‘CURRENT ACCORD’ CAPITAL REGULATIONS The regulation is quite detailed in order to cover all specific situations. Three sections covering credit risk, market risk and interest rate risk summarize the main orientations.

The Cooke Ratio and Credit Risk The 1988 Accord requires internationally active banks in the G10 countries to hold capital equal to at least 8% of weighted assets. Banks should hold at least half of its measured capital in Tier 1 form. The Cooke ratio addresses credit risk. Tier 1 capital is subject to a minimum constraint of 3% of total assets. The calculation of the ratio uses asset weights for differentiating the capital load according to quality in terms of credit standing. The weight scale starts from zero, for commitments with public counterparties, up to 100% for private businesses. Other weights are: 20% for banks and municipalities within OECD countries; 50% for residential mortgage backed loans. Off-balance sheet outstanding balances are weighted at 50%, in combination with the above weights scale. The 1988 Accord requires a two-step approach whereby banks convert their off-balance sheet positions into a credit equivalent amount through a scale of conversion factors, which are then weighted according to the counterparty’s risk weighting. The factor is 100% for direct credit substitutes such as guarantees and decreases for less stringent commitments. 2 If

the leverage is constant, capital being a fixed fraction of total liabilities, the growth rate of accounting capital is also the growth rate of the total liabilities. Leverage is the debt/equity ratio, or L = D/E where D is debt, E is equity plus retained earnings and L is leverage. The ROE is the ratio of net income (NI) to equity and retained earnings, or ROE = N I /E. ROE is identical to the sustainable growth rate of capital since it is the ratio of potential additional capital (net income NI) to existing capital. Since L is constant, total liabilities are always equal to (1 + L) × E. Hence, they grow at the same rate as E, which is identical to ROE under assumptions of no dividends and no outside capital. 3 In fact, both constraints are equivalent when the growth rate is exactly 15%, if this figure is representative of both accounting return on equity and the required market return to shareholders.

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The major strength of the Cooke ratio is its simplicity. Its major drawbacks are: • The absence of differentiation between the different risks of private corporations. An 8% ratio applying for both a large corporation rated ‘Aa’ (an investment grade rating) and a small business does not make much sense economically. This is obvious when looking at the default rates associated with high and low ratings. The accord is not risk-sensitive enough. • In addition, short facilities have zero weights, while long facilities have a full capital load, creating arbitrage opportunities to reduce the capital load. This unequal treatment leads to artificial arbitrage by banks, such as renewing short loans rather than lending long. • There is no allowance for recoveries if a default occurs, even in cases where recoveries are likely, such as for cash or liquid securities collateral. • Summing arithmetically capital charges of all transactions does not capture diversification effects. In fact, there is an embedded diversification effect in the 8% ratio since it recognizes that the likelihood of losses exceeding more than 8% of weighted assets is very low. However, the same ratio applies to all portfolios, whatever their degree of diversification. The regulators recognized these facts and reviewed these issues, leading to the New Basel Accord detailed in a subsequent dedicated section.

Market Risk A significant amendment was enacted in 1995–6, when the Committee introduced a measure whereby trading positions in bonds, equities, foreign exchange and commodities were removed from the credit risk framework and given explicit capital charges related to the bank’s open position in each instrument. Capital requirements extended explicitly to market risk. The amendment made explicit the notions of banking book and trading book, defined capital charges for market risk and allowed banks to use Tier 3 capital in addition to the previous two tiers. Scope

Market risk is the risk of losses in on- and off-balance sheet positions arising from movements in market prices. The risks subject to this requirement are: • The risks pertaining to interest rate-related instruments and equities in the trading book. • Foreign exchange risk and commodities risk throughout the bank. The 1995 proposal introduced capital charges to be applied to the current market value of open positions (including derivative positions) in interest rate-related instruments and equities in banks’ trading books, and to banks’ total currency and commodities positions.

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The extension to market risk provides two alternative techniques for assessing capital charges. The ‘standardized approach’ allows measurement of the four risks: interest rate, equity position, foreign exchange and commodity risks, using sets of forfeits. Under the standardized approach, there are specific forfeits and rules for defining to which base they apply, allowing some offsetting effects within portfolios of traded instruments. Offsetting effects reduce the base for calculating the capital charge by using a net exposure rather than gross exposures. Full netting effects apply only to positions subject to an identical underlying risk or, equivalently, a zero basis risk4 . For instance, it is possible to offset opposite positions in the same stocks or the same interest rates. The second method allows banks to use risk measures derived from their own internal risk management models, subject to a number of conditions, related to qualitative standards of models and processes. The April 1995 proposal allowed banks to use new ‘Tier 3’ capital, essentially made up of short-term subordinated debt to meet their market risks. Tier 3 capital is subject to a number of conditions, such as being limited to market risk capital and being subject to a ‘lock-in clause’, stipulating that no such capital can be repaid if that payment results in a bank’s overall capital being lower than a minimum capital requirement. The Basel extension to market risk relies heavily on the principles of sound economic measures of risks, although the standardized approach falls short of such a target by looking at a feasible compromise between simplicity and complexities of implementation. Underlying Economic Principles for Measuring Market Risks

The economics of market risk are relatively easy to grasp. Tradable assets have random variations of which distributions result from observable sensitivities of instruments and volatilities of market parameters. The standalone market risk of an instrument is a VaR (or maximum potential loss) resulting from Profit and Loss (P&L) distribution derived from the underlying market parameter variations. Things get more involved for portfolios because risks offset to some extent. The ‘portfolio effect’ reduces the portfolio risk, making it lower than the sum of all individual standalone risks. For example, long and short exposures to the same instrument are exactly offset. Long and short exposures to similar instruments are also partially offset. The price movements of a 4-year bond and a 5-year bond correlate because the 4-year and 5-year interest rates do. However, the correlation is not perfect since a 1% change in the former does not mechanically trigger a 1% change in the latter. There is a case for offsetting exposures of long and short positions in similar instruments, but only to a certain extent. A mechanical offset would leave out of the picture the residual risk resulting from the mismatch of the instrument characteristics. This residual risk is a ‘basis risk’ resulting from the mismatch between the reference rates applying to each instrument. Moving one step further, let us consider a common underlying flat rate across all maturities, with a maturity mismatch between opposing exposures in similar instruments. Any shift in the entire spectrum of interest rates by the same amount (a parallel shift of 1% for instance) does not result in the same price change for the two instruments because 4 ‘Basis

risk’ exists when the underlying sources of risk differ, even if only slightly.

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their sensitivities differ. The magnitudes of the two price changes vary across instruments as sensitivities to common parameters do. With different instruments, such as equity and bonds, a conservative view would consider that all prices move adversely simultaneously. This makes no sense because the market parameters—interest rates, foreign exchange rates and equity indexes—are interdependent. Some tend to move in the same direction, others in opposite directions. In addition, the strength of such ‘associations’ varies across pairs of market parameters. The statistical measure for such ‘associations’ is the correlation. Briefly stated, market parameters comply with a correlation structure, so that the assumption of simultaneous adverse co-movements is irrelevant. When trying to assess how prices change within a portfolio, the issue is: how to model co-movements of underlying parameters in conjunction with sensitivities differing across exposures. This leads to the basic concepts of ‘general’ versus ‘specific’ risk. General risk is the risk dependent on market parameters driving all prices. This dependence generates price co-movements, or price correlations, because all prices depend, to some extent, on a common set of market parameters. The price volatility unrelated to market parameters is the specific risk. By definition, it designates price variations unrelated to market parameter variations. Statistically, specific risk is easy to isolate over a given period. It is sufficient to relate prices to market parameters statistically to get a direct measure of general risk. The remaining fraction of the price volatility is the specific risk. The Basel Committee offer two approaches for market risk: • The standardized approach relies on relatively simple rules and forfeits for defining capital charges, which avoids getting into the technicalities of modelling price changes. • The proprietary model approach allows banks to use internal full-blown models for assessing market risk VaR, subject to qualifying conditions. The critical inputs include: • The current valuation of exposures. • Their sensitivities to the underlying market parameter(s)5 . • Rules governing offsetting effects between opposing exposures in ‘similar’ instruments and, eventually, across the entire range of instruments. Full-blown models should capture all valuation, sensitivity and correlation effects. The ‘standardized’ approach for market risk uses forfeits and allows partial diversification effects, striking a balance between complexities and measurement accuracy. The Standardized Approach

The standardized approach relies on forfeits for assessing capital charges as a percentage of current exposures, on grids for capturing the differences in sensitivities of various market instruments and on offsetting rules allowing netting of the risks within a portfolio 5 Sensitivities

are variations of values for a unit change in market parameter, as detailed in Chapter 6.

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whenever there is no residual ‘basis’ risk. Forfeits capture conservatively the potential adverse deviations of instruments. Forfeits vary across market compartments and products, such as bonds of different maturities, because their sensitivity changes. For stocks, forfeits are 8% of the net balances, after allowed offsetting of long and short comparable exposures. For bonds, forfeits are also 8%, but various weights apply to this common ratio, based upon maturity buckets. Adding the individual risks, without offsetting exposures, overestimates the portfolio risk because the underlying assumption is that all adverse deviations occur simultaneously. Within a given class of instruments, such as bonds, equity or foreign exchange, regulators allow offsetting risks to a certain extent. For instance, being long and short on the same stock results in zero risk, because the gain in the long leg offsets the loss in the short leg when the stock goes up, and vice versa. Offsetting is limited to exact matches of instrument characteristics. In other instances, regulators rely on the ‘specific’ versus ‘general’ risk distinction, following the principle of adding specific risk forfeits while allowing limiting offsetting effects for general risk. The rationale is that general risk refers to comovements of prices, while specific risk is unrelated to underlying market parameters. Such conservative measures provide a strong incentive to move to the modelling approach that captures portfolio effects. These general rules differ by product family. What follows provides only an overview of the main product families. All forfeit values and grids are available from the BIS documents. Interest Rate Instruments For interest rate instruments, the capital charge adds individual transaction forfeits, varying from 0% (government) to 8% for maturities over 24 months. Offsetting specific risks is not feasible except for identical debt issues. The capital requirement for general market risk captures the risk of loss arising from changes in market interest rates. There are two principal methods for measuring general risk: a ‘maturity’ method and a ‘duration’ method. The maturity method uses 13 time bands for assigning instruments. It aims at capturing the varying sensitivities across time bands of instruments. Offsetting long and short positions within time bands is possible, but there are additional residual capital charges applying because of intra-band maturity mismatches. Since interest rates across time bands correlate as well, partial offsets across time bands are possible. The procedure uses ‘zones’ of maturity buckets grouping the narrower time bands. Offsets decrease with the distance between the time bands. The accord sets percentages of exposures significantly lower than 1, to cap the exposures allowed to offset (100% implying a total offset of the entire exposure). The duration method allows direct measurement of the sensitivities, skipping the time band complexity by using the continuous spectrum of durations. It is necessary to assign sensitivities to a duration-based ‘ladder’, and within each slot, the capital charge is a forfeit. Sensitivities in values should refer to preset changes of interest rates, whose values are within the 0.6% to 1.00% range. Offsetting is subject to a floor for residual risk (basis risk, since there are duration mismatches within bands).

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Derivatives Derivatives are combinations of underlying exposures, and each component is subject to the same treatments as above. For example, a swap is a combination of two exposures, the receiving leg and the paying leg. Therefore, forfeit values add unless the absence of any residual risk allows full offsets. There is no specific risk for derivatives. For options, there are special provisions based on sensitivities to the various factors that influence their prices, and their changes with these factors. For options, the accord proposes the scenario matrix analysis. Each cell of the matrix shows the portfolio value given a combination of scenarios of underlying asset price and volatility (stressed according to regulators’ recommendations). The highest loss value in the matrix provides the capital charge for the entire portfolio of options. Equity The basic distinction between specific versus general risk applies to equity products. The capital charge for specific risk is 8%, unless the portfolio is both liquid and well diversified, in which case the charge is 4%. Offsetting long and short exposures is feasible for general risk, but not for specific risk. The general market risk charge is 8%. Equity derivatives should follow the same ‘decomposition’ rule as underlying exposures. Foreign Exchange Two processes serve for calculating the capital requirement for foreign exchange risk. The first is to measure the exposure in a single currency position. The second is to measure the risks inherent in a bank’s mix of long and short positions in different currencies. Structural positions are exempt of capital charges. They refer to exposures protecting the bank from movements of exchange rates, such as assets and liabilities in foreign currencies remaining matched. The ‘shorthand method’ treats all currencies alike, and applies the forfeited 8% capital charge over the greatest netted exposure (either long or short). Commodities Commodity risk is more complex than market instrument risk because it combines a pure commodity price risk with other risks, such as basis risk (mismatch of prices of similar commodities), interest rate risk (for carrying cost of exposures) and forward price risk. In addition, there is directional risk in commodities prices. The principle is to assign transactions to maturity buckets, to allow offsetting of matched exposures, and to assign a higher capital charge for risk. Proprietary Models of Market Risk VaR

The principles of the extension to proprietary market risk models are as follows:

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• Market risk is the risk of loss during the minimum period required to liquidate transactions in the market. • The potential loss is the 99th loss percentile6 (one-tailed) for market risk models. • This minimum period depends on the type of products, and on their sensitivities to a given variation of their underlying market parameters. However, a general 10-day period for liquidating position is the normal reference for measuring downside risk. • Potential losses depend on market movements during this period, and the sensitivities of different assets. Models should incorporate historical observation over at least 1 year. In addition, the capital charge is the higher of the previous day’s VaR or three times the average daily VaR of the preceding 60 days. A multiplication factor applies to this modelled VaR. It accounts for potential weaknesses in the modelling process or exceptional circumstances. In addition, the regulators emphasize: • Stress-testing, to see what happens under exceptional conditions. Stress-testing uses extreme scenarios maximizing losses to find out how large they can be. • Back-testing of models to ensure that models capture the actual deviations of the portfolios. Back testing implies using the model with past data to check whether the modelled deviations of values are in line or not with the historical deviations of values, once known. These additional requirements serve to modulate the capital charge, through the multiplier of VaR, according to the reliability of the model, in addition to fostering model improvements through time. Reliable models get a ‘premium’ in capital with a lower multiplier of VaR.

Derivatives and Credit Risk Derivatives are over-the-counter instruments (interest rate swaps, currency swaps, options) not liquid as market instruments. Theoretically, banks hold these assets until maturity, and bear credit risk since they exchange flows of funds with counterparties subject to default risk. For derivatives, credit risk interacts with market risk in that the mark-to-market (liquidation) value depends on market movements. It is the present value of all future flows at market rates. Under a ‘hold to maturity’ view, the potential future values over their life is the credit risk exposure because they are the value of all future flows that the defaulted counterparty will not pay. Future liquidation values are random because they are market-driven. To address random exposures over a long-term period, it is necessary to model the time profile of liquidation values. The principle is to determine the time profile of upper bounds that future values will not exceed with more than a preset low probability (confidence level) (see Chapter 39). From the regulatory standpoint, the issue is to define applicable rules as percentage forfeits of the notional of derivatives. The underlying principles for defining such forfeits are identical to those of models. The current credit risk exposure is the current liquidation value. There is the additional risk due to the potential upward deviations of liquidation 6A

loss percentile is the loss value not exceeded in more than the given preset percentage, such as 99%. See Chapter 7.

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value from the current value during the life of the instruments. Such drifts depend on the market parameter volatilities and on instrument sensitivities. It is feasible to calibrate forfeit values representative of such drifts on market observations. The regulators provided forfeits for capturing these potential changes of exposures. The forfeit add-ons are percentages of notional depending on the underlying parameter and the maturity buckets. Underlying parameters are interest rate, foreign exchange, equity and commodities. The maturity buckets are up to 1 year, between 1 and 5 years and beyond 5 years. The capital charge adds arithmetically individual forfeits. Forfeit add-ons are proxies and adding them does not allow offsetting effects, thereby providing a strong incentive for modelling derivative exposures and their offsetting effects. The G30 group, in 1993, defined the framework for monitoring derivative credit risk. The G30 report recommended techniques for assessing uncertain exposures, and for modelling the potential unexpected deviations of portfolios of derivatives over long periods. The methodology relies on internal models to capture worst-case deviations of values at preset confidence levels. In addition, it emphasized the need for enhanced reporting, management control and disclosure of risk between counterparties.

Interest Rate Risk (Banking Portfolio) Interest rate risk, for the banking portfolio, has generated debate and controversy. The current regulation does not require capital to match interest rate risk. Nevertheless, regulators require periodical internal reports to be made available. Supervising authorities take corrective actions if they feel them required. Measures of interest rate risk include the sensitivity of the interest income to shifts in interest rates and that of the net market value of assets and liabilities. The interest margin is adequate in the short-term. Market value sensitivities of both assets and liabilities capture the entire stream of future flows and provide a more long-term view. Regulators recommend using both. Subsequent chapters discuss the adequate models for such sensitivities, gaps and net present value of the balance sheet and its duration (see Chapters 21 and 22).

THE NEW BASEL ACCORD The New Basel Accord is the set of consultative documents that describes recommended rules for enhancing credit risk measures, extending the scope of capital requirements to operational risk, providing various enhancements to the ‘existing’ accord and detailing the ‘supervision’ and ‘market discipline’ pillars. The accord allows for a 3-year transition period before full enforcement, with all requirements met by banks at the end of 2004. Table 3.1 describes the rationale of the New Accord. The new package is very extensive. It provides a menu of options, extended coverage and more elaborate measures, in addition to descriptions of work in progress, with yet unsettled issues to be streamlined in the final package. The New Accord comprises three pillars: • Pillar 1 Minimum Capital Requirements.

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• Pillar 2 Supervisory Review Process. • Pillar 3 Market Discipline. TABLE 3.1 Rationale for a new accord: the need for more flexibility and risk sensitivity Existing Accord Focuses on a single risk measure One size fits all: only one option proposed to banks Broad brush structure (forfeits)

Proposed New Accord More emphasis on banks’ own internal methodologies, supervisory review and market discipline Flexibility, menu of approaches, incentives: banks have several options More credit risk sensitivity for better risk management

The Committee emphasizes the mutually reinforcing role of these three pillars—minimum capital requirements, supervisory review and market discipline. Taken together, the three pillars contribute to a higher level of safety and soundness in the financial system. Previous implementations of the regulations for credit and market risk, confirmed by VaR models for both risks, revealed that the banking book generates more risk than the trading book. Credit risk faces a double challenge. The measurement issues are more difficult to tackle for credit risk than for market risk and, in addition, the tolerance for errors should be lower because of the relative sizes of credit versus market risk.

Overview of the Economic Contributions of the New Accord The economic contributions of the New Accord extend to supervisory processes and market discipline. Economically, the New Accord appears to be a major step forward. On the quantitative side of risk measurements, the accord offers a choice between the ‘standardized’, the ‘foundation’ and the ‘advanced’ approaches, addresses and provides remedies for several critical issues that led banks to arbitrage the old capital requirements against more relevant economic measures. It remedies the major drawbacks of the former ‘Cooke ratio’, with a limited set of weights and a unique weight for all risky counterparties of 100%. Neither the definition of capital and Tier 1/Tier 2, nor the 8% coefficient change. The new set of ratios (the ‘McDonough ratios’) corrects this deficiency. Risk weights define the capital charge for credit risk. Weights are based on credit risk ‘components’, allowing a much improved differentiation of risks. They apply to some specific cases where the old weights appeared inadequate, such as for credit enhancement notes issued for securitization which, by definition, concentrate a major fraction of the securitized assets7 . Such distortions multiplied the discrepancies between economic risk-based policies and regulatory-based policies, leading banks to arbitrage the regulatory weights. In addition, banks relied on regulatory weights and forfeits not in line with risks, leading to distortion in risk assessment, prices, risk-adjusted measures of performance, etc. 7 The

mechanisms of securitizations are described in Chapter 34, and when detailing the economics of securitizations.

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The New Accord provides a more ‘risk-sensitive’ framework that should considerably reduce such distortions. In addition, it makes clear that data gathering on all critical credit risk drivers of transactions is a major priority, providing a major incentive to solve the ‘incompleteness’ of credit risk data. It also extends the scope of capital requirements to operational risk. Measures of operational risk remained in their ‘infancy’ for some time because of a lack of data, but the accord should stimulate rapid progress. From an economic standpoint, the accord suffers from the inaccuracy limitations of forfeit-based measures, a limitation mitigated by the need to strike a balance between accuracy and practicality. The accord does not go as far as authorizing the usage of credit risk VaR models, as it did for market risk. It mentions, as reasons for refraining from going that far, the difficulties of establishing the reliability of both inputs and outputs of such models. Finally, it still leaves aside interest rate risk for capital requirements and includes it in the supervision process (Pillar 2).

Pillar 1: Overall Minimum Capital Requirements The primary changes to the minimum capital requirements set out in the 1988 Accord are the approaches to credit risk and the inclusion of explicit capital requirements for operational risk. The accord provides a range of ‘risk-sensitive options’ for addressing both types of risk. For credit risk, this range includes the standardized approach, with the simplest requirements, and extends to the ‘foundation’ and ‘advanced’ Internal RatingsBased (IRB) approaches. Internal ratings are assessments of the relative credit risks of borrowers and/or facilities, assigned by banks8 . The Committee desires to produce neither a net increase nor a net decrease—on average—in minimum regulatory capital. With respect to the IRB approaches, the Committee’s ultimate goal is to improve regulatory capital adequacy for underlying credit risks and to provide capital incentives relative to the standardized approach through lower risk weights, on average, for the ‘foundation’ and the ‘advanced’ approaches. Under the New Accord, the denominator of the minimum total capital ratio will consist of three parts: the sum of all risk-weighted assets for credit risk, plus 12.5 times the sum of the capital charges for market risk and operational risk. Assuming that a bank has $875 of risk-weighted assets, a market risk capital charge of $10 and an operational risk capital charge of $20, the denominator of the total capital ratio would equal 875 + [(10 + 20) × 12.5] or $1250.

Risk Weights under Pillar 1 The New Accord strongly differentiates risk weights using a ‘menu’ of approaches designated as ‘standardized’, ‘foundation’ and ‘advanced’. 8 See

Chapters 35 and 36 on rating systems and credit data by rating class available from rating agencies.

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The Standardized Approach

In the standardized approach to credit risk, exposures to various types of counterparties—sovereigns, banks and corporates—have risk weights based on assessments (ratings) by External Credit Assessment Institutions (ECAIs or rating agencies) or Export Credit Agencies (ECAs) for sovereign risks. The risk-weighted assets in the standardized approach are the product of exposure amounts and supervisory determined risk weights. As in the current accord, the risk weights depend on the category of the borrower: sovereign, bank or corporate. Unlike in the current accord, there will be no distinction on the sovereign risk weighting depending on whether or not the sovereign is a member of the Organization for Economic Coordination and Development (OECD). Instead, the risk weights for exposures depend on external credit assessments. The treatment of off-balance sheet exposures remains largely unchanged, with a few exceptions. To improve risk sensitivity while keeping the standardized approach simple, the Committee proposes to base risk weights on external credit assessments. The usage of the supervisory weights is the major difference from the IRB approach, which relies on internal ratings. The approach is more risk-sensitive than the existing accord, through the inclusion of an additional risk bucket (50%) for corporate exposures, plus a 150% risk weight for low rating exposures. Unrated exposures have a 100% weight, lower than the 150% weight. The higher risk bucket (150%) also serves for certain categories of assets. The standardized approach does not allow weights to vary with maturity, except in the case of short-term facilities with banking counterparties in the mid-range of ratings, where weights decrease from 50% to 20% and from 100% to 50%, depending on the rating9 . The unrated class at 150% could trigger ‘adverse selection’ behaviour, by which lowrated entities give up their ratings to benefit from a risk weight of 100%, rather than 150%. On the other hand, the majority of corporates—and, in many countries, the majority of banks—do not need to acquire a rating in order to fund their activities. Therefore, the fact that a borrower has no rating does not generally signal low credit quality. The accord attempts to strike a compromise between these conflicting facts and stipulates that national supervisors have some flexibility in adjusting this weight. The 150% weight remains subject to consultation with the banking industry as of current date. For sovereign risks, the external ratings are those of an ECAI or ECA. For banks, the risk weights scale is the same, but the rating assignment might follow either one of two processes. Either the sovereign rating is one notch lower than the sovereign ratings, or it is the intrinsic bank rating. In Tables 3.2 and 3.3 we provide the grids of risk weights for sovereign ratings and corporates. The grid for banks under the first option is identical to that of sovereign ratings. TABLE 3.2

Risk weights

9 By

Risk weights of sovereigns AAA to AA−

A+ to A−

BBB+ to BBB−

BB+ to B−

Below B−

Unrated

0%

20%

50%

100%

150%

100%

contrast, the advanced IRB approach makes risk weights sensitive to maturity and ratings through the default probabilities.

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TABLE 3.3

Risk weights

Risk weights of corporates AAA to AA−

A+ to A−

BBB+ to BB−

Below BB−

Unrated

20%

50%

100%

150%

100%

Source: Basel Committee, January 2001.

Internal Ratings-based Framework

The IRB approach lays down the principles for evaluating economically credit risk. It proposes a treatment similar to the standardized approach for corporate, bank and sovereign exposures, plus separate schemes for retail banking, project finance and equity exposures. There are two versions of the IRB approach: the ‘foundation’ and the ‘advanced’ approaches. For each exposure class, the treatment uses three main elements called ‘risk components’. A bank can use either its own estimates for each of them or standardized supervisory estimates, depending on the approach. A ‘risk-weight function’ converts the risk components into risk weights for calculating risk-weighted assets. The risk components are the probability of default (PD), the loss given default (Lgd) and the ‘Exposure At Default’ (EAD). The PD estimate must represent a conservative view of a long-run average PD, ‘through the cycle’, rather than a short-term assessment of risk. Risk weights are a function of PD and Lgd. Maturity is a credit risk component that should influence risk weights, given the objective of increased risk sensitivity. However, the inclusion of maturity creates additional complexities and a disincentive for lending long-term. Hence, the accord provides alternative formulas for inclusion of maturity, which mitigate a simple mechanical effect on capital. The BIS proposes the Benchmark Risk Weight (BRW) for including the maturity effect on credit risk and capital weights. The function depends on the default probability DP. The benchmark example refers to the specific case of a 3-year asset, with various default probabilities and an Lgd of 50%. Three representative points show the sensitivity of risk weights to the annualized default probability (Table 3.4). TABLE 3.4 Sensitivity of risk weights with maturity: benchmark case (3-year asset, 50% Lgd) DP (%) BRW (%)

0.03 14

0.7 100

20 625

For DP = 0.7%, the BRW is 100% and the maximum risk weight, for DP = 20%, reaches 625%. This value is a cap for all maturities and all default probabilities. The weight profile with varying DP is more sensitive than the standardized approach weights, which vary in the range 20% to 150% for all maturities over 1 year. The weights increase less than proportionally with default probability until they reach the cap.

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The New Accord suggests using a forfeit maturity of 3 years for all assets for the ‘foundation’ approach, but leaves the door open to the usage of effective maturity. Risk weights adjusted for effective maturity apply to the ‘advanced’ approach. Retail Exposures

The Committee proposes a treatment of retail portfolios, based on the conceptual framework outlined above, modified to capture the specifics of retail exposures. Fixed rating scales and the assignment of borrower ratings are not standard practices for retail banking (see Chapters 37 and 38). Rather, banks commonly divide the portfolio into ‘segments’ made up of exposures with similar risk characteristics. The accord proposes to group retail exposures into segments. The assessment of risk components will be at the segment level rather than at the individual exposure level, as is the case for corporate exposures. In the case of retail exposures, the accord also proposes, as an alternative assessment of risk, to evaluate directly ‘expected loss’. Expected loss is the product of DP and Lgd. This approach bypasses the separate assessment, for each segment, of the PD and Lgd. The maturity (M) of the exposure is not a risk input for retail banking capital. Foundation Approach

The ‘foundation’ IRB approach allows banks meeting robust supervisory standards to input their own assessment of the PD associated with the obligor. Estimates of additional risk factors, such as loss incurred by the bank given a default (Lgd) and EAD, should follow standardized supervisory estimates. Exposures not secured by a recognized form of collateral will receive a fixed Lgd depending on whether the transaction is senior or subordinated. The minimum requirements for the foundation IRB approach relate to meaningful differentiation of credit risk with internal ratings, the comprehensiveness of the rating system, the criteria of the rating system and similar. There are a variety of methodologies and data sources that banks may use to associate an estimate of PD with each of its internal grades. The three broad approaches are: use of data based on a bank’s own default experience; mapping to external data such as those of ‘ECA’; use of statistical default models. Hence, a bank can use the foundation approach as long as it maps, in a sound manner, its own assessment of ratings with default probabilities, including usage of external data. Advanced Approach

A first difference with the ‘foundation’ approach is that the bank assesses the same risk components plus the Lgd parameter characterizing recoveries. The subsequent presentation of credit risk models demonstrates that capital is highly sensitive to this risk input. In general, banks have implemented ratings scales for some time, but they lack risk data on recoveries. The treatment of maturity also differs from the ‘foundation’ approach, which refers to a single benchmark for all assets. BRWs depend on maturity in the ‘advanced’ approach. The cap of 625% still applies as in the forfeit 3 years to maturity. The maturity effect depends on an annualized default probability as in the ‘foundation’ approach, plus a term b in a BRW function of DP and maturity depending on the effective maturity of assets. This

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is a more comprehensive response to the need to make capital sensitive to the maturity effect. It strikes a compromise between ‘risk sensitivity’ and the practical requirement of avoiding heavy capital charges for long-term commitments that would discourage banks from entering into such transactions. As a remark, credit risk models (see Chapter 42) capture maturity effects through revaluation of facilities according to their final risk class at a horizon often set at 1 year. The revaluation process at the horizon does depend on maturity since it discounts future cash flows from loans at risk-adjusted discount rates. Nevertheless, the relationship depends on the risk and excess return of the asset over market required yields, as explained in Chapter 8. Consequently, there is no simple relation between credit risk and maturity. Presumably, the BRW function combines contributions of empirical data of behaviour over time of historical default frequencies and relations between capital and maturity from models. However, the Committee is stopping short of permitting banks to calculate their capital requirements based on their own portfolio credit risk models. Reasons are the current lack of reliability of inputs required by such models, plus the difficulty of demonstrating the reliability of model capital estimates. However, by imposing the build-up of risk data for the next 3 years, the New Accord paves the way for a later implementation. Given the difficulty of assessing the implications of these approaches on capital requirements, the Committee imposes some conservative guidelines such as a floor on required capital. The Committee emphasizes the need for banks to anticipate regulatory requirements by performing stress testing and establishing additional capital cushions of their own (i.e. through Pillar 2) during periods of economic growth. In the longer run, the Committee encourages banks to consider the merits of building such stress considerations directly into their internal ratings framework. Guarantees

The New Accord grants greater recognition of credit risk mitigation techniques, including collateral, guarantees and credit derivatives, and netting. The new proposals provide capital reductions for various forms of transactions that reduce risk. They also impose minimum operational standards because a poor management of operational risks—including legal risks—would raise doubts with respect to the actual value of such mitigants. Further, banks are required to hold capital against residual risks resulting from any mismatch between credit risk hedges and the corresponding exposure. Mismatches refer to differences in amounts or maturities. In both cases, capital requirements will apply to the residual risks. Collateral The Committee has adopted for the standardized approach a definition of eligible collateral that is broader than that in the 1988 Accord. In general, banks can recognize as collateral: cash; a restricted range of debt securities issued by sovereigns, public sector entities, banks, securities firms and corporates; certain equity securities traded on recognized exchanges; certain mutual fund holdings; gold. For collateral, it is necessary to account for time changes of exposure and collateral values. ‘Haircuts’ define the required excess collateral over exposure to ensure effective

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credit risk protection, given time periods necessary for readjusting the collateral level (re-margining), recognizing the counterparty’s failure to pay or to deliver margin and the bank’s ability to liquidate collateral for cash. Two sets of haircuts have been developed for a comprehensive approach to collateral: those established by the Committee (i.e. standard supervisory haircuts); others based on banks’ ‘own estimates’ of collateral volatility subject to minimum requirements (see Chapter 41). There is a capital floor, denoted w, whose purpose is twofold: to encourage banks to focus on and monitor the credit quality of the borrower in collateralized transactions; to reflect the fact that, irrespective of the extent of over-collateralization, a collateralized transaction can never be totally without risk. A normal w value is 0.15. Guarantees and Credit Derivatives For a bank to obtain any capital relief from the receipt of credit derivatives or guarantees, the credit protection must be direct, explicit, irrevocable and unconditional. The Committee recognizes that banks only suffer losses in guaranteed transactions when both the obligor and the guarantor default. This ‘double default’ effect reduces the credit risk if there is a low correlation between the default probabilities of the obligor and the guarantor10 . The Committee considers that it is difficult to assess this situation and does not grant recognition to the ‘double default’ effect. The ‘substitution approach’ provided in the 1988 Accord applies for guarantees and credit derivatives, although an additional capital floor, w, applies. The substitution approach simply substitutes the risk of the guarantor for that of the borrower subject to full recognition of the enforceability of the guarantee. On-balance Sheet Netting On-balance sheet netting in the banking book is possible subject to certain operational standards. Its scope will be limited to the netting of loans and deposits with the same single counterparty. Portfolio Granularity

The Committee is proposing to make another extension of the 1988 Accord in that minimum capital requirements do not depend only on the characteristics of an individual exposure but also on the ‘concentration risk’ of the bank portfolio. Concentration designates the large sizes of exposures to single borrowers, or groups of closely related borrowers, potentially triggering large losses. The accord proposes a measure of granularity11 and incorporates this risk factor into the IRB approach by means of a standard supervisory capital adjustment applied to all exposures, except those in the retail portfolio. This treatment does not include industry, geographic or forms of credit risk concentration other than size concentration. The ‘granularity’ adjustment applies to the total risk-weighted assets at the consolidated bank level, based on the comparison of a reference portfolio with known granularity. 10 Chapter

41 provides details on the technique for assessing the default probability reduction resulting from ‘double’ or ‘joint’ default of the primary borrower and the guarantor. 11 An example is the ‘Herfindahl index’ calculation, measuring size concentration, given in Chapters 55–57.

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Other Specific Risks

The New Accord addresses various other risks: asset securitizations; project finance; equity exposures. The accord considers that asset securitizations deserve a more stringent treatment. It assigns risk weights more in line with the risks of structured notes issued by such structures and, notably, the credit enhancement note, subject to the first loss risk. These notes concentrate a large fraction of the risk of the pool of securitized assets12 . It also imposes the ‘clean break’ principle through which the non-recourse sale of assets should be unambiguous, limiting the temptation of banks to support sponsored structures for reputation motives (reputation risk13 ). Under the standardized approach, any invested amount in the credit enhancement note of securitization becomes deductible from capital. For banks investing in securitization notes, the Committee proposes to rely on ratings provided by an ECAI. Other issues with securitizations relate to operational risk. Revolving securitizations with early amortization features, or liquidity lines provided to structures (commitments to provide liquidity for funding the structure under certain conditions), generate some residual risks. There is a forfeited capital loading for such residual risk. The Committee considers that project finance requires a specific treatment. The accord also imposes risk-sensitive approaches for equity positions held in the banking book. The rationale is to remedy the possibility that banks could benefit from a lower capital charge when they hold the equity rather than the debt of an obligor. Interest Rate Risk

The accord considers it more appropriate to treat interest rate risk in the banking book under Pillar 2, rather than defining capital requirements. This implies no capital load, but an enhanced supervisory process. The guidance on interest rate risk considers banks’ internal systems as the main tool for the measurement of interest rate risk in the banking book and the supervisory response. To facilitate supervisors’ monitoring of interest rate risk across institutions, banks should provide the results of their internal measurement systems using standardized interest rate shocks. If supervisors determine that a bank is not holding capital commensurate with the level of interest rate risk, they can require that the bank reduces its risk, holds an additional amount of capital or combines the two.

Operational Risk The Committee adopted a standard industry definition of operational risk: ‘the risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events’. As a first approximation in developing minimum capital charges, the Committee estimates operational risk at 20% of minimum regulatory capital as measured under the 1988 12 See

details in subsequent descriptions of structures (Chapter 40). risk is the risk of adverse perception of the sponsoring bank if structure explicitly related to the bank suffers from credit risk deterioration or from a default event. 13 Reputation

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Accord. The Committee proposes a range of three increasingly sophisticated approaches to capital requirements for operational risk: basic indicator; standardized; internal measurement. The ‘basic indicator approach’ links the capital charge for operational risk to a single indicator that serves as a proxy for the bank’s overall risk exposure. For example, if gross income is the indicator, each bank should hold capital for operational risk equal to a fixed percentage (‘alpha factor’) of its gross income. The ‘standardized approach’ builds on the basic indicator approach by dividing a bank’s activities into a number of standardized business lines (e.g. corporate finance and retail banking). Within each business line, the capital charge is a selected indicator of operational risk times a fixed percentage (‘beta factor’). Both the indicator and the beta factors may differ across business lines. The ‘internal measurement approach’ allows individual banks to rely on internal data for regulatory capital purposes. The technique necessitates three inputs for a specified set of business lines and risk types: an operational risk exposure indicator; the probability that a loss event occurs; the losses given such events. Together, these components make up a loss distribution for operational risks. Nevertheless, the loss distribution might differ from the industry-wide loss distribution, thereby necessitating an adjustment, which is the ‘gamma factor’.

Pillar 2: Supervisory Review Process The second pillar of the new framework aims at ensuring that each bank has sound internal processes to assess the adequacy of its capital based on a thorough evaluation of its risks. Supervisors will be responsible for evaluating how well banks are assessing their capital needs relative to their risks. The Committee regards the market discipline through enhanced disclosure as a fundamental part of the New Accord. It considers that disclosure requirements and recommendations will allow market participants to assess key pieces of information for the application of the revised accord. The risk-sensitive approaches developed by the New Accord rely extensively on banks’ internal methodologies, giving banks more discretion in calculating their capital requirements. Hence, separate disclosure requirements become prerequisites for supervisory recognition of internal methodologies for credit risk, credit risk mitigation techniques and other areas of implementation. The Committee formulated four basic principles that should inspire supervisors’ policies: • Banks should have a process for assessing their overall capital in relation to their risk profile and a strategy for maintaining their capital levels. • Supervisors should review and evaluate banks’ internal capital adequacy assessments and strategies, as well as their ability to monitor and ensure their compliance with regulatory capital ratios. Supervisors should take appropriate supervisory actions if they are not satisfied with the results of this process. • Supervisors should expect banks to operate above the minimum regulatory capital ratios and should have the ability to require banks to hold capital in excess of this minimum.

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• Supervisors should intervene at an early stage to prevent capital from falling below the minimum levels required to support the risk of a particular bank, and should require corrective actions if capital is not maintained or restored.

Pillar 3: Market Discipline The third major element of the Committee’s approach to capital adequacy is market discipline. The accord emphasizes the potential for market discipline to reinforce capital regulations and other supervisory efforts in promoting safety and soundness in banks and financial systems. Given the influence of internal methodologies on the capital requirements established, it considers that comprehensive disclosure is important for market participants to understand the relationship between the risk profile and the capital of an institution. Accordingly, the usage of internal approaches is contingent upon a number of criteria, including appropriate disclosure. For these reasons, the accord is setting out a number of disclosure proposals as requirements, some of them being prerequisites to supervisory approval. Core disclosures convey vital information for all institutions and are important for market discipline. Disclosures are subject to ‘materiality’. Information is ‘material’ if its omission or misstatement could change or influence the assessment or decision of any user relying on that information. Supplementary disclosures may convey information of significance for market discipline actions with respect to a particular institution.

SECTION 3 Risk Management Processes

4 Risk Management Processes

The ultimate goal of risk management is to facilitate a consistent implementation of both risks and business policies. Classical risk practices consist of setting risk limits while ensuring that business remains profitable. Modern best practices consist of setting risk limits based on economic measures of risk while ensuring the best risk-adjusted performances. In both cases, the goal remains to enhance the risk–return profile of transactions and of the bank’s portfolios. Nevertheless, new best practices are more ‘risk-sensitive’ through quantification of risks. The key difference is the implementation of risk measures. Risks are invisible and intangible uncertainties, which might materialize into future losses, while earnings are a standard output of reporting systems complying with established accounting standards. Such differences create a bias towards an asymmetric view of risk and return, making it more difficult to strike the right balance between both. Characterizing the risk–return profile of transactions and of portfolios is key for implementing risk-driven processes. The innovation of new best practices consists of plugging new risk–return measures into risk management processes, enriching them and leveraging them with more balanced views of profitability and risks. The purpose of this chapter is to show how quantified risk measures feed the risk management processes. It does not address the risk measuring issue and does not describe the contribution of risk models, for which inputs are critical to enrich risk processes. Because quantifying intangible risks is a difficult challenge, concentrating on risk measures leaves in the shadow the wider view of risk processes implementing such risk measures. Since the view on risk processes is wider than the view on risk measuring techniques, we move first from a global view of risk processes before getting to the narrower and more technical view of risk measuring. The ‘risk–return profiles’ of transactions and portfolios are the centrepiece of the entire system and processes. For this reason, risk–return profiles are the interface between new

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risk models and risk processes. All risk models and measures converge to provide these profiles at the transaction, the business lines and the global portfolio levels. Risk models provide new risk measures as inputs for processes. Classical processes address risks without the full capability of providing adequate quantification of risks. Risk models provide new measures of return and risks, leveraging risk processes and extending them to areas that were previously beyond reach. New risk measures interact with risk processes. Vertical processes address the relationship between global goals and business decisions. The bottom-up and top-down processes of risk management allow the ‘top’ level to set up global guidelines conveyed to business lines. Simultaneously, periodical reporting from the business levels to the top allows deviations from guidelines to be detected, such as excess limits, and corrective actions to be taken, while comparing projected versus actual achievements. Transversal processes address risk and return management at ‘horizontal’ levels, such as the level of individual transactions, at the very bottom of the management ‘pyramid’, at the intermediate business line levels, as well as at the bank’s top level, for comparing risk and return measures to profitability target and risk limits. There are three basic horizontal processes: setting up risk–return guidelines and benchmarks; risk–return decision-making (‘ex ante perspective’); risk–return monitoring (‘ex post perspective’). The first section provides an overview of the vertical and horizontal processes. The subsequent sections detail the three basic ‘transversal’ processes (benchmarks, decisionmaking, monitoring). The last section summarizes some general features of ‘bank-wide risk management’ processes.

THE BASIC BUILDING BLOCKS OF RISK MANAGEMENT PROCESSES Processes cover all necessary management actions for taking decisions and monitoring operations that influence the risk and return profiles of transactions, subportfolios of business lines or the overall bank portfolio. They extend from the preparation of decisions, to decision-making and control. They include all procedures and policies required to organize these processes. Risk management combines top-down and bottom-up processes with ‘horizontal’ processes. The top-down and bottom-up views relate to the vertical dimension of management, from general management to individual transactions, and vice versa. The horizontal layers refer to individual transactions, business lines, product lines and market segments, in addition to the overall global level. They require to move back and forth from a risk–return view of the bank to a business view, whose main dimensions are the product families and the market segments.

Bottom-up and Top-down Processes The top-down process starts with global target earnings and risk limits converted into signals to business units. These signals include target revenues, risk limits and guidelines applicable to business unit policies. They make it necessary to allocate income and risks to business units and transactions. Otherwise, the global targets remain disconnected from

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operations. The monitoring and reporting of risks is bottom-up oriented, starting with transactions and ending with consolidated risks, income and volumes of transactions. Aggregation is required for supervision purposes and to compare, at all levels where decision-making occurs, goals with actual operations. In the end, the process involves the entire banking hierarchy from top to bottom, to turn global targets into signals to business units, and from bottom to top, to aggregate risks and profitability and monitor them. The pyramid image of Figure 4.1 illustrates the risk diversification effect obtained when moving up along the hierarchy. Each face of the pyramid represents a dimension of risk, such as credit risk or market risk. The overall risk is less than the simple arithmetic addition of all original risks generated by transactions (at the base of the pyramid) or subsets (subportfolios) of transactions. From bottom to top, risks diversify. This allows us to take more risks at the transaction level since risk aggregation diversifies away a large fraction of the sum of all individual transaction risks. Only post-diversification risk remains retained by the bank. Revenues and risk (capital) allocations

Global risk−return targets Poles, subsidiaries ... Business units Transactions

FIGURE 4.1

The pyramid of risk management

Risk models play a critical role in this ‘vertical’ process. Not only do they provide risk visibility at the transaction and business units level, but they also need to provide the right techniques for capturing risk diversification when moving up and down along the pyramid. Without quantification of the diversification effect, there are missing links between the sustainable risks, at the level of transactions, and the aggregated risk at the top of the pyramid that the bank capital should hedge. In other words, we do not know the overall risk when we have, say, two risks of 1 each, because the risk of the sum (1 + 1) is lower than 2, the sum of risks. There are missing links as well between the sustainable post-diversification risk, the bank’s capital and the risks tolerable at the bottom of the pyramid for individual transactions, business lines, market segments and product families. In other words, if a global post-diversification risk of 2 is sustainable at the top of the pyramid, it is compatible with a sum of individual risks larger than 2 at the bottom. How large can the sum of individual risks be (3, 4, 5 or more), compatible with an overall global risk limit of 2, remains unknown unless we have tools to allocate the global risk. The capital allocation system addresses these needs. This requires a unified risk management framework.

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Transversal Process Building Blocks Transversal processes apply to any horizontal level, such as business lines, product lines or market segments. The typical building transversal blocks of such processes (Figure 4.2) are: 1. Setting up risk limits, delegations and target returns. 2. Monitoring the compliance of risk–return profiles of transactions or subportfolios with guidelines, reporting and defining of corrective actions. 3. Risk and return decisions, both at the transaction and at the portfolio levels, such as lending, rebalancing the portfolio or hedging.

2. Decisionmaking Ex ante New transactions Hedging Portfolio R&R enhancement

1. Guidelines

Risk & Return (R&R)

3. Monitoring 3. Monitoring

FIGURE 4.2

Limits Delegations Target return

Ex post Follow-up Reporting R&R Corrective actions

The three-block transversal processes

These are integrated processes, since there are feedback loops between guidelines, decisions and monitoring. Risk management becomes effective and successful only if it develops up to the stage where it facilitates decision-making and monitoring.

Overview of Processes Putting together these two views could produce a chart as in Figure 4.3, which shows how vertical and transversal dimensions interact.

Risk Models and Risk Processes Risk models contribute to all processes because they provide them with better and richer measures of risk, making them comparable to income, and because they allow banks to enrich the processes using new tools such as risk-adjusted performance or valuing the risk reduction effects of altering the portfolio structure. Figure 4.4 illustrates how models provide the risk–return measures feeding transversal and vertical processes.

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Global Risk Management Guidelines

Decision-making Ex ante

Top Down

Limits Delegations Target returns ...

On- and offbalance sheet Excess limits Hedging Individual transactions Lending, trading ... Global portfolio Portfolio management

Monitoring Ex post Bottom - Up Business Unit Management

FIGURE 4.3

The three basic building blocks of risk management processes Risk Management 'Toolbox'

Risk Models

Risks Risks

Global

Return Return

Decisions

Guidelines

Business line Transaction Monitoring

FIGURE 4.4

Overall views of risk processes and risk–return

Risk Processes and Business Policy Business policy deals with dimensions other than risk and return. Attaching risk and returns to transactions and portfolios is not enough, if we cannot convert these views into the two basic dimensions of business policy: products and markets. This requires a third type of process reconciling the risk and return view with the product–market view. For

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Market Transactions

Specialized Finance

Off-balance Sheet

Corporate Lending

RFS

business purposes, it is natural to segment the risk management process across business lines, products and markets. The product–market matrices provide a synthethic business view of the bank. Broad business lines and market segments could look like the matrix in Figure 4.5. In each cell, it is necessary to capture the risk and return profiles to make economically meaningful decisions. The chart illustrates the combinations of axes for reporting purposes, and the need for Information Technology (IT) to keep in line with such multidimensional reports and analyses.

Consumers Corporate−Middle Market Large Corporate Firms Financial Institutions Specialized Finance

Risk & Return

FIGURE 4.5 Reporting credit risk characteristics within product–market couples RFS refers to ‘Retail Financial Services’. Specialized finance refers to structured finance, project finance, LBO or assets financing.

The next section discusses the three basic transversal processes: setting up guidelines; decision-making; monitoring.

PROCESS #1: SETTING UP RISK AND RETURN GUIDELINES Guidelines include risk limits and delegations, and benchmarks for return. The purpose of limits is to set up an upper bound of exposures to risks so that an unexpected event cannot impair significantly the earnings and credit standing of the bank. Setting up benchmarks of return refers to the target profitability goals of the bank, and how they translate into pricing. When setting up such guidelines, banks face trade-offs in terms of business volume versus risk and business volume versus profitability. Risk Benchmarks: Limits and Delegations

Traditional risk benchmarks are limits and delegations. For credit risk, limits set upper bounds to credit risk on obligors, markets and industries, or country risk. For market risk, limits set upper bounds to market risk sensitivity to the various market parameters.

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Delegations serve to decentralize the decision-making process. They mean business lines do not need to refer always to the central functions to make risk decisions as long as the risk is not too large. Classical procedures and policies have served to monitor credit risk for as long as banks have existed. Limit systems set caps on amounts at risk with any one customer, with a single industry or country. To set up limits, some basic rules are simple, such as: avoid any single loss endangering the bank; diversify the commitments across various dimensions such as customers, industries and regions; avoid lending to any borrower an amount that would increase debt beyond borrowing capacity. The equity of the borrower sets up some reasonable limit on debt given acceptable levels of debt/equity ratios and repayment ability. The capital of the bank sets up another limit to lending given the diversification requirements and/or the credit policy guidelines. Credit officers and credit committees reach a minimal agreement before making a credit decision by examining in detail credit applications. Delegations, at various levels of the bank, stipulate who has permission to take credit commitments depending on their size. Typical criteria for delegations are size of commitments or risk classes. Central reporting of the largest outstanding loans to customers serves to ensure that these amounts stay within reasonable limits, especially when several business units originate transactions with the same clients. This makes it necessary to have ‘global limit’ systems aggregating all risks on any single counterparty, no matter who initiates the risk. Finally, there are risk diversification rules across counterparties. The rationale for risk limits potentially conflicts with the development of business volume. Banking foundations are about trust and relationships with customers. A continuous relationship allows business to be carried on. The business rationale aims at developing such relationships through new transactions. The risk rationale lies in limiting business volume because the latter implies risks. ‘Name lending’ applies to big corporations, with an excellent credit standing. Given the high quality of risk and the importance of business opportunities with big customers, it gets harder to limit volume. This is the opposite of a limit rationale. In addition, allowing large credit limits is necessary in other instances. For instance, banks need to develop business relationships with a small number of big players, for their own needs, such as setting up hedges for interest rate risk. The same principles apply to controlling market risk. In these cases, to limit the potential loss of adverse markets moves, banks set upper bounds to sensitivities of instruments and portfolios. Sensitivities are changes in value due to forfeit shocks on market parameters such as interest rates, foreign exchange rates or equity indexes. Traders comply with such limits through trading and hedging their risks. Limits depend on expectations about market conditions and trading book exposures. To control the interest rate risk of the banking portfolio, limits apply to the sensitivities of the interest income or the Net Present Value (NPV, the mark-to-market valuation of assets minus liabilities) to shocks on interest rates. By bounding these values, banks limit the adverse movements of these target variables. Risk limits imply upper bounds on business volume, except when it is possible to hedge risks and avoid any excess exposure over limits. This is feasible for market transactions and for hedging interest rate risk, unless the cost of setting up the hedge is too high. For credit risk, there was no hedge until recently. Today, insurance, guarantees, credit derivatives and securitization offer a wide range of techniques for taking on more risk while still complying with risk limits.

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Return Benchmarks

The target profitability of the bank provides signals to business units for making business. Classical profitability measures include interest income and fees for the banking portfolio, Return On Equity (ROE) for the bank and for individual transactions, using the former Cooke ratio to determine the capital loading of a transaction or a portfolio, and Return On Assets (ROA) relating income to the size of the exposure. For the trading book, the profitability is in Profit and Loss (P&L), independent of whether sales of assets occur or not. Such measures have existed for a long time. Nevertheless, such measures fall short of addressing the issue of the trade-off between risk and return. However, only the risk and return profile of transactions or portfolios is relevant because it is easy to sacrifice or gain return by altering risk. Risk-based pricing refers to pricing differentiation based on risks. It implies that risks be defined at the transaction level, the subportfolio level and the entire bank portfolio level. Two systems are prerequisites for this process: the capital allocation system, which allocates risks; the funds transfer pricing system, which allocates income. The allocation of income and risks applies to transactions, business lines, market segments, customers or product lines. The specifications and roles of these two tools are expanded in Chapters 51 and 52 (capital allocation) and Chapters 26 and 27 (funds transfer pricing). Pricing benchmarks are based on economic transfer prices in line with the cost of funds. The Funds Transfer Pricing (FTP) system defines such economic prices and to which internal exchanges of funds they apply. Benchmark transfer prices generally refer to market rates. With internal prices, the income allocation consists simply of calculating the income as the difference between interest revenues and these prices. The target overall profitability results in target mark-ups over these economic prices. Risk allocation is similar to income allocation, except that it is less intuitive and more complex because risks do not add up arithmetically as income does. Capital allocation is a technique for allocating risks based on objective criteria. Risk allocations are capital allocations in monetary units. Capital refers to ‘economic capital’, or the amount of capital that matches the potential losses as measured by the ‘Value at Risk’ (VaR) methodology. Both transfer pricing and capital allocation systems are unique devices that allow interactions between risk management and business lines in a consistent bank-wide risk management framework. If they do not have strong economic foundations, they put the entire credibility of the risk management at stake and fail to provide the bottom-up and top-down links between ‘global’ targets and limits and ‘local’ business targets and limits.

PROCESS #2: DECISION-MAKING (EX ANTE PERSPECTIVE) The challenge for decision-making purposes is to capture risks upstream in the decision process, rather than downstream, once decisions are made. Helping the business decision process necessitates an ‘ex ante’ perspective, plus adequate tools for measuring and pricing risk consistently. Risk decisions refer to transactions or business line decisions, as well as portfolio decisions. New transactions, portfolio rebalancing and portfolio restructuring through securitizations (sales of bundled assets into the markets) are risk decisions. Hedging decisions effectively alter the risk–return profiles of transactions or of the entire portfolio. Decisions refer to both ‘on-balance sheet’, or business, decisions and

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‘off-balance sheet’, or hedging, decisions. Without a balanced risk–return view, the policy gets ‘myopic’ in terms of risks, or ignores the effect on income of hedging transactions.

On-balance Sheet Actions On-balance sheet actions relate to both new business and the existing portfolio of transactions. New business raises such basic questions as: Are the expected revenues in line with risks? What is the impact on the risk of the bank? Considering spreads and fees only is not enough. Lending to high-risk lenders makes it easy to generate margins, but at the expense of additional risks which might not be in line with income. Without some risk–return measures, it is impossible to solve the dilemma between revenues and volume, other than on a judgmental basis. This is what banks have done, and still do, for lack of better measures of risks. Banks have experience and perception of risks. Nevertheless, risk measures, combined with judgmental expertise, provide benchmarks that are more objective and help solve dilemmas such as volume versus profitability. Risk measures facilitate these decisions because they shed new light on risks. A common fear of focusing on risks is the possibility of discouraging risk taking by making risks explicit. In fact, the purpose is just the opposite. Risk monitoring can encourage risk taking by providing explicit information on risks. With unknown risks, prudence might prevail and prevent risk-taking decisions even though the profitability could well be in line with risks. When volume is the priority, controlling risks might become a second-level priority unless risks become more explicit. In both cases, risk models provide information for taking known and calculated risks. Similar questions arise ex post, once decisions are made. Since new business does not influence income and risks to an extent comparable to the existing portfolio, it is important to deal with the existing portfolio as well. Traditionally, periodical corrective actions, such as managing non-performing loans and providing incentives to existing customers to take advantage of new services, help to enhance the risk–return profile. Portfolio management extends this rationale further to new actions, such as securitizations that off-load risk in the market, syndications, loan trading or hedging credit risk, that were not feasible formerly. This is an emerging function for banks, which traditionally stick to the ‘originate and hold’ view of the existing portfolio, detailed further below.

Off-balance Sheet Actions Off-balance sheet recommendations refer mainly to hedging transactions. Asset–Liability Management (ALM) is in charge of controlling the liquidity and interest rate risk of the banking portfolio and is responsible for hedging programmes. Traders pursue the same goals when using off-balance instruments to offset exposures whenever they need to. Now, loan portfolio management and hedging policies also take shape, through credit derivatives and insurance. Hedging makes extensive use of derivative instruments. Derivatives include interest rate swaps, currency swaps and options on interest rates, if we consider interest rate only. Credit derivatives are new instruments. All derivatives shape both risk and return since they generate costs, the cost of hedging, and income as well because they capture the underlying market parameters or asset returns. For instance, a swap receiving the variable

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rate and paying a fixed rate might reduce the interest exposure and, simultaneously, generate variable rate revenues. A total return swap exchanges the return of a reference debt against a fixed rate, thereby protecting the buyer from capital losses of the reference asset in exchange for a return giving up the asset capital gains. Setting up such hedges requires a comprehensive view on how they affect both risk and return dimensions.

Loan Portfolio Management Even though banks have always followed well-known diversification principles, active management of the banking portfolio remained limited. Portfolio management is widely implemented with market transactions because diversification effects are obvious, hedging is feasible with financial instruments and market risk quantification is easy. This is not the case for the banking portfolio. The classical emphasis of credit analysis is at the transaction level, rather than the portfolio level, subject to limits defined by the credit department. Therefore, loan portfolio management is one of the newest fields of credit risk management. Incentives for the Development of Loan Portfolio Management

There are many incentives for developing portfolio management for banking transactions: • The willingness to make diversification (portfolio) effects more explicit and to quantify them. • The belief that there is a significant potential to improve the risk–return trade-off through management of the banking portfolio as a whole, rather than focusing only on individual banking transactions. • The growing usage of securitizations to off-load risk into the market, rather than the classical arbitrage between the on-balance sheet cost of funding versus the market cost of funds. • The emergence of new instruments to manage credit risk: credit derivatives. • The emergence of the loan trading market, where loans, usually illiquid, become tradable over an organized market. Such new opportunities generate new tools. Portfolio management deals with the optimization of the risk–return profile by altering the portfolio structure. Classical portfolio management relies more on commercial guidelines, on a minimum diversification and/or aims at limiting possible risk concentrations in some industries or with some big customers. New portfolio management techniques focus on the potential for enhancing actively the profile of the portfolio and on the means to achieve such goals. For example, reallocating exposures across customers or industries can reduce risk without sacrificing profitability, or increase profitability without increasing risks. This is the familiar technique of manipulating the relative importance of individual exposures to improve the portfolio profile, a technique well known and developed for market portfolios. The first major innovation in this area is the implementation of portfolio models providing measures of credit risk diversification. The second innovation is the new flexibility to shape the risk profile of portfolios through securitization, loan sales and usage of

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credit derivatives. Credit derivatives are instruments based on simple mechanisms. They include total return swaps, default options and credit spread swaps. Total return swaps exchange the return of any asset against some market index. Default options provide compensation in the event of default of a borrower, in exchange for the payment of a premium. Credit spread swaps exchange one market credit spread against another one. Such derivatives can be used, as other traditional derivatives, to hedge credit risk. The Loan Portfolio Management Function

The potential gains of more active portfolio management are still subject to debate. Manipulating the ‘weights’ of commitments for some customers might look like pure theory, given the practice of relationship banking. Banks maintain continuous relationships with customers that they know well, and are willing to keep doing business with. Volumes are not so flexible. Hence, a debate emerged on the relative merits of portfolio management and relationship banking. There might be an apparent conflict between stable relations and flexible portfolio management. What would be the benefit of trading loans if it adversely influenced ‘relationship banking’? In fact, the opposite holds. The ability to sell loans improves the possibility of generating new ones and develops, rather than restricts, relationship banking. Once we start to focus on portfolio management, the separation of origination from portfolio management appears a logical step, although this remains a major organizational and technical issue. The rationale of portfolio management is global risk–return enhancement. This implies moving away from the traditional ‘buy and hold’ policy and not sticking to the portfolio structure resulting from the origination business unit. Portfolio management requires degrees of freedom to be effective. They imply separation, to a certain extent, from origination. The related issues are numerous. What would be the actual role of a portfolio management unit? Should the transfer of transactions from origination to portfolio management be extensive or limited? What would be the internal transfer prices between both units? Can we actually trade intangible risk reductions, modelled rather than observed, against tangible revenues? Perhaps, once risk measures are explicit, the benefits of such trade-offs may become more visible and less debated. Because of these challenges, the development of the emerging portfolio management function is gradual, going through various stages: • • • • •

Portfolio risk reporting. Portfolio risk modelling. A more intensive usage of classical techniques such as syndication or securitizations. New credit derivatives instruments and loan trading. Separation of portfolio management and origination, since both functions differ in their perspective and both can be profit centres.

PROCESS #3: RISK–RETURN MONITORING (EX POST PERSPECTIVE) The monitoring and periodical reviews of risks are a standard piece of any controlling system. They result in corrective actions or confirmations of existing guidelines. For

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credit risk, the monitoring process has been in existence since banks started lending. Periodical reviews of risk serve to assess any significant event that might change the policy of the bank with respect to some counterparties, industries or countries. Monitoring systems extend to early warning systems triggering special reviews, including borrowers in a ‘watch list’, up to provisioning. Corrective actions avoid further deteriorations through restructuring of individual transactions. Research on industries and countries, plus periodical reviews result in confirmation of existing limits or adjustments. Reviews and corrective actions are also event-driven, for example by a sudden credit standing deterioration of a major counterparty. Analogous processes apply for market risk and ALM. A prerequisite for risk–return monitoring is to have measures of risk and return at all relevant levels, global, business lines and transactions. Qualitative assessment of risk is insufficient. The challenge is to implement risk-based performance tools. These compare ex post revenues with the current risks or define ex ante which pricing would be in line with the overall target profitability, given risks. The standard tools for risk-adjusted performance, as well as risk-based pricing, are the RaRoC (Risk-adjusted Return on Capital) and SVA (Shareholders Value Added) measures detailed in Chapters 53 and 54. Risk-based performance allows: • Monitoring risk–return profiles across business lines, market segments and customers, product families and individual transactions. • Making explicit the possible mispricing of subportfolios or transactions compared to what risk-based pricing would be. • Defining corrective or enhancement actions. Defining target risk-adjusted profitability benchmarks does not imply that such pricing is effective. Competition might not allow charging risk-based prices without losing business. This does not imply that target prices are irrelevant. Mispricing is the gap between target prices and effective prices. Such gaps appear as RaRoC ratios or SVA values not in line with objectives. Monitoring ex post mispricing serves the purpose of determining ex post what contributes to the bank profitability on a risk-adjusted basis. Without mispricing reports, there would be no basis for taking corrective actions and revising guidelines.

BANK-WIDE RISK MANAGEMENT The emergence of risk models allowed risk management to extend ‘bank-wide’. ‘Bankwide’ means across all business lines and risks. Bank-wide risk management implies using the entire set of techniques and models of the risk management ‘toolbox’. Risk management practices traditionally differ across risks and business lines, so that a bankwide scope requires a single unified and consistent framework.

Risk Management Differs across Risks Risk management appears more fragmented than unified. This contrasts with the philosophy of ‘bank-wide risk management’, which suggests some common grounds and common frameworks for different risks. Risk practice differs across business lines. The

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market risk culture and the credit culture seem not to have much in common yet. Market culture is quantitative in nature and model-driven. By contrast, the credit culture focuses on the fundamentals of firms and on the relationship with borrowers to expand the scope of services. Moreover, the credit culture tends to ignore modelling because of the challenge of quantifying credit risk. In the capital markets universe, trading risks is continuous since there is a market for doing so. At the opposite end of the spectrum, the ‘buy and hold’ philosophy prevails in the banking portfolio. ALM does not have much to do with credit and market risk management, in terms of goals as well as tools and processes. It needs dedicated tools and risk measures to define a proper funding and investment policy, and to control interest rate risk for the whole bank. Indeed, ALM remains very different from market and credit risk and expands over a number of sections in this book. Indeed, risk models cannot yet pretend to address the issues of all business lines. Risk-adjusted performance is easier to implement in the middle market than with project finance or for Leveraged Buy-Outs (LBOs), which is unfortunate because risk-adjusted performances would certainly facilitate decision-making in these fields. On the other hand, the foundations of risk measures seem robust enough to address most risks, with techniques based on the simple ‘potential loss’, or VaR, concept. Making the concept instrumental is quite a challenge, as illustrated by the current difficulties of building up data for credit risk and operational risk. But the concept is simple enough to provide visibility on which roads to follow to get better measures and understand why crude measures fail to provide sound bases for decision-making and influencing, in general, the risk management processes. Because of such limitations, new best practices will not apply to all activities in the near future, and will presumably extend gradually.

Different Risks Fit into a Single Framework The view progressively expanded in this book is that risk management remains differentiated, but that all risks tend to fit in a common basic framework. The underlying intuition is that many borders between market and credit risk tend to progressively disappear, while common concepts, such as VaR and portfolio models, apply gradually to all risks. The changing views in credit risk illustrate the transformation:

• Credit risk hedges now exist with credit derivatives. • The ‘buy and hold’ culture tends to recede and the credit risk management gets closer to the ‘trading philosophy’. • The portfolio view of credit risk gains ground because it brings some new ways of enhancing the risk–return profile of the banking portfolio. • Accordingly, the ‘model culture’ is now entering the credit risk sphere and increasingly interacts with the credit culture. • The building up of data has been productive for those banks that made it a priority. • Regulations using quantitative measures of risk perceived as intangibles gained acceptance in the industry and stimulate progress.

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Credit Risk

Interest Rate Risk

Global Integration

Liquidity Risk

FIGURE 4.6

Market Risk

+ + Other Risks

Global integration of differentiated risks and risk management techniques

A preliminary conclusion is that there are now sufficient common grounds across risks to capture them within a common framework. This facilitates the presentation of concepts, methodologies, techniques and implementations, making it easier to wrap around a framework showing how tools and techniques capture differentiated risks and risk management processes in ways that facilitate their global and comprehensive integration (Figure 4.6).

5 Risk Management Organization

The development of bank-wide risk management organization is an ongoing process. The original traditional commercial bank organization tends to be dual, with the financial sphere versus the business sphere. The business lines tend to develop volume, sometimes at the expense of risks and profitability, while the financial sphere tends to focus on profitability, with dedicated credit and market risk monitoring units. This dual view is fading away with the emergence of new dedicated functions implemented bank-wide. Bank-wide risk management has promoted the centralization of risk management and a clean break between risk-taking business lines and risk-supervising units. Technical and organizational necessities foster the process. Risk supervision requires separating the supervising units from the business units, since these need to take more risk to achieve their profitability targets. Risk centralization is also a byproduct of risk diversification. Only post-diversification risks are relevant for the entire bank’s portfolio, and required capital, once the aggregation of individual risks diversified away a large fraction of them. The risk department emerged from the need for global oversight on credit risk, market risk and interest rate risk, now extending to operational risk. Separating risk control from business, plus the need for global oversight, first gave birth to the dedicated Asset–Liability Management (ALM) function, for interest rate risk, and later on stimulated the emergence of risk departments, grouping former credit and market risk units. While risk measuring and monitoring developed, other central functions with different focuses, such as management control, accounting, compliance with regulations, reporting and auditing, differentiated. This chapter focuses on the risk management functions, and illustrates how they emerged and how they relate to other functions. The modern risk management organization separates risk management units from business units. The banking portfolio generates interest rate risk, transferred from business lines, which have no control over interest rate movements, to the ALM unit which is in charge

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of managing it. The banking portfolio and ALM, when setting up hedges with derivatives, generate credit risk, supervised by the credit risk unit. The market risk unit supervises the trading portfolio. The portfolio management unit deals with the portfolio of loans as a whole, for risk control purposes and for a more active restructuring policy through actions such as direct sales or securitization. The emerging ‘portfolio management’ unit might necessitate a frontier with origination through loans management post-origination. It might be related to the risk department, or be a profit-making entity as well. The first section maps risk origination with risk management units supervising. The second section provides an overview of central functions, and of the risk department. The third section details the ALM role. The fourth section focuses on risk oversight and supervision by the risk department entity. The fifth section details the emerging role of the ‘loan portfolio management’ unit. The last section emphasizes the critical role of Information Technology (IT) for dealing with a much wider range of models, risk data warehouses and risk measures than before.

MAPPING ORGANIZATION WITH RISK MANAGEMENT NECESSITIES Figure 5.1 shows who originates what risks and which central functions supervise them. Various banks have various organizations. There is no unique way of organizing the RISKS

Credit

Market

Liquidity

Interest Rate

Risk Management General Management

Guidelines and goals Guidelines and goals

Portfolio Management

Portfolio risk−return profile Portfolio risk- return profile

Risk Risks Department Department ALM ALM

Risk Origination Commercial Bank Investment Bank Market

FIGURE 5.1

Mapping risk management with business lines and central functions

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risk management processes. Available risk models and management tools and techniques influence organization schemes, as various stages of differentiation illustrate.

THE DIFFERENTIATION OF CENTRAL FUNCTIONS The functions of the different central units tend to differentiate from each other when going further into the details of risk and income actions. Figure 5.2 illustrates the differentiation process of central functions according to their perimeter of responsibilities. ALM Control

- Accounting - Regulatory compliance - Cost accounting - Budgeting and planning - Monitoring and control - Monitoring performances - Reporting to general management

- Liquidity and interest rate risk management - Measure and control of risks - Compliance - Hedging - Recommendations: balance sheet actions - Transfer pricing systems - ALCO - Reporting to general management

Credit Risk

- Credit policy - Setting credit risk limits and delegations - Assigning internal ratings - Credit administration (credit applications and documentation) - Credit decisions (credit committees) - Watch lists - Early warning systems - Reporting to general management

Risk Department

Risks Department

- Monitoring and control of all risks - Credit & market, in addition to ALM - Decision-making - Credit policy - Setting limits - Development of internal tools and risk data warehouses - Risk-adjusted performances - Portfolio actions - Reporting to general management

FIGURE 5.2

Market Risk

Portfolio Management

- Trading credit risk - Portfolio reporting - Portfolio restructuring - Securitizations - Portfolio actions - Reporting to general management

- Limits - Measure and control of risks - Compliance - Monitoring - Hedging - Business actions - Reporting to general management

Functions of central units and of the risk department

THE ALM FUNCTION ALM is the unit in charge of managing the interest rate risk and liquidity of the bank. It focuses essentially on the commercial banking pole, although the market portfolio also generates liquidity requirements.

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The ALM Committee (ALCO) is in charge of implementing ALM decisions, while the technical unit prepares all analyses necessary for taking decisions and runs the ALM models. The ALCO agenda includes ‘global balance sheet management’ and the guidelines for making the business lines policy consistent with the global policy. ALM addresses the issue of defining adequate structures of the balance sheet and the hedging programmes for liquidity and interest rate risks. The very first mission of ALM is to provide relevant risk measures of these risks and to keep them under control given expectations of future interest rates. Liquidity and interest rate policies are interdependent since any projected liquidity gap requires future funding or investing at an unknown rate as of today, unless setting up hedges today. ALM scope (Figure 5.3) varies across institutions, from a limited scope dedicated to balance sheet interest and liquidity risk management, to being a profit centre, in addition to its mission of hedging the interest rate and liquidity risks of the bank. In between, ALM extends beyond pure global balance sheet management functions towards a better integration and interaction with business policies.

FIGURE 5.3

Traditional

Tools, procedures and processes for managing globally the interest rate risk and the liquidity risk of commercial banking.

Wider scope

Tools, models and processes facilitating the implementation of both business and financial policy and an aid to decision-making at the global and business unit levels.

ALM policies vary across institutions

- Sometimes subordinated to business policy. - Sometimes active in business policy making. - Sometimes defined as a P&L unit with its own decisions, in addition to the hedging (risk reward) policy for the commercial bank.

ALM scope

The ALCO is the implementation arm of ALM. It groups heads or business lines together with the general bank management, sets up the guidelines and policies with respect to interest rate risk and liquidity risk for the banking portfolio. The ALCO discusses financial options for hedging risks and makes recommendations with respect to business policies. Both financial and business policies should ensure a good balance between ‘on-balance sheet’ actions (business policy and development) and ‘off-balance sheet’ actions (hedging policies). Financial actions include funding, investing and hedging. Since these are market transactions, they always have an influence on the interest income, creating a trade-off between risk and interest income. Technical analyses underlie the recommendations, usually conducted by a technical unit which prepares ALCO meetings and recommendations.

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Other units interact with ALM, such as a dedicated unit for funding, in charge of raising funds, or the treasury, which manages the day-to-day cash flows and derivatives, without setting the basic guidelines of interest rate risk management. Management control and accounting also interact with ALM to set the economic transfer prices, or because of the Profit and Loss (P&L) incidence of accounting standards for hedging operations. The transfer pricing system is in charge of allocating income between banking business units and transactions, and is traditionally under ALM control.

THE RISK DEPARTMENT AND OVERSIGHT ON RISKS Dedicated units normally address credit risk and market risk, while ALM addresses the commercial banking side of interest rate risk and funding. Risk supervisors should be independent of business lines to ensure that risk control is not under the influence of business and profit-making policies. This separation principle repeatedly appears in guidelines. Separation should go as far as setting up limits that preclude business lines from developing new transactions because they would impair the bank’s risk too much. The functions of the risk department include active participation in risk decision-making, with a veto power on transactions, plus the ability to impose restructuring of transactions to mitigate their risk. This empowers the risk-supervising unit to full control of risks, possibly at the cost of restricting risky business. Getting an overview of risks is feasible with separate entities dedicated to each main risk. However, when reaching this stage, it sounds natural to integrate the risk supervision function into a risk department, while still preserving the differentiation of risk measures and management. There are several reasons for integration in a single department, some of them organizational and others technical. From an organizational standpoint, integration facilitates the overview of all risks across all business lines. In theory, separate risk functions can do the same job. In practice, because of process harmonization, lack of interaction between information systems and lack of uniform reporting systems, it is worth placing the differentiated entities under the same control. Moreover, separate risk control entities, such as market and credit risk units, could possibly deal separately with business lines, to the detriment of the global policy. When transactions get sophisticated, separating functions could result in disruptions. A single transaction can trigger multiple risks, some of them not obvious at first sight. Market transactions and ALM hedges create credit risk. Investment banking activities or structured finance generate credit risk, interest rate risk and operational risk. Multiple views on risks might hurt the supervisory process, while a single picture of all risks is a simple practical way to have a broader view on all risks triggered by the same complex transactions. For such reasons the risk department emerged in major institutions, preserving the differentiation of different risks, but guaranteeing the integration of risk monitoring, risk analyses and risk reporting. Under this scheme, the risk department manages and controls risks, while business lines generate risks (Figure 5.4). Each major business pole can have its own risk unit, interacting with the risk department. Few banks set up a portfolio management unit, but many progress along such lines. The portfolio management unit is in charge of restructuring the portfolio, after origination, to enhance its risk–return profile. The first stages of portfolio management extend from

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Risk Department Credit risk

Market risk

Markets

ALM

Business Lines

Other risks Products

FIGURE 5.4

Risk department, risk units and business lines

reporting on risks and portfolio structure, to interaction with the limit setting process and the definitions of credit policy guidelines. At further stages, the portfolio management unit trades credit risk after origination through direct sales of loans, securitizations or credit risk hedging. Active portfolio management is a reality, with or without a dedicated unit, since all techniques and instruments allowing us to perform such tasks have been developed to a sufficient extent. The full recognition of the portfolio management unit would imply acting as a separate entity, with effective control over assets post-origination, and making it a profit centre. The goal is to enhance the risk–return profile of the entire portfolio. Transfers of assets from origination should not result in lost revenues by origination without compensation. The ultimate stage would imply full separation and transfer prices between origination and portfolio management, to effectively allocate income to each of these two poles. The risk department currently plays a major role in the gradual development of portfolio management. It has an overview of the bank’s portfolio and facilitates the innovation process of hedging credit risk. It has a unique neutral position for supervising rebalancing of the portfolio and altering of its risk–return profile. Nevertheless, the risk department cannot go as far as making profit, as a portfolio management unit should ultimately do, since this would negate its neutral posture with respect to risk. It acts as a facilitator during the transition period when risk management develops along these new dimensions.

INFORMATION TECHNOLOGY Information technology plays a key role in banks in general, and particularly in risk management. There are several reasons for this: • Risk data is continuously getting richer. • New models, running at the bank-wide scale, produce new measures of risk. • Bringing these measures to life necessitates dedicated front-ends for user decisionmaking. Risk data extends from observable inputs, such as market prices, to new risk measures such as Value at Risk (VaR). New risk data warehouses are required to put together data and to organize the data gathering process, building up historical data on new inputs, such as those required by the 2001 Basel Accord. VaR necessitates models that did not exist a few years ago, running at the entire bank scale to produce market risk and credit risk measures. IT scope extends to the implementation of these models, and to assembling their outputs in a usable form for

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end-users. Due to the scale of operations of bank systems, creating the required risk data warehouse with the inputs and outputs of these models, and providing links to front-ends and reporting modules for end-users, are major projects. Bringing the information to life is an IT challenge because it requires new generation tools capable of on-line queries and analyses embedded in front-ends and reporting. Without such aids, decision-makers might simply ignore the information because of lack of time. Since new risk measures are not intuitive, managers not yet familiar with them need tools to facilitate their usage. ‘On-Line Analysis and Processing’ (OLAP) systems are critical to forward relevant information to end-users whenever they need it. Multiple risk measures generate several new metrics for risks, which supplement the simple and traditional book exposures for credit risk, for example. The risk views now extend to expected and unexpected losses, capital and risk allocations, in addition to mark-to-market measures of loan exposures (the major building blocks of risk models are described in Chapter 9). Profitability measures also extend to new dimensions, from traditional earnings to risk-adjusted measures, ex ante and ex post, at all levels, transactions, subportfolios and the bank’s portfolio. Simultaneously, business lines look at other dimensions, transactions, product families, market segments or business unit subportfolios. Business Reporting

Risk−Return Reporting

Business units

Exposures

Markets

Expected loss Ratings

Product families Transactions

Capital RaRoC ...

.....

..... Reports + 'OLAP' Slicing & Dicing

Drill-Down What-If

FIGURE 5.5

IT and portfolio reporting

Combining several risk dimensions with profitability dimensions and business dimensions has become a conceptual and practical challenge. Multidimensional views of the bank’s portfolio have become more complex to handle. New generation IT systems can handle the task. However, IT still needs to design risk, profitability and business reports so that they integrate smoothly within the bank’s processes. Multidimensional reporting requires a more extensive usage of new tools:

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• Slicing and dicing the portfolio across any one of these dimensions, or combinations of them, such as reporting the risk-adjusted profitability by market segment, business unit or both. • A drilling-down function to find out which transactions are the source of risk for subsets of transactions. The simplest example would be to find which transactions and obligors contribute most to the risk of a business unit, or simply understanding what transactions make a risk metric (exposure, expected loss, risk allocation) higher than expected. • ‘What if’ simulation capabilities to find out the outcomes of various scenarios such as adding or withdrawing a transaction or a business line, conducting sensitivity analyses to find which risk drivers influence risk more, or when considering rebalancing the bank’s subportfolios. New risk softwares increasingly embed such functions for structuring and customizing reports. Figure 5.5 illustrates the multiple dimensions and related reporting challenges. To avoid falling into the trap of managing reports rather than business, on-line customization is necessary to produce the relevant information on time. Front-end tools with ‘what if’ and simulation functions, producing risk–return reports both for the existing portfolio and new transactions, become important for both the credit universe and the market universe.

SECTION 4 Risk Models

6 Risk Measures

Risk management relies on quantitative measures of risks. There are various risk measures. All aim at capturing the variation of a given target variable, such as earnings, market value or losses due to default, generated by uncertainty. Quantitative indicators of risks fall into three types: • Sensitivity, which captures the deviation of a target variable due to a unit movement of a single market parameter (for instance, an interest rate shift of 1%). Sensitivities are often market risk-related because they relate value changes to market parameters, which are value drivers. The interest rate gap is the sensitivity of the interest margin of the banking portfolio to a forfeit move of the yield curve. Sensitivities are variations due to forfeit moves of underlying parameters driving the value of target variables. • Volatility, which captures the variations around the average of any random parameter or target variable, both upside and downside. Unlike forfeit movements, volatility characterizes the varying instability of any uncertain parameters, which forfeit changes ignore. Volatility measures the dispersion around its mean of any random parameter or of target variables, such as losses for credit risk. • Downside measures of risk, which focus on adverse deviations only. They characterize the ‘worst-case’ deviations of a target variable, such as earnings, market values or credit losses, with probabilities for all potential values. Downside risk measures require modelling to have probability distributions of target variables. The ‘Value at Risk’ (VaR) is a downside risk measure. It is the adverse deviation of a target variable, such as the value of a transaction, not exceeded in more than a preset fraction of all possible future outcomes. Downside risk is the most ‘comprehensive’ measure of risk. It integrates sensitivity and volatility with the adverse effect of uncertainty (Figure 6.1). This chapter details related techniques and provides examples of well-known risk measures. In spite of the wide usage of risk measures, risks remain intangible, making

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Quantitative indicators of risk

FIGURE 6.1

1

Sensitivity

2

Volatility

3

Downside risk or VaR

Risk measures

the distinction between risks and risk measures important. The first section draws a line between intangible risks and quantitative measures. The next sections provide examples of sensitivity and volatility measures. The final section discusses VaR and downside risk.

MEASURING UNCERTAINTY Not all random factors that alter the environment and the financial markets—interest rates, exchange rates, stock indexes—are measurable. There are unexpected and exceptional events that radically and abruptly alter the general environment. Such unpredictable events might generate fatal risks and drive businesses to bankruptcy. The direct way to deal with these types of risks is stress scenarios, or ‘worst-case’ scenarios, where all relevant parameters take extreme values. These values are unlikely, but they serve the purpose of illustrating the consequences of such extreme situations. Stress testing is a common practice to address highly unlikely events. For instance, rating agencies use stress scenarios to assess the risk of an industry or a transaction. Most bank capital markets units use stress scenarios to see how the market portfolio would behave. Extreme Value Theory (see Embrechts et al., 1997 plus the discussion on stress testing market VaR models) helps in modelling extreme events but remains subject to judgmental assessment. Quantitative risk measures do not capture all uncertainties. They depend on assumptions, which can underestimate some risks. Risks depend on qualitative factors and on intangible events, which quantification does not fully capture. Any due diligence of risks combines judgments and quantitative risk assessment. Quantitative techniques address only measurable risks, without being substitutes for judgment. Still, the current trend is to focus on quantitative measures, enhancing them and extending their range throughout all types of risks. First, when data becomes available, risks are easier to measure and some otherwise intangible risks might become more prone to measurement. Second, when it is difficult to quantify a risk, it might be feasible to qualify it and rank comparable risks, as ratings agencies do. Finally, risk measures became more critical when regulators made it clear that they should provide the basis for capital requirements protecting the bank against unfavourable conditions. Quantitative measures gain feasibility and credibility for all these reasons. This chapter looks at the basic quantified risk measures, before tackling VaR in the next chapter.

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SENSITIVITY Sensitivities are ratios of the variation of a target variable, such as interest margin or change in mark-to-market values of instruments, to a forfeit shock of the underlying random parameter driving this change. This property makes them very convenient for measuring risks, because they link any target variable of interest to the underlying sources of uncertainty that influence these variables. Examples of underlying parameters are interest rates, exchange rates and stock prices. Market risk models use sensitivities widely, known as the ‘Greek letters’ relating various market instrument values to the underlying market parameters that influence them. Asset–Liability Management (ALM) uses gaps, which are sensitivities of the interest income of the banking portfolio to shifts of interest rates. Sensitivities have well-known drawbacks. First, they refer to a given forfeit change of risk drivers (such as a 1% shift of interest rates), without considering that some parameters are quite unstable while others are not. Second, they depend on the prevailing conditions, the value of the market parameters and of assets, making them proxies of actual changes. This section provides definitions and examples.

Sensitivity Definitions and Implications Percentage sensitivities are ratios of relative variations of values to the same forfeit shock on the underlying parameter. For instance, the sensitivity of a bond price with respect to a unit interest rate variation is equal to 5. This sensitivity means that a 1% interest rate variation generates a relative price variation of the bond of 5 × 1% = 5%. A ‘value’ sensitivity is the absolute value of the change in value of an instrument for a given change in the underlying parameters. If the bond price is 1000, its variation is 5% × 1000 = 50. Let V be the market value of an instrument. This value depends on one or several market parameters, m, that can be prices (such as indexes) or percentages (such as interest rates). By definition: s(% change of value) = (V /V ) × m S(value) = (V /V ) × V × m Another formula is s(% change of value) = (V /V ) × (m/m), if the sensitivity measures the ‘return sensitivity’, such as for stock return sensitivity to the index return. For example, if a stock return varies twice as much as the equity index, this last ratio equals 2. A high sensitivity implies a higher risk than a low sensitivity. Moreover, the sensitivity quantifies the change. The sensitivity is only an approximation because it provides the change in value for a small variation of the underlying parameter. It is a ‘local’ measure because it depends on current values of both the asset and the market parameter. If they change, both S and s do1 . 1 Formally,

the sensitivity is the first derivative of the value with respect to m. The next order derivatives take care of the change in the first derivative.

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Sensitivities and Risk Controlling Most of the random factors that influence the earnings of a bank are not controllable by the bank. They are random market or environment changes, such as those that increase the default probabilities of borrowers. A bank does not have any influence on market or economic conditions. Hence, the sources of uncertainty are beyond control. By contrast, it is possible to control the ‘exposure’ or the ‘sensitivities’ to these outside sources of uncertainties. There are two ways to control risk: through risk exposures and through sensitivities. Controlling exposures consists of limiting the size of the amount ‘at risk’. For credit risk, banks cap individual exposures to any obligor, industry or country. In doing so, they bound losses per obligor, industry or country. However, the obvious drawback lies in limiting business volume. Therefore, it should be feasible to increase business volume without necessarily increasing credit risk. An alternative technique for controlling risk is by limiting the sensitivity of the bank earnings to external random factors that are beyond its control. Controlling risk used to be easier for ALM and market risk, using derivatives. Derivatives allow banks to alter the sensitivities to interest rates and other market risks and keep them within limits. For market risk, hedging exposures helps to keep the various sensitivities (the ‘Greeks’) within stated limits (Chapter 30 reviews the main sensitivities). Short-selling bonds or stocks, for example, offsets the risk of long positions in stocks or bonds. For ALM, banks control the magnitude of ‘gaps’, which are the sensitivities of the interest margin to changes in interest rates. Sensitivities have straightforward practical usages. For example, a positive variable interest rate gap of 1000 implies that the interest margin changes by 1000 × 1% = 10 if there is a parallel upward shift of the yield curve. The same techniques now apply with credit derivatives, which provide protection against both the deterioration of credit standing and the default of a borrower. Credit derivatives are insurances sold to lenders by sellers of these protections. The usage of credit derivatives extends at a fast pace because there was no way, until recently, to limit credit risk without limiting size, with a mechanical adverse effect on business volume. Other techniques, notably securitizations, expand at a very fast pace when players realize they could off-load risk and free credit lines for new business.

VOLATILITY In order to avoid using a unique forfeit change in underlying parameters, independently of the stability or instability of such parameters, it is possible to combine sensitivities with measures of parameter instability. The volatility characterizes the stability or instability of any random variables. It is a very common statistical measure of the dispersion around the average of any random variable such as market parameters, earnings or mark-to-market values. Volatility is the standard deviation of the values of these variables. Standard deviation is the square root of the variance of a random variable (see Appendix).

Expectations, Variance and Volatility The mean, or the expectation, is the average of the values of a variable weighted by the probabilities of such values. The variance is the sum of the squared deviations around

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the mean weighted by the probabilities of such deviations. The volatility is the square root of the variance2 . When using time series of historical observations, the practice is to assign to each observation the same weight. The arithmetic mean is an estimator of the expectation. The arithmetic average of the squared deviations from this mean is the historical variance. Standard formulas apply to obtain these statistics. The appendix to this chapter illustrates basic calculations. Here, we concentrate on basic definitions and their relevance to volatility for risk measures.

Probability and Frequency Distributions The curve plotting the frequencies of occurrences for each of the possible values of the uncertain variable is a frequency distribution. It approximates the actual probability distribution of the random variable. The x-axis gives the possible values of the parameter. The y-axis shows either the number of occurrences of this given value, or the percentage over the total number of observations. Such frequency distributions are historical or modelled distributions. The second class of distributions uses well-known curves that have attractive properties, simplifying considerably the calculation of statistics characterizing these distributions. Theoretical distributions are continuous, rather than discrete3 . Continuous distributions often serve as good approximations of observed distributions of market values. A probability distribution of a random variable is either a density or a cumulative distribution. The density is the probability of having a value within any very small band of values. The cumulative function is the probability of occurrence of values between the lowest and a preset upper bound. It cumulates all densities of values lower than this upper bound. ‘pdf’ and ‘cdf’ designate respectively the probability density function and the cumulative density function. A very commonly used theoretical distribution is the normal curve, with its familiar bell shape (Figure 6.2). This theoretical distribution actually looks like many observed Probability The dispersion of values around the mean increases with the 'width' of the curve

Mean

FIGURE 6.2 2 Expectation

Values of the random variable

Mean and dispersion for a distribution curve

and variance are also called the first two ‘moments’ of a probability distribution. example of a simple discrete distribution is the distribution of loss under default risk. There are only two events: default or no default, with assigned probabilities such as 1% and 99%. There is a 1% chance that the lender suffers from the loss under default, while the loss remains zero under no default.

3 An

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distributions of a large number of random phenomena. The two basic statistics, mean and standard deviation, are sufficient to fully determine the entire normal distribution. The normal distribution N(µ, σ ) has mean µ and standard deviation σ . The pdf of the normal distribution is: √ Pr(X = X) = (1/σ 2π) exp[−(X − µ)2 /2σ 2 ] Bold letters (X) designate random variables and italic letters (X) a particular value of the random variable. The standardized normal distribution N(0, 1) has a mean of 0 and a variance of 1. The variable Z = (X − µ)/σ follows a standardized normal distribution N(0, 1), with probability density: √ Pr(Z = Z) = (1/ 2π) exp[−Z 2 /2] The cumulative normal distribution is available from tables or easily calculated with proxy formulas. The cumulative standardized normal distribution is (0, 1). The normal distribution is parametric, meaning that it is entirely defined by its expectation and variance. The normal distribution applies notably to relative stock price return, or ratio of price variations to the initial price over small periods. However, the normal distribution implies potentially extreme negative values, which are inconsistent with observed market prices. It can be shown that the lognormal distribution, which does not allow negative values, is more consistent with the price behaviour. The lognormal distribution is such that the logarithm of the random variable follows a normal distribution. Analytically, ln(X) follows N(µ, σ ), where ln is the Napierian logarithm and N(µ, σ ) the pdf of the normal distribution. The lognormal distribution is asymmetric, unlike the normal curve, because it does not allow negatives at the lower end of values, and permits unlimited positive values4 . Other asymmetric distributions have become popular when modelling credit risk. They are asymmetric because the frequency of small losses is much higher than the frequency of large losses within a portfolio of loans. The moments of a distribution characterize its shape. The moments are the weighted averages of the deviations from the mean, elevated to power 2, 3, 4, etc., using the discrete probabilities of discrete values, or the probability densities as weights. The first moment is the expectation, or mean, of the function. The second moment is the variance. It characterizes dispersion around the mean. The square root of the variance is the standard deviation. It is identical to the ‘volatility’. The third moment is skewness, which characterizes departure from symmetry. The fourth moment is kurtosis, which characterizes the flatness of the distribution.

Volatility The volatility measures the dispersion of any random variable around its mean. It is feasible to calculate historical volatility using any set of historical data, whether or not they follow a normal distribution. It characterizes the dispersion of market parameters, such as that of interest rates, exchange rate and equity index, because the day-to-day observations are readily available. Volatilities need constant updating when new observations are available. An alternative measure of volatility is the implicit volatility embedded in options prices. It derives 4 When

X takes extreme negative values, the logarithm tends towards zero but remains positive.

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the value of volatility from the theoretical relationship (the Black–Scholes formula) of observed option prices with all underlying parameters, one of them being the volatility of the underlying. A benefit of implicit volatilities is that they are forward looking measures, as prices are. A drawback of this approach is that implicit volatilities are fairly volatile, more than historical volatility. For other than market data, such as accounting earnings, the frequency of observations is more limited. It is always possible to calculate a standard deviation with any number of observations. Nevertheless, a limited number of observations might result in a distorted image of the dispersion. The calculation uses available observations, which are simply a sample from the entire distribution of values. If the sample size is large, statistics calculated over the sample are good proxies of the characteristics of the underlying distributions. When the sample size gets smaller, there is a ‘sampling error’. Obviously, the sampling error might get very large when we have very few observations. The calculation remains feasible but becomes irrelevant. The ‘Earnings at Risk’ (EaR) approach for estimating economic capital (Chapter 7) uses the observed volatility of earning values as the basis for calculating potential losses, hence the capital value capable of absorbing them. Hence, even though accounting data is scarcer than market data, the volatility of earnings might serve some useful purpose in measuring risk.

Historical Volatilities The calculation of historical mean and volatility requires time series. Defining a time series requires defining the period of observation and the frequency of observations. For example, we can observe daily stock returns for an entire year, roughly 250 working days. The choice of frequency determines the nature of volatility. A daily volatility results from daily observations, a weekly volatility from weekly observations, and so on. We are simply sampling from an underlying distribution a variable number of daily observations. When a distribution does not change over time, it is ‘stationary’. Daily volatilities are easier to calculate since we have more information. Because the size of the sample varies greatly according to the length of the period, the sampling error—the difference between the unobservable ‘real’ value of volatility and that calculated—is presumably greater with short observation periods such as 1 or 3 months. Convenient rules allow an easy conversion of daily volatilities into monthly or yearly volatilities. These simple rules rely on assumptions detailed in Chapter 30. In essence, they assume that the random process is stable through consecutive periods. For instance, for a random stock return, this would mean that the distribution of the stock return is exactly the same from one period to another. Monthly volatilities are larger than daily volatilities because the changes over 1 month are likely to be larger than the daily √ changes. A practical rule states that the volatility over horizon T , σT , is equal to σ1day T , when T is in days. For example, the daily volatility of a stock return is 1%, measured from 252√daily observations over a 1-year period. The monthly volatility would be σ1month = σ1day 30 = 1% × 5.477 = 5.477%. The volatility increases with time, but less than proportionally5 . According to this rule, the multiples used √ 5 This formula σ = σ t 1 t applies when the possible values of a variable at t do not depend on the value at t − 1.

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TABLE 6.1 Period Multiple

Time coefficients applicable to volatility 1

2

3

4

5

6

7

8

9

10

1.000

1.414

1.732

2.000

2.236

2.449

2.646

2.828

3.000

3.162

to convert a one-period (1 day, for instance) volatility into a multiple-period volatility (1 year, rounded to 250 days, for example) are as given in Table 6.1. The formula is known as the ‘square root of time rule’. It requires specifying the base period, whether 1 day or 1 year. For instance, the 1-year volatility when the yearly volatility is, say, 10% is 14.14%. The ‘square root’ rule is convenient but applies only under restrictive assumptions. Techniques for modelling volatilities as a function of time and across different periods have developed considerably. The basic findings are summarized in the market risk chapter (Chapter 30). Because the rule implies an ever-increasing volatility over time, it does not apply beyond the medium term to random parameters that tend to revert to a long-term value, such as interest rates.

VOLATILITY AND DOWNSIDE RISK Risk materializes only when earnings deviate adversely, whereas volatility captures both upside and downside deviations. The purpose of downside risk measures is to capture loss, ignoring the gains. Volatility and downside risk relate to each other, but are not equivalent. Volatility of earnings increases the chances of losses, and this is precisely why it is a risk measure. However, if downside changes are not possible, there is volatility but no downside risk at all. A case in point is options. The buyer of an option has an uncertain gain, but no loss risk, when looking forward, once he has paid the price for acquiring the option (the premium). A call option on stock provides the right to purchase the stock at 100. If the stock price goes up to 120, exercise generates a profit of 120 − 100 = 20. The potential profit is random just as the stock price is. However, the downside risk is zero since the option holder does not exercise his option when the price falls below 100, and does not incur any loss (besides the upfront premium paid). However, the seller of the call option has a downside risk, since he needs to sell the stock to the buyer at 100, even when he has to pay a higher price to purchase the stock, unless he holds it already. The downside risk actually has two components: potential losses and the probability of occurrence. The difficulty is to assess these probabilities. Worst-case scenarios serve to quantify extreme losses. However, the chances of observing the scenarios are subjective. If someone else has a different perception of the environment uncertainty, he or she might consider that another scenario is more relevant or more likely. The measure of risk changes with the perception of uncertainty. For these reasons, downside risk measures necessitate the prior modelling of the probability distributions of potential losses.

APPENDIX: STATISTICS There are well-known standard formulas for calculating the mean, variance and standard deviation. Some definitions may appear complex at first sight. Actually, these statistics

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are easy to calculate, as shown in the examples below6 . Let X be the random variable, with particular values X. The formulas for calculating the mean and standard deviation of discrete observed values of a random variable, the usual case when using time series of observations, are given below. The random variable is X, the mean is E(X). With historical time series, probabilities are the frequencies of the observed values, eventually grouped in narrow bands. The assigned probability to a single value among n is 1/n. The expectation becomes: E(X) = Xi /n i

The volatility, or standard deviation, is: σ (X) = (1/n) [Xi − E(X)]2 i

In general, probabilities have to be assigned to values, for instance by assuming that the distribution curve is given. The corresponding formulas are given in any statistics textbook. The mean and the standard deviation depend on the probabilities pi assigned to each value Xi of the random variable X. The total of all probabilities is 100% since the distribution covers all feasible values. The mean is: E(X) = pi Xi /n i

The variance is the weighted average by the probabilities of the squared deviations from the mean. The volatility is the square root of this value. The volatility is equal to: σ (X) = pi [Xi − E(X)]2 i

The variance V (X) is identical to σ 2 . With time series, all pi are equal to 1/n, which results in the simplified formulas above. The example in Table 6.2 shows how to calculate a yearly volatility of earnings over a 12-year time series of earnings observations. The expectation is the mean of all observed values. The variance is the sum of squared deviations from the mean, and the standard deviation is the square root. The table gives a sample of calculations using these definitions. Monthly observations of accounting earnings are available for 1 year, or 12 observed values. Volatilities are in the same unit as the random variable. If, for instance, the exchange rate of the euro against the dollar is 0.9 USD/EUR, the standard deviation of the exchange rate is also expressed in USD/EUR, for instance 0.09 USD/EUR. The percentage volatility is the ratio of the standard deviation to the current value of the variable. For instance, the above 0.09 USD/EUR volatility is also equal to 10% of the current exchange rate since 6 Since

the average algebraic deviation from the mean is zero by definition. Squared deviations do not cancel out. Dispersion should preferably be in the same unit as the random variable. This is the case with the standard deviation, making it directly comparable to the observed values of a random parameter.

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TABLE 6.2 Calculation of mean and volatility with a time series of observed data: example Dates

Earnings (dollars)

Deviations from mean

Squared deviations

1 2 3 4 5 6 7 8 9 10 11 12 Sum

15.00 12.00 8.00 7.00 2.00 −3.00 −7.00 −10.00 −5.00 0.00 5.00 11.00 35.00

12.08 9.08 5.08 4.08 −0.92 −5.92 −9.92 −12.92 −7.92 −2.92 2.08 8.08

146.01 82.51 25.84 16.67 0.84 35.01 98.34 166.84 62.67 8.51 4.34 65.34 712.92

Mean

2.92

Sum Statistics:a Variance Volatility

59.41 7.71

a The mean is the sum of observed values divided by the number of observations (12). The variance is the sum of squared deviations divided by 12. The volatility is the square root of variance.

0.09/0.9 = 10%. For accounting earnings, the volatility could either be in dollars or as a percentage of current earnings. For interest rates, the volatility is a percentage, as the interest rate, or a percentage of the current level of the interest rate. Continuous distributions are the extreme case when there is a probability that the variable takes a value within any band of values, no matter how small. Formally, when X is continuous, for each interval [X, X + dX], there is a probability of observing values, which depends on X. The probability density function provides this probability. There are many continuous distributions, which serve as proxies for representing actual phenomena, the most well known being the normal distribution. The pdf is such that all probabilities sum to 1, as with the frequency distribution above. A useful property facilitating the calculation of the variance is that: σ 2 (X) = E(X2 ) − [E(X)]2 Another property of the variance is: σ 2 (aX) = a 2 × σ 2 (X) This also implies that σ (aX) = a × σ (X).

7 VaR and Capital

The ‘Value at Risk’ (VaR) is a potential loss. ‘Potential losses’ can theoretically extend to the value of the entire portfolio, although everyone would agree that this is an exceptional event, with near-zero probability. To resolve this issue, the VaR is the ‘maximum loss’ at a preset confidence level. The confidence level is the probability that the loss exceeds this upper bound. Determining the VaR requires modelling the distribution of values at some future time point, in order to define various ‘loss percentiles’, each one corresponding to a confidence level. VaR applies to all risks. Market risk is an adverse deviation of value during a certain liquidation period. Credit risk materializes through defaults of migrations across risk classes. Defaults trigger losses. Migrations trigger risk-adjusted value changes. VaR for credit risk is an adverse deviation of value, due to credit risk losses or migrations, at a preset confidence level. VaR applies as long as we can build up a distribution of future values of transactions or of losses. The VaR methodology serves to define risk-based capital, or economic capital. Economic capital, or ‘risk-based capital’, is the capital required to absorb potential unexpected losses at the preset confidence level. The confidence level reflects the risk appetite of the bank. By definition, it is also the probability that the loss exceeds the capital, triggering bank insolvency. Hence, the confidence level is equivalent to the default probability of the bank. The VaR concept shines for three major reasons: it provides a complete view of portfolio risk; it measures economic capital; it assigns fungible values to risks. Unlike intuition would suggest, the average loss is not sufficient to define portfolio risk because portfolio losses vary randomly around this average. Because VaR captures the downside risk, it is the basis for measuring economic capital, the ultimate safety cushion for absorbing losses. Finally, instead of capturing risks through multiple qualitative indicators (sensitivities, ratings, watch lists, excess limits, etc.), VaR assigns a dollar value to risk. Valuation makes

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all risks fungible, whatever the sources of uncertainty. By contrast, classical indicators do not add up as dollar values do. ‘Earnings at Risk’ (EaR) is a simple and practical version of VaR. EaR measures, at preset confidence levels, the potential adverse deviations of earnings. EaR is not VaR but shares the same underlying concept, and has the benefit of being relatively easy to measure. Although similar to VaR, EaR does not relate the adverse deviations of earnings to the underlying risks because EaR aggregates the effects of all risks. By contrast, VaR requires linking losses to each risk. Relating risk measures to the sources of risk is a prerequisite for risk management, because the latter aims at controlling risk ex ante, rather than after its materialization into losses. VaR models the value of risk and relates it to the instrumental variables, allowing ex ante control of risk using such parameters as sensitivities to market risk, exposure limits, concentration for credit risk, and so on. The first section describes the potential uses of VaR, and shows how VaR ‘synthesizes’ other traditional measures of risk. The second section details the different levels of potential losses of interest. They include: expected loss; unexpected and exceptional losses from loss distributions. The next sections detail the loss distribution and relate VaR to loss percentiles (or ‘loss at a preset confidence level’), using the normal distribution as an example to introduce further developments. The last section discusses benefits and drawbacks of EaR.

VAR AND RISK MANAGEMENT VaR is a powerful concept for risk management because of the range and importance of its applications. It is also the foundation of economic capital measures, which underlie all related tools, from risk-based performance to portfolio management.

The Contributions of VaR-based Measures VaR provides the measure of economic capital defined as an upper bound of future potential losses. Once defined at the bank-wide level, the capital allocation system assigns capital, or a risk measure after diversification effect, to any subset of the bank’s portfolio, which allows risk-adjusted performances to be defined, using both capital allocation and transfer pricing systems. Economic capital is a major advance because it addresses such issues as: • Is capital adequate, given risks? • Are the risks acceptable, given available capital? • With given risks, any level of capital determines the confidence level, or the bank’s default probability. Both risks and capital should adjust to meet a target confidence level which, in the end, determines the bank’s risk and solvency.

VaR and Common Indicators of Risk VaR has many benefits when compared to traditional measures of risk. It assigns a value to risk, it is synthetic and it is fungible. In addition, the VaR methodology serves to define risk-based capital. The progress is significant over other measures.

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Figure 7.1 illustrates the qualitative gap between traditional risk measures and VaR. It describes the various indicators of risk serving various purposes for measuring or monitoring risks. Such indicators or quantified measures are not fungible, and it is not possible to convert them, except for market instrument sensitivities, into potential losses. By contrast, VaR synthesizes all of them and represents a loss, or a risk value. Because VaR is synthetic, it is not a replacement for such specific measures, but it summarizes them. Credit risk Market risk - Risk measures Volatilities Sensitivities 'Greek letters' - Market values

- Ratings / Maturities / Industries - Watch lists - Concentration - Portfolio monitoring

VaR

Interest rate risk - Gaps Liquidity Interest rate Duration gaps

FIGURE 7.1

Other risks

From traditional measures of risk to VaR

POTENTIAL LOSS This section further details the VaR concept. There are several types of potential losses: Expected Loss (EL); Unexpected Loss (UL); exceptional losses. The unexpected loss is the upper bound of loss not exceeded in more than a limited given fraction of all outcomes. Such potential loss is also a loss percentile defined with the preset confidence level. Since the confidence level might take various values, it is necessary to be able to define all of them. Hence, modelling the unexpected loss requires modelling the loss distribution of the bank portfolio, which provides the frequencies of all various possible values of losses. Obtaining such loss distributions is the major challenge of risk models. VaR is the unexpected loss set by the confidence level. The exceptional loss, or extreme loss, is the loss in excess of unexpected loss. It ranges from the unexpected loss, as a lower bound, up to the entire portfolio value, but values within this upper range have extremely low probability of occurrence.

Expected Loss The expected loss serves for credit risk. Market risk considers only deviations of values as losses, and ignores expected Profit and Loss (P&L) gains for being conservative. Expected loss represents a statistical loss over a portfolio of a large number of loans. The law of

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large numbers says that losses will sometimes be high or low. Intuition suggests that they revert to some long-term average. This is the foundation for economic provisioning and ‘expected loss risk management’. Intuition suggests that provisioning the expected loss should be enough to absorb losses. This might be true in the long-term. By definition, statistical losses average losses over a number of periods and yearly losses presumably tend to revert to some long-term mean. The intuition is misleading, however, because it ignores the transitory periods when losses exceed a long-term average. Lower than long-term average loss in good years could compensate, in theory, higher losses in bad years. There is no guarantee that losses will revert quickly to some long-run average. Deviations might last longer than expected. Therefore, economic provisioning will result in transitory excess losses over the long-term average. Unless there is capital to absorb such excesses, it cannot ensure bank solvency. The first loss above average would trigger default. However, the choice of reference period for calculating the average loss counts. Starting in good years, we might have an optimistic reference value for expected loss and vice versa. Regulators insist on measuring ‘through the cycle’ to average these effects. This is a sound recommendation, so that economic provisions, if implemented, do not underestimate average losses in bad years because they refer to loss observed during the expansion phase of the economic cycle. Statistical losses are more a portfolio concept rather than an individual transaction concept. For one single transaction, the customer may default or not. However, for a single exposure, the real loss is never equal to the average. On the other hand, for a portfolio, the expected loss is the mean of the distribution of losses. It makes sense to charge to each transaction this average, because each one should contribute to the overall required provision. The more diversified a portfolio is, the lower is the loss volatility and the closer losses tend to be to the average value. However, this does not allow us to ignore the unexpected loss. One purpose of VaR models is to specify precisely both dimensions of risk, average level and chances/magnitudes of deviations from this average. Focusing on only one does not provide the risk profile of a portfolio. In fact, characterizing this profile requires the entire loss distribution to see how likely are large losses of various magnitudes. The EL, as a long-term average, is a loss value that we will face it sooner or later. Therefore, it makes sense to deduct the EL from revenues, since it represents an overall averaged charge. If there were no random deviations around this average, there would be no need to add capital to economic provisions. This rationale implies that capital should be in excess of expected loss under economic provisioning.

Unexpected Loss and VaR Unexpected losses are potential losses in excess of the expected value. The VaR approach defines potential losses as loss percentiles at given confidence levels. The loss percentile is the upper bound of loss not exceeded in more than a given fraction of all possible cases, this fraction being the confidence level. It is L(α), where α is the one-tailed1 probability 1 Only

adverse deviations count as losses. Opposite deviations are gains and do not value risk.

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of exceeding L(α). For example, L(1%) = 100 means that the loss exceeds the value of 100 in no more than 1% of cases (one out of 100 possible scenarios, or two to three days within a year)2 . The purpose of VaR models is to provide the loss distribution, or the probability of each loss value, to derive all loss percentiles for various confidence levels. The unexpected loss is the excess of the loss percentiles over the expected loss, L(α) − EL. Economic capital is equal to unexpected loss measured as a loss percentile in excess of expected loss (under economic provisioning).

Exceptional Losses Unexpected loss does not include exceptional losses beyond the loss percentile defined by a confidence level. Exceptional losses are in excess of the sum of the expected loss plus the unexpected loss, equal to the loss percentile L(α). Only stress scenarios, or extreme loss modelling when feasible, help in finding the order of magnitude of such losses. Nevertheless, the probability of such scenarios is likely to remain judgmental rather than subject to statistical benchmarks because of the difficulty of inferring extreme losses which, by definition, are almost unobservable.

MEASURING EXPECTED AND UNEXPECTED LOSSES The two major ingredients for defining expected and unexpected losses are the loss distribution and the confidence level. The confidence level results from a management choice reflecting the risk appetite, or the tolerance for risk, of the management and the bank’s policy with respect to its credit standing. Modelling loss distributions raises major technical challenges because the focus is on extreme deviations rather than on the central tendency. Since downside risk characterizes VaR and economic capital, loss volatility and the underlying loss distribution are critical.

Loss Distributions In theory, historical loss distributions are observable historically. For market risk, loss distributions are simply the distributions of adverse price deviations of the instruments. Since there are approximately as many chances that values increase or decrease, such deviations tend to be bell-shaped, with some central tendency. Of course, the loss distribution is a distribution of negative earnings truncated at the zero level. The bell-shaped distribution facilitates modelling, especially when using the normal or the lognormal distributions as approximations. Unfortunately, historical data is scarce for credit risk and does not necessarily reflect the current risk of banks. Therefore, it is necessary to model loss distributions. For credit risk, losses are not negative earnings. They result from defaults, or loss of asset value because of credit standing deterioration. Such distributions are highly skewed to the left, because the most frequent losses are very small. Both types of distributions are shown in Figure 7.2. 2 An

alternative notation is L(99%), where 99% represents the probability that the loss value does not exceed the same upper bound.

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Market risk Probability

Probability

Credit risk

Gains

Losses

Losses - Large losses - Low probability

FIGURE 7.2

Fat tails and extreme losses

With distributions, the visual representation of losses is simple. In Figure 7.3, losses appear at the right-hand side of the zero level along the x-axis. The VaR at a given confidence level is such that the probability of exceeding the unexpected loss is equal to this confidence level. The area under the curve at the right of VaR represents this probability. The maximum total loss at the same confidence level is the sum of the expected loss plus unexpected loss (or VaR). Losses at the extreme right-hand side and beyond unexpected losses are ‘exceptional’. The VaR represents the capital in excess of expected loss necessary for absorbing deviations from average losses. Mode (most frequent)

Expected loss

Expected + Unexpected loss

Probability

Probability of loss < UL

Probability of loss > UL: Confidence level

Losses = 0

Expected loss

FIGURE 7.3

Losses

Unexpected loss = VaR

Exceptional loss

Unexpected loss and VaR

A well-known characteristic of loss distributions is that they have ‘fat tails’. Fat tails are the extreme sections of the distribution, and indicate that large losses, although unlikely because their probabilities remain low, still have some likelihood to occur that is not negligible. The ‘fatness’ of the tail refers to the non-zero probabilities over the long end of the distributions.

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The VaR is easy to determine under simplifying assumptions on the distribution curve of losses. With normal curves, the VaR is a multiple of loss volatility that depends on the confidence level. For example, the 2.5% one-tailed confidence level corresponds to a multiple of loss volatility of 1.96. Therefore, if the loss volatility is 100, the unexpected loss will not exceed the upper bound of 196 in more than two or three cases out of 100 scenarios. Unfortunately, such multiples do not apply when the distribution has a different shape, for instance for credit risk. When implementing techniques based on confidence levels and loss percentiles, there is a need for common benchmarks, such as confidence levels, for all players. With a very tight confidence level, the VaR could be so high that business transactions would soon become limited by authorizations, or not feasible at all. If competitors use different VaR models or confidence levels, banks will not operate on equal grounds. Tighter confidence levels than competitors’ levels would reduce the volume of business of the most prudent banks and allow competitors having more risk appetite to take advantage of an overly prudent policy.

LOSS PERCENTILES OF THE NORMAL DISTRIBUTION The normal distribution is a proxy for market random P&L over a short period, but it cannot apply to credit risk, for which loss distributions are highly asymmetrical. In this section, we use the normal distribution to illustrate the VaR concept and the confidence levels. The VaR at a confidence level α is the ‘loss percentile’ L(α). In Figure 7.4, the area under the curve, beyond the boundary value on the left-hand side, represents the probability that losses exceed this boundary value. Visually, a higher volatility means that the curve dispersion around its mean is wider. Hence, the chances that losses exceed a given boundary value grow larger. The confidence intervals are probabilities that losses exceed an upper bound (negative earnings, beyond the zero level). They are ‘one-tailed’ because only one-sided negative deviations materialize downside risk. Probability of earnings Probability 5%

Losses

FIGURE 7.4

Maximum loss at the 5% level

Gains

Volatility and downside risk

When both upside and downside deviations of the mean are considered, the confidence interval is ‘two-tailed’. With a symmetric distribution, the two-tailed probability is twice the one-tailed probability. When the probability of losses exceeding a maximum value is

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lev

els

2.5%, the probability that losses exceed either the lower or the upper bounds is 5%. Unless otherwise stated, we will stick to the ‘one-tailed’ rule for specifying confidence intervals. Confidence intervals are boundary values corresponding to a specified confidence level. In the case of the normal curve, confidence intervals are simply multiples of the volatility. Figure 7.5 shows their values. With the normal curve, the upper bounds of (negative) deviations corresponding to the confidence levels of 16%, 10%, 5%, 2.5% and all other values are in the normal distribution table. They correspond respectively to deviations from the mean of 1, 1.28, 1.65 and 1.96 times the standard deviation σ of the curve. Any other confidence interval corresponds to deviations expressed as multiples of volatilities for this distribution. 16% Normal distribution

id

en

ce

10%

Co

nf

5% 2.5%

1.0% 0

Mean 1.00 σ

Losses

Earnings

1.28 σ 1.65 σ 1.96 σ 2.33 σ

FIGURE 7.5

Confidence levels with the normal distribution

ISSUES AND ADVANCES IN MODELLING VAR AND PORTFOLIO RISKS When we characterize an individual asset, independent of a portfolio context, we adopt a ‘standalone’ view and calculate a ‘standalone’ VaR. This serves only as an intermediate step for moving from ‘standalone’ loss distributions of individual assets to the portfolio loss distribution, which combines losses from all individual assets held in the portfolio. Standalone loss distributions list all possible values of losses for an asset with their probabilities. For instance, a loan whose value is outstanding balance under no default and zero under default has a loss distribution characterized by a 0% loss with probability of, say, 98% and a 100% loss with probability of 2%. The obvious difficulty in VaR measures is the modelling of the loss distribution of a portfolio. The focus on high losses implies modelling the ‘fat tail’ of the distributions rather than looking at the central tendency. Even with market-driven P&L, the normal distribution does a poor job of modelling distribution tails. For credit risk, the issue is worse since the loss distributions are highly skewed. Loss distributions depend on portfolio structure, size discrepancies (concentration risk on big exposures) and the interdependencies between individual losses (the fact that a loss occurrence increases or

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decreases the likelihood of occurrence of other losses). In subsequent chapters, ‘portfolio models’ designate models providing the loss distribution of portfolios. Diversification and granularity effects within a portfolio were recognized long ago but banks could not quantify them. What is new with portfolio models is that they provide the ability to quantify concentration and diversification effects on the portfolio risk. The added complexity is the price to pay for this single most important value added of the new portfolio models. ‘Fat tails’ of actual distributions make the quantification of extreme losses and their probability of occurrence hazardous. The main VaR modelling drawback is that they are highly demanding in terms of data. Because of the technicalities of modelling loss distribution for market and credit risk, several dedicated chapters address the various building blocks of such models: for market risk, see Chapter 32; for credit risk, see Chapters 44–50.

RISK-BASED CAPITAL The VaR methodology applies to measure risk-based capital. The latter differs from regulatory capital or from available capital in that it measures actual risks. Regulatory capital uses forfeits falling short of measuring actual risks. Economic capital necessitates the VaR methodology, with the modelling of loss distribution, with all related complexities. The EaR concept is an alternative and simpler route to capital than VaR. For this reason, it is useful to detail the technique and to contrast the relative merits of EaR versus VaR.

The Limitations of Simple Approaches to Capital The simplest way to define the required capital is to use regulatory capital. This is a common practice in the absence of any simple and convincing measure of capital. In addition, the regulatory capital is a requirement. At first sight, there seems to be no need to be more accurate than the regulators. However, regulatory capital has many limitations, even after the enhancements proposed by the New Accord. Using regulatory capital as a surrogate for economic capital generates important distortions because of the divergence between the real risks and the forfeited risks of regulatory capital. For regulation purposes, credit risk is dependent on outstanding balances (book exposures) and on risk weights. Such forfeits are less risk-sensitive than economic measures. In addition, standardized regulatory approaches measure risk over a portfolio by a simple addition of individual risks for credit risk. This ignores the diversification effect and results in the same measure of risk for widely diversified portfolios and for highly concentrated portfolios. The shortcomings of forfeit measures have implications for the entire risk management system. What follows applies to credit risk, since market models are allowed. Visibility on actual risks remains limited. Credit risk limits remain based on book exposures since regulatory capital depends on these. The target performance also uses forfeit measures of capital. The allocation of this capital across business units does not depend on their ‘true’ risks. Any risk-based policy for measuring risk-adjusted performances, or for risk-based pricing, suffers from such distortions. The most important benefit of economic capital is to correct such distortions.

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Earnings at Risk EaR is an important methodology for measuring capital. A simple way of approaching risk is to use the historical distributions of earnings. The wider the dispersion of time series of earnings, the higher the risk of the bank. Principle

Several measures of earnings can be used to capture their instability over time: accounting earnings; interest margins; cash flows; market values, notably for the trading portfolio. The volatility is the adequate measure of such dispersion. It is the standard deviation of a series of observations. Such calculation always applies. For instance, even with few observations, it remains possible to calculate volatility. Of course, the larger the data set, the more relevant the measure will be. The concept applies to any subportfolio as well as to the entire bank portfolio. When adding the earning volatility across subportfolios, the total should exceed the loss volatility of the entire portfolio because of diversification. Once earnings distributions are obtained, it is easy to use loss volatility as the unit for measuring capital. Deriving capital follows the same principle as VaR. It implies looking for some aggregated level of losses that is not likely to be exceeded in more than a given fraction of all outcomes. Benefits

The major benefit of EaR is in providing a very easy overview of risks. This is the quicker approach to risk measurement. It is not time intensive, nor does it require much data. It relies on existing data since incomes are always available. EaR requires few Information Technology (IT) resources and does not imply major investments. There is no need to construct a risk data warehouse, since existing databases provide most of the required information and relate it easily to transactions. It is easy to track major variations of earnings to some specific events and to interpret them. EaR provides a number of outputs: the earnings volatility, the changes of earnings volatility when the perimeter of aggregation increases, measuring the diversification effect. The level of capital is an amount that losses are not likely to exceed. This is a simple method of producing a number of outputs without too much effort. This is so true that EaR has attracted attention everywhere, since it is a simple matter to implement it. Drawbacks

There are a number of drawbacks to this methodology. Some of them are purely technical. A volatility calculation raises technical issues, for instance when trends of time series increase volatility. In such cases, the volatility comes from the trend rather than instability. There is no need to detail further technical difficulties, since they are easy to identify. The technique requires assumptions, but the number of options remains tractable and easily managed. In general, the drawbacks of simplicity are that EaR provides only crude measures. However, the major drawbacks relate to risk management. It is not possible to define the sources of the risk making the earnings volatile. Presumably, various types of risks

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materialize simultaneously and create adverse deviations of earnings. The contributions of these risks to the final volatility remain unknown. Unlike VaR models, EaR captures risks as an outcome of all risks, not at their source. Without links to the sources of risk, market, credit or interest rates, EaR serves to define aggregated capital, but it does not allow us to trace back risks to their sources. A comprehensive and integrated risk management system links measures with specific sources of risk. The VaR for market risk and credit risk, and the ALM measures of interest rate, are specific to each risk. They fit better bank-wide risk management systems because they allow controlling each risk upstream, rather than after the fact. The EaR methodology does not meet such specifications. On the other hand, it is relatively easy to implement compared to full-blown systems. EaR appears to be an additional tool for risk management, but not a substitute.

8 Valuation

Under the traditional accounting framework applying to the banking portfolio, loans are valued at book value and earnings are interest income accrued over a period. Although traditional accounting has undisputed merits, it has economic drawbacks. Periodical measures of income ignore what happens in subsequent periods. The book value of a loan does not change when the revenues or the risks are higher or lower than average. Hence, book values are not revenue- or risk-adjusted, while ‘economic values’ are. In essence, an ‘economic value’ is a form of mark-to-market measure. It is a discounted value of future contractual cash flows generated by assets, using appropriate discount rates. The discounting process embeds the revenues from the assets in the cash flows, and embeds the risk also in the discount rates. This explains the current case for ‘fair value’. Marking-to-market does not imply that assets are actually tradable. The mark-to-market values of the trading portfolio are market prices. However, a loan has a mark-to-market value even though it is not tradable. Since the risk–return trade-off is universal in the banking universe, a major drawback of book values is that they are not faithful images of such risks and returns. Another drawback is that book values are not sensitive to the actual ‘richness’ or ‘poorness’ of transactions, and of their risks, which conflicts with the philosophy of a ‘faithful’ image. Risk models also rely on mark-to-market valuations. ‘Mark-to-model’ valuations are similar to mark-to-market, but exclude some value drivers. For example, isolating the effect of credit risk on value does not require using continuously adjusted interest rates. ‘Mark-to-future’ is a ‘mark-to-model’ valuation at future time points, differentiated across scenarios characterizing the random outcomes from current date up to a future time point. Value at Risk (VaR) risk models use revaluations of assets at future dates for all sets of random outcomes, to provide the value distribution from which VaR derives. This makes the ‘revaluation block’ of models critical for understanding VaR.

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However, moving to ‘fair value’ is a quantum leap because of the implication in terms of profitability. The relevant measure of profitability becomes Profit and Loss (P&L) rather than accrual income. Traditional measures of performance for the banking portfolio are interest income plus fees. For the market portfolio they are P&L or the change of markto-market values of assets traded between any two dates. Moving to fair values would imply P&L measures, and P&L volatility blurring the profitability view of the bank. The debate on accounting rules follows. Whatever the outcome on accounting standards, economic values will be necessary for two main reasons: they value both risks and returns; valuation is a major building block of credit models because all VaR models need to define the distribution of future values over the entire range of their future risk states. Because of the current growing emphasis on ‘fair value’, we review here valuation issues and introduce ‘mark-to-model’ techniques. The first section is a reminder of accounting standards. The next section details mark-to-market calculations and market yields. The third section summarizes why economic valuation is a building block of risk models. The fourth discusses the relative merits of book versus economic valuations. The fifth section introduces risk-adjusted performance measures, which allow separation of the risk and return components of economic value, rather than bundling them in a single fair value number.

ACCOUNTING STANDARDS Accounting standards have evolved progressively following the guidelines of several boards. The Financial Accounting Standard Board (FASB) promotes the Generally Accepted Accounting Practices (GAAP) that apply to all financial institutions. The International Accounting Standard (IAS) Committee promotes guidelines to which international institutions, such as the multinational banks, abide. All committees and boards promote the ‘fair value’ concept, which is a mark-to-market value. ‘Fair value’ is identical to market prices for traded instruments. Otherwise, it is an economic value. It is risk-adjusted because it uses risky market yields for valuation. It is revenue-adjusted because it discounts all future cash flows. Economic values might be either above or below face value depending on the gap between asset returns and the market required yield applicable to assets of the same risk class. The main implication of fair value accounting is that earnings would result from the fluctuations of values, generating volatile P&L. The sensitivity of value to market movements is much higher for interest-earning assets with fixed rates than with floating rates, as subsequent sections illustrate. Fair value is implemented partially. The IAS rules recommended that all derivatives should be valued at mark-to-market prices. The rule applies to derivatives serving as hedges to bank exposures. The implication is that a pair made up of a banking exposure valued at book value matched with a derivative could generate a value fluctuating because of the valuation of the derivative while the book value remains insensitive to market movements. The pair would generate a fluctuating value and, accordingly, a volatile P&L due to the valuation of the derivative. This is inconsistent with the purpose of the hedge, which is to stabilize the earnings for the couple ‘exposure plus hedge’. To correct the effect of the valuation mismatch, the rule allows valuation of both exposure and hedge at the fair value.

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It is too early to assess all implications of fair values. Moreover, many risk models use mark-to-model valuation for better measuring risk, notably credit risk models that apply to the banking portfolio. Nevertheless, it is necessary to review fair value calculations in order to understand them. The subsequent sections develop gradually all concepts for determining mark-to-market ‘fair values’.

MARK-TO-MARKET VALUATION This section develops the basic mark-to-market model step-by-step. First, it explains the essentials of principles for discounting, the interpretation of the discounting process and the implication for selecting a relevant discount rate. Second, it introduces the identity between market interest rates, or yields, and required rates of return on market investments in assets providing interest revenues. Third, it explains the relationship between market prices and yield to maturity, and explains how this applies to non-marketable assets as well, such as loans. For the sake of clarity, we use a one-period example. Then, using the yield to maturity (Ytm) concept, we extend the same conclusions to a multiple-period framework with simple examples.

The Simple ‘Discounted Cash Flow’ Valuation and the ‘Time Value of Money’ The familiar Discounted Cash Flow (DCF) model explains how to convert future values into present values and vice versa, using market transactions. Therefore, it explains which are the discount rates relevant for such actual time translations. It is also the foundation of mark-to-market approaches, although it is not a full mark-to-market because the plain DCF model does not embed factors embedded in actual mark-to-market values, such as the credit spreads specific to each asset. The present value of an asset is the discounted value of the stream of future flows that it generates. When using market rates as discount rates, the present value is a mark-to-market value. The present value of a stream of future flows Ft is: Ft /[1 + y(t)]t V = t

The market rates are rates applying to a single flow at date t, or the zero-coupon rates y(t). The formula applies to any asset that generates a stream of contractual and certain flows. Cash flows include, in addition to interest and principal repayments, fees such as upfront flat fees, recurring fees and non-recurring fees (Figure 8.1). With floaters, interest payments are indexed to market rates, and the present value is equal to the face value. For instance, an asset with a face value of 100 generates a flow over the next period that includes both the interest and the principal repayment. This final flow is equal to 100(1 + r), r also being the market rate that applies to the period. When this flow is discounted with these market rates, the present value is constant and equal to 100(1 + r)/(1 + r) = 100. The same result applies when the horizon extends beyond one period with the same assumptions. Nevertheless, when the asset pays more than the discount rate, say y + m, with m being a constant percentage, while the discount rate is y, the one-period flow becomes 100(1 + y + m) with discounted value at y: 100(1 + y +

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Discounting at y (t)

Ft

0

FIGURE 8.1

t

Time

The present value of assets with the DCF model

m)/(1 + y) = 100[1 + m/(1 + y)]. The value becomes higher than face value, by a term proportional to m/(1 + y), and sensitive to the yield y. The relevant discount rates are those that allow flows to be transferred across time through borrowing and lending. This is why market rates are relevant. For instance, transferring a future flow of 1000 to today requires borrowing an amount equal to 1000/(1 + yb ), where yb is the borrowing rate. Transferring a present flow of 1000 to a future date requires lending at the market rate yl . The final flow is 1000(1 + yl ). Hence, market rates are relevant for the DCF model (Figure 8.2).

0

1

Borrow X/(1+yb)

FIGURE 8.2

Repay X

0

1

Lend X

Receive X(1+yl)

Discounting and borrowing or lending at market rates

The simple DCF model refers to borrowing and lending rates of an entity. The actual market values of traded assets use different discount rates, which are market rates given the risk of the asset. Full mark-to-market valuation implies using the DCF model with these rates.

Continuous and Discrete Compounding or Discounting Pricing models use continuous compounding instead of discrete compounding, even though actual product valuation uses discrete compounding. Because some future examples use continuous calculations, we summarize the essentials here. The basic formulas for future and present values are very simple, with a discount rate y (continuous) and a horizon n: FVc (y, n) = exp(yn) = eyn

PVc (y, n) = exp(−yn) = e−yn

The index c stands for continuous. Using discounted cash flow formulas to value the present value of a stream of cash flows, simply substitutes e−rt in the discount factor

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1/(1 + y)t . The formulas do not give the same numerical results, although there is a continuous rate that does. If the discrete rate is y, the discrete time and the continuous time formulas are respectively: PVc (y) = PVc (y) =

n t=1

n

CFt /(1 + y)t CFt exp(−yt)

t=1

We provide details and, notably, the rule for finding continuous rates equivalent to discrete rates in the appendix to this chapter.

Market Required Rates and Asset Yields or Returns The yield of a tradable asset is similar to a market interest rate. However, banking assets and liabilities generally do not provide or pay the market rates because of positive spreads over market rates for assets and costs of deposits lower than market rates on the liability side. Valuation depends on the spread between asset and market yields. The asset fixed return is r = 6% and y is the required market yield for this asset. The market required yield depends on maturity and the risk of the asset. We assume both are given, and the corresponding y is 7%. We receive a contractual cash flow, 1 year from now, equal to 1 + r in 1 year, r being the interest payment. For instance, an asset of face value 1000 provides contractually 6% for 1 year, with a bullet repayment in 1 year. The contractual risk-free flow in 1 year is 1060. The 1-year return on the asset is 6%, if we pay the asset at face value 1000 since r = (1060 − 1000)/1000 = 6%. If the market rate y is 7%, the investors want 7%. If they still pay 1000 for this asset, they get only 6%, which is lower. Therefore, they need to pay less, say a price P unknown. The price P is such that the return should be 7%: (1060 − P )/P = 7%. This equation is identical to: P = 1060/(1 + 7%) = 990.65 If y = 7% is the 1-year market required rate, the present value, or mark-to-market value, of the future flow is 990.65 = 1060(1 + y) = 1000(1 + r)/(1 + y). If we pay the face value, the 1-year return becomes equal to the market yield because [(1 + y) − 1]/1 = y. Hence, any market interest rate has the dimension of a return, or a yield. Getting 1 + y, one period from now, from an investment of 1 means that its yield is y. Market rates are required returns from an investor perspective. Since the market provides y, a rate resulting from the forces of supply and demand for funds, any investor requires this return (Figure 8.3). The implication is that the investor pays the price that provides the required market return whatever the actual contractual flows are. Paying 990.65 generates the required market return of 7%. This price is lower than the face value. If the market return is identical to the contractual asset rate of 6%, the investor actually gets exactly 6% by paying the face value 1000 since (1060 − P )/P = 6%. If the market return is lower than the contractual asset rate of 6%, say 5%, the investor actually gets more than 5%

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Asset

Cash flow = 1060

Face value = 1000 0

1

Value of asset cash flow given market required yields

V(0) = 990.65 0

FIGURE 8.3

Time

Return r = 6%

Time

Market required yield y = 7%

1

Asset return versus market required yield

by paying 1000 since (1060 − P )/P = 6%. For the investor to get exactly 5%, he pays exactly P = 1060/(1 + 5%) = 1009.52. The price becomes higher than the face value. The general conclusions are that the value is identical to the face value only when the asset return is in line with the required market return. Table 8.1 summarizes these basic conclusions. TABLE 8.1

Asset return versus market required yield

Asset return r > market required return y Asset return r < market required return y Asset return r = market required return y

value > book value value > book value value = book value

Asset Return and Market Rate Since the asset value is the discounted value of all future cash flows at market rates, there is an inverse relationship between interest rate and asset value and the market rate. This inverse relationship appears above when discounting at 6% and 7%. The asset value drops from 1000 to 990.65. The value–interest rate profile is a downward sloping curve. This relationship applies to fixed rate assets. It serves constantly when looking at the balance sheet economic value (Net Present Value, NPV) in Asset–Liability Management (ALM) (Figure 8.4). The value of a floater providing interest revenues calculated at a rate equal to the discounting rate does not depend on prevailing rates. This is obvious in the case of a single period, since the asset provides 1 + r at the end of the period and this flow discounted at i has a present value of (1 + r)/(1 + r) = 1. Although this is not intuitive, the property extends to any type of floater, for instance, an amortizing asset generating revenue calculated with the same rate as the discount rate. This can be seen through examples. An amortizing loan of 1000, earning 10%, repaid in equal amounts of 200 for 5 years, provides the cash flow stream, splitting flows into capital and interest: 200 + 100, 200 + 80, 200 + 60, 200 + 40, 200 + 20. The discounted value of these flows at 10% is equal to 1000. If the rate floats, and becomes 12% for instance, the cash flow stream then

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Asset Value Fixed rate asset

Variable rate asset

Interest Rate

FIGURE 8.4

Fixed rate asset value and interest rate

becomes: 200 + 120, 200 + 96, 200 + 72, 200 + 48, 200 + 241 . Discounting these flows at 12% results in a present value of 1000 again. When the asset yield is above the discount rates, there is an ‘excess spread’ in the cash flows. For example, if the asset provides 10.5% when rates are 10%, the spread is 0.5%. When rates are 12%, the spread also remains at 0.5% because the asset provides 12.5%. A floater is the sum of a zero-spread floater (at 12%) whose value remains equal to the face value whatever the rates, plus a smaller fixed rate asset providing a constant 0.5% of the outstanding balance, which is sensitive to rate changes. Since the amount is much smaller than the face value, the value change of such a variable rate asset is much smaller than that of a similar fixed rate asset.

Interest Rates and Yields to Maturity These results extend to any number of periods through the yield to maturity concept. When considering various maturities, we have different market yields for each of them. The entire spectrum of yields across maturities is the ‘yield curve’, or the term structure of interest rates2 . In practice, two different types of yields are used: zero-coupon yields and yields to maturity. The zero-coupon rates apply to each individual future flow. The yields to maturity are ‘compounded averages’ across all periods. We index current yields by the maturity t, the notation being y(t). The date t is the end of period t. For instance, y(1) is the yield at the end of period 1. If the period is 1 year, and if we are at date 0 today, y(1) is the spot rate for 1 year, y(2) the spot rate for 2 years, etc. Using zero-coupon rates serves to price an asset starting from the contractual stream of cash flows and the market zero-coupon rates. Market prices of bonds are the discounted cash flows of various maturities using these yields. Instead of using these various rates, it is convenient to use a single discount rate, called the yield to maturity (or ‘Ytm’). Using the Ytm addresses the issue of finding the return for an investor between now and 1 The

interest flows are 12% times the outstanding balance, starting at 1000 and amortized by 200 at each period. 2 See Chapter 12 for details on the term structure of rates and zero-coupon rates.

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maturity, given the contractual stream of cash flows and the price of the asset3 . The Ytm is the unique discount rate making the present value of all future flows identical to the observed price. Yield to Maturity and Asset Return

For a loan, there is also a yield to maturity. It is the discount rate making the borrowed amount, net of any fees, identical to the discounted contractual cash flows of the loan. The yield to maturity derives from the stream of contractual cash flows and the current price of a bond or of the net amount borrowed. The following example illustrates the concept with a bullet bond (principal repayment is at maturity) generating coupons equal to 6% of face value, with a face value equal to 1000, and maturing in 3 years. The stream of cash flows is therefore 60, 60 and 1060, the last one including the principal repayment. In this example, we assume that cash flows are certain (risk-free) and use market yields to maturity applicable to the 3year maturity. The asset rate is 6%. Discounting all flows at 6% provides exactly 1000, implying that the yield to maturity of the bond valued at 1000 is 6%. It is equal to its book return of 6%. This is a general property. When discounting at a rate equal to the asset contractual return r = 6%, we always find the face value 1000 as long as the bond repays without any premium or discount to the principal borrowed. If the market yield to maturity for a 3-year asset rises to 7%, the value of the asset cash flow declines to 973.77. This is identical to what happens in our former example. If the required yield is above the book return of 6%, the value falls below par. Back to the 7% market yield to maturity, the asset mark-to-market value providing the required yield to maturity to investors has to be 973.77. An investor paying this value would have a 7% yield to maturity equal to the market required yield. The reverse would happen with a yield to maturity lower than 6%. The value would be above the book value of 1000. These conclusions are identical to those of the above example of a 1-year asset (Table 8.2). TABLE 8.2

Value and yield to maturity

Date Rate Cash flows Discounted cash flows Current value Rate Discounted cash flows Current value

3 The

0

1

2

3

6% 60 56.60

6% 60 53.40

6% 1060 890.00

7% 56.07

7% 52.41

7% 865.28

1000

973.77

second application relies on assumptions limiting the usage of the Ytm, which we ignore at this stage. It is easy to show that the Ytm is the effective yield obtained by the investor if he holds the asset to maturity and if all intermediate flows received, whatever their nature, interest or principal repayments, are reinvested in the market at the original yield to maturity. The second assumption is unrealistic since there is no way to guarantee that the future market rate prevailing will be in line with the original Ytm calculation, except by an extraordinary coincidence.

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Value and Term Structure of Yields

When using different yields to maturity, the discounting calculations serve to determine the mark-to-market value of any asset, including loans, even though these are not traded in the market. In this latter case, the value is a theoretical mark-to-market, rather than an actual price in line with market rates. We can compare the theoretical price with actual prices for traded assets, which we cannot do for non-traded assets. Using the market zero-coupon yields of Table 8.3, we find that the discounted value of the bond flows is: V = 60/(1 + 5.00%) + 60/(1 + 6.00%)2 + 1060/(1 + 7.00%)3 V = 57.14 + 53.40 + 865.28 = 975.82 TABLE 8.3 rates

Term structure of ‘zero-coupon’ market

End of period Market rate, date t

1

2

3

5.00%

6.00%

7.00%

The value has no reason to be identical to 1000. It represents a mark-to-market valuation of a contractual and certain stream of cash flows. There is a yield to maturity making the value identical to the discounted value of cash flows, using it as a unique discount rate across all periods. It is such that: 975.82 = 60/(1 + y) + 60/(1 + y)2 + 1060/(1 + y)3 The value is y = 6.920%4 . It is higher than the 6% on the face value, because we acquire this asset at a value below par (975.82). The 6.92% yield is the ‘averaged’ return of an investor buying this asset at its market price, with these cash flows and holding it to maturity.

Risk-free versus Risky Yields Some yields are risk-free because the investors are certain of getting the contractual flows of assets, whereas others are risky because there is a chance that the borrower will default on his obligations to pay debt. Risk-free debt is government debt. The risky yield and the risk-free yield at date t are y and yf . The risky required yield has to be higher than the risk-free rate for the same maturity to compensate the investors for the additional risk borne by acquiring risky debt. For each maturity t, there is a risk-free yield yf (t) and a risky yield y(t) for each risk class of debt. The market provides both yield curves, derived from observed bond prices. The difference is the credit spread. Risky yields are the sum of the risk-free yield plus a ‘credit spread’ corresponding to the risk class of the asset and the maturity. The credit spread is cs(t) = y(t) − yf (t). Both zero-coupon yields and 4 It

is easy to check that discounting all contractual flows at this rate, we actually find the 949.23 value. The rate y is the internal rate of return of the stream of flows when using 949.23 as initial value.

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yields to maturity are such that y(t) = yf (t) + cs(t). When using full mark-to-market, the discount rates depend on both the maturity of flows and their credit risk5 (Table 8.4). Credit spreads vary with maturity and other factors. The spreads theoretically reflect the credit risk of assets under normal conditions. Since the prices of risky bonds might also depend on other factors such as product and market liquidity, spreads might not depend only on credit risk. Credit spreads are observable from yields and increase with credit risk, a relationship observed across rating classes. TABLE 8.4 risky yields

Term structure of risk-free and

End of period Risky yield Risk-free yield Credit spreads

1

2

3

5.00% 4.50% 0.50%

6.00% 5.00% 1.00%

7.00% 5.90% 1.10%

There are two basic ways to discount future flows: using zero-coupon yields for each date, or using a unique yield to maturity. Zero-coupon yields embed credit spreads for each maturity. The yield to maturity embeds an average credit spread for risky debts. V (0) is the current value of an asset at date 0. By definition of the risky yield to maturity, the mark-to-market value of the entire stream of contractual cash flows is: V (0) =

T

Ft /(1 + y)t

V (0) =

T

Ft /[1 + yf (t) + cs(t)]t

t=1

t=1

Risk and Relative Richness of Facilities Marking-to-model differentiates facilities according to relative richness and credit risk irrespective of whether assets are traded or illiquid banking facilities. Rich facilities providing revenues higher than market have a value higher than book, and conversely with low revenue facilities. Richness results from both revenues and risk. The ‘excess spread’ concept characterizes richness. It is the spread between the asset return over the sum of the risk-free rate plus the credit spread corresponding to the risk class of the asset. To make this explicit, let us consider a zero-coupon maturing at T , with a unit face value, providing the fixed yield r to the lender, and a risk such that the risky discount rate is y, including the credit spread: V = (1 + r)T /(1 + y)T It is obvious that V > 1 and V ≤ 1 depending on whether r > y or r ≤ y. From the above, valuation is at par value (book value) when the excess spread is zero. Deviations 5 There

is an alternative valuation technique for risky debt using so-called risk-neutral probabilities of default. In addition, credit spreads provide information on these probabilities. The details are in Chapter 42 dedicated to the valuation of credit risk.

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from par result from both revenues and risks. Mark-to-model valuation reflects risk and excess revenues compared to the market benchmark. Figure 8.5 shows two cases, with fair value above and below book value, from these two return and risk adjustments. The risk adjustment results from discounting at a rate y > yf , with a risk premium over the risk-free rate. The return adjustment corresponds to the excess return r − yf of the asset above the risk-free rate. The overall adjustment nets the two effects and results in the excess return of the asset over the risky discount rate, or r − y. If both adjustments compensate, implying y = r, the value remains at par. Excess return adjustment

Book value

Fair value

Book value

Fair value

Risk adjustment

FIGURE 8.5

From book value to fair value: excess return and risk adjustments

The excess spread is r − y. The gap between r and y is the ‘relative richness’ of the asset, relative to the required market return. A rich asset provides a higher return than the market requires given risk. Its mark-to-market value is above book value. A poor asset pays less than the required market return y. Its value is below book value. These properties apply for all risky assets, except that, for each risk class, we need to refer to market rates corresponding to this specific risk class. As an example, we assume that a risky asset provides a 6% yield, with a principal of 100 and a 1-year maturity. The market required risky yield is 5.5% and the risk-free yf required yield is 5.5%. The excess spread is 6% −5.5% = 0.5%. The value of the asset at an excess spread of zero implies a zero excess spread, or an asset return of 5.5%. With this return, the market value of the asset is obviously the face value, or 1000. The value of risk is the difference between the value at the required risky rate and the value at the risk-free yield, or 1060/(1+5.5%) − 1060/(1+5%) = 1004.74 − 1009.52 = −4.78. The value of the excess spread is the difference between the value of an asset providing exactly the required risky yield, or 100, and the value of the actual asset with a positive excess spread, or 1004.74, which is +4.74.

MARK-TO-MODEL VERSUS FULL MARK-TO-MARKET VALUATION In many risk models, the valuation building block plays a critical role: • For ALM models, the NPV of the balance sheet is the present value of assets minus the value of liabilities. It serves as a target variable of ALM models because it captures

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the entire stream of cash flows generated by assets and liabilities (see Chapter 22). Intuitively, if assets provide excess return over market yields, and liabilities cost less than market yields, assets are above par and liabilities below par value. The difference should capture the expected profitability of the balance sheet in terms of present value. • For market risk VaR models, the loss percentile derives from the distribution of future values of all assets at a future time point. The future value is random because of market movements and the distribution implies revaluation of assets at the future time point as a function of random market moves. • For credit risk models, the same principle applies, except that we focus on random credit risk events to find the distribution of uncertain values at a future horizon. The credit events include defaults and changes of credit standing, implying a value adjustment with market risky yields. The revaluation building block provides the spectrum of future values corresponding to all future credit states. The valuation calculation depends on various options. Full mark-to-market valuation uses all parameters influencing prices to find out the distribution of random values at future time points. This is not so for ALM and credit risk. The NPV of the balance sheet uses a single set of market rates differentiated by maturity, but not by risk class. The calculation differs from mark-to-market and fair value, in that it does not differentiate the discount rates according to the risk of each individual asset. The interpretation of the NPV is straightforward if we consider the bank as ‘borrowing’ from the market at its average cost of funding, which depends on its risk class. The bank could effectively bring back to present all future asset cash flows to present by borrowing against them. Borrowing ‘against’ them means borrowing the exact amount that would require future repayments, inclusive of interest, identical to the cash flows of all assets at this future date. This is a mark-to-model rather than a full mark-to-market. Mark-to-model calculations eliminate some of the drawbacks of the full MTM. For instance, when isolating credit risk, day-to-day variations due to interest rate fluctuations are not necessarily relevant because the aim is to differentiate future values according to their future random credit states. One option is to price the credit risk using the current ‘crystallized’ rates allowing credit spreads only to vary according to the future credit states at the horizon. The process avoids generating unstable values due to interest rate risk, which NPV scenarios capture for ALM. Mark-to-future refers to the forward valuation of the assets. It is different from markto-market because it addresses the issue of unknown future values using models to determine the possible future risk states of assets. The valuation block of VaR models uses both current valuation and mark-to-future to generate the value distribution at a future time point.

BENEFITS AND DRAWBACKS OF ECONOMIC VALUES The theoretical answer to the problem of considering both revenues and risk in exposures is to mark-to-market transactions. Pure mark-to-market applies to market transactions only. Economic values, or ‘fair values’, are similar and apply to all assets, including banking portfolio loans. The economic values discount all future flows with rates that reflect the risk of each transaction.

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The arguments for accounting standards for the banking book relate to stability of earnings. A major drawback of mark-to-market is that the P&L would change every day, as the market does, thereby generating earnings instability. This is more the case, however, with fixed rate assets than with floating rate assets. The latter have values closer to par when interest rates vary. Moreover, it is not possible to trade the assets and liabilities of the banking portfolio, so that marking them to market does not make much sense. Lack of liquidity could result in large discounts from the theoretical mark-to-market values. In spite of their traditional strengths, accounting values suffer from a number of deficiencies. Accounting flows capture earnings over a given period. They do not give information about the long-term profitability of existing facilities. They ignore the market conditions, which serve as an economic benchmark for actual returns. For example, a fixed rate loan earning an historical interest rate has less value today if interest rates increased. It earns less than a similar loan based on current market conditions. Therefore, accounting measures of performance do not provide an image of profitability relative to current interest rates. In addition, different spreads apply to different credit risks. Hence, without risk adjustment, the interest incomes are not comparable from one transaction to another. Contractual interest income is not risk-adjusted. The same applies to book value. Whether a loan is profitable or not, it has the same accounting value. Whether it is risky or not, it also has the same value. On the other hand, ‘fair value’ is economic and richer in terms of information. A faithful image should provide the ‘true’ value of a loan, considering its richness as well as its risk. Book values do not. Economic or fair values do. They have informational and reporting value added for both the bank and outside observers.

RISK-ADJUSTED PERFORMANCES AND ECONOMIC VALUES Risk-adjusted performances are a compromise between accounting measures of performance and the necessity of adjusting income for risk. Although not going the full way to economic valuation, since they are accrual income-based measures, they do provide a risk adjustment. The popular risk-adjusted measures of accounting profitability are the Risk-adjusted Return on Capital (RaRoC) and the Shareholders Value Added (SVA)6 . Both combine profitability and risk. RaRoC nets expected losses from accrued revenues and divides them by the capital allocated to a transaction, using a capital (risk) allocation system. The RaRoC ratio should be above an appropriate hurdle rate representative of a minimum profitability. SVA nets both the expected loss from revenues and the cost of the required capital, valued with an appropriate price in line with shareholder’s required return. A positive SVA implies creation of value, while a negative SVA implies destruction of value from the bank’s shareholders’ standpoint. The same principle applies for market transactions, except for expected loss. Capital results from either market risk or credit risk. It is either regulatory capital or economic capital, the latter being conceptually a better choice. Both calculations use the accrued income flow of a period, instead of discounting to present all future flows. Economic valuation, or ‘mark-to-market’, is not a substitute for risk-adjusted performances because it combines both the revenue and the risk adjustments 6 Both

are detailed in Chapters 53 and 54. Here, we focus on the difference between such measures and valuation.

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in a single figure. The excess spread, or spread between contractual return and the market required yield, also combines both adjustments. Instead of bundling these two adjustments in a single value, RaRoC or SVA make both revenues and risk adjustments explicit.

APPENDIX: DISCRETE TIME AND CONTINUOUS TIME COMPOUNDING This appendix details the calculations of continuous time compounding and discounting and how to derive the continuous rate equivalent to a discrete time rate. Continuous time compounding serves for the pricing model literature.

Future Values The future value FV(y, n, 1) of an investment of 1 at y for n years, compounding only once per year, is (1 + y)n . Dividing the year into k subperiods, we have FV(y, n, k) = (1 + y/k)kn . For instance, at y = 10% per year, the future value is: FV(10%, 1, 1) = (1 + 10%) = 1.1000 for one subperiod

FV(10%, 1, 2) = (1 + 10%/2)2 = 1.1025 FV(10%, 1, 4) = (1 + 10%/4)4 = 1.1038

for two subperiods for four subperiods

The limit when k tends towards infinity provides the ‘continuous compounding’ formula. When k increases, kn grows and y/k decreases. Mathematically, the limit becomes: FVc (y, n) = exp(yn) = eyn The exponential function is exp(x) = ex . For n = 1 and y = 10%, the limit is: FVc (10%, 1) = exp(10% × 1) = 1.1052

Present Values The present value PVc (y, n, k) of a future flow in n years results from FVc (y, n, k). If FVc (n) = eyn , the present value of 1 dated n at the current date is: PVc (y, n) = exp(−yn) = e−yn For one period: PV(10%, 1) = (1 + 10%)−1 = 0.90909 PVc (10%, 1) = exp(−10% × 1) = 0.90484 When the period n comprises k distinct subperiods, the present value formula uses the actual number of subperiods for n. If n = 1 year, the present value for one subperiod is PVc (10%, 1) and becomes PVc (10%, 0.5) if k = 2.

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Equivalence between Continuous and Discrete Rates There is a discrete rate y such that the future values are identical when compounding with discrete subperiods and continuously. It is such that the future value at the continuous rate yc is identical to the future value at the discrete rate yd . The equivalence results from: (1 + y/k)kn = exp(yc n) This rate y does not depend on n: (1 + y/k)k = exp(yc ). Hence, taking the Napierian logarithm (ln) of both sides: yc = k ln(1 + y/k) y = k[exp(yc /k) − 1] As an example, y = 10% is equivalent for 1 year and k = 1 to a continuous rate such that: 10% = exp(yc ) − 1 or

exp(yc ) = 1 − 10% = 0.9

It is equivalent to state that: yc = k ln(1 + y/k) = ln(1 + 10%) = 9.531% This allows us to move back and forth from continuous to discrete rates.

9 Risk Model Building Blocks

The development of risk models for banks has been extremely fast and rich in the recent period, starting with Asset–Liability Management (ALM) models, followed by market Value at Risk (VaR) models for risk and the continuous development of credit risk models, to name the main trends only. The risk models to which we refer in this book are ‘instrumental’ models, which help in implementing better risk management measures and practices in banks. Looking at the universe of models—the variety of models, the risks they address and their techniques—raises real issues over how to structure a presentation of what they are, what they achieve and how they do it. Looking closely at risk models reveals a common structure along major ‘blocks’, each of them addressing essential issues such as: What are the risk drivers of a transaction? How do we deal with risk diversification within portfolios? How do we obtain loss distributions necessary for calculating VaR? The purpose of this chapter is to define the basic ‘building blocks’ of risk models. The definition follows from a few basic principles: • The primary goal of risk management is to enhance the risk–return profiles of transactions and portfolios. The risk–return profiles of transactions or portfolios are a centrepiece of risk management processes. Traditional accounting practices provide returns. By contrast, risks raise measurement challenges because they are intangible and invisible until they materialize. • Risk–return measures appear as the ultimate goal of risk models. Their main innovation is in risk measurement. Models achieve this goal by assembling various techniques and tools that articulate to each other. Together, they make up the risk management ‘toolbox’. • Risk models differ along two main dimensions: the nature of the risk (interest rate risk, market, credit, etc.) that they address; whether they apply to isolated transactions or portfolios, because portfolios raise the issue of measuring diversification in addition

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to measuring the standalone risk of transactions. Accordingly, techniques to address various risks for standalone transactions or portfolios differ. Such principles suggest a common architecture for adopting a ‘building block’ structure of risk models. Summarizing, we find that risk models combine four main building blocks: I. Risk drivers and standalone risk of transactions. Risk drivers are all factors influencing risks, which are necessary inputs for measuring the risk of individual transactions. When considered in isolation, the intrinsic of a transaction is ‘standalone’. II. Portfolio risk. Portfolio models aim to capture the diversification effect that makes the risk of a portfolio of transactions smaller than the sum of the risks of the individual transactions. They serve to measure the economic capital under the VaR methodology. III. Top-down and bottom-up links. These links relate global risk to individual transaction risks, or subportfolio risks. They convert global risk and return targets into risk limits and target profitability for business lines (top-down) and, conversely, for facilitating decision-making at the transaction level and for aggregating business line risks and returns for global reporting purposes (bottom-up). IV. Risk-adjusted performance measuring and reporting, for transactions and portfolios. Both risk–return profiles feed the basic risk processes: profitability and limit setting, providing guidelines for risk decisions and risk monitoring. The detailed presentation uses these four main blocks as foundations for structuring the entire book. Each main block subdivides into smaller modules, dedicated to intermediate tasks, leading to nine basic blocks. The book presents blocks I and II for the three main risks: interest rate risk, market risk and credit risk. On the other hand, tools and techniques of blocks III and IV appear transversal to all risks and do not require a presentation differentiated by risk. After dealing with blocks I and II separately for each main risk, we revert to a transversal presentation of blocks III and IV. The first section focuses on the double role of risk–return profiles: as target variables of risk policies for decision-making; as a key interface between risk models and risk processes. The second section reviews the progressive development stages of risk models. The third section provides an overview of the main building blocks structure. The next sections detail each of the blocks. The last section provides a view of how sub-blocks assemble and how they map to different risks.

RISK MODELS AND RISK PROCESSES Risk–return is the centrepiece of risk management processes. All bank systems provide common measures of income, such as accrual revenues, fees and interest income for the banking portfolio, and Profit and Loss (P&L) for the market portfolio. Measuring risk is a more difficult challenge. Without a balanced view on both risk and return, the management of banks is severely impaired since these two dimensions are intertwining in every process, setting guidelines, making risk decisions or monitoring performance. Therefore, the major contributions of models are to capture risks in all instances where it is necessary to set targets, make risk decisions, both at the transaction level and at the portfolio level.

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There are multiple risk models. ALM models aims to model the interest rate risk of banks at the global balance sheet level. Market risk and credit risk models serve both to measure the risk of individual transactions (standalone risk) and to measure the portfolio risk. To reach this second goal, they need to address the intrinsic risk of individual transactions, independent of the portfolio context. This requires identifying all risk drivers and modelling how they alter the risk of transactions. When reaching the portfolio risk stage, complexities appear because of the diversification effect, making the risk of the portfolio—the risk of the sum of individual transactions—much less than the sum of all individual risks. The issue is to quantify such diversification effects that portfolio models address. The applications of portfolio models are critical and numerous. They serve to determine the likelihood of various levels of potential losses of a portfolio. The global portfolio loss becomes the foundation for allocating risk to individual transactions, after diversification effects. Once this stage is reached, it becomes relatively simple to characterize the risk–return profile of the portfolio as well as those of transactions, since we have the global risk and the risk allocations, plus the income available from the accounting systems. Moving to portfolio models is a quantum leap because they provide access to the risk post-diversification effect for any portfolio, the global portfolio or any subset of transactions. This is a major innovation, notably for credit risk. Measuring risks allows the risk–return view to be extended bank-wide, across business lines and from transactions to the entire bank portfolio. This faculty considerably leverages the efficiency of the bank processes that previously dealt with income without tangible measures of portfolio risks. The building blocks of bank processes respond to common sense needs: setting guidelines, making decisions, monitoring risk and return, plus the ‘vertical’ processes of sending signals to business lines, complying with overall targets and obtaining a consolidated view of risk and return. Risk models provide risk, the otherwise ‘missing’ input to processes. Figure 9.1 summarizes the overview of interaction between risk models and risk processes.

Risk Models

Risk drivers

Risk & Return View

Risk & Return Processes

Risks Return Transaction risk models

Portfolio risk models

From Risk Models to Risk Processes through Risk & Return Measures From Risk Models to Risk Processes through Risk & Return Measures

FIGURE 9.1

From risk models to risk processes through risk and return measures

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GENERATIONS OF BANK RISK MODELS There are a range of banking risk models differing in their purposes and techniques, and the risks that they address. The ALM models were the first to appear. They aimed to capture the interest rate risk of the banking book of commercial banks. Later on, they developed into several building blocks, some of them addressing the interest rate risk of the interest income, others the interest rate risk of the mark-to-market value of the balance sheet, and others aiming to value embedded prepayment options in banking products. Regulations focused on capital requirements and banks focused on economic capital. The capital adequacy principle led to the VaR concept and to VaR models for market risk and credit risk. Market risk models aim to capture the market portfolio risk and find the capital requirement adequate for making this risk sustainable. They address the behaviour of risk drivers, interest rates, foreign exchange rates and equity indexes and, from this first block, derive market risk for the portfolio as VaR. There are several generations of risk models. The academic community looked at modelling the borrowers’ risk, captured through ratings or observed default events, from observable characteristics such as financial variables a long time ago. There is renewed interest in such models because of the requirements of the New Basel Accord to provide better measures of credit risk drivers at the borrower and the transaction levels. The newest generation of models addresses both transaction risk and portfolio risk with new approaches. Risk drivers are exposures, default probabilities and recoveries. Modelling default probability became a priority. New models appeared relating observable attributes of firms to default and ratings, using new techniques, such as the neural network approach. The implementation of the conceptual view of default as an option held by stockholders1 is a major advance that fostered much progress. Various techniques address the portfolio diversification issue, trying to assess the likelihood of simultaneous adverse default or risk deterioration events depending on the portfolio structure. Modelling credit risk and finding the risk of portfolios remained a major challenge because of the scarcity of data, in contrast to market models, which use the considerable volume of market price information. Recent models share common modules: identifying and modelling risk drivers in order to model the standalone risk of individual transactions; modelling diversification effects within portfolios to derive the portfolio risk. Other tools serve for linking risk measures with the business processes, by allocating income and risks to transactions and portfolios. Allocating income is not a pure accounting problem since it is necessary to define the financial cost matching the transaction revenues. Risks raise a more difficult challenge. The Funds Transfer Pricing (FTP) system and the capital—or risk—allocation system perform these two tasks. Once attached to risk models, they allow us to reach the ultimate stage of assigning a risk–return profile to any transaction or portfolio. Then, the bank processes can take over and use these risks–returns as inputs for management and business purposes. 1 This

is the Merton (1974) model, later implemented by the KMV Corporation to measure the ‘expected default frequency’ (Edf ). The default option is the option held by stockholders to give up the assets of the firm if their value falls below that of debt. See Chapter 38.

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RISK MODEL BUILDING BLOCKS The risk management toolbox comprises four main building blocks that Figure 9.2 maps with nine basic risk modelling building blocks, each related to a specific issue. The four main blocks are: I. Risk measurement of transactions on a standalone basis, or ‘intrinsic risk’, without considering the portfolio context that diversifies away a large fraction of the sum of individual risks. II. Risk modelling of portfolios, addressing the issue of portfolio risk diversification and its measurement through basic characteristics such as expected (statistical) losses and potential losses. III. Risk allocation and income allocation within portfolios. The risk allocation mechanisms should capture the diversification effect within the portfolio. The diversification effect results in a lower risk of the portfolio than the sum of the intrinsic risks of transactions. Since risks do not add arithmetically as income does, the mechanism is more complex than earnings allocations to subsets of the portfolio. IV. Determination of the risk–return profiles of transactions, subsets of the bank portfolio, such as business lines, and the overall bank portfolio. The risk–return profile is the ultimate aim of modelling, which risk management attempts to enhance for individual transactions and portfolios.

FIGURE 9.2

I

Risk drivers and transaction risk (standalone)

II

Portfolio risk

III

Top-down & bottom-up tools

IV

Risk & return measures

The main modelling building blocks

This basic structure applies to all risks. The contents of the first two blocks differ with risks, although all modelling techniques need similar inputs and produce similar outputs. The third and fourth blocks apply to all risk, and there is less need to differentiate them for credit and market risks, for instance. When considering all risk models, it appears they share the same basic structure. The four main building blocks each differentiate into sub-blocks dedicated to specific tasks. Various specific ‘modules’ provide the necessary information to progress step-by-step towards the ultimate goal of assigning risk–return profiles to transactions and portfolios. Figure 9.3 breaks down these building blocks into modules addressing intermediate issues. The top section describes the main blocks. The bottom section illustrates the linkages between risk models and risk processes. All modules need to fit with each other, using

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MODELS Basic Modelling Building Blocks Basic Modeling Building Blocks

Main Building Blocks Main Building Blocks

Standalone Risk

I

I.

II

II

III III

IV IV

Portfolio Portfolio Risk Risk Models Models

1

Risk Drivers

2

Risk Exposures and Values

3

Risk Valuation (Standalone)

4

Correlations between Risk Events

5

Portfolio Risk Profile (Loss Distribution)

6

Capital (Economic)

7

Funds Transfer Pricing

8

Capital Allocation

II.

Top-down & Top-Down & Bottom-up Bottom Tools - up Tools

III.

Risk−Return Risk- Return

IV.

9

Risk-adjusted Performance

PROCESSES Risk−Return Measuring and Monitoring

Risks Return

FIGURE 9.3

Risk−Return Decisions, Hedging & Enhancement

Main risk modelling building blocks

the inputs of the previous ones, and providing outputs that serve as inputs to the next ones. This structure conveniently organizes the presentation of various modelling contributions. Vendors’ models bundle several modules together. Other modules serve to fill out gaps in inputs that other building blocks need to proceed further, such as the modelling of the market risk or the credit drivers of transactions. This ‘four/nine’ block structures the entire presentation expanded in this book. The book expands this structure for each of the three risks: interest rate risk for ALM, market risk and credit risk. What follows describes in detail the basic modelling blocks.

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BLOCK I: STANDALONE RISK Block I requires risk and transaction data to capture individual transaction risks.

Risk Drivers Block I starts by identifying ‘risk drivers’, those parameters whose uncertainty alters values and revenues. These random risk drivers are identical to value drivers because measures of risk are variations of value. For ALM, the main risk drivers are the interest rates, which drive the interest income. For market risk, risk drivers are all the market parameters that trigger changes in value of the transactions. They differ across the market compartments: fixed income, foreign exchange or equity. The credit risk drivers are all parameters that influence the credit standing of counterparties and the loss for the bank if it deteriorates. They include exposure, recoveries under default, plus the likelihood of default, or default probability (DP) and its changes (migration risk) when time passes. Later on, we make a distinction between risk drivers and risk factors. Risk factors alter risk drivers, which then directly influence risk events, such as default or the change in value of transactions. The distinction is purely technical. In risk models, risk drivers relate directly to risk events, while risk factors serve for modelling these direct drivers. The distinction is expanded in Chapters 31 and 44 addressing correlation modelling with ‘factor’ models, hence the above distinction. The basic distinction for controlling risk is between risk drivers, which are the sources of uncertainty, and the exposure and sensitivities of transaction values to such sources. Risk drivers remain beyond the control of the bank, while exposures and sensitivities to risk drivers result from the bank’s decisions.

Exposures Block I’s second sub-block relates to exposure size and economic valuation. Book value, or notional amount (for derivatives), characterizes size. Economic valuation is the ‘fair value’, risk and revenue-adjusted. Economic valuation requires mapping exposures to the relevant risk drivers and deriving values from them. Because it is risk-adjusted, any change in the risk drivers materializes in a value change, favourable or unfavourable. Exposure serves as a basis for calculating value changes. For market risk, exposures are mark-to-market and change continuously with market movements. Exposures for credit risk are at book values for commercial banking. Since book values do not change when the default probabilities change, economic values serve for full valuation of credit risk. Exposures data include, specifically, the expected size under a default event (Exposure At Default, EAD), plus the recoveries, which result in loss given default (Lgd) lower than exposure. For ALM, exposures are the ‘surfaces’ serving to calculate interest revenues and costs. ALM uses both book values and economic values. Note that ALM’s first stage is to consolidate all banking portfolio exposures by currency and type of interest rates, thereby capturing the entire balance sheet view from the very beginning. By contrast, credit and market risks modelling starts at the transaction level, to characterize the individual risk, and the global portfolio risk is dealt with in other blocks.

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Standalone Risk and Mark-to-Future Sub-block 3 is forward looking. It looks at future events materializing risks today for each individual transaction, or on a ‘standalone’ basis. Such events include deviations of interest rates for the banking portfolio, all market parameters for market risk and all factors influencing the credit standing of borrowers for credit risk. Sub-block 3 assesses risk as any downside deviation of value at a future time point. Since future values are uncertain, they follow a distribution at this horizon. Adverse deviations are the losses materializing risk. ‘Marking-to-future’ designates the technical processes of calculating potential values at a future date, and deriving losses from this distribution. The principle is the same for transactions and portfolios. The difference is the effect of diversification within a portfolio, which standalone risk ignores.

Obtaining the Risk Driver Inputs Risk drivers and exposure are ingredients of risk models. Sometimes, it is possible to observe them, such as interest rates whose instability over time is readily observable. Sometimes, they are assigned a value because they result from a judgment on risk quality, such as ratings from rating agencies or from internal rating systems. A rating is simply a grade assigned to the creditworthiness combining a number of factors that influence the credit standing of borrowers. Several models help in modelling the uncertainty of the risk drivers. For instance, interest rate models allow us to derive multiple scenarios of interest rates that are consistent with their observed behaviour. Rating models attempt to mimic ratings from agencies through some observable characteristics of firms, such as financial data. Default probability models, such as KMV’s Credit Monitor or Moody’s RiskCalc, provide modelled estimates of default probabilities.

BLOCK II: PORTFOLIO RISK Block I focuses on a standalone view of risk, ignoring the portfolio context and the diversification effects. Block II’s goal is to capture the risk profile of the portfolio rather than the risk of single transactions considered in isolation.

Diversification Effects Diversification requires new inputs characterizing the portfolio risk, mainly the sizes of individual exposures and the correlations between risk events of individual transactions. The idea is intuitive. Should all transactions suffer losses simultaneously, the portfolio loss would be the sum of all simultaneous losses. Fortunately, not all losses materialize at the same time, and the portfolio loss varies according to the number and magnitude of individual losses occurring within a given period. Intuitively, a small number of losses are much more frequent than a simultaneous failure of many of them. This is the essence of diversification: not all risks materialize together. Which losses are going to materialize and

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how many will occur within the period are uncertain. This is the challenge of quantifying diversification effects, which requires ‘portfolio models’. The technical measure of association between transaction risks is the ‘correlation’. For instance, the adverse deviations of interest income, a usual target variable for ALM policies, result from interest rate variations. The correlation between interest rates tends to reduce the diversification effect. All interest revenues and costs tend to move simultaneously in the same direction. Asset and liability exposures to interest rates are offset to the extent that they relate to the same interest rate references. The adverse deviations of the market value of the trading portfolio correlate because they depend on a common set of market parameters, equity indexes, interest rates and foreign exchange rates. However, diversification effects are important because some of these risk drivers are highly correlated, but others are not, and sometimes they vary in opposite directions. Moreover, positions in opposite directions offset. Credit defaults and migrations tend to correlate because they depend on the same general economic conditions, although with different sensitivities. Models demonstrated that portfolio risk is highly sensitive to such correlations. In spite of such positive correlations, diversification effects are very significant for credit risk. The opposite of ‘diversification’ is ‘concentration’. If all firms in the same region tend to default together, there is a credit risk concentration in the portfolio because of the very high default correlation. A risk concentration also designates large sizes of exposures. A single exposure of 1000 might be riskier than 10 exposures of 100, even if borrowers’ defaults correlate, because there is no diversification for this 1000, whereas there is some across 10 different firms. ‘Granularity’ designates risk concentration due to size effect. It is higher with a single exposure of 1000 than with 10 exposures of 100.

Correlation Modelling Obtaining correlations requires modelling. Correlations between risk events of each individual transaction result from the correlation between their risk and value drivers. Correlations for market risk drivers, and for interest rates for ALM, are directly observable. Correlations between credit risk events, such as defaults, suffer from such a scarcity of data that it is difficult to infer correlations from available data on yearly defaults. In fact, credit risk remains largely ‘invisible’. Credit risk models turn around the difficulty by inferring correlations between credit risk events from the correlations of observable risk factors that influence them. For all main risks, correlation modelling is critical because of the high sensitivity of portfolio risk to correlations and size discrepancies between exposures.

Portfolio Risk and VaR Once correlations are obtained, the next step is to characterize the entire spectrum of outcomes for the portfolio, and its downside risk. Since a portfolio is subject to various levels of losses, the most comprehensive way of characterizing the risk profile is with the distribution of losses, assigning a probability to each level of portfolio loss. Portfolio losses are declines in value triggered by various correlated risk events. From a risk perspective, the main risk statistics extracted from loss distributions are the expected loss, the loss volatility characterizing the dispersion around the mean and

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the loss percentiles at various confidence levels for measuring downside risk and VaR. The VaR is also the ‘economic capital’ at the same preset confidence level.

BLOCK III: REVENUE AND RISK ALLOCATION Blocks I and II concentrate on individual and global risk measures. For the bank as a whole, only portfolio risk and overall return count. However, focusing only on these aggregated targets would not suffice to develop risk management practices within business lines and for decision-making purposes. Moving downwards to risk processes requires other tools. They are the top-down links, sending signals to business lines, and the bottomup links, consolidating and reporting, making up block III. Two basic devices serve for linking global targets and business policy (Figure 9.4): • The FTP system. The system allocates income to individual transactions and to business lines, or any other portfolio segment, such as product families or market segments. • The capital allocation system, which allocates risks to any portfolio subset and down to individual transactions. Global Risk Management

Transfer prices

Risk allocation

Business Policy

FIGURE 9.4

From global risk management to business lines

Funds Transfer Pricing Transfer pricing systems exist in all institutions. Transfer prices, are internal references used to price financial resources across business units. Transfer pricing applies essentially to the banking book. The interest income allocated to any transaction is the difference between the customer rate and the internal reference rate, or ‘transfer price’, that applies for this transaction. Transfer rates also serve as a reference for setting customer rates. They should represent the economic cost of making the funds available. Without such references, there is no way to measure the income generated by transactions or business units (for commercial banking) and no basis for setting customer rates.

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Transfer prices serve other interrelated purposes in a commercial bank. Chapters 26 and 27 detail the specifications making the income allocation process consistent with all risk and return processes.

Capital Allocation Unlike revenues and costs, risks do not add arithmetically to portfolio risk because of diversification. Two risks of 1 each do not add up to 2, but add up in general to less than 2 because of the diversification effect, a basic pillar for the banking industry. Portfolio risk models characterize the portfolio risk as a distribution of loss values with a probability attached to each value. From this distribution, it is possible to derive all loss statistics of interest after diversification effects. This represents a major progress but, as such, it does not solve the issue of moving back to the risk retained by each individual transaction, after diversification effects. Since we cannot rely on arithmetic summation for risks, there is no obvious way to allocate the overall risk to subportfolios or transactions. To perform this task, we need a capital (risk) allocation model. The capital allocation model assigns to any subset of a portfolio a fraction of the overall risk, called the ‘risk contribution’. Risk contributions are much smaller than ‘standalone’ risk measures, or measures of the intrinsic risk of transactions considered in isolation, pre-diversification effects, because they measure only the fraction of risk retained by a transaction post-diversification. The capital allocation system performs this risk allocation and provides risk contributions for all individual transactions as well as for any portfolio subsets. The underlying principle is to turn the non-arithmetic properties of risks into additive risk contributions that sum up arithmetically to the overall bank’s portfolio risk. The capital allocation system allocates capital in proportion to the risk contributions provided by portfolio models. For example, using some adequate technique, we can define the standalone risk of three transactions, for example 20, 30 and 30. The sum is 80. However, the portfolio model shows that VaR is only 40. The capital allocation module tells us how much capital to assign to each of the three transactions, for example 10, 15 and 25. They sum up to the overall risk of 40. These figures are ‘capital allocations’, or ‘risk contributions’ to the portfolio risk. The FTP and capital allocation systems provide the necessary links between the global portfolio management of the bank and the business decisions. Without these two pieces, there would be missing links between the global policy of the bank and the business lines. The two pieces complement each other for obtaining risk-adjusted performance measures. Block III leads directly to the ultimate stage of defining risk and return profiles and risk-based performance and pricing.

Aggregating VaR for Different Risks Note that the overall portfolio management for the bank includes integrating all risks in a common unified framework. The VaR concept solves the issue by valuing risks in dollar value. Adding the VaR of different risks remains a problem, however, since the different

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main risks may correlate more or less, or be independent. A sample global report of a VaR implementation for a bank illustrates the overview (Table 9.1). TABLE 9.1

Example of a VaR report

Risks Credit Market Interest rate (ALM) Operational risk Total

VaR capital 260 50 50 80 400

65.0% 12.5% 12.5% 20.0%

In this example, total potential losses are valued at 40. The reports aggregate the VaR generated by different risks, which is a prudent rule. However, it is unlikely that unexpected losses will hit their upper bounds at the same time for credit risk, market risk and other risks. Therefore, the arithmetic total is an overestimate of aggregated VaR across risks.

BLOCK IV: RISK AND RETURN MANAGEMENT Block IV achieves the ultimate result of providing the risk–return profiles of the overall bank portfolio, of transactions or subportfolios. It also allows breaking down aggregated risk and return into those of transactions or subportfolios. Reporting modules take over at this final stage for slicing risks and return along all relevant dimensions for management. The consistency of blocks I to IV ensures that any breakdown of risks and returns reconciles with the overall risk and return of the bank. At this stage, it becomes possible to plug the right measures into the basic risk management processes: guidelines-setting, decision-making and monitoring. In order to reach this final frontier with processes, it is necessary to add a risk-based performance ‘module’. The generic name of risk-adjusted profitability is RaRoC (Risk-adjusted Return on Capital). Risk-Adjusted Performance Measurement (RAPM) designates the risk adjustment of individual transactions that allows comparison of their profitability on the same basis. It is more an ex post measure of performance, once the decision of lending is made, that applies to existing transactions. Risk-Based Pricing (RBP) relates to ex ante decision-making, when we need to define the target price ensuring that the transaction profitability is in line with both risk and the overall target profitability of the bank. For reasons detailed later in Chapters 51 and 52, we differentiate the two concepts, and adopt this terminology. RBP serves to define ex ante the risk-based price. RAPM serves to compare the performance of existing transactions. RAPM simply combines the outputs of the previous building blocks: the income allocation system, which is the FTP system, and the capital allocation system. Once both exist, the implementation is very simple. We illustrate the application for ex post RAPM. Transaction A has a capital allocation of 5 and its revenue, given the transfer price, is 0.5. Transaction B has a capital allocation of 3 and a revenue of 0.4. The RaRoC of transaction A is 0.5/5 = 10% and that of transaction B is 0.4/3 = 13.33%. We see that the less risky

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transaction B has a better risk-adjusted profitability than the riskier transaction A, even though its revenue is lower.

THE RISK MANAGEMENT TOOLBOX: AN OVERVIEW This section summarizes: the nature of risk and return decisions that models aim to facilitate; how the several building blocks assemble to provide such risk–return profiles; how they differentiate across the main risks, interest rate risk, credit risk and market risk.

Risk Modelling and Risk Decisions All model blocks lead ultimately to risk and return profiles for transactions, subportfolios of business units, product families or market segments, plus the overall bank portfolio. The purpose of characterizing this risk–return profile is to provide essential inputs for decision-making. The spectrum of decisions includes: • • • •

New transactions. Hedging, both individual transactions (micro-hedges) and subportfolios (macro-hedges). Risk–return enhancement of transactions, through restructuring for example. Portfolio management, which essentially consists of enhancing the portfolio risk–return profile.

Note that this spectrum includes both on-balance sheet and off-balance sheet decisions. On-balance sheet actions refer to any decisions altering the volume, pricing and risk of transactions, while off-balance sheet decisions refer more to hedging risk.

How Building Blocks Assemble In this section, we review the building block structure of risk models and tools leading to the ultimate stage of modelling the risk–return profiles of transactions and portfolios. It is relatively easy to group together various sub-blocks to demonstrate how they integrate and fit comprehensively to converge towards risk and return profiles. An alternative view is by nature of risk, interest rate risk for commercial banking (ALM), market risk or credit risk, plus other risks. The structure by building blocks proves robust across all risks because it distinguishes the basic modules that apply to all. The differences are that modules differ across risks. For instance, market risk uses extensively data-intensive techniques. Credit risk relies more on technical detours to remedy the scarcity of data. ALM focuses on the global balance sheet from the very start simply because the interest rate risk exposure does not raise a conceptual challenge, but rather an information technology challenge. We provide below both views of the risk models: • The building blocks view, across risks. • The view by risk, with the three main risks ‘view’ developed in this text: interest rate risk in commercial banking (ALM), market risk and credit risk.

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Figure 9.5 summarizes how basic blocks link together to move towards the final risk and return block. Making available this information allows the main risk management processes to operate. The figure uses symbols representative of the purpose of risk modelling blocks and of risk process blocks.

II. Portfolio Risk

III. TopIII. Topdown & Down & Bottom-up Bottom Tools - up tools

IV. Risk− IV. RiskReturn Return Decisions Decisions

ALM

Risk Drivers

Market Risk

Exposures

Credit Risk

Standalone Risk

Correlations

Frequency

I. Risk Data and Risk Measures

Portfolio Risk Loss

Capital

Global Business Line Transaction

Capital Allocation Funds Transfer Pricing

R&R Measures R&R Decisions R&R Monitoring

Risks Risks

Return Return

Risk Processes

FIGURE 9.5

From modelling building blocks to risk and return profiles

How Building Blocks Map to the Main Risks All building blocks map to different risks: interest rate risk, market risk and credit risk. The book uses this mapping as the foundation for structuring the outline of the subsequent detailed presentations. In practice, the organization by risk type applies essentially for building blocks I and II. The FTP system and the capital allocation system are ‘standalone modules’. The ALM is in charge of the FTP module. The capital (or risk) allocation module applies to market and credit risk with the same basic rules. Finally, the ‘risk–return’ module uses

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all contributions of upper blocks. The book uses the matrix in Table 9.2, developing the modelling for each main risk. The book structure is shown by the cells of the table. TABLE 9.2

Cross-tabulating model building blocks with main risks ALM

I. II. III. IV.

Market risk

Credit risk

Standalone risk of transactions Portfolio risk Top-down and bottom-up tools Risk and return

Figure 9.6 combines the risk view with the building block view, and shows how all blocks articulate to each other. Credit Risk

Market Risk

Interest Rate Risk (Banking book - ALM)

I

Credit risk

Market risk

IR risk

II

Portfolio risk models

ALM models

III

Capital allocation

FTP

Risk Contributions

IV

Market P&L

Banking Spreads

Risk & Return

Risk Processes

FIGURE 9.6

Building blocks view and risks view of the risk management toolbox

SECTION 5 Asset–Liability Management

10 ALM Overview

Asset–Liability Management (ALM) is the unit in charge of managing the Interest Rate Risk (IRR) and the liquidity of the bank. Therefore, it focuses essentially on the commercial banking pole. ALM addresses two types of interest rate risks. The first risk is that of interest income shifts due to interest rate movements, given that all balance sheet assets and liabilities generate revenues and costs directly related to interest rate references. A second risk results from options embedded in banking products, such as the prepayment option of individual borrowers who can renegotiate the rate of their loans when the interest rates decline. The optional risk of ALM raises specific issues discussed in a second section. These two risks, although related, need different techniques. ALM policies use two target variables: the interest income and the Net Present Value (NPV) of assets minus liabilities. Intuitively, the stream of future interest incomes varies in line with the NPV. The main difference is that NPV captures the entire stream of future cash flows generated by the portfolio, while the interest income relates to one or several periods. The NPV view necessitates specific techniques, discussed in different chapters from those relating to the interest income sensitivity to interest rates. This chapter presents a general overview of the main building blocks of ALM. For ALM, the portfolio, or balance sheet, view dominates. It is easier to aggregate the exposures and the risk of individual transactions in the case of ALM than for other risks, because all exposures share interest rates as common risk drivers.

ALM BUILDING BLOCKS In the case of ALM, the global view dominates because it is easier to aggregate the exposures and the risk of individual transactions than for other risks. Figure 10.1

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summarizes the main characteristics of ALM building blocks using the common structure proposed for all risks. The next sections detail blocks I: standalone risk and II: banking book—portfolio—risk. Building Blocks: ALM Building Blocks: ALM I.

I. 1

2

3

II. II.

FIGURE 10.1

4

Risk Drivers

Interest rates by currency, entire term structure. Mapped to all assets and liabilities.

Risk Exposures

Outstanding balances of assets and liabilities, at book or mark-to-model values. Banking products also embed implicit options.

Standalone Risk

Valuation of adverse deviations of interest margin and NPV, inclusive of option risk.

Correlations

Gaps fully offset exposures to the same interest rates.

5.1

Portfolio Risk

5.2

Capital (Economic)

Using projected portfolios, valuation of margins and NPV under both interest rate and business scenarios. Forward NPV and interest income distributions from scenarios. Not required. EaR results from adverse deviations of interest margins. An NPV VaR captures the interest rate risk, given business risk.

Major building blocks of ALM

BLOCK I: STANDALONE RISK The risk drivers are all interest rates by currency. Exposures in the banking portfolio are the outstanding balances of all assets and liabilities, except equity, at book value. To capture the entire profile of cash flows generated by all assets and liabilities, the NPV of assets and liabilities is used at market rates. Risk materializes by an adverse change of either one of these two targets. The standalone risk of any individual exposure results from the sensitivity to interest rate changes of the interest income or of its value. The downside risk is that of adverse variations of these two target variables due to interest rate moves. Hence, the valuation of losses results from revaluation at future dates of interest income or of the NPV for each scenario of interest rates. There are two sources of interest rate exposure: liquidity flows and interest rate resets. Each liquidity flow creates an interest rate exposure. For a future inflow, for instance from a future new debt or a repayment of loans, the interest rate risk results from the unknown rate of the utilization of this inflow for lending or investing. Interest rate risk arises also from interest rate resets in the absence of liquidity flows. A floating rate long-term debt

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has several reset dates. For example, a 10-year bullet loan with a floating LIBOR 1-year rate has rate resets every year, while the main cash flow occurs at maturity. Interest rate risk depends on the nature, fixed or variable, of interest rates attached to transactions. When a transaction earns a variable rate, rate resets influence interest revenues both positively and negatively. When it earns a fixed rate, the asset has less value if rates rise above its fixed rate, which makes sense since it generates revenues lower than the market does, and vice versa. When a debt pays a fixed rate above market rate, there is an economic cost resulting from borrowing at a rate higher than the rate. This is an ‘opportunity cost’, the cost of giving up more profitable opportunities. Interest rate variations trigger changes of either accounting revenues or costs, or opportunity (economic) costs. The exposures are sensitive to the entire set of common interest rates. Each interest rate is a common index for several assets and liabilities. Since they are sensitive to the same rate, ‘gaps’ offset these exposures to obtain the net exposure for the balance sheet for each relevant rate. ALM also deals with the ‘indirect’ interest rate risk resulting from options embedded in retail banking products. The options are mainly prepayment options that individuals hold, allowing them to renegotiate fixed rates when rates move down. These options have a significant value for all retail banking lines dealing with personal mortgage loans. Should he exercise his option, a borrower would borrow at a lower rate, thereby reducing interest revenues. Depositors also have options to shift their demand deposits to higher interest-earning liabilities, which they exercise when rates increase. The value of options is particularly relevant for loans since banks can price them rather than give them away to customers. The expected value of options is their price. The potential risk of options for the lender is the potential upside of values during their life, depending on how rates behave. Dealing with optional risk addresses two questions: What is the value of the option for pricing it to customers? What is the downside risk of options given all possible time paths of rates until expiration? The techniques for addressing these questions differ from those of classical ALM models. They are identical to the valuation techniques for market options. The main difference with market risk is that the horizon is much longer. Technical details are given in Chapters 20 and 21.

BLOCK II: PORTFOLIO RISK ALM exposures are global in nature. They result from mismatches of the volumes of assets and liabilities or from interest rate mismatches. The mismatch, or ‘gap’, is a central and simple concept for ALM. A liquidity gap is a difference between assets and liabilities, resulting in a deficit or an excess of funds. An interest rate gap is a mismatch between the size of assets and liabilities indexed to the same reference interest rate. The NPV is the mismatch between liability values and asset values. The sensitivity of the NPV to interest rate changes results from a mismatch of the variations of values of assets and liabilities. For these reasons, ALM addresses from the very beginning the aggregated and netted exposure of the entire balance sheet. Implicit options embedded in banking products add an additional complexity to ALM risk. The interest rate risk of the balance sheet depends on correlations between interest rates. Often, these correlations are high, so that there is no need to differentiate the movements of similar interest rates, although this is only an approximation. In addition, only netted

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exposures are relevant for assets and liabilities indexed to the same, or to similar, interest rates. Treasurers manage netted cash flows, and ALM controls net exposures to interest rate risk. Measuring risk consists of characterizing the downside variations of target variables under various scenarios. ALM uses the simulation methodology to obtain the risk–return profile of the balance sheet using several interest rate scenarios. The expected return is the probability weighted average of profitability across scenarios, measured by interest income or NPV. The downside risk is the volatility of the target variables, interest income or NPV, across scenarios, or their worst-case values. ALM models serve to revalue at future time points the target variables as a function of interest rate movements. To achieve this purpose, ALM models recalculate all interest revenues and costs, or all cash flows for NPV, for all scenarios. The process differs from the market risk and the credit risk techniques in several ways: • The revaluation process focuses on interest income and NPV, instead of market values for market risk and credit risk-adjusted values at the horizon for credit risk. • Since the ALM horizon is medium to long-term, ALM does not rely only on the existing portfolio as a ‘crystallized’ portfolio over a long time. Simulations also extend to projections of the portfolio, adding new assets and liabilities resulting from business scenarios. By contrast, market risk and credit models focus on the crystallized portfolio as of today. • Because of business projections, ALM scenarios deal with business risk as well as interest rate risk, while the market risk and credit risk models ignore business risk. • Because ALM discusses periodically the bank position with respect to interest and liquidity risk in conjunction with business activities, it often relies on a discrete number of scenarios, rather than attempting to cover the entire spectrum of outcomes as market and credit models do. In addition, it is much easier to define worst-case scenarios for ALM risk than it is for market risk and credit risk, because only interest rates influence the target variables in ways relatively easy to identify. • Since there is no regulatory capital for ALM risk, there is no need to go as far as fullblown simulations to obtain a full distribution of the target variables at future horizons. However, simulations do allow us to derive an NPV ‘Value at Risk’ (VaR) for interest rate risk. An ALM VaR is the NPV loss percentile at the preset confidence level. In the case of interest rate risk, large deviations of interest rates trigger optional risk, from the implicit options (prepayment options) embedded in the banking products. This makes it more relevant to extend simulations to a wide array of possible interest rate scenarios. Distribution of forward values of interest income or NPV

Expected value Current value

FIGURE 10.2 NPV)

Loss

Forward valuation of target variables (interest income or balance sheet

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135

To value options for the banking portfolio, the same technique as for single options applies. The NPV is more convenient in order to have a broader view of both loans and options values as of today. Visualizing the portfolio risk–return profile is easy because there is a one-to-one correspondence between interest rate scenarios and the interest income or the NPV, if we ignore business risk (Figure 10.2). Moreover, assigning probabilities to interest rate scenarios allows us to define the Earnings at Risk (EaR) for interest income, as the lower bound of interest income at a preset confidence level, or the NPV VaR as the lower bound of the NPV at the same preset confidence level.

11 Liquidity Gaps

Liquidity risk results from size and maturity mismatches of assets and liabilities. Liquidity deficits make banks vulnerable to market liquidity risk. Liquid assets protect banks from market tensions because they are an alternative source of funds for facing the near-term obligations. Whenever assets are greater than resources, deficits appear, necessitating funding in the market. When the opposite occurs, the bank has available excess resources for lending or investing. Both deficits and excesses are liquidity gaps. A time profile of gaps results from projected assets and liabilities at different future time points. A bank with long-term commitments and short-term resources generates immediate and future deficits. The liquidity risk results from insufficient resources, or outside funds, to fund the loans. It is the risk of additional funding costs because of unexpected funding needs. The standard technique for characterizing liquidity exposure relies on gap time profiles, excesses or deficits of funds, starting from the maturity schedules of existing assets and liabilities. This time profile results from operating assets and liabilities, which are loans and deposits. Asset–Liability Management (ALM) structures the time schedule of debt issues or investments in order to close the deficits or excess liquidity gaps. Cash matching designates the reference situation such that the time profile of liabilities mirrors that of assets. It is a key benchmark because it implies, if there are no new loans and deposits, a balance sheet without any deficit or excess of funds. Future deficits require raising funds at rates unknown as of today. Future excesses of funds require investing in the future. Hence, liquidity risk triggers interest rate risk because future funding and investment contracts have unknown rates, unless ALM sets up hedges. The next chapters discuss the interest rate risk. This chapter focuses on the liquidity position of the banking portfolio. The first section introduces the time profile of liquidity gaps, which are the projected differences between asset and liability time profiles. The second section discusses ‘cash matching’ between assets and liabilities. The third

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section discusses the limitations of liquidity gaps, which result mainly from lines without contractual time profiles.

THE LIQUIDITY GAP Liquidity gaps are the differences, at all future dates, between assets and liabilities of the banking portfolio. Gaps generate liquidity risk, the risk of not being able to raise funds without excess costs. Controlling liquidity risk implies spreading over time amounts of funding, avoiding unexpected important market funding and maintaining a ‘cushion’ of liquid short-term assets, so that selling them provides liquidity without incurring capital gains and losses. Liquidity risk exists when there are deficits of funds, since excess of funds results in interest rate risk, the risk of not knowing in advance the rate of lending or investing these funds. There are two types of liquidity gaps. Static liquidity gaps result from existing assets and liabilities only. Dynamic liquidity gaps add the projected new credits and new deposits to the amortization profiles of existing assets.

The Definition of Liquidity Gaps Liquidity gaps are differences between the outstanding balances of assets and liabilities, or between their changes over time. The convention used below is to calculate gaps as algebraic differences between assets and liabilities. Therefore, at any date, a positive gap between assets and liabilities is equivalent to a deficit, and vice versa.

L A

Gap

Time profile of gaps

Time

A = Assets L = Liabilities

FIGURE 11.1

Liquidity gaps and time profile of gaps

Marginal, or incremental, gaps are the differences between the changes in assets and liabilities during a given period (Figure 11.1). A positive marginal gap means that the algebraic variation of assets exceeds the algebraic variation of liabilities. When assets and liabilities amortize over time, such variations are negative, and a positive gap is equivalent to an outflow1 . The fixed assets and the equity also affect liquidity gaps. Considering equity and fixed assets given at any date, the gap breaks down as in Figure 11.2, isolating the gap between 1 For

instance, if the amortization of assets is 3, and that of liabilities is 5, the algebraic marginal gap, as defined above, is −3 − (−5) = +2, and it is an outflow. Sometimes the gaps are calculated as the opposite difference. It is important to keep a clear convention, so that the figures can easily be interpreted.

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Liabilities (50)

Fixed Assets − Equity = −15

Assets (95) Equity (30)

Fixed Assets (15) Total = 110

FIGURE 11.2

Liquidity Gap Liquidity Gap 110 − 80 = 30 110 - 80 = 30 Total = 110

The structure of the balance sheet and liquidity gap

fixed assets and equity and the liquidity gap generated by commercial transactions. In what follows, we ignore the bottom section of the balance sheet, but liquidity gaps should include the fixed assets equity gap.

Static and Dynamic Gaps Liquidity gaps exist for all future dates. The gaps based on existing assets and liabilities are ‘static gaps’. They provide the image of the liquidity posture of the bank. ‘Static gaps’ are time profiles of future gaps under cessation of all new business, implying a progressive melt-down of the balance sheet. When new assets and liabilities, derived from commercial projections, cumulate with existing assets and liabilities, the gap profile changes completely. Both assets and liabilities, existing plus new ones, tend to increase, rather than amortize. This new gap is the ‘dynamic gap’. Gaps for both existing and new assets and liabilities are required to project the total excesses or deficits of funds. However, it is a common practice to focus first on the existing assets and liabilities to calculate the gap profile. One reason is that there is no need to obtain funds in advance for new transactions, or to invest resources that are not yet collected. Instead, funding the deficits or investing excesses from new business occurs when they appear in the balance sheet. Another reason is that the dynamic gaps depend on commercial uncertainties.

Example As mentioned before, both simple and marginal gaps are calculated. The marginal gaps result from variations of outstanding balances. Therefore, the cumulated value over time of the marginal gaps is equal to the gap based on the outstanding balances of assets and liabilities. Table 11.1 is an example of a gap time profile.

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TABLE 11.1 Time profiles of outstanding assets and liabilities and of liquidity gaps Dates Assets Liabilities Gapa Asset amortization Liability amortization Marginal gapb Cumulated marginal gapc

1

2

3

4

5

6

100 100 0

900 800 100 −10 −20 100 100

700 500 200 −20 −30 100 200

650 400 250 −50 −10 50 250

500 350 150 −15 −50 −10 150

300 100 200 −20 −25 50 200

a Calculated

as the difference between assets and liabilities. A positive gap is a deficit that requires funding. A negative gap is an excess of resources to be invested. b Calculated as the algebraic variation of assets minus the algebraic variation of liabilities between t and t − 1. With this convention, a positive gap is an outflow and a negative gap is an inflow. c The cumulated marginal gaps are identical to the gaps calculated with the outstanding balances of assets and liabilities.

In this example, assets amortize quicker than liabilities. Therefore, the inflows from repayments of assets are less than the outflows used to repay the debts calculated with the outstanding balances of assets and liabilities. Hence, a deficit cumulates from one period to the next, except in period 5. Figures 11.3 and 11.4 show the gaps time profile. Assets

1000 800

Liabilities

Gaps

600 400 200 0 1

2

3

4 Dates

FIGURE 11.3

5

6

Gaps

Time profile of liquidity gaps

Liquidity Gaps and Liquidity Risk When liabilities exceed assets, there is an excess of funds. Such excesses generate interest rate risk since the revenues from the investments of these excess assets are uncertain. When assets exceed liabilities, there is a deficit. This means that the bank has long-run commitments, which existing resources do not fund entirely. There is a liquidity risk generated by raising funds in the future to match the size of assets. The bank faces the

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Assets 300 200

Gaps

100 0 −100

1

2

3

4

5

6

−200 Marginal gaps

−300 −400 Liabilities amortization

FIGURE 11.4

Dates

Time profile of marginal liquidity gaps

risk of not being able to obtain the liquidity on the markets, and the risk of paying higher than normal costs to meet this requirement. In addition, it has exposure to interest rate risk. Liquidity gaps generate funding requirements, or investments of excess funds, in the future. Such financial transactions occur in the future, at interest rates not yet known, unless hedging them today. Liquidity gaps generate interest rate risk because of the uncertainty in interest revenues or costs generated by these transactions. In a universe of fixed rates, any liquidity gap generates an interest rate risk. A projected deficit necessitates raising funds at rates unknown as of today. A projected excess of resources will result in investments at unknown rates as well. With deficits, the margin decreases if interest rates increase. With excess resources, the margin decreases when interest rates decrease. Hence, the liquidity exposure and the interest rate exposure are interdependent in a fixed rate universe. In a floating rate universe, there is no need to match liquidity to cancel out any interest rate risk. First, if a short debt funds a long asset, the margin is not subject to interest rate risk since the rollover of debt continues at the prevailing interest rates. If the rate of assets increases, the new debt will also cost more. This is true only if the same reference rates apply on both sides of the balance sheet. However, any mismatch between reset dates and/or a mismatch of interest references, puts the interest margin at risk. For instance, if the asset return indexed is the 3-month LIBOR, and the debt rate refers to 1-month LIBOR, the margin is sensitive to changes in interest rate. It is possible to match liquidity without matching interest rates. For instance, a long fixed rate loan funded through a series of 3-month loans carrying the 3-month LIBOR. In this case, there is no liquidity risk but there is an interest rate risk. Sometimes, future excesses or deficits of liquidity motivate early decisions, before they actually appear. It might be advisable to lock in an interest rate for these excesses and deficits without waiting. Such decisions depend on interest rate expectations, plus the bank policy with respect to interest rate risk.

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Gap Profiles and Funding In the example of deficits, gaps represent the cumulated needs for funds required at all dates. Cumulated funding is not identical to the new funding required at each period because the debt funding gaps of previous periods do not necessarily amortize over subsequent periods. For instance, a new funding of 100, say between the dates 2 and 3, is not the cumulated deficit between dates 1 and 3, for example 200, because the debt contracted at date 2, 100, does not necessarily amortize at date 3. If the debt is still outstanding at date 3, the deficit at date 3 is only 100. Conversely, if the previous debt amortizes after one period, the total amount to fund would be 200, that is the new funding required during the last period, plus any repayment of the previous debt. The marginal gaps represent the new funds required, or the new excess funds available for investing. A positive cumulated gap is the cumulated deficit from the starting date, without taking into consideration the debt contracted between this date and today. This is the rationale for using marginal gaps. These are the amounts to raise or invest during the period, and for which new decisions are necessary. The cumulated gaps are easier to interpret, however.

Liquidity Gaps and Maturity Mismatch An alternative view of the liquidity gap is the gap between the average maturity dates of assets and liabilities. If all assets and liabilities have matching maturities, the difference in average maturity dates would be zero. If there is a time mismatch, the average maturity dates differ. For instance, if assets amortize slower than liabilities, their average maturity is higher than that of liabilities, and vice versa. The average maturity date calculation weights maturity with the book value of outstanding balances of assets and liabilities (Figure 11.5).

Time profile of gaps: Assets > Liabilities

FIGURE 11.5 liabilities

Time

Asset Average Maturity

Liability Average Maturity Gap

Liquidity gaps and gaps between average maturity dates of assets and

In a universe of fixed rates, the gap between average maturities provides information on the average location of assets and liabilities along the yield curve, which provides the market rates for all maturities. Note that averages can be arithmetic or, better, weighted by the outstanding volumes of assets and liabilities.

CASH MATCHING Cash matching is a basic concept for the management of liquidity and interest rate risks. It implies that the time profiles of amortization of assets and liabilities are identical. The

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nature of interest applicable to assets and maturities might also match: fixed rates with fixed rates, floating rates revised periodically with floating rates revised at the same dates using the same reference rate. In such a case, the replication of assets’ maturities extends to interest characteristics. We focus below on cash matching only, when the repayment schedule of debt replicates the repayment schedule of assets. Any deviation from the cash matching benchmark generates interest rate risk, unless setting up hedges.

The Benefits of Cash Matching With cash matching, liquidity gaps are equal to zero. When the balance sheet amortizes over time, it does not generate any deficit or excess of funds. If, in addition, the interest rate resets are similar (both dates and nature of interest rate match), or if they are fixed interest rates, on both sides, the interest margin cannot change over time. Full matching of both cash and interest rates locks in the interest income. Cash matching is only a reference. In general, deposits do not match loans. Both result from the customers’ behaviour. However, it is feasible to structure the financial debt in order to replicate the assets’ time profile, given the amortization schedule of the portfolio of commercial assets and liabilities. Cash matching also applies to individual transactions. For instance, a bullet loan funded by a bullet debt having the same amount and maturity results in cash matching between the loan and the debt. However, there is no need to match cash flows of individual transactions. A new deposit with the same amount and maturity can match a new loan. Implementing cash matching makes sense after netting assets and liabilities. Otherwise, any new loan will necessitate a new debt, which is neither economical nor necessary.

The Global Liquidity Posture of the Balance Sheet The entire gap profile characterizes the liquidity posture of a bank. The zero liquidity gaps are the benchmark. In a fixed rate universe, there would not be any interest rate risk. In a floating rate universe, there is no interest risk either if reset dates and the interest index are identical for assets and liabilities. Figures 11.6 to 11.8 summarize various typical situations. The assumption is that any excess funds, or any deficits, at the starting date are fully funded or invested. Therefore, Outstanding balances

Assets

Liabilities

Time

FIGURE 11.6

Gaps profile close to zero

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Outstanding balances Assets

Liabilities Time

FIGURE 11.7

Deficits Outstanding balances

Liabilities Assets

Time

FIGURE 11.8

Excess funds

the current gap is zero, and non-zero gaps appear only in the future. In the figures, the liability time profile hits the level of capital once all debts amortize completely. The figures ignore the gap between fixed assets and equity, which varies across time because both change.

New Business Flows With new business transactions, the shapes of the time profiles of total assets and liabilities change. The new gap profile is dynamic! Total assets can increase over time, as well as depositors’ balances. Figure 11.9 shows the new transactions plus the existing assets and liabilities. The amortization schedules of existing assets and liabilities are close to each other. Therefore, the static gaps, between existing assets and liabilities, are small. The funding requirements are equal to netted total assets and liabilities. Since total assets increase more than total resources from customers, the overall liquidity gap increases with the horizon. The difference between total assets and existing assets, at any date, represents the new business. To be accurate, when projecting new business, volumes should be net from the amortization of the new loans and the new deposits, since the amortization starts from the origination date of these transactions. When time passes, the gap profile changes: existing assets and liabilities amortize and new business comes in. The gap, which starts from 0 at t, widens up to some positive

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Total assets Outstanding balances

New assets Total liabilities New liabilities Existing assets

Existing liabilities Time

FIGURE 11.9

Gap profile with existing and new business

value at t + 1. The positive gap at t + 1 is a pre-funding gap, or ex ante. In reality, the funding closes the gap at the same time that new business and amortization open it. The gap at t + 1 is positive before funding, and it is identical to the former gap projected, at this date, in the prior periods (Figure 11.10). Gap Assets

Liabilities

t

FIGURE 11.10

t+1

t+1

Time

The gap profile when time passes

LIQUIDITY MANAGEMENT Liquidity management is the continuous process of raising new funds, in case of a deficit, or investing excess resources when there are excesses of funds. The end of this section briefly discusses the case of investments. The benchmark remains the cash matching case, where both assets and liabilities amortize in parallel.

Funding a Deficit In the case of a deficit, global liquidity management aims at a target time profile of gaps after raising new resources. In Figure 11.10, the gap at date t is funded so that, at the current date, assets and liabilities are equal. However, at date t + 1, a new deficit appears, which requires new funding. The liquidity decision covers both the total amount required to bridge the gap at t + 1 and its structuring across maturities. The funding decision reshapes the entire amortization profile of liabilities. The total amount of the new debt is the periodic gap. Nevertheless, this overall amount piles up ‘layers’ of bullet debts with varying maturities. Whenever a layer amortizes, there is a downside step in the amortization profile of the debt.

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The amount of funds that the bank can raise during a given period is limited. There is some upper bound, depending on the bank’s size, credit standing and market liquidity. In addition, the cost of liquidity is important, so that the bank does not want to raise big amounts abruptly for fear of paying the additional costs if market liquidity is tight. If 1000 is a reasonable monthly amount, the maximum one-period liquidity gap is 1000. When extending the horizon, sticking to this upper bound implies that the cumulated amount raised is proportional to the monthly amount. This maximum cumulated amount is 2000 for 2 months, 3000 for 3 months, and so on. The broken line shows what happens when we cumulate a constant periodical amount, such as 1000, to the existing resources profile. The length of vertical arrows increases over time. Starting from existing resources, if we always use the ‘maximum’ amount, we obtain an upper bound for the resources profiles. This is the upper broken line. The upper bound of the resources time profile should always be above the assets time profile, otherwise we hit the upper bound and raising more funds might imply a higher cost. In Figure 11.11, the upper bound for resources exceeds the assets profile. Hence there is no need to raise the full amount authorized.

Outstanding balances

Maximum resource profile

Total assets

Total resources New New funding funding Time

FIGURE 11.11

The gap and resources time profiles

Cash Matching in a Fixed Rate Universe This subsection refers to a situation where resources collected from customers are lower than loans, generating a liquidity deficit. The target funding profile depends on whether ALM wishes to close all liquidity gaps, or maintain a mismatch. Any mismatch implies consistency with expectations on interest rates. For instance, keeping a balance sheet underfunded makes sense only when betting on declining interest rates, so that deferring funding costs less than now. Making the balance sheet overfunded implies expectations of raising interest rates, because the investment will occur in the future. Sometimes the balance sheet remains overfunded because there is an excess of deposits over loans. In such a case, the investment policy should follow guidelines with respect to target interest revenue and interest rate risk on future investments. Structuring the Funding

Below we give two examples of profiles of existing assets and liabilities. In both cases, the resources amortize quicker than assets. The amortization of the balance sheet generates

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new deficits at each period. In a universe of fixed rates, such a situation could make sense if the ALM unit decided to try to bet on decreasing interest rates. We assume that ALM wishes to raise the resources profile up to that of assets because expectations have changed, and because the new goal is to fully hedge the interest margin. The issue is to define the structuring of the new debts consistent with such a new goal. The target resources profile becomes that of assets. The process needs to define a horizon. Then, the treasurer piles up ‘layers’ of debts, starting from the longest horizon. Figure 11.12 illustrates two different cases. On the left-hand side, the gap decreases continuously until the horizon. Then, layers of debts, raised today, pile up, moving backwards from the horizon, so that the resources’ amortization profile replicates the asset profile. Of course, there are small steps, so this hedge is not perfect. In the second case (right-hand side), it is not feasible to hit the target assets’ profile with debt contracted as of today (spot debt). This is because the gap increases with time, peaks and then narrows up to the management horizon. Starting from the end works only partially, since filling completely the gap at the horizon would imply raising excess funds today. This would be inconsistent with cash matching. On the other hand, limiting funds to what is necessary today does not fill the gap in intermediate periods between now and the horizon. Horizon

Amount

Horizon

Amount Assets

Assets

Resources

Resources Time

FIGURE 11.12

Time

Neutralizing the liquidity gap (deficit)

In a fixed rate universe, this intermediate mismatch generates interest rate risk. Hence, ‘immunization’ of the interest margin requires locking in as of today the rates on the future resources required. This is feasible with forward hedges (see Chapter 14). The hedge has to start at different dates and correspond to the amounts of debts represented in grey. Numerical Example

Table 11.2 shows a numerical example of the first case above with three periods. The existing gap is 200, and the cash matching funding necessitates debts of various maturities. The ‘layer 1’ is a bullet debt extending from now to the end of period 3. Its amount is equal to the gap at period 3, or 50. A second bullet debt from now to period 2, of amount 50, bridges the gap at period 2. Once these debts are in place, there is still a 100 gap left for period 1. In the end, the treasurer contracts three bullet debts: 50 for three periods, 50 for two periods, and 100 for one period. Once this is done, the time profile of resources becomes identical to that of assets over the total horizon. In the second case of the preceding subsection, the same process does not apply because the gap increases and, after a while, decreases until the horizon. It is not possible to reach

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TABLE 11.2

Bridging the liquidity gap: example

Period Assets Resources Gap New funding: Debt 1 Debt 2 Debt 3 Total funding Gap after funding

1

2

3

1000 800 200

750 650 100

500 450 50

50 50 100 200 0

50 50

50

100 0

50 0

cash matching with resources raised today. One bullet debt, of 100, bridges partially the gaps from now up to the final horizon. A second 100 bullet debt starts today until the end of period 2. Then, the treasurer needs a third forward starting debt of 150. In a fixed rate universe, a forward contract should lock in its rate as of today. However, effective raising of liquidity occurs only at the beginning of period date 2 up to the end of period 2 (Table 11.3). TABLE 11.3

Bridging the liquidity gap: example

Period Assets Resources Gap New funding: Debt 1 Debt 2 Debt 3 Total funding Gap after funding

1

2

3

1000 800 200

750 400 350

500 400 100

100 100 0 200 0

100 100 150 350 0

100

100 0

Structural Excesses of Liquidity Similar problems arise when there are excesses of funds. Either some funds are re-routed to investments in the market, or there are simply more resources than uses of funds. ALM should structure the timing of investments and their maturities according to guidelines. This problem also arises for equity funds because banks do not want to expose capital to credit risk by lending these funds. It becomes necessary to structure a dedicated portfolio of investments matching equity. The expectations about interest rates are important since the interest rates vary across maturities. Some common practices are to spread securing investments over successive periods. A fraction of the available funds is invested up to 1 year, another fraction for 2 years, and so on until a management horizon. Spreading each periodical investment over all maturities avoids locking in the rate of a single maturity and the potential drawback of renewing entirely the investment at the selected maturity at uncertain rates. Doing so for periodical investments avoids crystallizing the current yield curve in the investment

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portfolio. This policy smooths the changes of the yield curve shape over time up to the selected longest maturity. There are variations around this policy, such as concentrating the investments at both short and long ends of the maturity structure of interest rates. Policies that are more speculative imply views on interest rates and betting on expectations. We address the issue again when dealing with interest rate risk.

THE COST OF THE LIQUIDITY RATIO The cost of liquidity for banks often refers to another concept: the cost of maintaining the liquidity ratio at a minimum level. The liquidity ratio is the ratio of short-term assets to short-term liabilities, and it should be above one. When a bank initiates a combination of transactions such as borrowing short and lending long, the liquidity ratio deteriorates because the short liabilities increase without a matching increase in short-term assets. This is a typical transaction in commercial banking since the average maturity of loans is often longer than the average maturity of resources. In order to maintain the level of the liquidity ratio, it is necessary to have additional short-term assets, for instance 1000 if the loan is 1000, without changing the short-term debt, in order to restore the ratio to the value before lending (1000). Borrowing long and investing short is one way to improve the liquidity ratio. The consequence is a mismatch between the long rate, of the borrowing side, and the short rate, of the lending side. The cost of restoring the liquidity ratio to its previous level is the cost of neutralizing the mismatch. Bid–ask spreads might increase that cost, since the bank borrows and lends at the same time. Finally, borrowing implies paying a credit spread, which might not be present in the interest of short-term assets, such as Treasury bills. Therefore, the cost of borrowing includes the bank’s credit spread. The first cost component is the spread between long-term rates and short-term rates. In addition, borrowing long at a fixed rate to invest short also generates interest rate risk since assets roll over faster than debt. Neutralizing interest rate risk implies receiving a fixed rate from assets rather that a rate reset periodically. A swap receiving the fixed rate and paying the variable rate makes this feasible (see Chapter 14). The swap eliminates the interest rate risk and exchanges the short-term rate received by assets for a long-term rate, thereby eliminating also any spread between long-term rates and short-term rates. The cost of the swap is the margin over market rates charged by the issuer of the swap. Finally, the cost of the liquidity ratio is the cost of the swap plus the credit spread of borrowing (if there is none on investments) plus the bid–ask spread. The cost of maintaining the liquidity ratio is a component of the ‘all-in’ cost of funds, that serves for pricing loans. Any long-term loan implies swapping short-term assets for a loan, thereby deteriorating the liquidity ratio.

ISSUES FOR DETERMINING THE LIQUIDITY GAP TIME PROFILE The outstanding balances of all existing assets and liabilities and their maturity schedules are the basic inputs to build the gap profile. Existing balances are known, but not necessarily their maturity. Many assets and liabilities have a contractual repayments schedule.

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However, many others have no explicit maturity. Such assets without maturity are overdrafts, credit card consumer loans, renewed lines of credit, committed lines of credit or other loans without specific dates. Demand deposits are liabilities without maturity. Deriving liquidity gaps necessitates assumptions, conventions or projections for the items with no explicit maturity. These are: • Demand deposits. These have no contractual maturity and can be withdrawn instantly from the bank. They can also increase immediately. However, a large fraction of current deposits is stable over time, and represents the ‘core deposit base’. • Contingencies such as committed lines of credit. The usage of these lines depends on customer initiative, subject to the limit set by the lender. • Prepayment options embedded in loans. Even when the maturity schedule is contractual, it is subject to uncertainty due to prepayment options. The effective maturity is uncertain.

Demand Deposits There are several solutions to deal with deposits without maturity. The simplest technique is to group all outstanding balances into one maturity bucket at a future date, which can be the horizon of the bank or beyond. This excludes the influence of demand deposit fluctuations from the gap profile, which is neither realistic nor acceptable. Another solution is to make conventions with respect to their amortization, for example, using a yearly amortization rate of 5% or 10%. This convention generates an additional liquidity gap equal to this amortization forfeit every year which, in general, is not in line with reality. A better approach is to divide deposits into stable and unstable balances. The core deposits represent the stable balance that remains constantly as a permanent resource. The volatile fraction is treated as short-term debt. Separating core deposits from others is closer to reality than the above assumptions, even though the rule for splitting deposits into core deposits and the remaining balance might be crude. The last approach is to make projections modelled with some observable variables correlated with the outstanding balances of deposits. Such variables include the trend of economic conditions and some proxy for their short-term variations. Such analyses use multiple regression techniques, or time series analyses. There are some obvious limitations to this approach. All parameters that have an impact on the market share deposits are not explicitly considered. New fiscal regulations on specific tax-free interest earnings of deposits, alter the allocation of customers’ resources between types of deposits. However, this approach is closer to reality than any other.

Contingencies Given (Off-balance Sheet) Contingencies generate outflows of funds that are uncertain by definition, since they are contingent upon some event, such as the willingness of the borrower to use committed lines of credit. There are many such lines of credit, including rollover short-term debts, or any undrawn fraction of a committed credit line. Only the authorization and its expiry date are fixed. Statistics, experience, knowledge of customers’ accounts (individuals or corporate borrowers) and of their needs help to make projections on the usage of such

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lines. Otherwise, assumptions are required. However, most of these facilities are variable rate, and drawings are necessarily funded at unknown rates. Matching both variable rates on uncertain drawings of lines eliminates interest rate risk.

Amortizing Loans When maturities are contractual, they can vary considerably from one loan to another. There are bullet loans, and others that amortize progressively. Prepayment risk results in an effective maturity different from the contractual maturity at origination. The effective maturity schedule is more realistic. Historical data help to define such effective maturities. Prepayment models also help. Some are simple, such as the usage of a constant prepayment ratio applicable to the overall outstanding balances. Others are more sophisticated because they make prepayments dependent on several variables, such as the interest rate differential between the loan and the market, or the time elapsed since origination, and so on. The interest income of the bank is at risk whenever a fixed rate loan is renegotiated at a lower rate than the original fixed rate. Section 7 of this book discusses the pricing of this risk.

Multiple Scenarios Whenever the future outstanding balances are uncertain, it is tempting to use conventions and make assumptions. The problem with such conventions and assumptions is that they hide the risks. Making these risks explicit with several scenarios is a better solution. For instance, if the deposit balances are quite uncertain, the uncertainty can be captured by a set of scenarios, such as a base case plus other cases where the deposit balances are higher or lower. If prepayments are uncertain, multiple scenarios could cover high, average and low prepayment rate assumptions. The use of multiple scenarios makes more explicit the risk with respect to the future volumes of assets and liabilities. The price to pay is an additional complexity. The scenarios are judgmental, making the approach less objective. The choices should combine multiple sources of uncertainty, such as volume uncertainty, prepayment uncertainty, plus the uncertainty of all commercial projections for new business. In addition, it is not easy to deal with interest rate uncertainty with more than one scenario. There are relatively simple techniques for dealing with multiple scenarios, embedded in ALM models detailed in Chapter 17.

12 The Term Structure of Interest Rates

The yield curve is the entire range of market interest rates across all maturities. Understanding the term structure of interest rates, or yield curve, is essential for appraising the interest rate risk of banks because: • Banks’ interest income is at risk essentially because of the continuous movements of interest rates. • Future interest rates of borrowing or lending–investing are unknown, if no hedge is contracted before. • Commercial banks tend to lend long and borrow short. When long-term interest rates are above short-term interest rates, this ‘natural exposure’ of commercial banks looks beneficial. Often, banks effectively lend at higher rates than the cost of their debts because of a positive spread between long-term rates and short-term rates. Unfortunately, the bank’s interest income is at risk with the changes of shape and slope of the yield curve. • All funding or investing decisions resulting from liquidity gaps have an impact on interest rate risk. Banks are net lenders, when they have excess funds, or net borrowers, when they have future deficits. As any lender or borrower, they cannot eliminate interest rate risk. A variable rate borrower faces the risk of interest rate rises. A fixed rate borrower faces the risk of paying a fixed rate above declining rates. The exposure of the lender is symmetrical. The consequence is that there is no way to neutralize interest rate risk. The only available options are to choose the best exposure according to management criteria. The purpose of controlling interest rate risk is to adopt this ‘best’ exposure. The prerequisite is measuring the interest rate exposure. Defining this ‘best’ exposure requires

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understanding the mechanisms driving the yield curve and making choices on views of what future rates could be. This chapter addresses both issues. Subsequent chapters describe the management implications and the techniques for controlling interest rate risk. The expectation theory of interest rates states that there are no risk-free arbitrage opportunities for investors and lenders ‘riding’ the spot yield curve, which is the term structure of interest rates. The theory has four main applications: • Making lending–borrowing decisions based on ‘views’ of interest rates compared to ‘forward rates’ defined from the yield curve. • Locking in a rate at a future date, as of the current date. • Extending lending–borrowing opportunities to the forward transactions, future transactions of which terms are defined today, based on these forward rates. The first section discusses the nature of interest rate references, such as variable rates, fixed rates, market rates and customer rates. The second section discusses interest rate risk and shows that any lender–borrower has exposure to this risk. The third section discusses the term structure of market interest rates, or the ‘spot’ yield curve. It focuses on the implications of the spreads between short-term rates and long-term rates for lenders and borrowers. The fourth section defines forward rates, rates set as of today for future transactions, and how these rates derive from the spot yield curve. The fifth section presents the main applications for taking interest rate exposures. The last section summarizes some practical observations on economic drivers of interest rates, providing a minimum background underlying ‘view’ on future interest rates, which guide interest rate policies.

INTEREST RATES To define which assets and liabilities are interest-sensitive, interest rates qualify as ‘variable’ or ‘fixed’. This basic distinction needs some refining. Variable rates change periodically, for instance every month if they are indexed to a 1-month market rate. They remain fixed between any two reset dates. The distinction, ‘fixed’ or ‘variable’, is therefore meaningless unless a horizon is specified. Any variable rate is fixed between now and the next reset date. A variable rate usually refers to a periodically reset market rate, such as the 1-month LIBOR, 1-year LIBOR or, for the long-term, bond yields. The frequency of resets can be as small as 1 day. The longer the period between two reset dates, the longer the period within which the value of the ‘variable’ rate remains ‘fixed’. In some cases, the period between resets varies. The prime rate, the rate charged to the best borrowers, or the rates of regulated saving accounts (usually with tax-free interest earnings) can change at any time. The changes depend on the decision of banks or regulating bodies. Such rates are not really fixed. In some cases, they change rather frequently, such as the prime rate of banks. In others, the changes are much less frequent and do not happen on a regular basis. In the recent period, it has become permitted to make regulated rates variable according to market benchmarks. The motivation is that rates in Euroland have decreased to unprecedented levels, comparable to regulated rates. This phenomenon impairs the profitability of banks collecting this kind of resource.

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Finally, among rates periodically reset to market, some are determined at the beginning of the period and others at the end. Most rates are set at the beginning of the period. However, in some countries, some market indexes are an average of the rates observed during the elapsed period. For instance, a 1-month rate can be the average of the last month’s daily rates. In such cases, the uncertainty is resolved only at the end of the current period, and these rates remain uncertain until the end of the period. On the other hand, predetermined rates are set at the beginning of the current period and until the next reset date. From an interest rate risk perspective, the relevant distinction is between certain and uncertain rates over a given horizon. A predetermined variable rate turns fixed after the reset date and until the next reset date. The ‘fixed rate’ of a future fixed rate loan is actually interest rate-sensitive since the rate is not set as of today, but some time in the future. All rates of new assets and liabilities originated in the future are uncertain as of today, even though some will be those of fixed rate transactions. The only way to fix the rates of future transactions as of today is through hedging. The nature of rates, fixed or variable, is therefore not always relevant from an interest risk standpoint. The important distinction is between unknown rates and interest rates whose value is certain over a given horizon. As shown in Figure 12.1, there are a limited number of cases where future rates are certain and a large number of other rates indexed to markets, or subject to uncertain changes. In the following, both distinctions ‘fixed versus variable’ and ‘certain versus uncertain’ are used. Without specifying, we use the convention that fixed rates are certain and variable rates are uncertain as of today.

FIGURE 12.1

Today

Horizon until next reset

Fixed

Fixed

Variable

Variable

To maturity Certain

Uncertain

Interest rates, revision periods and maturity

INTEREST RATE RISK Intuition suggests that fixed rate borrowers and lenders have no risk exposure because they have certain contractual rates. In fact, fixed rate lenders could benefit from higher rates. If rates rise, lenders suffer from the ‘opportunity cost’ of not lending at these higher rates. Fixed rate borrowers have certain interest costs. If rates decline, they suffer from not taking advantage of lower rates. They also face the ‘opportunity cost’ because they do not take advantage of this opportunity. Variable rate borrowers and lenders have interest costs or revenues indexed to market rates. Borrowers suffer from rising rates and lenders from declining rates. Future (forward) borrowers benefit from declining rates, just as variable rate borrowers do. Future (forward) lenders suffer from declining rates, just as variable rate lenders do. The matrix in Table 12.1 summarizes these exposures, with ‘+’

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TABLE 12.1 Rate

Exposure to Interest Rate Risk (IRR) Change of rates

Lender

Borrower

Existing exposure Floating rate Fixed ratea

Rate ↑ Rate ↓ Rate ↑ Rate ↓

+ − − +

− + + −

Projected exposure (same as current + floating) — —

Rate ↑ Rate ↓

+ −

− +

a Uncertainty

with respect to opportunity cost of debt only. + gain, − loss.

representing gains and ‘−’ losses, both gains and losses being direct or opportunity gains and costs. A striking feature of this matrix is that there is no ‘zero’ gain or loss. The implication is that there is no way to escape interest rate risk. The only options for lenders and borrowers are to choose their exposure. The next issue is to make a decision with respect to their exposures.

THE TERM STRUCTURE OF INTEREST RATES Liquidity management decisions raise fundamental issues with respect to interest rates. First, funding or investing decisions require a choice of maturity plus the choice of locking in a rate, or of using floating rates. The choice of a fixed or a floating rate requires a comparison between current and expected future rates. The interest rate structure is the basic input to make such decisions. It provides the set of rates available across the maturities range. In addition, it embeds information on expected future rates that is necessary to make a decision on interest rate exposure. After discussing the term structure of interest rates, we move on to the arbitrage issue between fixed and variable rates.

The Spot Yield Curve There are several types of rates, or yields. The yield to maturity of a bond is the discount rate that makes the present value of future cash flows identical to its price. The zerocoupon rates are those discount rates that apply to a unique future flow to derive its present value. The yield to maturity embeds assumptions with respect to reinvestment revenues of all intermediate flows up to maturity1 . The yield to maturity also changes 1 The yield to maturity is the value of y that makes the present value of future flows equal to the price of a bond: price = nt=1 CFt /(1 + y)t where CFt are the cash flows of date t and y is the yield to maturity. This equation can also be written as: price = nt=1 CFt (1 + y)n−t /(1 + y)n . The equation shows that the price is also the present value of all flows reinvested until maturity at the yield y. Since the future reinvestment of

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Rates

with the credit standing of the issue. The spread between riskless yields, such as those applicable to government bonds, and the yields of private issues is the credit spread. Yields typically derive from bond prices, given their future cash flows. The zero-coupon yields can be derived from yields to maturity observed for a given risk class using an iterative process2 . The set of yields from now to any future maturity is the yield curve. The yield curve can have various shapes, as shown in Figure 12.2.

Maturity

FIGURE 12.2

The term structure of interest rates

Interest Rate Arbitrage for Lenders and Borrowers The difference in rates of different maturities raises the issue of the best choice of maturity for lending or borrowing over a given horizon. An upward sloping yield curve means that long rates are above short rates. In such a situation, the long-term lender could think that the best solution is to lend immediately over the long-term. The alternative choice would be to lend short-term and renew the lending up to the horizon of the lender. The loan is rolled over at unknown rates, making the end result unknown. This basic problem requires more attention than the simple comparison between short-term rates and long-term rates. We make the issue more explicit with the basic example of a lender whose horizon is 2 years. Either he lends straight over 2 years, or he lends for 1 year and rolls the loan over after 1 year. The lender for 2 years has a fixed interest rate over the horizon, which is above the current 1-year rate. If he lends for 1 year, the first year’s revenue will be lower since the current spot rate for 1 year is lower than the 2-year rate. At the beginning of the second year, he reinvests at the prevailing rate. If the 1-year rate, 1 year from now, increases sufficiently, it is conceivable that the second choice beats the first. There is a minimum value of the yearly rate, 1 year from now, that will provide more revenue at the end of the 2-year horizon than the straight 2-year loan. intermediate flows will not occur at this yield y, but at the prevailing rates at the date when reinvestments occur, this assumption is unrealistic. 2 Zero-coupon rates apply to a single flow. Such rates derive from the observed bond yields. The one-period yield to maturity is a zero-coupon rate since the first flows occur at the end of the first period. In the following, this period is 1 year. Starting with this first zero-coupon rate, we can derive the others. A 2-year instrument generates two flows, one at date 1, CF1 , and one at date 2, CF2 . The present value of the flow CF1 uses the 1-year rate z1 . This value is CF1 /(1 + z1 ). By borrowing this amount at date 0, we cancel the flow CF1 of the bond. Then, the present value of CF2 discounts this flow at the unknown zero-coupon rate for date 2. This present value should be equal to the price (for example 1) minus the present value of the flow CF1 . Using the equality: 1 − CF1 /(1 + z1 ) = CF2 /(1 + z2 )2 , we obtain the value of the zero-coupon rate for date 2, z2 .

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The same reasoning applies to a borrower. A borrower can borrow short-term, even though he has a long-term liquidity deficit because the short-term rates are below the long-term rates. Again, he will have to roll over the debt for another year after 1 year. The outcome depends on what happens to interest rates at the end of the first year. If the rates increase significantly, the short-term borrower who rolls over the debt can have borrowing costs above those of a straight 2-year debt. In the above example, we compare a long transaction with a series of short-term transactions. The two options differ with respect to interest rate risk and liquidity risk. Liquidity risk should be zero to have a pure interest rate risk. If the borrower needs the funds for 2 years, he should borrow for 2 years. There is still a choice to make since he can borrow with a floating or a fixed rate. The floating rate choice generates interest rate risk, but no liquidity risk. From a liquidity standpoint, short-term transactions rolled over and long-term floating rate transactions are equivalent (with the same frequency of rollover and interest rate reset). The subsequent examples compare long to short-term transactions. The issue is to determine whether taking interest risk leads to an expected gain over the alternative solution of a straight 2-year transaction. The choice depends first on expectations. The short-term lender hopes that interest rates will increase. The short-term borrower hopes that they will decrease. Nevertheless, it is not sufficient that interest rates change in the expected direction to obtain a gain. It is also necessary that the rate variation be large enough, either upward or downward. There exists a break-even value of the future rate such that the succession of short-term transactions, or a floating rate transaction—with the same frequency of resets as the rollover transaction—is equivalent to a long fixed rate transaction. The choice depends on whether the expected rates are above or below such break-even rates. The break-even rate is the ‘forward rate’. It derives from the term structure of interest rates.

INTEREST RATE EXPECTATIONS AND FORWARD RATES The term structure of interest rates used hereafter is the set of zero-coupon rates derived from government, or risk-free, assets. The zero-coupon rates apply to single future flows and are pure fixed rate transactions. The expectation theory considers that rates of various maturities are such that there is no possible risk-free arbitrage across rates. Forward rates result directly from such an absence of risk-free arbitrage opportunities. A risk-free arbitrage generates a profit without risk, based upon the gap between observed rates and their equilibrium value. Today, we observe only current spot rates and future spot rates are uncertain. Even though they are uncertain, they have some expected value, resulting from the expectations of all market players. This principle allows us to derive the relation between today’s spot rates and future expected rates. The latter are the implicit, or forward, rates. The standard example of a simple transaction covering two periods, such as investing for 2 years, illustrates the principle. For simplicity, we assume this is a risk-neutral world.

Risk-neutrality Risk-neutrality is a critical concept for arbitrage issues. When playing heads and tails, we toss a coin. Let’s say we gain if we have heads and lose in the case of tails. There is a 50% chance of winning or losing. The gain is 100 if we win and −50 if we lose. The expected gain is 0.5 × 100 + 0.5 × (−50) = 25. A risk-neutral player is, by definition,

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indifferent, between a certain outcome of 25 and this uncertain outcome. A risk-adverse investor is not. He might prefer to hold 20 for sure rather than betting. The difference between 25 and 20 is the value of risk aversion. Since investors are risk averse, we need to adjust expectations for risk aversion. For the sake of simplicity, we explain arbitrage in a risk-neutral world.

Arbitrage across Maturities Two alternative choices are investigated: investing straight for 2 years or twice for 1 year. These two choices should provide the same future value to eliminate arbitrage opportunities. The mechanism extends over any succession of periods for any horizon. Forward rates depend on the starting date and the horizon after the starting date, and require two indexes to define them. We use the following notations: • The future uncertain spot rate from date t up to date t + n is t it+n 3 . The current spot rates for 1 and 2 years are known and designated by 0 i1 and 0 i2 respectively. Future spot rates are evidently random. • The forward rate from date t up to date t + n is t Ft+n 4 . This rate results from current spot rates, as shown below, and has a certain value as of today. The forward rate for 1 year and starting 1 year from now is 1 F2 , and so on. For instance, the spot rates for 1 and 2 years could be 10% and 11%. The future value, 2 years from now, should be the same using the two paths, straight investment over 2 years or a 1-year investment rolled over an additional year. The condition is: (1 + 11%)2 = (1 + 10%)[1 + E(1 i2 )]. Since 1 i2 remains uncertain as of today, we need the expectation of this random variable. The rate 1 year from now for 1 year 1 i2 is uncertain, and its expected value is E(1 i2 ). According to this equation, this expected value is around 12% (the accurate value is 12.009%). If not, there is an arbitrage opportunity. The reasoning would be the same for borrowing. If the 1-year spot rate is 10% and the future rate is 12% (approximately), it is equivalent to borrow for 2 years directly or to borrow for 1 year and roll over the debt for another year. This applies under risk-neutrality, since we use expectations as if they were certain values, which they are not. The expected value that makes the arbitrage unprofitable is the expected rate by the market, or by all market players. Some expect that the future rate 1 i2 will be higher, others that it will be lower, but the net effect of expectations is the 12% value. It is also the value of 1 i2 making arbitrage unprofitable between a 1-year transaction and a 2-year transaction in a risk-neutral world. The determination of forward rates extends to any maturity through an iterative process. For instance, if the spot rate 0 i1 is 10% and the spot rate 0 i2 is 11%, we can derive the forward rate 2 F3 from the spot rate for date 3, or 0 i3 . The return over the last year would be the forward rate 2 F3 = E(2 i3 ). Therefore, the condition of non-profitable arbitrage becomes: (1 + 0 i3 )3 = (1 + 0 i2 )2 [1 + E(2 i3 )] = (1 + 0 i2 )2 (1 + 2 F3 ) It gives 2 F3 from the two spot rates. 3 Bold

letter i indicates a random variable, italic letter i indicates any particular value it might have. alternative notation would be n it , n being the period and t the starting date. We prefer to use date notations, t and t + n, to avoid any ambiguity between dates and periods. 4 An

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Expected Rates

Rates

Rates

The implications of the expectation theory of the term structure are important to choose the maturity of transactions and/or the nature, fixed or variable, of interest rates. The long-run rates are geometric averages of short-run rates5 following the rule of no riskfree arbitrage across spot rates. If the yield curve is upward sloping, the forward rates are above the spot rates and expectations are that interest rates will rise. If the yield curve is downward sloping, forward rates are below the spot rates and expectations are that interest rates will fall (Figure 12.3).

Time Increasing rates

FIGURE 12.3

Time Decreasing rates

Term structure of interest rates and future expected rates

If the market expectations are true, the outcome is identical for long transactions and a rollover of short transactions. This results from the rule of no profitable risk-free arbitrage. The future rates consistent with this situation are the forward rates, which are also the expected values of uncertain future rates. The limit of the expectation theory is that forward rates are not necessarily identical to future spot rates, which creates the possibility of profitable opportunities. The expectation theory does not work too well when expectations are heterogeneous. The European yield curve was upward sloping in recent years, and the level of rates declined for all maturities over several years, until 1999. This demonstrates the limitations of the predictive power of the expectation theory. It works necessarily if expectations are homogeneous. If everyone believes that rates will increase, they will, because everyone tries to borrow before the increase. This is the rationale of the ‘self-fulfilling prophecy’. On the other hand, if some believe that rates might increase because they seem to have reached a bottom line and others predict further declines, the picture becomes fuzzy. In the late nineties, while Europe converged towards the euro, expectations were unclear.

FORWARD RATES APPLICATIONS The forward rates, embedded in the spot yield curves, serve a number of purposes: • They are the break-even rates for comparing expectations with market rates and arbitraging the market rates. • It is possible to effectively lock in forward rates for a future period as of today. 5 This

is because the reasoning can extend from two periods to any number of periods. The average is a geometric average.

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159

• They provide lending–borrowing opportunities rather than simply lending–borrowing cash at the current spot rates.

Forward Rates as Break-even Rates for Interest Rate Arbitrage The rule is to compare, under risk-neutrality, the investor’s expectation of the future spot rate with the forward rate, which serves as a break-even rate. In the example of two periods: • If investor’s expectation is above the forward rate 1 F1 , lending short and rolling over the loan beats the straight 2-year investment. • If investor’s expectation is below the forward rate 1 F1 , lending long for 2 years beats the short-term loan renewed for 1 year. • If investor’s expectation matches the break-even rate, investors are indifferent between going short and going long. If a lender believes that, 1 year from now, the rate will be above the forward rate for the same future period, he expects to gain by rolling over short transactions. If the borrower expects that, 1 year from now, the rate will be below the forward rate, he expects to gain by rolling over short-term transactions. Whether the borrower or the lender will actually gain or not depends on the spot rates that actually prevail in the future. The price to pay for expected gains is uncertainty. For those willing to take interest rate risk, the benchmarks for their decisions are the forward rates. The lender makes short transactions if expected future rates are above forward rates. The lender prefers a long transaction at the spot rate if expected future rates are below the forward rates. Similar rules apply to borrowers. Hence, expectations of interest rises or declines are not sufficient for making a decision. Expectations should state whether the future rates would be above or below the forward rates. Otherwise, the decision to take interest rate risk or not remains not fully documented (Figure 12.4). These are simplified examples, with two periods only. Real decisions cover multiple periods of varying lengths6 . Spot yield curve

Forward rates

Rates

Long or Short

Expectations Time

FIGURE 12.4 6 For

Expectations and decisions

instance, over a horizon of 10 years, the choice could be to use a 1-year floating rate, currently at 4%, or to borrow at a fixed rate for 10 years at 6%. Again, the choice depends on expectations with respect to the 9-year rate 1 year from now. The break-even rate, or forward rate, for 9 years 1 year from now is such that: (1 + 4%)(1 + 1 F9 %)9 = (1 + 6%)10 . The yearly forward rate 1 F9 is 6.225%. If the borrower believes

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Locking in a Forward Rate as of Today It is possible to lock in a rate for a future date. This rate is the forward rate derived from the spot yield curve. We assume that an investor has 1000 to invest 1 year from now and for 1 year. He will receive this flow in 1 year. The investor fears that interest rates will decline. Therefore, he prefers to lock in a rate as of today. The trick is to make two transactions that cancel out as of today and result, by netting, in a forward investment 1 year from now and for 1 year. The spot rates are 4% for 1 year and 6% for 2 years. In addition, borrowing and lending are at the same rate. Since the investor wants to lend, he lends today for 2 years. In order to do so, he needs to borrow the money invested for 1 year, since he gets an inflow 1 year from now. The amount borrowed today against the future inflow of 1000 is 1000/(1 + 4%) = 961.54. This results in a repayment of 1000 in 1 year. Simultaneously, he lends for 2 years resulting in an inflow of 961.54(1 + 6%)2 = 1080.385 2 years from now. The result is lending up to date 2 a flow of 1000 at date 1 to get 1080.385. This is a yearly rate of 8.0385%. This rate is also the forward rate resulting from the equation: (1 + 4%)(1 + 1 F2 ) = (1 + 6%)2 Since (1 + 1 F2 ) = (1 + 6%)2 /(1 + 4%) = 1.1236/1.04 = 1.80385. Therefore forward rates ‘exist’ and are more than the result of a calculation (Figure 12.5).

Dates

961.54

1000

961.54

1000

0

1

Forward Rate = 8.0385%

FIGURE 12.5

2

1080.385

Locking in a forward rate 1 year from now for 1 year

that 1 year from now the yearly rate for a maturity of 9 years will be above 6.225%, he or she should borrow fixed. Otherwise, he or she should use a floating rate debt. With a spread of 2% between short-run and long-run rates, intuition suggests that borrowing fixed today is probably very costly. A fixed debt has a yearly additional cost over short-run debt, which is 6% − 4% = 2%. However, the difference in yearly rate does not provide a true picture because we are comparing periods of very different lengths. The 2% additional cost is a yearly cost. If the long rate (9 years) actually increases to 6.225%, the additional cost of borrowing fixed 1 year later will be 0.225% per year. This additional cost cumulates over the 9 years. In other words, the loss of 2% over 1 year should be compared to a potential loss of 0.225% over 9 years.

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The Spot and Forward Yield Curves There is a mechanical relationship between spot rates and forward rates of future dates. The Relative Positions of the Spot and Forward Yield Curves

Spot yield curve

Spot yield curve Forward yield curve

Maturity

FIGURE 12.6

Interest Rates

Forward yield curve

Interest Rates

Interest Rates

When spot yield curves are upward sloping, the forward rates are above the spot yield curve. Conversely, with an inverted spot yield curve, the forward rates are below the spot yield curve. When the spot yield curve is flat, both spot and forward yields become identical (Figure 12.6).

Maturity

Forward yield curve Spot yield curve Maturity

The relative positions of the spot and forward yield curves

When investing and borrowing, it is important to consider whether it is more beneficial to ride the spot or the forward yield curves, depending on expectations about future spot rates. Since banks often have non-zero liquidity gaps, they are net forward borrowers when they have positive liquidity gaps (assets higher than liabilities) and net forward lenders when they have negative liquidity gaps (assets lower than liabilities). The spot yield curve plus the forward yield curves starting at various future dates chart all funding–investment opportunities. Example of Calculating the Forward Yield Curves

The spot yield curve provides all yields by maturity. Forward yield curves derive from the set of spot yields for various combinations of dates. The calculations in Table 12.2 provide an example of a forward yield curve 1 year from now and all subsequent maturities, as well as 2 years from now and all subsequent maturities. In the first case, we plot the forward rates starting 1 year from now and for 1 year (date 2), 2 years (date 3), 3 years (date 4), and so on. In the second case, the forward rates are 2 years from now and for maturities of 1 year (date 3), 2 years (date 4), and so on. The general formulas use the TABLE 12.2 Date t Spot yield Forward t F1 Forward t+2 F1

The spot and forward yield curves 0

1

2

3

4

5

3.50%

4.35% 5.21%

5.25% 6.14% 7.98%

6.15% 7.05% 8.28%

6.80% 7.64% 8.57%

7.30% 8.08% 8.82%

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term structure relationships. For instance, the forward rates between dates 1 and 3 (1 year from now for 2 years) and between dates 1 and 4 (1 year from now for 3 years) are such that: (1 + 1 F3 )2 = (1 + 0 i3 )3 /(1 + 0 i1 )

(1 + 1 F4 )3 = (1 + 0 i4 )4 /(1 + 0 i1 )

If we deal with yields 2 years from now, we divide all terms such as (1 + 0 i1 ) by (1 + 0 it )2 to obtain 1 + 2 Ft , when t ≥ 2. All forward rates are above the spot rates since the spot yield curve is upward sloping (Figure 12.7). In addition, the further we look forward, the lower is the slope of the forward curve. 10% 9% 8% 7% 6% 5% 4% 3% 2% 1% 0%

Forward t + 2 Forward t + 1

Spot t

0

FIGURE 12.7

1

2

3

4

5

6

Spot and forward yield curves

Forward yield curves start later than the spot yield curves because they show forward investment opportunities. The graph provides a broad picture of spot and forward investment opportunities.

VIEWS ON INTEREST RATES A major Asset–Liability Management (ALM) choice is in forming views about future rates. Interest rate models do not predict what future rates will be. However, most ALM managers will face the issue of taking views of future rates. Not having views on future rates implies a ‘neutral’ ALM policy considering all outcomes equally likely and smoothing variations in the long-term. Any deviations from that policy imply an opinion on future interest rates. Various theories on interest rates might help. However, forming ‘views’ on future rates is more a common sense exercise. The old segmentation theory of interest rates is a good introduction. The segmentation approach considers that rates of different maturities vary independently of each other because there are market compartments, and because the drivers of rates in each different compartment are different. For instance, short-term rates depend on the current economic conditions and monetary policy, while long-term rates are representative of the fundamentals of the economy and/or depend on budget deficits. This view conflicts with the

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163

arbitrage theory and with practice, which makes it obvious that rates of different maturities are interdependent. Drifts of interest rates follow the law of supply and demand. Large borrowings shift yields upwards, and conversely. Hence, large budget deficits tend to increase rates. Still, arbitraging long and short interest rates happens continuously, interlinking all yields across maturities. The shape of the yield curve changes. Monetary policy mechanisms tend to make short interest rates more volatile. They work through the ‘open market’ policy. When the central bank wants to raise interest rates, it simply increases its borrowing, drying up the market. This explains the high levels of interest rates in some special situations. Traditional policies fight inflation through higher interest rates in the United States. In Europe, some countries used short-term interest rates to manage exchange rates before ‘Euroland’ formed as a monetary entity. If a local currency depreciates against the USD, for example, a rise in interest rates makes it more attractive to investors to buy the local currency for investing. This increases the demand for the local currency, facilitating its appreciation, or reducing its depreciation. The capability of the central banks to influence strongly interest rates results from their very large borrowing capacity. Long-term interest rates are more stable in general. They might relate more to the fundamentals of the economy than short-term rates. Such observations help us to understand why the yield curve is upward sloping or downward sloping. Whenever short rates jump to high levels because of monetary policy, the yield curve tends to be ‘inverted’, with a downward slope. Otherwise, it has a flat shape or a positive slope. However, there is no mechanism making it always upward sloping. Unfortunately, this is not enough to predict what is going to happen to the yield curve. The driving forces might be identifiable, but not their magnitude, or the timing of central interventions. All the above observations are consistent with arbitraging rates of different maturities. They also help in deciding whether there is potential for future spot rates to be above or below forward rates. Expectations and comparisons with objective benchmarks remain critical for making sound decisions.

13 Interest Rate Gaps

The interest ‘variable rate gap’ of a period is the difference between all assets and liabilities whose interest rate reset dates are within the period. There are as many interest rate gaps as there are interest rate references. The ‘fixed rate gap’ is the difference between all assets and liabilities whose interest rates remain fixed within a period. There is only one fixed rate gap, which is the mirror image of all variable rate gaps. Gaps provide a global view of the balance sheet exposure to interest rates. The underlying rationale of interest rate gaps is to measure the difference between the balances of assets and liabilities that are either ‘interest rate-sensitive’ or ‘insensitive’. If assets and liabilities have interest revenues and costs indexed to the same rate, this interest rate drives the interest income mechanically. If the balances of these assets and liabilities match, the interest income is insensitive to rates because interest revenues and costs vary in line. If there is a mismatch, the interest income sensitivity is the gap. The rationale for interest rate gaps is that they link the variations of the interest margin to the variations of interest rates. This single property makes interest rate gaps very attractive. The ‘gap’ concept has a central place in Asset–Liability Management (ALM) for two main reasons: • It is the simplest measure of exposure to interest rate risk. • It is the simplest model relating interest rate changes to interest income. Interest rate derivatives directly alter the interest rate gaps, providing easy and straightforward techniques for altering the interest rate exposure of the banking portfolio. However, gaps have several drawbacks due to: • Volume and maturity uncertainties as for liquidity gaps: the solution is multiple scenarios, as suggested for liquidity gaps.

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• The existence of options: implicit options on-balance sheet and optional derivatives off-balance sheet. A fixed rate loan is subject to potential renegotiation of rates, it remains a fixed rate asset whether or not the likelihood of renegotiation is high or low. Obviously, when current rates are well below the fixed rate of the loan, the asset is closer to a variable rate asset than a fixed rate one. Such options create ‘convexity risk’, developed later. • Mapping assets and liabilities to selected interest rates rather than using the actual rates of individual assets and liabilities. • Intermediate flows within time bands selected for determining gaps. The first section introduces the gap model. The second section explains how gaps link interest income to the level of interest rates. The third section explains the relationship between liquidity and interest rate gaps. The fourth section details the direct calculation of interest income and of their variations with interest rates. The fifth section discusses hedging. The last section discusses the drawbacks and limitations of interest rate gaps.

THE DEFINITION OF INTEREST RATE GAPS The interest rate gap is a standard measure of the exposure to interest rate risk. There are two types of gaps: • The fixed interest rate gap for a given period: the difference between fixed rate assets and fixed rate liabilities. • The variable interest rate gap: the difference between interest-sensitive assets and interest-sensitive liabilities. There are as many variable interest rate gaps as there are variable rates (1-month LIBOR, 1-year LIBOR, etc.). Both differences are identical in absolute value when total assets are equal to total liabilities. However, they differ when there is a liquidity gap, and the difference is the amount of the liquidity gap. The convention in this text is to calculate interest rate gaps as a difference between assets and liabilities. Therefore, when there is no liquidity gap, the variable rate gap is the opposite of the fixed rate gap because they sum up to zero. Specifying horizons is necessary for calculating the interest rate gaps. Otherwise, it is not possible to determine which rate is variable and which rate remains fixed between today and the horizon. The longer the horizon, the larger the volumes of interest-sensitive assets and liabilities, because longer periods allow more interest rate resets than shorter periods (Figure 13.1). An alternative view of the interest rate gap is the gap between the average reset dates of assets and liabilities. With fixed rate assets and liabilities, the difference between the reset dates is the difference between the average maturity dates of assets and liabilities. If the variable fraction of assets grows to some positive value, whether all liabilities remain fixed rate or not, the average reset maturity of assets shortens because rate resets occur before asset maturity. With variable rate assets and liabilities, the variable interest rate gap, for a given variable rate reference, roughly increases with the gap between reset dates of assets and liabilities, if average reset dates are weighted by sizes of assets and

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Fixed rate assets (FRA) 400

Variable rate assets (VRA) 600

Fixed rate liabilities (FRL) 200

Variable rate liabilities (VRL) 800

Fixed IR gap = FRA − FRL = 400 − 200 = + 200 Variable IR gap = VRA − VRL = 600 − 800 = − 200

Fixed IR Gap = − Variable IR Gap

FIGURE 13.1

Interest rate gap with total assets equal to total liabilities

liabilities. The gap between average reset dates of assets and maturities is a crude measure of interest rate risk. Should they coincide, there would be no interest rate risk on average since rate resets would occur approximately simultaneously. In Figure 13.2 the positive variable rate gaps imply that variable rate assets are larger than variable rate liabilities. This is equivalent to a faster reset frequency of assets. The average reset dates show approximately where asset and liability rates are positioned along the yield curve.

IR gaps: Assets reset dates > Liabilities reset dates

IR Gap Time

Asset average reset date

Liability average reset date

FIGURE 13.2 Interest rate gaps and gaps between average reset dates Note: For a single variable rate reference for assets and liabilities.

INTEREST RATE GAP AND VARIATIONS OF THE INTEREST MARGIN The interest rate gap is the sensitivity of the interest income when interest rates change. When the variable rate gap (interest rate-sensitive assets minus interest rate-sensitive liabilities) is positive, the base of assets that are rate-sensitive is larger than the base of liabilities that are rate-sensitive. If the index is common to both assets and liabilities, the interest income increases mechanically with interest rates. The opposite happens when the variable rate gap is negative. When the interest rate gap is zero, the interest income is insensitive to changes in interest rates. In this specific case, the interest margin is ‘immune’ to variations of rates. In the above example, the gap between interest-sensitive assets and liabilities is +200. The interest income is the difference between interest revenues and charges over the period. The calculation of the variation of the margin is simple under the set of assumptions below:

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167

• The gap is constant over the period. This implies that the reset dates are identical for assets and liabilities. • The rates of interest-sensitive assets and liabilities are sensitive to a common rate, i. We use the notation: IM, Interest Margin; VRA, Variable Rate Assets; VRL, Variable Rate Liabilities; i, interest rate. The change in IM due to the change in interest rate, i, is: IM = (VRA − VRL)i = +200i VRA and VRL are the outstanding balances of variable rate assets and liabilities. The variation of the margin in the above example is 2 when the rate changes by 1%. The basic formula relating the variable rate gap to interest margin is: IM = (VRA − VRL)i = (interest rate gap)i The above formula is only an approximation because there is no such thing as a single interest rate. It applies however when there is a parallel shift of all rates. The formula applies whenever the variable interest rate gap relates to a specific market rate. A major implication for hedging interest rate risk is that making the interest margin ‘immune’ to interest rate changes simply implies neutralizing the gap. This makes the sensitivity of the margin to interest rate variations equal to zero.

INTEREST GAP CALCULATIONS Interest rate and liquidity gaps are interrelated since any future debt or investment will carry an unknown rate as of today. The calculation of gaps requires assumptions and conventions for implementation purposes. Some items of the balance sheet require a treatment consistent with the calculation rules. Interest rate gaps, like liquidity gaps, are differences between outstanding balances of assets and liabilities, incremental or marginal gaps derived from variations of volumes between two dates, or gaps calculated from flows.

Fixed versus Variable Interest Rate Gaps The ‘fixed rate’ gap is the opposite of the ‘variable rate’ gap when assets and liabilities are equal. A future deficit implies funding the deficit with liabilities whose rate is still unknown. This funding is a variable rate liability, unless locking in the rate before funding. Accordingly, the variable rate gap decreases post-funding while the fixed rate gap does not change. The gap implications are symmetrical when there is an excess of liquidity: the interest rate of the future investment is unknown. The variable rate gap increases while the fixed rate gap remains unchanged. The fixed rate gap is consistent with an ex ante view, when no hedge is yet in place. Hedging modifies the interest rate structure of the balance sheet ex post. In addition, the fixed rate calculation is convenient because there is no need to deal with the variety of interest rates that are ‘variable’. It excludes all interest rate-sensitive assets and liabilities within the period whatever the underlying variable rates. However, this is a drawback of the fixed rate calculation since interest rate-sensitive assets and liabilities are

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sensitive to different indexes. Therefore, deriving the interest margin sensitivity from the fixed rate gap is only a crude proxy because it synthesizes the variations of all variable rates into a single index. In reality, accuracy would require calculating all variable interest rate gaps corresponding to each one of these market rates. The interest rate gap is similar to the liquidity gap, except that it isolates fixed rate from variable rate assets and liabilities. Conversely, the liquidity gap combines all of them, whatever the nature of rates. Another difference is that any interest rate gap requires us to define a period because of the fixed rate–variable rate distinction. Liquidity gaps consider amortization dates only. Interest rate gaps require all amortization dates and all reset dates. Both gap calculations require the prior definition of time bands.

Time Bands for Gap Calculation Liquidity gaps consider that all amortization dates occur at some conventional dates within time bands. Interest rate gaps assume that resets also occur somewhere within a time band. In fact, there are reset dates in between the start and end dates. The calculation requires the prior definition of time bands plus mapping reset and amortization dates to such time bands. Operational models calculate gaps at all dates and aggregate them over narrow time bands for improving accuracy. In addition, calculations of interest revenues and interest costs require exact dates, which is feasible with the detailed data on individual transactions independent of the necessity of providing usable reports grouping dates in time bands.

Interdependency between Liquidity Gaps and Interest Rate Gaps For future dates, any liquidity gap generates an interest rate gap. A projected deficit of funds is equivalent to an interest-sensitive liability. An excess of funds is equivalent to an interest-sensitive asset. However, in both cases, the fixed rate interest gap is the same. This is not the case with variable rate gaps if they isolate various interest rates. Liquidity gaps have no specific variable rate ex ante, since the funding or the investment is still undefined. In the example below, there is a projected deficit. Variable interest rate gaps pre-funding differ from post-funding gaps. The interest rate gap derived from interest-sensitive assets and liabilities is the gap before funding minus the liquidity gap, as Figure 13.3 shows. Specific items deserve some attention, the main ones being equity and fixed assets. These are not interest-earning assets or liabilities and should be included from interest rate gaps, but not from liquidity gaps. They influence the interest rate gaps through the liquidity gaps. Some consider equity as a fixed rate liability, perhaps because equity requires a fixed target return. In fact, this compensation does not have much to do with a fixed rate. Since it depends on the interest margin, the effective return is not the target return and it is interest rate-sensitive. We stick to the simple rule that interest rate gaps use only interest rate assets and liabilities and exclude non-interest-bearing items. We illustrate below the detailed calculations of both liquidity and interest rate gaps:

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Fixed Assets (10)

Fixed Assets − Equity = −10

Fixed Rate Assets (75)

Equity (20)

Fixed Rate Liabilities (30) Variable Rate Liabilities (40)

Variable Rate Gap (VRA − VRL) = −5 Pre-funding

Variable Rate Assets (35)

Liquidity Gap (Variable Rate) (30)

Total = 120

Total = 120

Variable Rate Gap Post-funding = −35

FIGURE 13.3

Variable Rate Gap Pre-funding = −5

=

−

Fixed Rate Gap (FRA − FRL) = 45

Liquidity Gap 120 − 90 = + 30

Liquidity Gap = + 30

The balance sheet structure and gaps

Liquidity gaps = total assets − total liabilities = 10 + 75 + 35 − (20 + 30 + 40) = 120 − 90 = +30 (a deficit of liquidity) a = (interest rate assets − interest rate liabilities) + (fixed assets − equity) = (75 + 35 − 30 − 40) + (10 − 20) = +30 Fixed interest rate gap = fixed rate assets − fixed rate liabilities (excluding non-interest-bearing items) = 75 − 30 = 45 Variable interest rate gap = variable rate assets − variable rate liabilities = 35 − 40 = −5 without the liquidity gap = −5 − 30 = −35 with the liquidity gap Therefore, the general formula applies: Variable interest rate gap post-funding = fixed interest rate gap − liquidity gap In the example, −35 = −5 − 30.

Cumulative and Marginal Gaps Gaps are differences between outstanding balances at one given date, or differences of variations of those balances over a period. Gaps calculated from variations between two

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dates are ‘marginal gaps’, or differences of flows rather than volumes. For a unique given period, the variation of the interest margin is equal to the gap times the interest rate variation. For a succession of periods, it becomes necessary to break down the same types of calculations by period. The gaps change with time. When the gap is assumed constant over one period, it means that the volume of interest-sensitive assets and liabilities does not change. If not, splitting the period into smaller subperiods makes the constant gap assumption realistic. The example below demonstrates the impact of varying gaps over time (Table 13.1). There are three balance sheets available at three dates. The variations of assets and liabilities are differences between two consecutive dates. The data are as follows: TABLE 13.1

Interest rate gaps, cumulative and marginal

End of period

Month 1

Month 2

Month 3

0 200

250 200

300 200

−200 −200

+50 +250

+100 +50

Variable rate assets Variable rate liabilities Cumulative gaps Marginal gaps

• The total period is 3 months long. • The total assets and liabilities are 1000 and do not change. All items mature beyond the 3-month horizon and there is no new business. The liquidity gap is zero for all 3 months. • The interest rate increases during the first day of the first month and remains stable after. • The interest-sensitive liabilities are constant and equal to 200. • The interest-sensitive assets change over time. The fluctuation of interest-sensitive assets is the only source of variation of the gap.

MARGIN VALUES AND GAPS All the above calculations aim to determine variations of margins generated by a variation of interest rates. They do not give the values of the margins. The example below shows the calculation of the margins and reconciles the gap analysis with the direct calculations of margins. The margins, expressed in percentages of volumes of assets and liabilities, are the starting point. They are the differences between the customer rates and the market rates. The market rate is equal to 10% and changes by 1%. If the commercial margins are 2% for loans and −4% for deposits, the customer rates are, on average, 10 + 2 = 12% for loans and 10 − 4 = 6% for liabilities. The negative commercial margins for liabilities mean that rates paid to customers are below market rates. The values of margins are given in Table 13.2. The initial margin in value is 1000(10% + 2%) − 1000(10% − 4%) = 60. If the interest rate changes by 1%, only the rates of interest rate-sensitive assets and liabilities change, while others keep their original rates. New customer rates result from the variation of the reference rate plus constant percentage margins. The change in interest margin is +1. This is consistent with an interest rate gap of 100, combined with a 1% variation of interest rate since 100 × 1% = 1 (Table 13.3).

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TABLE 13.2

TABLE 13.3

Balance sheets and margins: example

Assets

Outstanding balances

Margins

Fixed rate Variable rate Total assets Liabilities

700 300 1000 Outstanding balances

+2% +2% Margins

Fixed rate Variable rate Total liabilities

800 200 1000

−4% −4%

Variable rate gap

+100

Calculation of interest margin before and after an interest rate rise

Rate

10%

Fixed rate Variable rate

700 300

Total assets Fixed rate Variable rate Total liabilities Margin

800 200

11%

Assets: 700 × 12% 84 300 × 12% 36 120 Liabilities: 800 × 6% 48 200 × 6% 12 60 60

700 × 12% 300 × 13%

800 × 6% 200 × 7%

Variation 84 39

0 +3

123

+3

48 14

0 +2

62 61

+2 +1

The Interest Margin Variations and the Gap Time Profile Over a given period, with a constant gap and a variation of interest rate i, the revenues change by VRA × i and the costs by VRL × i. Hence the variation in IM is the difference: IM = (VRA − VRL)i = gap × i Over a multi-periodic horizon, subperiod calculations are necessary. When interest rates rise, the cost increase depends on the date of the rate reset. At the beginning of month 1, an amount of liabilities equal to 200 is interest rate-sensitive. It generates an increase in charges equal to 200 × 1%1 (yearly value) over the first month only. At the beginning of month 2, the gap becomes 50, generating an additional revenue of 50 × 1% = 0.5 for the current month. Finally, the last gap generates a change in margin equal to 100 × 1% for the last month. The variation of the cumulative margin over the 3-month period is the total of all monthly variations of the margin. Monthly variations are 1/12 times the yearly variations. This calculation uses cumulative gaps (Table 13.4). In practice, the gaps should be available on a monthly or a quarterly basis, to capture the exposure time profile with accuracy. Table 13.4 also shows that the calculation of the cumulative variation of the margin over the total 3-month period can follow either one of two ways. The first calculation is 1 Calculations

are in yearly values. The actual revenues and charges are calculated over a quarter only.

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TABLE 13.4 Cumulative gaps and monthly changes in interest margins Month 1 2 3

Cumulative gaps −200 +50 +100

Monthly variation of the margin −200 × 1% +50 × 1% +100 × 1%

−2.0 +0.5 +1.0

Total

−0.5

as above. Using cumulative gaps over time, the total variation is the sum of the monthly variations of the margins. Another equivalent calculation uses marginal gaps to obtain the margin change over the residual period to maturity. It calculates the variation of the margin over the entire 3-month period due to each monthly gap. The original gap of −200 has an impact on the margin over 3 periods. The second marginal gap of +250 alters the margin during 2 periods. The third marginal gap influences only the last period’s margin. The cumulative variation of the 3-period margin is the product of the periodical marginal gaps times the change in rate, and times the number of periods until maturity2 . Monthly calculations derive from the above annual values.

INTEREST RATE GAPS AND HEDGING One appealing feature of gaps is the simplicity of using them for hedging purposes. Hedging often aims at reducing the interest income volatility. Since the latter depends linearly on the interest rate volatility through gaps, controlling the gaps suffices for hedging or changing the exposure using derivatives. Simple examples and definitions suffice to illustrate the simplicity of gap usage.

Hedging over a Single Period It is possible to neutralize the interest rate gap through funding, without using any interest rate derivative. We assume that the liquidity gap is +30 and that the variable rate gap is 2 The

calculation with marginal gaps would be as follows. The marginal gaps apply to residual maturity: [(−200 × 3) + (250 × 2) + (50 × 1)] × 1% = −0.5

With cumulated gaps, the calculation is: [(−200) + (+250 − 200) + (+50 + 250 − 200)] × 1% = −0.5 These are two equivalent calculations, resulting from the equality of the sum of marginal gaps, weighted by residual maturities, with the sum of cumulated gaps of each period. The periodic gaps are g1 , g2 and g3 . The cumulated gaps are G1 = g1 , G2 = g1 + g2 , G3 = g1 + g2 + g3 . The total of cumulated gaps is: G1 + G2 + G3 = g1 + (g1 + g2 ) + (g1 + g2 + g3 ) which is identical to: 3 × g1 + 2 × g2 + 1 × g3

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+20 before funding and −10 after funding with a variable rate debt. The bank wishes to hedge its sensitivity to interest rates. If the bank raises floating rate funds, it has too many floating rate liabilities, exactly 10. If the bank raises floating rate debt for 20 and locks in the rate for the remaining 10, as of today, the interest rate gap post-funding and hedging becomes 0. The interest rate margin is ‘immune’ to interest rate changes. In order to lock in a fixed rate for 10 at the future date of the gap calculation, it is necessary to use a Forward Rate Agreement (FRA). If we change the data, this type of hedge does not work any more. We now assume that we still have a funding gap of +30, but that variable interest rate gap is +45 pre-funding and +15 post-funding with floating rate debt. The bank has too many floating rate assets, and even though it does not do anything to lock in a rate for the debt, it is still interest rate-sensitive. The excess of floating rate assets post-funding is +15. The bank receives too much floating rate revenue. To neutralize the gap, it is necessary to convert the 15 variable rate assets into fixed rate assets. Of course, the assets do not change. The only way to do this conversion is through a derivative. The derivative should receive the fixed rate and pay the floating rate. The asset plus the derivative generates a neutralized net floating rate flow and a net fixed rate flow. This derivative is an Interest Rate Swap (IRS) receiving the fixed rate and paying the floating rate (Figure 13.4). Receive floating Floating rate Asset

Receive fixed

Fixed rate Asset

Receive fixed IRS Pay floating

FIGURE 13.4

Hedging interest rate risk with an interest rate swap

Hedging over Multiple Periods Figure 13.5 shows the time profiles of both marginal and cumulated gaps over multiple periods for the previous example. The cumulated gap is negative at the end of the first month, and becomes positive later. Marginal gap

300

Cumulated gap

200

Gap

100 0 −100

Month 1

Month 2

−200 −300

FIGURE 13.5

Interest rate gap profiles

Dates

Month 3

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Immunization of the interest margin requires setting the gap to zero at all periods. For instance, at period 1, a swap with a notional of 200, paying the fixed rate and receiving the floating rate, would set the gap to zero at period 1. The next period, a new swap paying the floating rate and receiving the fixed rate, with a notional of 250, neutralizes the interest rate gap. Actually, the second swap should be contracted from date 0 and take effect at the end of month 1. This is a forward swap3 starting 1 month after date 0. A third swap starting at the end of month 2, and maturing at the end of month 3, neutralizes the gap of month 3. Since the forward swap starting at the beginning of month 2 can extend to hedge the risk of month 3, the third new swap has a notional of only 50. The marginal gaps represent the notionals of the different swaps required at the beginning of each period, assuming that the previous hedges stay in place. The usage of marginal interest rate gaps is similar to that of marginal liquidity gaps, these representing the new debts or investments assuming that the previous ones do not expire. The gap model provides the time profile of exposures to interest rate risk. Very simple rules suffice for hedging these exposures. The start dates and the termination dates of such hedges result from the gap profile. The hedge can be total or partial. The residual exposure, after contracting a hedge for a given amount, is readily available from the gap profile combined with the characteristics of the hedge.

LIMITATIONS OF INTEREST RATE GAPS Gaps are a simple and intuitive tool. Unfortunately, gaps have several limitations. Most of them result from assumptions and conventions required to build the gap profile. The main difficulties are: • Volume and maturity uncertainties as for liquidity gaps: the solution lies in assumptions and multiple scenarios, as suggested for liquidity gaps. • Dealing with options: implicit options on-balance sheet and optional derivatives offbalance sheet. These options create ‘convexity risk’, developed later. • Mapping assets and liabilities to selected interest rates as opposed to using the actual rates of individual assets and liabilities. • Dealing with intermediate flows within time bands selected for determining gaps. We deal with the latter three issues, the first one having been discussed previously with liquidity gaps.

Derivatives and Options Some derivative instruments do not raise any specific difficulties. A swap substitutes a variable rate for a fixed rate. The impact on the interest rate gap follows immediately. Other optional instruments raise issues. Caps and floors set maximum and minimum values of interest rates. They change the interest sensitivity depending on whether the level 3A

forward swap is similar to a spot swap, but starts at a forward date.

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of the interest rate makes them in-the-money or not. When they are out-of-the-money, options do not change the prevailing rates. When they are in-the-money, caps make assets interest rate-insensitive. In other words, options behave sometimes as variable rate items and sometimes as fixed rate items, when the interest rates hit the guaranteed rates. The interest rate gap changes with the level of interest rates. This is a serious limitation to the ‘gap picture’, because the gap value varies with interest rates! There are many of embedded or implicit options in the balance sheet. Prepayments of fixed rate mortgage loans when interest rates decrease are options held by the customers. Some regulated term deposits offer the option of entering into a mortgage with a subsidized rate. Customers can shift funds from non-interest-bearing deposits to interest-earning deposits when rates increase. When sticking to simple gap models, in the presence of options, assumptions with respect to their exercise are necessary. The simplest way to deal with options is to assume that they are exercised whenever they are in-the-money. However, this ignores the value of waiting for the option to get more valuable as time passes. Historical data on the effective behaviour might document these assumptions and bring them close to reality. The prepayment issue is a well-known one, and there are techniques for pricing implicit options held by customers. Another implication is that options modify the interest rate risk, since they create ‘convexity risk’. Measuring, modelling and controlling this optional risk are dealt with in related chapters (see Chapters 20 and 21 for pricing risk, Chapter 24 for measuring optional risk at the balance sheet level).

Mapping Assets and Liabilities to Interest Rates Interest rate gaps assume that variable rate assets and liabilities carry rates following selected indexes. The process requires mapping the actual rates to selected rates of the yield curve. It creates basis risk if both rates differ. Sensitivities, relating actual rates to selected rates, correct basis risk. The technique serves for calculating ‘standardized gaps’4 . The alternative solution is to use directly the contractual rate references of contracts at the level of individual transactions, raising Information Technology (IT) issues. Sensitivities

The average rate of return of a subportfolio, for a product family for instance, is the ratio of interest revenues (or costs) to the total outstanding balance. It is feasible to construct time series of such average rates over as many periods as necessary. A statistical fit to observed data provides the relationship between the average rate of the portfolio and the selected market indexes. A linear relation is such that: Rate = a0 + a1 × index1 + ‘random residual’ The coefficient a1 is the sensitivity of the loan portfolio rate with respect to index1 . The residual is the random deviation between actual data and the fitted model. A variation of 4 Sensitivities

measure correlations between actual and selected rates. A basis risk remains because correlations are never perfect (equal to 1). See Chapter 28.

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the market index of 1% generates a variation of the rate of the loan portfolio of a1 %. The same methodology extends notably to transaction rates, which are not market rates, such as the prime rate. The prime rate variations depend on the variations of two market rates. The loan balances indexed with the prime rate behave as a composite portfolio of loans indexed with the market index1 and a portfolio of loans indexed with the market index2 . The statistical relationship becomes: Prime rate = a0 + a1 × index1 + a2 × index2 + ‘random residual’ Such statistical relations can vary over time, and need periodical adjustments. The sensitivities are the coefficients of the statistical model. Standardized Gaps

With sensitivities, the true relationships between the interest margin and the reference interest rates are the standardized gaps, rather than the simple differences between assets and liabilities. Standardized gaps are gaps calculated with assets and liabilities weighted by sensitivities. For instance, if the sensitivity of a loan portfolio is 0.5, and if the outstanding balance is 100, the loan portfolio weighted with its sensitivity becomes 50. With interest rate-sensitive assets of 100 with a sensitivity of 0.5, and interest rate-sensitive liabilities of 80 with a sensitivity of 0.8, the simple gap is 100 − 80 = 20 and the standardized gap is: 0.5 × 100 − 0.8 × 80 = 50 − 64 = −14 This is because a variation of the interest rate i generates a variation of the interest on assets which is 0.5i and a variation of the rate on liabilities which is 0.8i. The resulting variation of the margin is: IM = 100 × 0.5 + 80 × 0.8i = (0.5 × 100 − 0.8 × 80)i IM = (standardized gap)i In this example, the simple gap and the standardized gap have opposite signs. This illustrates the importance of sensitivities when dealing with aggregated portfolios of transactions rather than using the actual rate references.

Intermediate Flows and Margin Calculations The gap model does not date accurately the flows within a period. It does not capture the effect of the reinvestment or the funding of flows across periods. In some cases, both approximations can have a significant impact over margins. This section expands possible refinements. Intra-periodic Flows

Gaps group flows within time bands as if they were simultaneous. In reality, there are different reset dates for liquidity flows and interest rates. They generate interest revenues

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Reinvestment at the rate at t t′ t Borrowing at the rate at t′

Reference period for the gap calculation

FIGURE 13.6

Interest rate risk depends on the positioning of flows across time

or costs, which are not valued in the gap–margin relationship. Figure 13.6 shows that revenues and costs assigned to intermediate reinvestments or borrowings depend on the length of the period elapsed between the date of a flow and the final horizon. A flow with reset date at the end of the period has a negligible influence on the current period margin. Conversely, when the reset occurs at the beginning of the period, it has a significant impact on the margin. The simple gap model considers that these flows have the same influence. For instance, a flow of 1000 might appear at the beginning of the period, and another flow of 1000, with an opposite sign, at the end of the period. The periodic gap of the entire period will be zero. Nevertheless, the interest margin of the period will be interest-sensitive since the first flow generates interest revenues over the whole period, and such revenues do not match the negligible interest cost of the second flow (Figure 13.7). 1000 Reinvestment at the rate at 0

End of period (1) Beginning of period (0) 1000

Borrowing at the rate at 1

Reference period for the gap calculation

FIGURE 13.7

Zero gap combined with an interest-sensitive margin

The example below illustrates the error when the goal is to immunize the interest margin. In the preceding example, the gap was zero, but the margin was interest-sensitive. In the example below, we have the opposite situation. The gap differs from zero, but the interest margin is immune to interest rate changes. The gap is negative, which suggests that the margin should increase when the interest rate decreases, for example from 10% to 8%. However, this ignores the reinvestment of the positive intermediate flow of date 90 at

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a lower rate for 270 days. On the other hand, the negative flow generates a debt that costs less over the remaining 180 days. The margin is interest rate-sensitive if we calculate interest revenues and costs using the accurate dates of flows (Figure 13.8). 1000 270 days

180 days 0

1

90 days 180 days −1536

FIGURE 13.8

Negative cumulative gap and rate-insensitive margin

The changes of interest revenues and costs, due to a shift of rates from 8% to 10%, are: Inflow at day 90 : Outflow at day 180 :

1000(1.10270/360 − 1.08270/360 ) = −14.70

−1536(1.10180/360 − 1.08180/360 ) = +14.70

In this example, the decline of funding costs matches the decline of interest revenues. This outcome is intuitive. The interest revenues and costs are proportional to the size of flows and to the residual period over which reinvestment or funding occurs. The first flow is less important than the second flow. However, it generates interest revenues over a longer period. The values are such that the residual maturity differential compensates exactly the size differential. The outflow is ‘equivalent’ to a smaller outflow occurring before, or the inflow is ‘equivalent’ to a smaller inflow occurring after. The example shows how ‘gap plugging’, or direct gap management, generates errors. In the first example, the margin is interest-sensitive, which is inconsistent with a zero gap. In the second example, the gap model suggests plugging another flow of 536 to hedge the margin. Actually, doing so would put the margin at risk! The exact condition under which the margin is insensitive to interest rates is relatively easy to derive for any set of flows. It differs from the zero gap rule (see appendix to Chapter 23). The condition uses the duration concept, discussed in Chapter 22. Interest Cash Flows and Interest Rate Gaps

The flows used to calculate interest rate gaps are variations of assets and liabilities only. However, all assets and liabilities generate interest revenues (or costs), received (or paid), at some dates. Only the cash payments of interest count, not the accrued accounting interest flows. Gap profiles based on capital flows only ignore these flows for the sake of simplicity. Gaps should include such interest payments, as the mark-to-market value of the balance sheet does.

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Floating rate assets or liabilities generate interest flows as well. A common practice is to project the stream of interest flows using the current value of the floating rate. It ‘crystallizes’ the current rates up to the horizon. Forward rates would serve the same purpose and are conceptually more appropriate since they are the image of the market expectations. Nevertheless, when the interest rate changes, floating rate flows change as well, and change both the liquidity and interest rate gap profiles.

14 Hedging and Derivatives

Derivatives alter interest rate exposures and allow us to hedge exposures and make interest income independent of rates. This does not eliminate risk, however, since hedges might have opportunity costs because they set the interest income at lower levels than rates could allow. There are two main types of derivatives: forward instruments are forward rate agreements or swaps; optional instruments allow capping the interest rate (caps) or setting a minimum guaranteed rate (floors). Forward instruments allow us to change and reverse exposures, such as turning a floating rate exposure into a fixed rate one, so that the bank can turn an adverse scenario into a favourable one. However, hedging with forward instruments implies a bet on the future and still leaves the bank unprotected against adverse interest rate movements if the bet goes wrong. Optional instruments allow us to capture the best of both worlds: getting the upside of favourable interest rate movements and having protection against the downside of adverse deviations, with a cost commensurable with such benefits. Derivatives serve to alter the exposure, to resize or reverse it, and to modify the risk–return profile of the balance sheet as a whole. Futures are instruments traded on organized markets. They perform similar functions. Since there is no way to eliminate interest rate risk, the only option is to modify the exposure according to the management views. To choose an adequate hedging policy, Asset–Liability Management (ALM) needs to investigate the payoff profiles of alternative policies under various interest rate scenarios. Examples demonstrate that each policy implies a bet that rates will not hit some break-even value depending on the particular hedge examined and its cost. This break-even value is such that the payoff for the bank is identical with and without the hedge. Decision-making necessitates comparing the break-even value of rates with the management views on interest rates.

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Foreign exchange derivatives perform similar functions, although the mechanisms of forward contracts differ from those of interest rate contracts. There is a mechanical connection between interest rates in two currencies, spot and forward exchange rates. This allows shifting exposures from one currency to the other, and facilitates multicurrency ALM. This chapter provides the essentials for understanding the implementation of these hedging instruments. The first section discusses alternative ‘hedging’ policies. The next two sections explain the mechanisms that allow derivatives to alter the exposure profile, starting with forward instruments and moving on to options. Both sections detail the payoffs of hedges depending on interest rate scenarios. The fourth section introduces definitions about future markets, explaining how they serve for hedging. The last section deals with currency risk, using the analogy with interest rate hedging instruments. It details the mechanisms of foreign exchange forward contracts and of options on currency exchange rates.

INTEREST RATE RISK MANAGEMENT This section discusses interest rate exposure and hedging policies. There are two alternative objectives of hedging policies: • To take advantage of opportunities and take risks. • A prudent management aimed at hedging risks, totally or partially (to avoid adverse market movements and benefit others). In the second case, policies aim to: • Lock in interest rates over a given horizon using derivatives or forward contracts. • Hedge adverse movements only, while having the option to benefit from other favourable market movements. There are two types of derivative instruments: • Forward hedges lock in future rates, but the drawback lies in giving up the possibility of benefiting from the upside of favourable movements in rates (opportunity cost or risk). • Optional hedges provide protection against adverse moves and allow us to take advantage of favourable market movements (no opportunity cost), but their cost (the premium to pay when buying an option) is much higher. In addition, futures also allow us to implement hedging policies. The futures market trades standard contracts, while derivatives might be ‘plain vanilla’ (standard) or customized1 . 1 See

Figlewski (1986) for both theoretical and practical aspects of hedging with futures for financial institutions.

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FORWARD INTEREST RATE INSTRUMENTS Forward instruments aim at locking the rate of exposure. Whether the hedge is profitable or not depends on the subsequent variations of interest rates. Forward hedges lock in rates, but they do not allow borrowers and lenders to benefit from the upside, as options do. Therefore, there are benchmarks to consider when setting up such hedges. The universal principle that applies is to make sure, as far as possible, that expected interest rates would be above, or below, some break-even values. These break-even values are the interest rates making the hedge unfavourable, since contracting a hedge implies contracting an interest rate. We present the basic mechanisms with examples only. The most common instruments are forward rate agreements and swaps2 .

Interest Rate Swaps The objective is to exchange a floating rate for a fixed rate, and vice versa, or one floating rate for another (a 1-month rate against a 1-year rate). The example is that of a firm that borrows fixed rate and thinks rates will decline. The firm swaps its fixed rate for a floating rate in order to take advantage of the expected decline. The fixed rate debt pays 11%, originally contracted for 5 years. The firm would prefer to borrow at the 1-year LIBOR. The residual life of the debt is 3 years. The current 3-year spot rate is 10%. The 1-year LIBOR is 9%. The swap exchanges the current 3-year fixed rate for the 1-year LIBOR. The treasurer still pays the old fixed rate of 11%, but receives from the counterparty (a bank) the current 3-year fixed rate of 10% and pays LIBOR to the bank. Since the original debt is still there, the borrower has to pay the agreed rate. The cost of the ‘original debt plus swap’ is: 1-year LIBOR (paid to bank) + 11% (original debt) − 10% (received from bank) The borrower now pays LIBOR (9%) plus the rate differential between the original fixed rate (5 years, 11%) and the current spot rate (3 years, 10%), or 1%. The total current cost is LIBOR + 1% = 10%. The treasurer has an interest rate risk since he pays a rate reset every 3 months. As long as LIBOR is 9%, he now saves 1% over the original cost of the debt. However, if LIBOR increases over 10%, the new variable cost increases beyond the original 11%. The 10% value is the break-even rate to consider before entering into the swap. The treasurer bets that LIBOR will remain below 10%. If it moves above, the cost of the joint combination ‘original debt plus swap’ increases above the original cost. If this happens, and lasts for a long period, the only choice left is to enter into a reverse swap. The reverse swap exchanges the floating rate with the current fixed rate. Forward hedges are reversible, but there is a cost in doing so. The cost of the swap is the spread earned by the counterparty, a spread deducted from the rate received by the swap. The above swap receives the fixed rate and pays the floating rate. Other swaps do the reverse. In addition, swaps might exchange one floating rate for another. There are 2 See

Hull (2000) for an overview of derivative definitions and pricing.

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also differed swaps, or forward starting swaps. These swaps use the forward market rates, minus any retained spread by the counterparty. Swaptions are options on swaps, or options of entering into a swap contract at a deferred date. For a bank, the application is straightforward. Banks have both assets and liabilities, which result in interest rate gaps. The interest rate gap is like a debt or an investment. When the variable rate gap is positive, the bank is adversely exposed to interest rate declines and conversely. If the variable rate gap is negative, the bank fears an increase in interest rates. This happens when the variable debt is larger than the variable rate assets. The exposure is similar to that of the above borrower. Therefore, banks can swap fixed rates of assets or liabilities to adjust the gap. They can swap different variable rates and transform a floating rate gap, say on 1-month LIBOR, into a floating rate gap, for instance 3-month LIBOR.

Forward Contracts Such contracts allow us to lock in a future rate, for example locking in the rate of lending for 6 months in 3 months from now. These contracts are Forward Rate Agreements (FRAs). The technology of such contracts is similar to locking in a forward rate through cash transactions, except that cash transactions are not necessary. To offer a guaranteed rate, the basic mechanism would be that the bank borrows for 9 months, lends to a third party for 3 months, and lends to the borrower when the third party reimburses after 3 months. The cost of forward contracts is the spread between the 9-month and 3-month interest rates. Since this would imply cash transactions increasing the size of the bank balance sheet, they are lending costs for the bank because of regulations. Hence, the technology for FRAs is to separate liquidity from interest rate exposure, and the bank saves the cost of carrying a loan. A future borrower might buy an FRA if he fears that rates will increase. A future lender might enter into an FRA if he fears that rates will decline. As usual, the FRA has a direct cost and an opportunity cost. When the guaranteed rate is 9%, the bank pays the difference between the current rate, for instance 9.25%, and 9% to the borrower. Should the rate be 8%, the borrower will have to pay to the counterparty the 1% differential. In such a case, the borrower loses with the FRA. The break-even rate is that of the FRA.

OPTIONAL INTEREST RATE INSTRUMENTS An option is the right, but not the obligation, to buy (call) or sell (put) at a given price a given asset at (European option), or up to (American option), a given date. Options exist for stocks, bonds, interest rates, exchange rates and futures.

Terminology The following are the basic definitions for understanding options. A call option is the right, but not the obligation, to buy the underlying at a fixed exercise price. The exercise price is the price to pay the underlying if the holder of the option exercises it.

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The payoff, or liquidation value, of the option is the gain between the strike price and the underlying price, netted from the premium paid. The option value is higher by its ‘time value’, or the value of waiting for better payoffs. A put option is the right, but not the obligation, to sell the underlying. Options are in-the-money when they provide a gain, which is when the liquidation value is higher than the exercise price. Options are out-of-the-money when the underlying value is below the exercise price. Such options have only time value, since their payoff would be negative. Options have various underlyings: stock, interest rate or currency. Hence, optional strategies serve for hedging interest rate risk and foreign exchange risk, market risk and, more recently, credit risk3 . Figure 14.1 illustrates the case of a call option on stock that is in-the-money. Option Value Time value

Liquidation value Underlying value

Premium Strike Strike Price Price

FIGURE 14.1

Option value, liquidation value and payoff

Interest Rate Options Common interest rate options are caps and floors. A cap sets up a guaranteed maximum rate. It protects the borrower from an increase of rates, while allowing him to take advantage of declines of rates. The borrower benefits from the protection against increasing rates without giving up the upside gain of declining rates. This benefit over forward hedges has a higher cost, which is the premium to pay to buy the cap. A floor sets up a minimum guaranteed rate. It protects the lender from a decline of rates and allows him to take advantage of increasing rates. As for caps, this double benefit has a higher cost than forwards do. Figure 14.2 visualizes the gains and the protection. Since buying caps and floors is expensive, it is common to minimize the cost of an optional hedge by selling one and buying the other. For instance, a floating rate borrower buys a cap for protection and sells a put to minimize the cost of the hedge. Of course, the rate guaranteed by the cap has to be higher than the rate that the borrower guarantees to the buyer of the floor. Hence, there is no benefit from a decline in rates below the guaranteed rate of the put sold. Figure 14.2 visualizes the collar together with the cap and floor guaranteed rates. 3 See

Hull (2000), Cox and Rubinstein (1985) and Chance (1990) for introductions to options.

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Interest Rate

Cap Collar Floor

Time

FIGURE 14.2

Cap, floor and collar

Optional Strategies Optional strategies might be directional, on volatility, or aim to minimize the cost of hedges. Directional policies are adequate when a trend appears likely. Buying a cap is consistent with a bet on interest rate increases. Betting on volatility makes sense when interest rates are unstable and when the current information does not permit identification of a direction. Both an increase and a decrease of rates might occur beyond some upper and lower bounds. For instance, buying a cap at 10% combined with the purchase of a put at 8% provides a gain if volatility drives the interest rate beyond these boundary values. The higher the volatility, the higher are the chances that the underlying moves beyond these upper and lower bounds. Minimizing costs requires collars. The benefit is a cost saving since X buys one option (cap) and sells the other (floor). The net cost is: premium of cap–premium of floor. The drawback is that the interest rate might decline beyond the strike of the floor. Then, the seller of the floor pays the difference between the interest rate and the strike of the floor.

Example: Hedging an Increase of Rates with a Cap The borrower X has 200 million USD floating rate debt LIBOR 1-month + 1.50% maturing in 1 year. X fears a jump in interest rates. He buys a cap with a guaranteed maximum rate of 8% yearly, costing a premium of 1.54%. After 1 year, the average value of LIBOR becomes 10%. Then X pays 11.50% to the original lender and 1.54% to the seller of the cap. He receives from the seller of the cap: 10% − 8% = 2%, the interest rate differential between the guaranteed rate of 8% and the current rate of 10%. The effective rate averaged is 10% + 1.5% + 1.54% − 2% = 11.04% instead of 11.50% without the cap. This is a winning bet. Payments are the notional 200 million USD times the percentages above. Now assume that after 1 year the average value of LIBOR becomes 7%. X pays 8.50% to the lender and 1.54% to the seller of the cap. X does not receive anything from the seller of the cap. The effective rate averaged over a year is: 7% + 1.50% + 1.54% = 10.04% instead of 8.50% without the cap! The break-even value of the yearly average of the LIBOR is 9.54% to have a winning optional hedge, or ‘guaranteed rate

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8% + premium 1.54%’. Before contracting the cap, the borrower should bet that rates would be above 9.54%.

Example: Hedging an Increase of Rates with a Collar The collar minimizes the direct cost of the optional hedge. The drawback is that the rates should not drop down to the minimum guaranteed rate to the buyer of the put, since the seller has to pay this minimum rate. Hence, there is no gain below that rate. For example, the debt rate is the LIBOR 1-year, with a residual maturity of 5 years, and an amount of 200 million USD. X buys a 5-year cap, with a strike at 10%, costing 2% and sells a floor with a strike at 7%, receiving the premium of 1%. The net cost is 1%. If the interest rate declines below 7%, the seller of the floor pays the buyer 7%. The maximum (but unlikely) loss is 7% if rates fall close to zero.

Exotic Options There are plenty of variations around ‘plain vanilla’ derivatives, called ‘exotic’ derivatives. There is no point in listing them all. Each one should correspond to a particular view of the derivative user. For instance, ‘diff swaps’, or ‘quanto swaps’, swap the interest percentage in one currency for an interest rate percentage in another currency, but all payments are in the local currency. Such options look very exotic at first, but it makes a lot of sense in some cases. Consider the passage to the euro ex ante. Sometimes before, this event was uncertain. We consider the case of a local borrower in France. Possible events were easy to identify. If the euro did not exist, the chances are that the local French interest rate would increase a lot, because the FRF tended to depreciate. In the event of depreciation, the French rate would jump to very high levels to create incentives for not selling FRF. With the euro, on the other hand, the rates would remain stable and perhaps converge to the relatively low German rates, since the DEM was a strong currency. Without the euro, a floating rate borrower faced the risk of paying a very high interest rate. Swapping to a fixed rate did not allow any advantage to be taken of the decline in rates that prevailed during the years preceding the euro. Swapping a French interest rate for a German interest rate did not offer real protection, because, even though the German rate was expected to remain lower than the French rate, the interest payment in depreciated FRF would not protect the borrower against the euro risk. The ‘diff swap’ solves this issue. It swaps the French rate into the German rate, but payments are in FRF. In other words, the French borrower pays the German percentage interest rate in FRF, using as notional the FRF debt. The diff swap split the currency risk from the interest rate risk. It allowed to pay in FRF the low German rate without suffering from the potential FRF depreciation. Hence, the view on risks makes the choice of this exotic product perfectly clear. Of course, should the euro come into force, the borrower does not have to use the diff swap. The general message on derivatives, whether plain vanilla or exotic, is that such hedges require taking ‘views’ on risks. Plain vanilla derivatives have implied break-even values of rates turning potential gains into losses if expectations do not materialize. This applies also to exotic products.

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187

FUTURES MARKETS Unlike over-the-counter derivatives, futures are standard contracts sold in organized markets. The major difference between over-the-counter derivatives and futures is the customization of derivatives versus standardized future contracts.

Future Contracts A future contract is a double transaction consisting of: • A commitment over a price and a delivery at a given maturity for a given asset. • Settling the transaction at the committed date and under the agreed terms. For instance, it is possible to buy today a government debt delivered in 1 year at a price of 90 with a future. If, between now and 1 year, the interest rate declines, the transaction generates a profit by buying at 90 a debt valued higher in the market. The reverse contract is to sell today a government debt at a price of 90 decided today. If, between now and 1 year, the interest rate increases, the transaction generates a profit by selling at 90 a debt valued lower in the market. The main features of futures markets4 are: • • • •

Standard contracts. Supply and demand centralized. All counterparties face the same clearinghouse. The clearinghouse allocates the gains and losses among counterparties and hedges them with the collaterals (margin calls) provided by counterparties.

A typical contract necessitates the definition of the ‘notional’ underlying the contract. The notional is a fictitious bond, or Treasury bill, continuously revalued. Maturities extend over variable periods, with intermediate dates (quarterly for instance). Note that forward prices and futures should relate to each other since they provide alternative ways to provide the same outcome, allowing arbitrage to bring them in line for the same reference rate and the same delivery date5 .

Hedging with Futures The facility to buy or sell futures allows us to make a profit or loss depending on the interest rate movements. Adjusting this profit to compensate exactly for the initial exposure makes the contract a hedge. The adjustment involves a number of technicalities, such as defining the number of contracts that matches as closely as possible the amount to be hedged and limiting basis risk, the risk that the underlying of the contract does not track perfectly the underlying risk of the transactions to be hedged (interest rates are different for example). 4 See 5 See,

Hull (2000) for a review of the futures market. for example, French (1983) for a comparison of forward and futures prices.

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Since a contract consists of buying or selling a notional debt at a future date, the value of the debt changes with interest rates. These changes of values should match the changes of the transaction to hedge. For instance, for a borrower fearing interest rate increases, buying a contract at a preset price is a hedge because the value of the notional declines if interest rates increase. For a lender fearing that interest rates will decline, selling a contract is a hedge because the value of the notional increases if they do, providing a gain (Table 14.1). TABLE 14.1

Hedging interest rate transactions on the futures market Hedge againsta :

Increase of rates Lender buys a contract • Buy forward at 100 what should be valued more (110) if interest rate decreases • Hence, there is a gain if interest rate decreases a Lender

Decrease of rates Borrower sells a contract • Sell forward at 100 what should be valued less (90) if interest rate increases • Hence, there is a gain if interest rate increases

and borrower have variable rate debts.

CURRENCY RISK As with interest rates, there are forward and optional hedges. Most of the notions apply, so that we simply highlight the analogies, without getting into any further detail.

The Mechanisms of Forward Exchange Rates The forward exchange rate is the exchange rate, for a couple of currencies, with market quote today for a given horizon. Forward rates serve to lock in an exchange rate for a future commercial transaction. Forward rates can be above or below spot exchange rates. There is a mechanical relationship between spot and forward exchange rates. The mechanisms of the ‘forex’ (foreign exchange) and the interest rates markets of two countries create these relationships between the spot and forward exchange rates, according to the interest rate differential between currencies. The only major difference with interest rate contracts is that there is a market of forward exchange rates, resulting from the interaction of interest rate differentials differing from the mechanisms of forward interest rates (Figure 14.3).

Hedging Instruments Forward rates lock in the exchange rate of a future flow. Swaps exchange one currency for another. This is equivalent to changing the currency of debt or investment into another one. Options allow us to take advantage of the upside together with protection against downside risk. In the case of currencies, the receiver of a foreign currency flow will buy a put option to sell this foreign currency against the local currency at a given strike price. Being long

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189

Lend at 4%

1 USD

−1.04 USD

+1.04 USD

Convert USD in FRF spot

5 FRF

FIGURE 14.3

+

Convert 5.25 FRF in USD forward (0.198 FRF per USD)

Lend at 5%

5.25 FRF

Spot, forward exchange rates and interest rates

in a currency means receiving the foreign currency, while being short means having to pay in a foreign currency. Hedging with derivatives is straightforward. For instance, X is long 200 million USD. A decline of USD is possible. The spot rate is 0.95 EUR/USD. The forward rate is 0.88 EUR/USD. The alternative choices are: • Sell forward at 0.88 EUR/USD. This generates an opportunity cost if the USD moves up rather than down because the sale remains at 0.88. • Buy a put (sale option) with a premium of 2% at strike 6.27 EUR/USD. The put allows the buyer to benefit from favourable upward movements of USD without suffering from adverse downside moves. On the other hand, it costs more than a forward hedge (not considering the opportunity cost of the latter). Embedded in the put are break-even values of the exchange rate. In order to make a gain, the buyer of the put needs to make a gain higher or equal to the premium paid. The premium is 2%, so the USD has to move by 2% or more to compensate the cost. Otherwise, X is better off without the put than with it. As usual, X needs a view on risk before entering into a hedge. There are plenty of variations around these plain vanilla transactions. We leave it to the reader to examine them.

SECTION 6 Asset–Liability Management Models

15 Overview of ALM Models

Asset–Liability Management (ALM) decisions extend to hedging, or off-balance sheet policies, and to business, or on-balance sheet policies. ALM simulations model the behaviour of the balance sheet under various interest rate scenarios to obtain the risk and the expected values of the target variables, interest income or the mark-to-market value of the balance sheet at market rates. Without simulations, the ALM Committee (ALCO) would not have a full visibility on future profitability and risk, making it difficult to set up guidelines for business policy and hedging programmes. ALM simulation scope extends to both interest rate risk and business risk. ALM policies are medium-term, which implies considering the banking portfolio changes due to new business over at least 2 to 3 years. Simulations have various outputs: • They provide projected values of target variables for all scenarios. • They measure the exposure of the bank to both interest rate and business risk. • They serve to ‘optimize’ the risk and return trade-off, measured by the expected values and the distributions of the target variables across scenarios. Simulation techniques allow us to explore all relevant combinations of interest rate scenarios, business scenarios and alternative hedging policies, to find those which enhance the risk–return profile of the banking portfolio. Three chapters detail the simulation methodology applied to ALM. The first chapter (Chapter 17) explains the specifics of typical exposures to interest rate risk of commercial banks. The second chapter (Chapter 18) explains how to derive risk and expected profitability profiles from simulations, ignoring business risk. A simplified example illustrates the entire process. The third chapter (Chapter 19) extends the approach to business risk,

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building on the same example. The examples use the interest income as target variable, but they apply to the balance sheet mark-to-market value (‘NPV’) as well. These chapters ignore optional risk, dealt with subsequently. This chapter provides an overview of the process. The first section describes the various steps of simulations. The second section discusses the definition of interest rate scenarios, which are critical inputs to the simulation process. The third section discusses the joint view on interest rate risk and business risk. The fourth section explains how ALM simulations help enhance the risk–return profile of the bank’s portfolio, by combining interest rate scenarios, business scenarios and alternative hedging policies. Finally, the last section briefly describes the inputs for conducting ALM simulations.

OVERVIEW OF ALM SIMULATIONS Simulations serve to construct the risk–return profile of the banking portfolio. Therefore, full-blown ALM simulations proceed through several steps: • Select the target variables, interest income and the balance sheet NPV. • Define interest rate scenarios. • Build business projections of the future balance sheets. The process either uses one single base business scenario, or extends to several business scenarios. For each business scenario, the goal is to project the balance sheet structure at different time points. • Project margins and net income, or the balance sheet NPV, given interest rates and balance sheet scenarios. The process necessitates calculations of interest revenues and costs, with all detailed information available on transactions. • When considering optional risk, include valuing options using more comprehensive interest rate scenarios than for ‘direct’ interest rate risk. Simulations consider optional risks by making interest income, or balance sheet NPV, calculations dependent on the time path of interest rates. • Combine all steps with hedging scenarios to explore the entire set of feasible risk and return combinations. • Jointly select the best business and hedging scenarios according to the risk and return goals of the ALCO. Once the risk–return combinations are generated by multiple simulations, it becomes feasible to examine which are the best hedging policies, according to whether or not they enhance the risk–return profile, and whether they fit the management goals. The last step requires an adequate technique for handling the large number of simulations resulting from this process. A simple and efficient technique consists of building up through simulations the expected profitability and its volatility, for each hedging policy, under business risk, in order to identify the best solutions. The same technique applies to both interest income and NPV as target variables. Figure 15.1 summarizes the process.

OVERVIEW OF ALM MODELS

195

Business scenarios

Interest rate scenarios

Target Target variables: variables: Interest Interest income & income & NPV NPV

Simulations Simulations

Target variable (Income & NPV) distributions

Current value

Loss

Funding and hedging scenarios Feedback loop for optimizing hedging

FIGURE 15.1

Risk Return

Overview of the ALM simulation process

Multiple simulations accommodate a large number of scenarios. Manipulating the large number of scenarios requires a methodology. This chapter and the next develop such methodology gradually using examples. The examples are simplified. Nevertheless, they are representative of real world implementations. The latter involve a much larger amount of information, but the methodology is similar. The technical tools for exploring potential outcomes range from simple gaps to fullblown simulations. Gaps are attractive but have drawbacks depending on uncertainties with respect to the volumes of future assets and liabilities, which relate to business risks. There are as many gaps as there are multiple balance sheet scenarios. Gaps do not capture the risk from options embedded in banking products. Optional risks make asset liabilities sensitive to interest rates in some ranges of values and insensitive in others. Simulations correct the limitations of the simple gap model in various ways. First, they extend to various business scenarios if needed, allowing gaps to depend on the volume uncertainties of future assets and liabilities. Second, they calculate directly the target variable values, without necessarily relying on gaps only to derive new values of margins when interest rates change. This is feasible by embedding in the calculation any specific condition that influences the interest revenues and costs. Such conditions apply to options, when it is necessary to make rates dependent on the possibility of exercising prepayment options. ALM simulations focus on selected target variables. They include the interest margins over several periods, the net income, the mark-to-market value of the balance sheet (NPV). For a long-term view, the NPV is the best target variable. Medium to long-term horizons make the numerous periodical interest margins more difficult to handle. Because the NPV summarizes the entire stream of these interest incomes over future years (see Chapter 21), it provides an elegant and simple way to handle all future margins. Since both short and long-term views are valuable, it makes sense to use interest income as a target variable for

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the near future, and the NPV for the long-term view. Technically, this implies conducting both calculations, which ALM models excel in doing.

INTEREST RATE SCENARIOS AND ALM POLICIES Interest rate scenarios are a critical input for ALM simulations since they aim to model the interest income or the balance sheet NPV. The discussion on interest rate scenarios is expanded further in subsequent chapters, through examples. The current section is an overview only. ALM simulations might use only a small number of yield curve scenarios, based on judgmental views of future rates, or consider a large number of values to have a comprehensive view of all possible outcomes. The definitions of interest rate scenarios depend on the purpose. Typical ALM simulations aim to model the values of the target variables and discuss the results, the underlying assumptions and the views behind interest rate scenarios. This does not necessitate full-blown simulation of rates. On the other hand, valuing interest rate options or finding full distributions of target variable values requires generating a large number of scenarios. Discrete scenarios imply picking various yield curve scenarios that are judgment based. The major benefit of this method is that there are few scenarios to explore, making it easier to understand what happens when one or another materializes. The drawback is that the method does not provide a comprehensive coverage of future outcomes. Increasing the number of scenarios is the natural remedy to this incomplete coverage. Multiple simulations address valuation of options, which requires modelling the entire time path of interest rates. Options embed a hidden risk that does not materialize until there are large variations of interest rates. Valuing implicit options necessitates exploring a large spectrum of interest rate variations. Models help to generate a comprehensive array of scenarios. Chapter 20 illustrates the valuation process of implicit options using a simple ‘binomial’ model of interest rates. Finding an ALM ‘Value at Risk’ (VaR), defined as a downside variation of the balance sheet NPV at a preset confidence level, also requires a large number of interest rate values for all maturities, in order to define the downside variation percentiles. When considering changes of interest rates, it is also important to consider shifts over all maturities and changes of slopes. ‘Parallel shifts’ assume that the yield curve retains the same shape but its level varies. Parallel shifts are a common way to measure sensitivity to interest rate variations. Changing slopes of the yield curve drive the spread between long and short rates, a spread that is critical in commercial banking, since banks borrow short to lend long. During the recent history of rates in Euroland, up to the date when the euro substituted all currencies in the financial industry, the yield curve shifted downwards and its slope flattened. These moves adversely affected the interest income of positive gap banks, because it embeds the spread between the long and short rates. Interest rate modelling raises a number of issues, ranging from model specifications to practical implementations. First-generation ALM models used a small number of interest rate scenarios to project the ALM target variables. New models integrate interest rate models, allowing us to perform a full valuation of all possible outcomes, while complying with the consistency of rates across the term structure and the observed rate volatilities. Interest rate models serve to provide common interest rate scenarios for ALM simulations,

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197

for the valuation of implicit options and consistent with market risk models used for calculating market risk VaR1 . Models help because they make the entire structure of interest rates a function of one or several factors, plus a random error term measuring the discrepancy between modelled values and actual values. The high sensitivity to both the level and slope of the yield curve of the interest margin rules out some models that tend to model rates based on the current curve2 . In many instances, as indicated above, the shape of the yield curve changes significantly. Some models help by generating a large number of yield curve scenarios without preserving the current yield curve structure3 . On the other hand, they do not allow us to use yield curve scenarios that match the management views on interest rate, which is a drawback for ALCO discussions of future outcomes. In addition, business risk makes the volumes of assets and liabilities uncertain, which makes it less important to focus excessively on interest rate uncertainty. For such reasons, traditional ALM models use judgmental yield curve scenarios as inputs rather than full-blown simulations of interest rates. Models excel for valuing implicit options, and for exploring a wide array of interest outcomes, for instance when calculating an ALM VaR. Historical simulations use interest data from observed time series to draw scenarios from these observations. Models allow the generation of random values of interest rates for all maturities, mimicking the actual behaviour of interest rates. Correlation matrices allow the associations between rates of different maturities to be captured. Principal component analysis allows modelling of the major changes of the yield curves: parallel shifts, slopes and bumps. Chapter 30 provides further details on Monte Carlo simulations of rates. Chapters 20 and 21 show how models help in valuing implicit options. In what follows, we stick to discrete interest rate scenarios, which is a common practice.

JOINT VIEWS ON INTEREST RATES AND BUSINESS UNCERTAINTIES Restricting the scope to pure ‘financial’ policies leads to modelling the risk–return profile of the banking portfolio under a given business policy, with one base scenario for the banking portfolio. Simulations allow us to explore the entire set of risk–return combinations under this single business scenario. They show all feasible combinations making up the risk–return profile of the banking portfolio without hedging, or without modifying the existing hedging programme. Obtaining this risk–return profile is a prerequisite for defining hedging policies. When considering a single business scenario, and a single period, the interest margin is a linear function of the corresponding gap if there are no options. For a single gap value, the interest income varies linearly with the interest rates. The distribution of net income values provides a view on its risk, while its expected value across all interest rate scenarios is the expected profitability. Modifying the gap through derivatives changes the interest 1 Market

risk VaR models need to simulate interest rates, but it is not practical to use full-blown models with all interest rates. The principal component technique, using a factor model of the yield curve, is easier to handle (Chapter 30). The same technique would apply to ALM models. 2 An example is the Ho and Lee (1986) model. 3 See Hull (2000) for a review of all interest rate models.

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income distribution, as well as its expected profitability. Each gap value corresponds to a ‘risk plus expected return’ combination. It becomes possible to characterize the risk with simple measures, such as the volatility of the target variable, or its downside deviation at a preset confidence level. This converts the distribution of target variable values resulting from interest rate uncertainty into a simple risk–return pair of values. Varying the gap allows us to see how the risk–return combination moves within this familiar space. The last step consists in selecting the subset of ‘best’ combinations and retaining the corresponding hedging solution (Figure 15.2). A full example of this technique is developed in Chapter 17, dealing with business risk. Distribution of forward values of margins or NPV

Expected value Current value

FIGURE 15.2

Loss

Risk−return space Expected value

Future time point

Risk

Forward valuation of ALM target variables

When the gap becomes a poor measure of exposure because there is optional risk, simulations require calculating the interest income for various interest rate scenarios, without using gaps. The calculation requires assumptions with respect to the behaviour of borrowers in exercising their right to reset their borrowing rate. The ALCO might find none of these risk–return profiles acceptable, because of the risk–return trade-off. Hedging risk might require sacrificing too much expected profitability, or the target profitability might require taking on too much risk, or simply remain beyond reach when looking at a single business scenario and altering risk and return through hedging policies only. The ALCO might wish to revise the business policy in such cases. Extending the simulation process to business scenarios has a twofold motivation: • Addressing ALCO demand for exploring various business scenarios, and allowing recommendations to be made for on-balance sheet policies as well as for off-balance sheet policies. Under this view, ALCO requires additional ‘what if’ simulations for making joint on and off-balance sheet decisions. • Addressing the interaction between business risk and interest rate risk by exploring business uncertainties with several business scenarios. Under this second view, the goal is to find the optimal hedging solutions given two sources of uncertainties, interest rate and business.

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199

ALM simulations address business risk through projections of the banking portfolio. This is a specific feature of ALM, compared to similar techniques applied to market risk and credit risk. When dealing with market or credit risk, simulations apply to a ‘crystallized’ portfolio, as of current date, ignoring medium-term business scenarios. Because of the ALCO willingness to explore various projections, and because of business risk in the medium and long-term, it is necessary to extend simulations to explicit projections of the banking portfolio over 2 to 3 years. Scenarios differ because they consider various new business assumptions and various product–market mixes. Since there are many business scenarios, there are many distributions of the target variables. Exposure uncertainty is another motive for multiple balance sheet scenarios. Existing assets and liabilities have no contractual maturities making gaps undefined. New business originates new assets and liabilities, whose volume is uncertain. Dealing with a single projection of the balance sheet for future periods ‘hides’ such uncertainties. A simple way to make them explicit consists of defining various scenarios. Some of them differ because they make different assumptions, for example to deal with lines without maturity.

ENHANCING AND OPTIMIZING RISK–RETURN PROFILES When the ALCO considers alternative business policies, ALM simulations address both on and off-balance sheet management. The goal becomes the joint optimization of hedging policies and on-balance sheet actions. This requires combining interest rate scenarios with several balance sheet scenarios and hedging solutions. The methodology for tackling the resulting increase in the number of simulations as illustrated in Chapter 18. For each pair of business and interest rate scenarios, ALM models determine the value of the target profitability variables. When the interest rate only varies, there is one distribution of values, with an expected value and the dispersion around this average. When there are several balance sheet projections, there are several distributions. This addresses the first goal of conducting ‘what if’ simulations to see what happens and make sure that the profitability and the risk remain in line with goals. However, it is not sufficient to derive a hedging policy. This issue requires testing various hedging policies, for instance various values of gaps at different time points, to see how they alter the distribution of the target variable. The process generates new distributions of values of the target variable, each one representing the risk–return profile of the balance sheet given the hedging solution. Each hedging solution value corresponds to a ‘risk plus expected return’ combination. The last step consists in selecting the subset of ‘best’ combinations and retaining the corresponding hedging solutions. The technique accommodates various measures of risks. Since we deal with distributions of values of the target variables at various time points, the volatility or the percentiles of downside deviations of target variables quantify the risk. This allows us to determine an ALM VaR as a maximum downside deviation of the target variable at a preset confidence level. Changing the hedging policy allows us to move in the ‘risk–return space’, risk being volatility or downside deviations and return being the expected value of interest income or the NPV.

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SIMULATIONS AND INFORMATION Actual projections of the balance sheet require a large set of information. In order to project margins, assumptions are also required on the interest rate sensitivity of transactions and the reference rates. The availability of such information is a critical factor to build an ALM information system. Table 15.1 summarizes the main information types, those relating to the volume of transactions and rates, plus the product–market information necessary to move back and forth from the purely financial view to the business view of future outcomes. TABLE 15.1

Information by transaction

Transaction

Type of transaction and product

Volume

Initial outstanding balance Amortization schedule Renewal and expected usage Optionsa Currency

Rates

Reference rate Margin, guaranteed historical rate, historical target rate Nature of rate (calculation mode) Interest rate profileb Sensitivity to reference rates Options characteristics

Customers

Customers, product–market segments

a Prepayment, b Frequency

etc. and reset dates.

The nature of the transactions is necessary to project exposure and margins. Amortizing loans and bullet loans do not generate the same gap profiles. Projected margins also depend on differentiated spreads across bank facilities. In addition, reporting should be capable of highlighting the contributions of various products or market segments to both gaps and margins. The information on rates is obviously critical for interest rate risk. Margin calculations require using all formulas to calculate effective rates (pre-determined, postdetermined, number of days, reset dates, options altering the nature of the rate, complex indexation formulas over several rates, and so on). The rate level is important for margin calculations and to assess the prepayment, or renegotiation, likelihood, since the gap with current rates commands the benefit of renegotiating the loan for mortgages. Sensitivities are required when balance sheet items map to a restricted set of selected reference rates. Options characteristics refer to strike price and the nature of the options for off-balance sheet hedging and embedded options in banking products, such as contractual caps on variable rates paid by borrowers.

16 Hedging Issues

Typically, banks behave as net lenders when they have positive variable rate gaps and as net borrowers when they have negative variable rate gaps. A net lender, like a lender, prefers interest rate increases and needs to hedge against declines. A net borrower, like a borrower, prefers interest rate declines and needs protection against increases. However, the sensitivity to variations of rates is not sufficient to characterize the exposure of a ‘typical’ commercial bank and to decide which hedging policy is the best. A ‘typical’ commercial bank pays short-term rates, or zero rates, on deposits and receive longer-term rates from loans. An upward sloping curve allows these banks to earn more from assets than they pay for liabilities, even if commercial margins are zero. Commercial banks capture a positive maturity spread. Simultaneously, the interest income is sensitive to both shifts and steepness of the yield curve. This variable gap depends on the nature of the rate paid on the core deposit base. If the rates are zero or fixed (regulated rates), the variable rate gap tends to be positive. If the rate is a short-term rate, the variable rate gap tends to become negative. In both cases, banks still capture the positive maturity spread, but the sensitivity to parallel shifts of the curve changes. The joint view of variable rate gap plus the gap between liability average rates and asset average rates characterizes the sensitivity of interest income of banks to shifts and slope changes of the yield curve. Deposits blur the image of liquidity and interest rate gaps because they have no maturity. Core deposits remain for a long period, earn either a short-term or a zero rate and tend to narrow the liquidity gap. Zero-rate deposits widen the variable rate gap, and interest-earning deposits narrow it. Such an economic view contrasts with the legal (zero maturity) or conventional (amortizing deposits) views. Net lender banks, with a large deposit base, are adversely exposed to declining rates, and vice versa. High long-term rates increase income. They are also adversely exposed to a decline in steepness of the yield curve. When rates decline, net lenders face the dilemma

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of waiting versus hedging. Early hedges protect them while late hedges are inefficient because they lock the low rates into interest income. Moreover, banks with excess funds face arbitrage opportunities between riding the spot or the forward yield curves, dependent on both the level and slope of yield curves. For such reasons, hedging policies are highly dependent on interest rate ‘views’ and goals. This chapter addresses these common dilemmas of traditional commercial banks. The first section details the joint view on interest rate gaps and yield curve, with a joint sensitivity to shifts of yield curve level and steepness. The second section shows how conventions on deposits alter the economic view of the bank’s exposure. The third section details hedging issues.

INTEREST RATE EXPOSURE The sensitivity to parallel shifts of the yield curve depends on the sign of the variable interest rate gap, or the gap between the average reset period of assets and that of liabilities. With a positive variable rate gap, banks behave as net lenders. With a negative variable rate gap, banks behave as net borrowers. Variable rate gaps increase with the gap between the asset reset period and the liability reset period. With more variable rate assets than variable rate liabilities, the reset period of assets becomes shorter than the average reset period of liabilities. With more variable rate liabilities than variable rate assets, liability resets are faster than asset resets. The relation between interest rate gaps and average reset dates is not mechanical. We assume, in what follows, that a positive variable rate gap implies a negative gap between the asset average reset date and the liability average reset date, and conversely. The difference between the two views is that the gap between average reset dates positions the average rates of assets and liabilities along the yield curve. The average dates indicate whether the bank captures a positive or negative yield curve spread, depending on the slope of the yield curve and how sensitive it is to this slope. Hence, the two views of gaps, variable rate gap and average reset dates gap, complement each other. The first refers to parallel shifts of the curve, while the second relates to its steepness. Figure 16.1 shows the two cases of positive and negative variable rate gaps. If variable rate gaps and reset date gaps have the same sign, the posture of the bank is as follows: • With a positive variable rate gap, the assets average reset period is shorter than the liabilities average reset period. The bank borrows long and lends short. With an upward sloping yield curve, the interest income of the bank suffers from the negative market spread between short and long rates. • With a negative variable rate gap, the liabilities average reset period is shorter than the assets average reset period. The bank borrows short and lends long. With an upward sloping yield curve, the interest income of the bank benefits from the positive market spread between short and long rates. The sensitivity to a change in slope of the yield curve depends on the gap between average reset periods of assets and liabilities. The hedging policy results from such views of the bank’s position. Banks with positive variable rate gaps have increasing interest income with increasing rates and behave as net

HEDGING ISSUES

FRA

203

Positive Positive variable variable rate gap rate gap

FRL

FRA

Negative Negative variable rate gap variable rate gap

FRL VRL

VRA VRA VRL

VRA VRL

VRA VRL

VR Gap > 0 i A

VR Gap < 0

IM

i L

L

IM

A

Average reset period of assets Average reset period assets Average reset period of of liabilities Average reset period of liabilities

FIGURE 16.1 liabilities

Interest rate gaps and gaps between average reset periods of assets and

Forward Yield Curve

Spot Yield Curve

A

FIGURE 16.2

L

Maturity

Interest Rate

Interest Rate

lenders. If they expect rates to rise, they benefit from maintaining their gaps open. If not, they should close their gaps. Banks with negative variable rates have increasing interest income with decreasing rates and behave as net borrowers. The average reset period of assets is longer than the average reset period of liabilities. If they expect rates to rise, they benefit from closing their gaps. If not, they should open their gaps. Rules that are more specific follow when considering the forward rates. A bank behaving as a net lender benefits from higher rates. Therefore, if it does not hedge fully its exposures, views on rates drive the hedging policies (Figure 16.2):

Spot Yield Curve Forward Yield Curve A

L

Maturity

Bank as net lender: asset reset period is shorter than liability reset period

• With an upward sloping yield curve, forward rates are above spot rates. The bank needs to lock in the current forward rates for future investments if it believes that future spot rates will be below the current forward rates, since these are the break-even rates.

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• With an inverted yield curve, forward rates are below spot rates. The bank profits from forward rates for future investments only if it believes that the future spot rates will decrease to a level lower than the current forward rates. Obviously, if banks expect short rates to be equal to future spot rates are equal to forward rates, they become indifferent since locking in a forward rate or not doing anything are, by definition, equivalent. When dealing with banks having large deposit bases, the situation differs. Essentially, this is because core deposits are long-term sources of funds earning a short-term rate or a zero rate. The discrepancy between the legal maturity of deposits and their effective maturity blurs the image of the bank’s exposure to interest rates.

THE ‘NATURAL EXPOSURE’ OF A COMMERCIAL BANK The ‘natural liquidity exposure’ of a bank results from the maturity gap between assets and liabilities because banks transform maturities. They collect short and lend longer. However, the core demand deposit base is stable. Hence, overfunded banks tend to have a large deposit base. Underfunded banks tend to have a small deposit base. Since deposits have no maturity, conventions alter the liquidity image. Deposits are stable and earn either a small fixed rate or a short-term rate. Therefore, their rate does not match their effective long maturity. This section discusses the ‘natural exposure’ of commercial banks having a large deposit base and draws implications for their hedging policies.

Liquidity

Volume

Core deposits

Volume

Volume

The views on the liquidity posture depend on the conventions used to determine liquidity gaps. Figure 16.3 shows three different views of the liquidity profile of a bank that has a large deposit base. Core deposits have different economic, conventional and legal maturities. Conventional rules allow for progressive amortization, even though deposits do not amortize. The legal maturity is short-term, as if deposits amortized immediately. This is also unrealistic. The economic maturity of core deposits is long by definition. This is the only valid economic view and we stick to it. Note that this economic view is the

Assets

Assets Deposits

Assets

Deposits Time Economic View

FIGURE 16.3

Time Conventional View

Liquidity views of a commercial bank

Time Legal View

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205

Liabilities + core deposits

Assets Time Overfunded

FIGURE 16.4

Assets

Time Underfunded

Volume

Liabilities + core deposits

Volume

Volume

reverse of the view that banks transform maturities by lending long and collecting short resources. Figure 16.4 shows two possible economic views: an overfunded bank that collects excess deposits and liabilities; an underfunded bank collecting fewer deposits and liabilities than it lends. The situation is either stable through time or reverses after a while. We assume that the total liabilities decrease progressively and slowly because there are some amortizing debts in addition to the large stable deposit base.

Assets

Liabilities + core deposits Time Mixed: underfunded then overfunded

Underfunded versus overfunded commercial bank

Banks with a large core deposit base have excess funds immediately or become progressively overfunded when assets amortize faster than the core deposit base. Banks with a small deposit base are underfunded, at least in the near term, and might become overfunded if assets amortize faster. The interest rate view refers to how the bank captures the maturity spread of interest rates and to its sensitivity to interest rate changes.

Interest Rate Exposure The exposure to interest rate shifts depends on the variable rate gap or, equivalently, on the gap between the average reset period of assets and the average reset period of liabilities. The larger the variable rate gap, the higher the favourable sensitivity to a parallel upward shift of the yield curve. Stable deposits do not earn long-term interest rates because their legal maturity is zero, or remains short for term deposits. Many demand deposits have a zero rate. Regulated savings accounts earn a fixed regulated rate in some countries. Demand deposits, subject to certain constraints, earn a short-term rate or a lower rate. In all cases, the stable deposit base has a short-term rate, a fixed rate close to short-term rates or a zero rate. In terms of the yield curve, they behave as if they were on the short end of the curve, rather than on the long end corresponding to their effective maturity. Hence, banks earn the market spread between the zero rate and the average maturity rate of assets, or the spread between the short-term rate of demand deposits and the average maturity rate of assets. For an upward sloping curve, the maturity spread contributes positively to the interest income, and even more so for zero-rate demand deposits.

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However, the interest rate gap, or the gap between reset dates of deposits and assets, depends on the rate of deposits. When the rate is a short-term rate, deposits are ratesensitive liabilities. When the rate is fixed or zero, they are rate-insensitive liabilities. Given their mass in the balance sheet, this changes the interest rate gap of the bank. Banks with zero-rate (or fixed-rate) deposits tend to have a positive variable rate gap, while banks with interest rate-sensitive deposits tend to have a negative variable rate gap. The zero-rate, or fixed-rate, deposits make banks favourably sensitive to a parallel upward shift of the yield curve, while the interest rate-sensitive deposits make banks adversely sensitive to the same parallel upward shift of the yield curve. In both cases, banks benefit from the positive maturity spread of an upward sloping curve (Figure 16.5). FRA

FRL

Deposit base: Deposit zero ratebase at zero rate

VRA

VRA VRL

FRA

FRL

Deposit base: Deposit base: short-term rate short-term rate VRA VRL

VRL VRA

VRL

VR Gap > 0 i L

FIGURE 16.5

A

VR Gap < 0

IM

i L

IM

A

Interest rate exposure of commercial banks

HEDGING POLICIES This section makes explicit some general rules driving the hedging policies. Then, we discuss the case of banks behaving as net lenders in continental Europe during the period of declining rates. This is a period of continuous downward shifts of upward sloping yield curves. For several players, it was a period of missed opportunities to take the right exposures and of failures to timely hedge exposures to unfavourable market movements. The context also illustrates why the trade-off between profitability and risks is not the same when rates are low and when rates are high. In times of low rates, hedging locks in a low profitability and increases the opportunity cost of giving up a possible upward move of rates. In times of high rates, the issue might shift more towards reducing the volatility of the margin rather than the level of the profitability.

General Principles Banks with a large zero-rate deposit base are net lenders, have positive variable rate gaps, and their interest income improves with upward shifts of the yield curve and with an

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increasing slope of the yield curve. Banks with a large short-term-rate deposit base are also net lenders but have negative variable rate gaps. Their interest income deteriorates with upward shifts of the yield curve and improves with an increasing slope of the yield curve. We now focus only on ‘net lender’ banks for simplicity. With the upward sloping yield curves of European countries in the nineties, forward rates were above spot rates. Banks behaving as net lenders faced a progressive decline of the yield curve. They needed to lock in the current forward rates for future investments only if they believed that future spot rates would be below current forward rates. If they expected rates to rise again, the issue would be by how much, since maintaining exposure implies betting on a rise beyond the current forward rates.

Implication for On-balance Sheet Hedging A net lender bank with a large base of zero-rate deposits tends to have a positive variable rate gap, a negative liquidity gap (excess of funds) and an average reset date of assets shorter than that of liabilities, but they benefit from a positive slope of the yield curve: • In times of increasing rates, they benefit both from the positive variable rate gap and the asset rate higher than liabilities. • In times of declining rates, they tend to close the gap, implying looking for fixed rates, lending longer or swapping the received variable rate of assets against the fixed rate. Note that declining rates right be an opportunity to embed higher margins over market rates in the price of the variable rate loans. In this case, variable rate loans would not reduce the bank’s risk but would improve its profitability. Upward sloping curves theoretically imply that future interest rates will increase to the level of forward rates and that these are the predictors of future rates because of arbitrage. In practice, this is not necessarily true. The euro term structure remained upward sloping for several years while continuously moving down for years in the late nineties, until short-term rates hit repeatedly historical lows. Rates declined while the yield curve got flatter. Going early towards fixed rates on the asset side was beneficial. After the decline, such hedges became useless because banks could only lock in low rates.

Hedging and the Level of Interest Rates Hedging policies typically vary when interest rates are low and when they are high. Starting with high interest rates, commercial banks faced a continuous decline of the yield curve, and finally the bottom level, before new hikes in interest levels began to take place in the year 2000. With high interest rates, interest rate volatility might be higher. This is when hedging policies are most effective. Derivatives reduce the volatility by closing gaps. There are timing issues, since locking fixed rates in revenues is a bet that rates will not keep increasing, otherwise the bank with favourable exposures to interest rate increases would suffer an opportunity cost. Before reaching low levels, rates decline. Banks acting as net lenders prefer to receive more fixed rates when rates decline and pay variable rates on the liabilities side. Both on-balance sheet and off-balance sheet actions become difficult:

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• Banks are reluctant to hedge their current exposures because that would lock in their revenues at the current rates, for instance through swaps, when there is still a chance of hitting a turning point beyond which rates could increase again. This temptation postpones the willingness to set up hedges in time. • Simultaneously, customers might be reluctant to accept long fixed rates in line with the profitability targets of the bank if they expect rates to keep declining. Once rates get low, not having set up hedges on time makes them useless later, shifting the pressure from off-balance sheet actions to on-balance sheet actions again. The likelihood of a rise increases, but crystallizing the low rates in the revenues does not make much sense, because the profitability reaches such a low level that it might not suffice any more to absorb the operating costs. There is a break-even value of the interest revenue such that profits net of operating costs hit zero and turn into losses if revenues keep falling. Late forward hedges do not help. Optional hedges are too costly. The European situation illustrates well this dilemma. Many banks waited before setting up hedges because of a fear of missing the opportunity of rates bouncing back up. After a while, they were reluctant to set up forward hedges because low interest revenues from hedges were not economical. Moreover, institutions collecting regulated deposits pay a regulated rate to customers. This regulated rate remains fixed until regulations change. This makes it uneconomical to collect regulated resources, while depositors find them more attractive because of low short-term rates. The decline in rates puts pressure on deregulating these rates. Because of this pressure, regulated rates become subject to changes, at infrequent intervals, using a formula linking them to market rates.

Riding the Spot and Forward Yield Curves Riding an upward sloping yield curve offers the benefit of capturing the positive maturity spread, at the cost of interest rate risk. Riding the forward yield curve, with an upward sloping yield curve, offers the benefit of capturing the positive spread between the forward and spot yields. Steeper slopes imply higher spreads between long-term and short-term interest rates. The spot yield curve is not necessarily the best bet for institutions having recurring future excess funds. The difference between forward and spot rates is higher when the slope is steeper. This is an opportunity for forward investments to lock in rates significantly higher than the spot rates. There are ‘windows of opportunity’ for riding the forward yield curve, rather than the spot yield curve. A case in point is the transformation of the yield curve in European countries in the late nineties. The slope of the spot yield curve became steep, before decreasing when approaching the euro. When the slope was very steep, there was an opportunity to shift from lending to investing forward. Banks that faced declining interest rates for lending did not hedge their gaps if they bet that rates were close to the lowest point and would start rising again. An alternative policy to lending was to invest future excess funds at forward rates when the yield curve slope was very steep. These rates ended much higher than the spot rates prevailing later at the forward investment dates, because the yield curves kept shifting downward. Riding

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Invest forward 1 year

Rates

Forward yield curve

Invest spot 2 years Spot yield curve Invest spot 1 year Time 1 year

FIGURE 16.6

2 years

Comparing spot and forward investments

the forward yield curve was more beneficial than riding the spot yield curve. Figure 16.6 summarizes this policy. Of course, when interest rates reached the bottom line in late 1999 and early 2000, the yield curve had become lower and flatter. The spread between forward and spot rates narrowed. Consequently, the ‘window of opportunity’ for investing in forward rates higher than subsequent spot rates closed.

17 ALM Simulations

ALM simulations focus on the risk and expected return trade-off under interest rate risk. They also extend to business risk because business uncertainties alter the gap profile of the bank. Return designates the expected value of the target variables, interest income or NPV, and their distribution across scenarios, while volatility of target variables across scenarios characterizes risk. This chapter focuses on interest income. Simulations project the balance sheet at future horizons and derive the interest income under different interest rate scenarios. When the bank modifies the hedge, it trades off risk and expected profitability, eventually setting the gap to zero and locking in a specific interest income value. For each gap value, the expected interest income and risk result from the set of income values across scenarios. Altering the gap generates various combinations. Not all of them are attractive. Some combinations dominate others because they provide a higher expected income at the same risk or have a lower risk for the same value of expected income. These are the ‘efficient’ combinations. Choosing one solution within the efficient set, and targeting a particular gap value, becomes a management decision depending on the willingness to take risk. Simulations with simple examples illustrate these mechanisms. The next chapter extends the scope to business risk. Simple examples involve only one variable rate, as they all vary together. In practice, there are several interest rate scenarios. Multiple variablerate exposures generate complexity and diversification effects because not all rates vary simultaneously in the same direction. EaR (‘Earnings at Risk’), in this case ‘interest income at risk’, results from the interest rate distribution given gap and multiple interest rate scenarios. The management can set an upper bound to EaR, for limiting the interest income volatility or imposing a floor to a potential decline. Such gap limits alter the expected interest income. The first section details balance sheet projections, all necessary inputs, and the projected gaps. The second section details the direct calculations of interest income under different

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interest rate scenarios. The third section makes explicit the risk–return profile of the balance sheet given interest rate uncertainty and exposure. It also introduces the case of multiple variable-rate gaps. The fourth section shows how controlling the exposure through hedging modifies this profile and implies a trade-off between expected interest income and uncertainty. The last section discusses the hedging policy, from ‘immunization’ of the interest income to taking exposures depending on the bank’s views on interest and setting risk limits.

BALANCE SHEET AND GAP PROJECTIONS Projections of balance sheets are necessary to project liquidity and interest rate gap profiles and the values of the target variables. Business projections result from the business policy of the bank. Projections include both existing assets and liabilities and new business.

Projections of the Existing Portfolio and of New Transactions The projection of all existing assets and liabilities determines the liquidity gap time profile. The projection of interest rate gaps requires breaking down assets and liabilities into ratesensitive items and fixed-rate items, depending on the horizon and starting as of today. The existing portfolio determines the static gaps. The new business increases the volume of the balance sheet while existing assets and liabilities amortize. The new assets and liabilities amortize as soon as they enter the balance sheet. Projections should be net of any amortization of new assets and liabilities. Projecting new business serves to obtain the total volume of funding or excess funds for each future period, as well as the volume of fixed-rate and variable-rate balance sheet items. Projections include both commercial assets and liabilities and financial ones, such as debt and equity. Equity includes retained earnings derived from the projected income statements.

Interest Rate Sensitivity of New and Existing Assets and Liabilities Interest-sensitive assets and liabilities are those whose rate is reset during each subperiod. Real world projections typically use monthly periods up to 1 or 2 years, and less frequent ones after that. In our example, we consider only a 1-year period. An item is interest rate-sensitive if there is a rate reset before the end of the year. For new business, all items are interest rate-sensitive, no matter whether the expected rate is fixed or variable. The variable–fixed distinction is not relevant to determine which assets or liabilities are interest-sensitive. A future fixed-rate loan is interest-sensitive, even if its rate remains fixed for its entire life at origination, because origination occurs in the future. The future fixed rate will depend on prevailing market conditions. On the other hand, the fixed–variable distinction, and the breakdown of variable rates for existing assets and liabilities, are necessary to obtain fixed-rate gaps and as many variablerate gaps as distinct market references. Therefore, interest-sensitive assets and liabilities include variable-rate existing transactions plus all new transactions. Interest-insensitive items include only those existing transactions that have a fixed rate. For simplicity, all sensitivities to the single variable-rate reference are 100% for all balance sheet items in our example.

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Table 17.1 provides a sample set of data for a simplified balance sheet, projected at date 1, 1 year from now. There is no need to use the balance sheet at date 0 for gap calculations. All subsequent interest revenue and cost calculations use the end of year balance sheet. We assume there is no hedge contracted for the forthcoming year. TABLE 17.1 Balance sheet projections for the banking portfolio Dates Banking portfolio Interest rate-insensitive assets (a) Interest rate-sensitive assets (b) Total assets (c = a + b) Interest rate-insensitive resources (d) Interest rate-sensitive resources (e) Total liabilities (f = d + e)

1 19 17 36 15 9 24

Projected Gap Profiles The gaps result from the 1-year balance sheet projections. All gaps are algebraic differences between assets and liabilities. The liquidity gap shows a deficit of 12. The variablerate gap before funding is +8, but the deficit of 12 counts as a variable-rate liability as long as its rate is not locked in advance, so that the post-funding variable interest rate gap is −4 (Table 17.2). TABLE 17.2

Gap projections

Dates Banking portfolio Interest-insensitive assets (a) Interest-sensitive assets (b) Total assets (c = a + b) Interest-insensitive resources (d) Interest-sensitive resources (e) Total liabilities (f = d + e) Liquidity gap (c − f )a Variable interest rate gap (b − e) Total balance sheet Variable interest rate gap after funding (b − e) − (c − f )b Gaps Liquidity gapc Interest rate gapb

1 19 17 36 15 9 24 +12 +8 −4 +12 −4

a Liquidity gaps are as algebraic differences between assets and liabilities. The +12 value corresponds to a deficit. b Funding is assumed to be variable-rate before any hedging decision is made. c Interest rate gaps are interest-sensitive assets minus interest-sensitive liabilities, or ‘variable-rate’ interest rate gaps.

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INTEREST INCOME PROJECTIONS The interest margin is the target of financial policy in our example. The margins apply to different levels. The interest margin generated by the banking portfolio (operating assets and liabilities only) is the ‘commercial margin’. The interest income is after funding costs, inclusive of all interest revenues or costs from both operating and financial items. In order to calculate margins, we need to relate market rates to customers’ rates. The spread between the customers’ rates and the market rate is the percentage commercial margin. Such spreads are actually uncertain for the future. Assigning values to future percentage margins requires assumptions, or taking them as equal to the objectives of the commercial policy. Value margins are the product of volume and percentage margins. In this example, we use the market rate as an internal reference rate to calculate the commercial margins. Normally, the Funds Transfer Pricing (FTP) system defines the set of internal transfer prices (see Chapter 26).

Interest Rate Scenarios In our example, we stick to the simple case of only two yield curve scenarios. The example uses two scenarios for the flat yield curve, at the 8% and 11% levels (Table 17.3). TABLE 17.3

Interest rate scenarios

Scenarios 1 (stability) 2 (increase)

Rate (%) 8 11

Note: Flat term structure of interest rates.

Commercial and Accounting Margins The assumptions are simplified. The commercial margins are 3% for assets and −3% for liabilities. This means that the average customer rate for assets is 3% above the market rate, and that the customer rate for demand and term deposits is, on average, 3% less than the market rate. When the market rate is 8%, these rates are 11% and 5%1 . In practice, percentage margins differ according to the type of asset or liability, but the calculations will be identical. The commercial margin, before the funding cost, results from the customer rates and the outstanding balances of assets and liabilities in the banking portfolio. It is: 36 × 11% − 24 × 5% = 3.96 − 1.20 = 2.76 The accounting interest margin is after cost of funding. The cost of funding is the market rate2 . Since the yield curve is flat, the cost of funds does not depend on maturity in 1 These

figures are used to simplify the example. The actual margins obviously differ for various items of assets and liabilities. 2 Plus any credit spread that applies to the bank, assumed to be included in the rates scenarios.

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this example. The interest margin after funding is the commercial margin minus the cost of funding a deficit of 12. This cost is 12 × 8% = 0.96. The net accounting margin is therefore 2.76 − 0.96 = 1.80.

The Sensitivity of Margins In this example, we assume that the percentage commercial margins remain constant when the interest rate changes. This is not a very realistic assumption since there are many reasons to relate percentage margins to interest rate levels or competition. For instance, high rates might not be acceptable to customers, and imply percentage margin reductions. For all interest-insensitive items, the customers’ rates remain unchanged when market rates move. For interest-sensitive items, the customers’ rate variation is identical to the market rate variation because of the assumption of constant commercial margins. The value of the margins after the change in interest rate from 8% to 11% results from the new customers’ rates once the market rate rose. It should be consistent with the interest rate gap calculated previously. Detailed calculations are given in Table 17.4. TABLE 17.4

Margins and rate sensitivity

Interest-insensitive assets Interest-sensitive assets Revenues Interest-insensitive resources Interest-sensitive resources Costs Commercial margin Liquidity gap Net interest margin

Volume

Initial rate

Revenues/ costs

Final rate

Revenues/ costs

19 17

11% 11%

11% 14%

15 9

5% 5%

12

8%

2.09 1.87 3.96 0.75 0.45 1.20 2.76 0.96 1.80

2.09 2.38 4.47 0.75 0.72 1.47 3.00 1.32 1.68

5% 8% 11%

The values of the commercial margins, before and after the interest rate rise, are 2.76 and 3.00. This variation is consistent with the gap model. The change in commercial margin is 0.24 for a rate increase of 3%. According to the gap model, the change is also equal to the interest rate gap times the change in interest rate. The interest rate gap of the commercial portfolio is +8 and the variation of the margin is 8 × 3% = 0.24, in line with the direct calculation. The net interest margin, after funding costs, decreases by 1.80 − 1.68 = 0.12. This is because the funding cost of the liquidity gap is indexed to the market rate and increases by 12 × 3% = 0.36. The commercial margin increase of 0.24 minus the cost increase results in the −0.12 change. Alternatively, the interest gap after funding is that of the commercial portfolio alone minus the amount of funding, or +8 − 12 = −4. This gap, multiplied by 3%, also results in a −0.12 change in margin. This assumes that the funding cost is fully variable.

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THE RISK–RETURN PROFILE The gaps summarize the balance sheet image and provide a simple technique to derive all possible variations of interest margins. If we know the original interest margin, gaps and interest rate scenarios provide all information necessary to have the risk–return profile of each future time point. The risk–return profile of the banking portfolio is the image of all attainable combinations of risk and expected return given all possible outcomes. The process requires selecting a target variable, whose set of possible values serves to characterize risk. In the example of this chapter, the interest income is the target variable, but the same approach could use the Net Present Value (NPV) as well. The interest rate changes drive the margin at a given gap. The full distribution of the interest margin values when rates vary characterizes the risk–return profile. Making explicit this trade-off serves to set limits, such as maximum gap values, and assess the consequences on expected net income.

The Risk–Return Profile of the Portfolio given Gaps Characterizing the risk–return profile is straightforward using the basic relationship: IM = gap × i The margin at date t is random, IMt , the period is from 0 to t, and the relationship between the variation of the margin and that of interest rates is: IM = gap × i = gap × (it − i0 ) To project the interest margin, we need only to combine the original IM0 at date 0 with the above relations: IMt = IM0 + IM = IM0 + gap × i The expected variation of margin depends on the expected interest rate variation. The final interest rate is a random variable following a probability distribution whose mean and standard deviation are measurable. With a given gap value, the probability distribution of the margin results from that of the interest rate3 . The change of margin is a linear function of the interest rate change with a preset gap. The expected value of the interest margin at date t, E(IMt ), is the summation of the fixed original margin plus the expected value of its variation between 0 and t. The volatility of the final margin is simply that of the variation. They are: E(IM) = gap × E(i)

and E(IMt ) = IM0 + gap × E(i)

σ (IM) = |gap| × σ (i) and σ (IMt ) = |gap| × σ (i) The vertical bars stand for absolute value, since the volatility is always positive. Consequently, the maximum deviation of the margin at a preset confidence level results directly 3 With the usual notation, the expected rate is E(i ) and its volatility is σ (i ). When X is a random variable with t t expectation E(X) and standard deviation σ (X), any variable Y = aX, a being a constant, follows a distribution with expectation a × E(X) and volatility a × σ (X). The above formula follows: E(IM) = gap × E(it ) and σ (IM) = |gap| × σ (it ).

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e rg La

p ga

Sm

Interest rate variation Confidence Level

FIGURE 17.1 preset gap

IM IM Distribution Distribution

Interest margin variation all

gap

Probability

Probability

Interest Rate Interest Rate Distribution Distribution

Probability

from the maximum deviation of the interest rate at the same confidence level. However, if the gap changes, the interest income distribution also changes. This is the case when the bank modifies the gap through its hedging policy. Figure 17.1 shows the distribution of the interest margin with two different values of the gap. The probabilities of exceeding the upper bound variations are shaded.

IM Distribution

Interest margin variation

Distribution of Interest Margin (IM) with random interest rates and a

Interest Income at Risk and VaR The usual Value at Risk (VaR), or EaR, framework applies to the interest margin. However, it is possible that no straight loss occurs if we start with a positive interest margin, because the downside deviation might not be large enough to trigger any loss. The margin volatility results from that of interest rates: σ (IM) = |gap| × σ (i). The historical volatility of interest rates is observable from time series. In this example, we assume that the yearly volatility is 1.5%. Using a normal distribution, we can define confidence levels in terms of multiples of the interest rate volatility. The interest rate will be in the range defined by ±2.33 standard deviations from the mean in 1% of all cases. Since the interest margin follows a distribution curve derived from the distribution of interest rates multiplied by the gap, similar intervals apply to the margin distribution of values. With a gap equal to 2000, the margin volatility is: σ (IM) = 2000 × 1.5% = 30. The upper bound of the one-sided deviation is 2.33 times this amount at the 1% confidence level, or 2.33 × 30 = 70 (rounded). The EaR should be the unexpected loss only. This unexpected loss depends on the expected value of the margin at the future time point. Starting from an expected margin at 20, for example, the unexpected loss is only 20 − 70 = −50.

THE RISK–RETURN TRADE-OFF WHEN HEDGING GAPS With a given interest rate distribution, the bank alters its risk–return profile by changing its gap with derivatives. Unlike the previous section, the gap becomes variable rather than preset. When altering its exposure, the bank changes both the expected margin and the

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risk, characterized by margin volatility or a downside deviation at a preset confidence level. In what follows, we ignore the cost of derivatives used to modify the gap and assume that they generate interest payments or revenues with the current flat rate.

The Risk–Return Profile The basic relationships are σ (IM) = |gap| × σ (i) and E(IM) = IM0 + gap × E(i). The gap value, controlled by the bank, appears in both equations and is now variable. Since the volatility is an absolute value, we need to specify the sign of the gap before deriving the relationship between σ (IM) and E(IM) because we eliminate the gap between the two equations to find the relationship between expected margin and margin volatility. With a positive gap, the relationship between E(IM) and σ (IM) is linear, since gap = |gap| = σ (IM)/σ (i) and: E(IM) = IM0 + [E(i)/σ (i)] × σ (IM) This relation shows that the expectation of the margin increases linearly with the positive gap. When the gap is negative, |gap| = −gap and we have a symmetric straight line with respect to the σ (IM) axis. Therefore, with a given gap value, there is a linear relationship between the expected interest margin and its volatility. The risk–return profile in the ‘expected margin–volatility of margin’ space is a set of two straight lines. If the bank selects a gap value, it also selects an expected interest income. If the bank sets the gap to zero, it fully neutralizes the margin risk. We illustrate such risk–return trade-offs with a simple example when there are only two interest rate scenarios.

Altering the Risk–Return Profile of the Margin through Gaps In general, there are a large number of interest rate scenarios. In the following example, two possible values of a single rate summarize expectations. The rate increases in the first scenario and remains stable in the second scenario. The probability of each scenario is 0.5. If the upward variation of the interest rate is +3% in the first scenario, the expected value is 1.5% (Figure 17.2). Variation Variation Initial rate Initial = 8% rate = 8%

FIGURE 17.2

Probability Probability

Final rate Final rate

+ 3%

0.5

11%

+ 0%

0.5

10%

Expected variation = +1.5%

Expected rate = 11.5%

Interest rate expectations

Swaps serve to adjust the gap to any desired value. For each value of the gap, there are two values of the margin, one for each of the interest rate scenarios, from which

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the average and the volatility of the margin derive. Gap changes generate combinations of average margin and volatility. The variations of the margins, when the interest rate changes, are equal to the gap times the variation of interest rate: IM = gap × i. If the gap is −4, as in the above example, the variations of the margin are −4 × 3% = −0.12 and −4 × 0% = 0, since +3% and 0% are the only possible deviations of interest rate. Their probabilities are both 0.5. The expected value and the variance are respectively: 0.5 × (−0.12) + 0.5 × 0 = −0.06 and 0.5 × (−0.12 − 0.06)2 + 0.5 × (−0.12 + 0.06)2 = 0.0036. The volatility is the square root, or 0.06. With various values of the gap, between +10 and −10, the average and the volatility change according to Table 17.5. TABLE 17.5 Interest rate gap −10 −8 −6 −4 −2 0 +2 +4 +6 +8 +10

Expected margin and margin volatility Volatility of the margin variation

Expected variation of margin

0.15 0.12 0.09 0.06 0.03 0.00 0.03 0.06 0.09 0.12 0.15

−0.15 −0.12 −0.09 −0.06 −0.03 0.00 +0.03 +0.06 +0.09 +0.12 +0.15

By plotting the expected margin against volatility, we obtain all combinations in the familiar risk–return space. The combinations fall along two straight lines that intersect at zero volatility when the gap is zero. If the gap differs from zero, there are two attainable values for the expected margin at any given level of volatility. This is because two gaps of opposite sign and the same absolute value, for instance +6 and −6, result in the same volatility (0.09) and in symmetrical expected variations of the margin (+0.09 and −0.09). The expected variation of the margin is positive when the gap is positive. This is intuitive since the expectation of interest rate, the average of the two scenarios, is above the current rate. A rise in interest rate results in an increase of the expected margin when the gap is positive. Negative gaps are irrational choices with this view on interest rates. The expected interest rate would make the margin decrease with such gaps. When, at a given risk, the expected margin is above another combination, the risk–return combination is ‘efficient’. In this case, only positive values of the gap generate efficient combinations that dominate others, and negative values of the gap generate inefficient combinations. Therefore, only the upper straight line represents the set of efficient combinations. It corresponds to positive gaps only (Figure 17.3). The optimum depends on the bank’s preference. One way to deal with this issue is to set limits on the margin volatility or its maximum downside variation. Such limits set the gap and, therefore, the expected margin. This result extends over multiple interest rate scenarios since the risk–return profile depends only on the expectation and volatility of interest rate changes. However, the

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Expected variation of margin

0.20 0.10

Gap = +10 Gap

0.05 0.00 −0.05

0.00

0.05

0.10

0.15

0.20

−0.10 Gap = −10

−0.15 −0.20

FIGURE 17.3

Efficient combinations

0.15

Margin volatility

Risk–return relationships

above calculation applies only with a preset gap controlled by the bank. It does not hold when the gap is random. This is precisely what happens when considering business risk, because it makes future balances of assets and liabilities uncertain. To deal with this double uncertainty, interest rate and business risks, we need a different technique. The multiple interest rate and business scenarios approach addresses this issue (Chapter 18).

Risk–Return Profiles with Multiple Interest Rates In general, there are multiple gaps for different interest rates. Since all rates vary together, adding up the interest income volatilities due to each exposure, or the worst-case exposures for each gap, would overestimate the risk. This issue is the same for market risk and credit risk when dealing with diversification. The correlation4 between risk drivers determines the extent of diversification. The two rates might not vary mechanically together, so the chances are that one varies less than the other or in the opposite direction. This reduces the overall exposure of the bank to an exposure lower than the sum of the two gaps. Section 11 of this book provides the extended framework for capturing correlation effects. In the case of interest rates, the analysis relies on gaps. The following example illustrates these issues. The balance sheet includes two subportfolios that depend on different interest indexes. These indexes are i1 and i2 . Their yearly volatilities are 3% and 2%. The correlation between these two interest rates can take several values (0, +1, +0.3). The sensitivities of the interest income are the gaps, 100 and 200, relative to those indexes (Table 17.6). The interest margin variation IM becomes a function of both gaps and interest rate changes: IM = 100 × i1 + 200 × i2 It is the summation of two random variables, with constant coefficients. The expected variation is simply the weighted summation of the expected variation of both rates, using the gaps as weights. The volatility of the interest margin results from both interest rate volatilities and the correlation. The standalone risk for the first gap is the margin volatility generated by the variation of the first index only, or 100 × 3% = 3. The standalone risk 4 The

correlation is a statistical measure of association between two random variables, which is extensively discussed in Chapter 28.

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TABLE 17.6

Overall volatility of two interest rate exposures Market volatilities

Rate i1 Rate i2 Exposures Exposure 1 Exposure 2 Portfolio volatilitya Correlation = 0 Correlation = 1 Correlation = +0.3

σ (i1 ) = 3% σ (i2 ) = 2% Gap1 = 100 Gap2 = 200 5.00 7.00 5.67

values are calculated with the general formula: [(100 × 3%)2 + (200 × 2%)2 + 2ρ(100 × 3%)(200 × 2%)] where ρ takes the values 0.1 and 0.3 (see Chapter 28).

a These

for the second exposure is 200 × 2% = 4. The sum of these two volatilities is 7, which overestimates the true risk unless rates are perfectly correlated. The actual volatility is that of a weighted summation of two random variables: [gap1 × σ (i1 )]2 + [gap2 × σ (i2 )]2 + 2ρ[gap1 × σ (i1 )][gap2 × σ (i2 )] σ (IM) = (100 × 3%)2 + (200 × 2%)2 + 2ρ(100 × 3%)(200 × 2%)

σ (IM) =

With a zero correlation the volatility is 5, and with a correlation of 0.3 it becomes 5.67. It reaches the maximum value of 7 when the correlation is 1. The margin volatility is always lower than in the perfect correlation case (7). The difference is the effect of diversification of exposures on less than perfectly correlated rates. The extreme case of a perfect correlation of 1 implies that the two rates are the same. In such a case, the gap collapses to the sum of the two gaps. When characterizing the risk–return profile of the portfolio of two exposures by the expected margin and its volatility, the latter depends on the correlation between rates. Using a normal distribution as a proxy for rates is acceptable if rates are not too close to zero. In this case, it is a simple matter to generate the distribution of the correlated rates to obtain that of the margin5 . When considering the hedging issue, one possibility is to neutralize all gaps for full immunization of interest income. Another is to use techniques taking advantage of the correlation between interest rates. In any case, when rates correlation is lower than 1, variable-rate gaps do not add together. Note that using fixed-rate gaps is equivalent to an arithmetic addition of all variable-rate gaps, as if they referred to the same index. This corresponds to a correlation of 1 and overestimates the exposure. 5 Alternatively,

interest rates follow a lognormal, more realistic distribution. The changes in interest rates still follow a stochastic process with a random normal factor. It is possible to correlate the rates by correlating their random factor. See Chapter 30 for a description of the main stochastic processes driving rates. Chapter 28 discusses techniques for generating the distribution of a sum of correlated normal variables.

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INTEREST RATE POLICY The ALM unit controls the interest rate exposure by adjusting the gap after funding. This section discusses, through simple examples, various hedging issues: immunization of the interest income; trade-off of the hedging policy when the bank has ‘views’ on future interest rates; usage of multiple gap exposures; setting limits for interest rate risk.

Hedging In order to obtain immunization, the post-funding gap should be zero. In other words, the funding and hedging policy should generate a gap offsetting the commercial portfolio positive gap. The funding variable-rate gap is equal to the fraction of debt remaining at variable rate after hedging, with a minus sign since it is a liability. Since the gap is +8 before funding, the variable-rate debt should be set at 8 to fully offset this gap. This means that the remaining fraction of debt, 12 − 8 = 4, should have an interest rate locked in as of today. An alternative and equivalent approach is to start directly from the post-funding gap. This gap is +8 − 12 = −4. Setting this gap to zero requires reducing the amount of floating rate debt from 12 to 8. Therefore, we find again that we need to lock in a rate today for the same fraction of the debt, or 4, leaving the remaining fraction of the debt, 8, with a floating rate. In the end, the interest rate gap of the funding solution should be the mirror image of the interest rate gap of the commercial portfolio. Any other solution makes the net margin volatile. The solution ‘floating rate for 8 and locked rate for 4’ neutralizes both the liquidity gap and the interest rate gap (Table 17.7). TABLE 17.7 rate gaps

Hedging the liquidity and interest

Liquidity gap Interest rate gap of the banking portfolio Fixed-rate debt Floating-rate debt Total funding Liquidity gap after funding Interest rate gap after funding and hedging

+12 +8 4 8 12 0 0

In order to lock in the rate for an amount of debt of 4, various hedges apply. In this case, we have too many variable-rate liabilities, or not enough variable-rate assets. A forward swap converting 4 of the floating-rate debt into a fixed-rate debt is a solution. The swap would therefore pay the fixed rate and receive the floating rate. Alternatively, we could increase the variable-rate fraction of assets by 4. A swap paying the fixed rate and receiving the variable rate would convert 4 of fixed-rate assets into 4 of variable-rate assets. The same swap applies to both sides.

Hedging Policy If the amount of floating-rate debt differs from 8, there is an interest rate exposure. The new gap differs from zero. For instance, locking in the rate for an amount of debt of 2

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results in a gap which is +2. The margin volatility results directly from the remaining gap after funding. Any increase in rate increases the margin since the variable-rate gap is positive. However, the price to pay is the risk that rates decrease, thereby reducing the margin. Maintaining open a gap makes sense only if it is consistent with an expected gain. In this example, the expectation embedded in the scenarios is a rise of rate, since rates are either stable or increase. A positive variable-rate gap makes sense, but a negative variable-rate gap would be irrational. The level of interest rates is an important consideration before making a decision to close the gap. A swap would lock in a forward rate. In this case, the locked rate is on the cost side, since the swaps pay the fixed rate. If the forward rate is low, the hedge is acceptable. If the forward rate is high, it will reduce the margin. Using a full hedge for a short horizon only (1 year) provides added flexibility, while later maturities remain open. This allows for the option to hedge at later periods depending on the interest rate moves. Of course, if interest rates keep decreasing, the late hedge would become ineffective. The exposure to interest rate risks has to be consistent with risk limits. The limits usually imply that adverse deviations of the interest margin remain within given bounds. This sets the volume of the necessary hedge for the corresponding period. Setting the maximum adverse deviation of the target variable, given a confidence level, is equivalent to setting a maximum gap since the margin volatility is proportional to the gap. This maximum gap is the limit.

Setting Limits for Interest Rate Risk Setting limits involves choosing the maximum volatility, or the maximum downward variation of the margin at a given confidence level. Limits would effectively cap the gap, thereby also influencing the expected income, as demonstrated earlier. With a given volatility of the margin, the maximum acceptable gap is immediately derived: σ (IM) = |gap| × σ (i). If the management caps the volatility of the margin at 20, and the interest rate volatility is 1.5% yearly, the maximum gap is 20/1.5% = 1333. Alternatively, it is common to set a maximum downward variation of the interest margin, say −100. Assuming operating costs independent of interest rates, the percentage variation of pre-tax and post-operating costs income is higher than the percentage change in interest margin at the top of the income statement. For instance, starting with a margin of 200, a maximum downside of 100 represents a 50% decrease. If operating costs are 80, the original pre-tax income is 200 − 80 = 120. Once the margin declines to 100, the remaining pre-tax income becomes 100 − 80 = 20. The percentage variation of pre-tax income is −100/120 = −80%, while that of margin is −50%. If interest rate volatility is 1.5% and confidence level is 2.5%, the upper bound of the interest rate deviation, using the normal distribution proxy for rates, is 1.96 × 1.5%, or approximately 3%. Starting from a maximum decline of the margin set at 100, at the 2.5% confidence level, the maximum sustainable gap is 100/3% = 3333. If the interest rate decreases by 3%, the margin declines by 3333 × 3% = 100, which is the maximum.

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Hedging Multiple Variable-rate Gaps For hedging purposes, the simplest way to obtain margin immunization is to neutralize both gaps with respect to the two interest rates. Another option would be to hedge both gaps at the same time using the interest rate correlation. Since rates correlate, a proxy hedge would be to take a mirror exposure to only one of the rates used as a unique driver of the interest rate risk, the size of the hedge being the algebraic summation of all gaps. This is the well-known problem of hedging two correlated exposures, when there is a residual risk (basis risk). When considering the two variable-rate gaps, respectively 100 and 200 for two different and correlated rates, the basic relationship for the interest margin is: IM = 100 × i1 + 200 × i2 Since the two rates correlate, with ρ12 being the correlation coefficient, there is a statistical relationship between the two variations of rates, such that: i1 = βi2 + ε12 IM = 100 × (βi2 + ε12 ) + 200 × i2 The value of the second gap becomes a variable G, since the bank can control it in order to minimize the margin volatility: IM = 100 × (βi2 + ε12 ) + G × i2 IM = (100 × β + G) × i2 + ε12 The variance of IM is the sum of the variance of the interest rate term and the residual term since they are independent: σ 2 (IM) = (100 × β + G)2 × σ 2 (i2 ) + σ 2 (ε12 ) There is a value X of the gap G such that this variance is minimum. Using β = 0.5 and σ 2 (ε12 ) = 1%, we find that X = −50 results in a variance of 1. This also has a direct application for setting limits to ALM risk, since the volatility is an important parameter for doing so. Note that hedging two exposures using a single interest rate results in a residual volatility. On the other hand, neutralizing each variable rate gap separately with perfect hedges (same reference rates) neutralizes basis risk. This is a rationale for managing gaps separately.

18 ALM and Business Risk

Since both interest rates and future business volumes and margins are random, banks face the issue of how to jointly optimize, through financial hedges, the resulting uncertainty from these multiple sources of risk. There are practical simulation-based solutions to this joint optimization problem, using financial hedges. The starting point of the method requires defining a number of interest rate scenarios and business scenarios. The principle consists of simulating all values of the target variable, the interest income or the Net Present Value (NPV) of the balance sheet, across all combinations of interest rate and business scenarios. The resulting set of values is organized in a matrix cross-tabulating business and interest rate scenarios. Such sets of target variable values depend on any hedge that affects interest income or NPV. Since gaps become subject to business risk, it is not feasible any more to fully hedge the interest income. When hedging solutions change, the entire set of target variable values changes. For turning around the complexity of handling too many combinations of scenarios, it is possible to summarize any set of values of the target of variables within the matrix, by a couple of values. The first is the expected value of the target variable, and the second is its volatility, or any statistics representative of its risk, both calculated across the matrix cells. For each hedging solution, there is such a couple of expected profitability and risk across scenarios. When hedging changes, both expectation and risk change, and move in the ‘risk–return’ space. The last step consists of selecting the hedging solutions that best suit the goals of the Asset–Liability Management Committee (ALCO), such as minimizing the volatility or targeting a higher expected profitability by increasing the exposure to interest rate risk. Those solutions that maximize expected profitability at constant risk or minimize risk at constant expected profitability make up a set called the ‘efficient frontier’. All other hedging solutions are discarded. It is up to the management to decide

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what level of risk is acceptable. This methodology is flexible and accommodates a large number of simulations. The technique allows us to investigate the impact on the risk–return profile of the balance sheet under a variety of assumptions, for example: What is the impact on the risk–return profile of assumptions on volumes of demand deposits, loans with prepayment risk or committed lines whose usage depends on customer initiatives? Which funding and hedging solutions minimize the risk when only interest rate risk exists, when there is business risk only and when both interact? How can the hedging solutions help to optimize the risk–return combination? The Asset–Liability Management (ALM) simulations also extend to optional risks because the direct calculation of interest income or NPV values allows us to consider the effect of caps and floors on interest rates, as explained in Section 7 of this book, which further details these risks. Finally, the ALM simulations provide a unique example of joint control of interest rate and business risks, compared to market risk and credit risk measures, which both rely on a ‘crystallized’ portfolio as of today. The first section of this chapter illustrates the methodology in a simple example combining two simple interest rate scenarios with two business scenarios only. It provides the calculation of interest income, and of its summary statistics, expectation and volatility, across the matrix of scenarios. The second section details the calculations when changing the hedging solution, considering only two different hedging solutions for simplicity. The third section summarizes the findings in the risk–return space, demonstrating that extending simulations to a larger number of scenarios does not raise any particular difficulty. It defines the efficient frontier and draws some general conclusions.

MULTIPLE SCENARIOS WITH BUSINESS AND INTEREST RATE RISKS The previous methodology applies to a unique gap value for a given time point. This is equivalent to considering a unique business scenario as if it were certain, since the unique value of the gap results directly from asset and liability volumes. Ignoring business risk is not realistic, and amounts to hiding risks rather than revealing them when defining an ALM policy. Moreover, the ALM view is medium-term, so that business uncertainty is not negligible. To capture business risk, it is necessary to define multiple business scenarios. Multiple scenarios result in several projections of balance sheets and liquidity and interest rate gap profiles. For hedging purposes, the risk management issue becomes more complex, with interest income volatility resulting from both interest rate risk and business risk. Still, it is possible to characterize the risk–return profile and the hedging policies when both risks influence the interest income and the NPV. Business risk influences the choice of the best hedging policy.

The Risk–Return Profile of the Balance Sheet When using discrete scenarios for interest rates and for business projections, it is possible to summarize all of them in a matrix cross-tabulating all interest rate scenarios with all business scenarios. For each cell, and for each time point, it is feasible to calculate the

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margin or the NPV. The process is not fully mechanical, however, because there is a need to make assumptions on funding and interest rate management. Constructing a risk–return profile and hedging are two different steps. For the first step, we consider that there are no new hedges other than the existing ones. Hence, all future liquidity gaps result in funding or investing at prevailing rates. The ultimate goal of the simulations remains to define the ‘best’ hedge. If we calculate an interest income value or NPV for each cell of the matrix, we still face the issue of summarizing the findings in a meaningful way in terms of return and risk. A simple way to summarize all values resulting from all combinations of scenarios is to convert the matrix into a couple of parameters: the expected interest income (or NPV) and the interest income (or NPV) volatility across scenarios. This process summarizes the entire matrix, whatever its dimensions, into a single pair of values (Figure 18.1). The volatility measures the risk and the expectation measures the return. This technique accommodates any number of scenarios. It is possible to assign probabilities to each scenario instead of considering that all are equally probable, and both expected margin and margin volatility would depend on such probabilities. If all combinations have the same probability, the arithmetic average and the volatility are sufficient. Business / Interest rate

FIGURE 18.1 scenarios

A

B

Rate 1

IM & NPV IM & NPV

Rate 2

IM & NPV IM & NPV

Expected IM & NPV Expected + IM & NPV + Volatility Volatility

Matrix of scenarios: cross-tabulating business scenarios and interest rate

However, since the interest income or NPV depends on the funding, investing and hedging choices, there are as many matrices as there are ways of managing the liquidity and the interest rate gaps. We address this issue in a second step, when looking for, and defining, the ‘best’ policies.

The Matrix Approach A simple example details the methodology. When using two scenarios, there are only two columns and two rows. It is possible to summarize each business scenario by a gap value. Referring to the example detailed, we use two such gap values: −4 and −8 crosstabulated with two interest rate scenarios, with +8% and +11% as flat rates. The original value of the margin with rate at 8% is 1.80 with a gap equal to −4. When rates vary, it is easy to derive the new margin value within each column, since the gap is constant along a column. The matrix in this very simple case is as given in Table 18.1, with the corresponding expectation and volatility on the right-hand side, for a single time point 1 year from now. Using two business scenarios, with net gaps of −4 and −8, and two interest rate scenarios, at +8% and +11%, we find the expectations of the margin change and of the margin volatility in Table 18.1. The variations are respectively −0.12 and −0.24 when the rate moves up from 8% to 11%. Starting from a margin of 1.80, the corresponding final

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TABLE 18.1 Matrix of margins: two interest rate scenarios and a single business scenario Rate

8% 11%

Scenario Gap = −4 Gap = −8 +1.80 +1.68

1.80 1.56

Risk–return profile

E(IM) σ (IM)

1.7100 0.1149

margins are 1.68 and 1.56. The expected margin and its volatility across the four cells are respectively 1.7100 and 0.11491 . In the range of scenarios considered, the worst-case value of the margin is 1.56. By multiplying the number of scenarios, we could get many more margin values, and find a wider distribution than the three values above. In the current simple set of four cases, we have a 2 × 2 matrix of margins for each financing solution. If we change the financing solution, we obtain a new matrix. There are as many matrices as there are financing solutions.

Handling Multiple Simulations and Business Risk The matrix approach serves best for handling multiple business scenarios as well as multiple yield curve scenarios. There is no limitation on the number of discrete scenarios. The drawback of the matrix approach is that it summarizes the risk–return profile of the portfolio using two parameters only, which is attractive but incomplete. To characterize the full portfolio risk, we would prefer to have a large distribution of values for each future time point. In addition, probabilities are subjective for business scenarios. This approach contrasts with the market risk ‘Value at Risk’ (VaR) and the credit risk VaR techniques, because they use only a ‘crystallized’ portfolio as of today. With a single business scenario, we could proceed with the same technique to derive an ALM VaR with full probability distributions of all yield curves. This would allow us to generate distributions of margins, or of the NPV, at all forward time points fully complying with observed rates. The adverse deviations at a preset confidence level derive from such simulations. We use this technique when detailing later the specifics of NPV VaR with given asset and liability structures. However, ignoring business risk is not realistic with ALM. It does not make much sense to generate a very large number of interest rate scenarios if we ignore business risk, since there is no point in being comprehensive with rates if we ignore another significant risk. The next chapter details how the matrix methodology helps in selecting the ‘best’ hedging solutions.

HEDGING BOTH INTEREST RATE AND BUSINESS RISKS For any given scenario, there is a unique hedging solution immunizing the margin against changes in interest rates. This hedging solution is the one that offsets the interest rate gap. With several scenarios, it is not possible to lock in the margin for all scenarios by 1 The

standard deviation uses the formula for a sample, not over the entire population of outcomes (i.e. it divides the squared deviations from the mean by 3, not 4).

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hedges, because gaps differ from one business scenario to another. To determine hedging solutions, we need to simulate how changing the hedge alters the entire risk–return profile corresponding to each business scenario.

The Matrix Technique, Business Risk and Hedging When ignoring new hedges, we generate a number of values for the target variables identical to the number of combinations of interest rate and business scenarios, conveniently summarized into a couple of statistics, the ‘expected value’ and ‘volatility’. When modifying the hedge, we change the entire matrix of values. There are as many matrices of values of the target variables, interest income or NPV, as there are hedging solutions. Summarizing each set of new values in a risk–return space allows us to determine which hedges result in better risk–return profiles than others. Figure 18.2 illustrates the process. Hedging 2 Hedging 2 Hedging 3

Business / A Interest rate Business / 1 Taux A Marge B Interest rate 2 Marge Business / Taux Taux 1 A Marge B Interest rate Taux 2 Marge IM & NPV IM & NPV Rate 1 Rate 2

IM & NPV IM & NPV

B

Expected IM & NPV / Volatility Expected IM & NPV / Volatility Expected IM & NPV / Volatility

FIGURE 18.2 scenarios

Matrix of scenarios: cross-tabulating business scenarios and interest rate

Simulations offer the maximum flexibility for testing all possible combinations of assumptions. Multiplying the number of scenarios leads to a large number of simulations. Two business scenarios and two interest rate scenarios generate four combinations. With two hedging scenarios, there are eight combinations. The number of simulations could increase drastically, making the interpretation of the results too complex. To handle this complexity, summarizing each matrix using a pair of values helps greatly. Only the risk–return combinations generated for each hedging solution serve, independently of the size of the matrix. The approach facilitates the identification of the best solutions within this array of combinations. In what follows, we ignore probabilities assigned to business scenarios. Using probabilities would require assigning probabilities to each interest rate scenario and to each business scenario. Therefore, each cell of the matrix has a probability of occurrence that becomes the joint probability of the particular pair of business and interest rate scenarios. In the case of uniform probabilities, there is no need to worry about different likelihoods of occurrence when using the matrix technique. Otherwise, we should derive these joint

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probabilities. Under independence between business scenarios and interest rate scenarios, these would be the product of the probabilities assigned to the particular business scenario and the particular interest rate scenario, as explained in Chapter 28.

Business Scenarios The interest rate scenarios are those of the first analysis with only one base case for business projections (interest scenarios 1 and 2). One business scenario (A) is identical to that used in the previous chapter. A second business scenario (B) is considered. Each business scenario corresponds to one set of projected balance sheet, liquidity gap and interest rate gap. The scenario B is shown in Table 18.2. The starting points are the liquidity gaps and the interest rate gaps of the banking portfolio for scenarios A and B. TABLE 18.2

Two business scenarios

Projected balance sheet Interest-insensitive assets Interest-sensitive assets Interest-insensitive resources Interest-sensitive resources Total assets Total liabilities Gaps Liquidity gaps Interest rate gap—banking portfolio Interest rate gap—total balance sheet

Scenario A

Scenario B

19 17 15 9 36 24

22 17 14 13 39 27

12 8 −4

12 4 −8

We use the same 3% commercial margins to obtain customer rates and margins in value. For instance, with scenario B, at the 8% rate, the value of the commercial margin is 39 × 11% − 27 × 5% = 4.29 − 1.35 = 2.94, and the net margin, after funding costs, is 2.94 − 0.96 = 1.98. The margins are different between scenarios A and B because the volumes of assets and liabilities generating these margins are different (Table 18.3). TABLE 18.3 scenario 1

Interest margins: interest rate

Interest rate gap (banking portfolio) Initial commercial margina Liquidity gap (a) Cost of funding (a) × 8% Net interest margin after funding a The

A

B

+8 2.76 +12 0.96 1.80

+4 2.94 +12 0.96 1.98

initial margins correspond to an interest rate of 8%.

When the interest rate changes, the variations of margins result from the interest rate gaps. The cost of funding depends on the hedging solution. The fraction of the total funding whose rate is locked in through swaps defines a hedging solution.

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Hedging Scenarios There are two hedging scenarios. The first solution locks in the interest rate for a debt of 4. The remaining debt, 12 − 4 = 8, required to bridge the liquidity gap is a floating rate debt. The second solution locks in the interest rate for an amount of 8, the remaining 4 being floating rate debt. These are the H1 and H2 hedging solutions. It is possible to think of the first hedging scenario as the existing hedge in place and of H2 as an alternative solution (Table 18.4). TABLE 18.4

Funding scenarios

Scenario Interest rate gap (banking portfolio) Liquidity gap Hedginga Hedging H1 Hedging H2 Interest rate gap after hedgingb Hedging H1 Hedging H2

A

B

+8 −12

+8 −12

8vr + 4fr 4vr + 8fr 0 +4

+4 0

a The

hedging solution is the fraction of total funding whose rate is locked in. ‘fr’ and ‘vr’ designate respectively the fractions of debt with fixed rate and variable rate. b The interest rate gap is that of the banking portfolio less the floating rate debt.

The interest rate gap after funding is the difference between the commercial portfolio gap and the gap resulting from the funding solution. This gap is simply the floating rate debt with a minus sign. With the hedging solution H1, the floating rate debt is 8. The interest rate gap post-funding is therefore +8 − 8 = 0. In scenario B, the solution H1 results in an interest rate gap post-funding of +8 − 4 = +4.

The Matrices of Net Interest Margins The banking portfolio margins and the net interest margins derive from the initial values of margins using the gaps and the variation of the interest rate. For scenario A, the initial value of the net margin, with interest of 8%, is 1.80. For scenario B, the net margin with initial interest rate of 8% is 1.98. The margin variation with a change in interest rates of i = +3% is: (Net interest margin) = gap × i It is easier to derive the matrix of net margins from initial margins and gaps than to recalculate all margins directly. A first matrix cross-tabulates the business scenarios A and B with the interest rate values of 8% and 11%, given the hedging solution H1. The second matrix uses the hedging solution H2. For instance, with the business scenario A and the hedging solution H1, both the initial net margin and the final margin, after an interest rate increase of 3%, are equal. With the hedging solution H2 and the scenario B, the initial margin is 1.98. Since the interest rate gap after hedging is −4, the final margin is 1.98 − 4 × 3% = 1.86 when the rate increases by 3% (Table 18.5).

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TABLE 18.5

The net margin matrix after hedging

Business scenario

A

B

Fixed rate amount: 4 Net margin Rate 1: 8% Rate 2: 11% Business scenario

Hedging H1 1.80 + 0 × i 1.80 1.80 A

Hedging H2 1.80 − 4 × i 1.98 1.86 B

Fixed rate amount: 8 Net margin Rate 1: 8% Rate 2: 11%

Hedging H1 1.80 + 4 × i 1.80 1.92

Hedging H2 1.98 + 0 × i 1.98 1.98

In order to define the best hedging solutions, we need to compare the risk–return combinations.

RISK–RETURN COMBINATIONS First, the risk–return profiles generated by the two hedging solutions are calculated. By comparing them, we see how several funding and hedging solutions compare, and how to optimize the solution given a target risk level. For each matrix, the average value of the margins and the volatility2 across the cells of each matrix are calculated. The Sharpe ratio, the ratio of the expected margin to the margin volatility, is also calculated. This ratio is an example of a risk-adjusted measure of profitability3 . The results are as given in Table 18.6. TABLE 18.6 Net combinations Hedging H1 Rate 1 Rate 2 Hedging H2 Rate 1 Rate 2

A 1.80 1.80 A 1.80 1.92

margin B 1.98 1.86 B 1.98 1.98

matrix

and

risk–return

Mean Volatility Mean/volatility Mean Volatility Mean/volatility

1.860 0.085 21.920 1.920 0.085 22.627

The first solution generates a lower average margin with the same risk. The second generates a higher margin, with the same risk. The best solution is therefore H2. The Sharpe ratio is greater with H2. 2 The

volatility is the square root of the sum of the squared deviations of the margin from the mean. All values in the matrix have the same probability. The expected value is the arithmetic mean. 3 The Sharpe ratio is a convenient measure of the risk-adjusted performance of a portfolio. It also serves when modelling the credit risk of the portfolio.

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In general, a large number of hedging solutions are considered. When graphed in the risk–return space, they generate a cloud of points4 . Each dot summarizes a matrix combining several target variable values corresponding to each pair of interest rate scenarios and business scenarios. In spite of the high number of simulations, changing hedging scenarios simply moves the dots in the risk–return space. Some solutions appear inefficient immediately. They are those with an expected return lower than others, at the same risk level or, alternatively, those with the same expected return as others, but with a higher risk. These solutions are inefficient since others dominate them. The only solutions to consider are those that dominate others. The set of these solutions is the ‘efficient frontier’ (Figure 18.3). Increasing return and constant risk

Expected margin Efficient frontier

With new hedge H2 Without new hedge H1

Decreasing risk, constant return Minimum risk solution Margin volatility

FIGURE 18.3

Risk–return combinations

A risk–return combination is efficient if there is no other better solution at any given level of risk or return. The efficiency criterion leads to several combinations rather than a single one. The optimization problem is as follows: • Minimize the volatility of the margin subject to the constraint of a constant return. • Maximize the expected margin subject to the constraint of a constant risk. For each level of profitability, or of return, there is an optimal solution. When the risk, or the return, varies, the optimum solution moves along the efficient frontier. In order to choose a solution, a risk level has to be set first. For instance, the minimum risk leads to the combination located at the left and on the efficient frontier. When both interest rate and business risks interact, solutions neutralizing the margin risk do not exist. Only a minimum risk solution exists.

4 The

graph shows the risk–return profiles in a general case. In the example, the only variable that changes the risk–return combination is the gap after funding. When this gap varies continuously, the risk–return combinations move along a curve. The upper ‘leg’ of this curve is the efficient frontier of this example.

19 ALM ‘Risk and Return’ Reporting and Policy

The Asset–Liability Management (ALM) reporting system should provide all elements for decision-making purposes to the ALM Committee (ALCO). The reporting should provide answers to such essential questions as: Why did the interest income vary in the last period? Which interest income drivers caused the variations? Funding, investment and hedging issues necessitate reporting along the two basic financial dimensions, earnings and risks. To move back and forth from a financial perspective, risk and return, to a business policy view, it is also necessary to break down risk and performance measures by transaction, product family, market segment and business line. ALM reporting links gaps, interest incomes and values of transactions to business units, products, markets and business lines. Reports slice gaps, incomes and gaps along these dimensions. To analyse the interest income variations, ALM reports should break down the variations due to the interest income drivers: changes in the structure of the existing by level of interest earned or paid, due to both amortization effects plus new business; changes of the yield curve shape. Moving back and forth from financial to business dimensions, plus the necessity of relating interest margin changes to identified drivers, creates high demands on the ALM information system, which should record all business and financial data. This chapter provides examples of reporting related to common ALCO issues. The first section lists the typical ALCO agenda for a commercial bank. The second section describes the reporting system specifications. The third and fourth sections illustrate breakdowns of gaps and interest incomes by product family and market segment. The last section focuses on explaining interest income variations.

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ALM MISSIONS AND AGENDA Table 19.1 lists the basic missions and agenda of ALCO. The next subsections discuss them further. TABLE 19.1

ALM scope and missions

Liquidity management

Funding deficits Investing excess funds Liquidity ratios

Measuring and controlling interest rate risk Recommendations off-balance sheet

Gaps and NPV reporting Hedging policy and instruments Hedging programmes

Recommendations on-balance sheet

Flows: new business Existing portfolio

Transfer pricing system

Economic prices and benchmarks Pricing benchmarks

Preparing the ALCO

Recent history, variance analysis (projections versus realizations) GAP and NPV reports ‘What if’ analyses and simulations . . .

Risk-adjusted pricing

Mark-up to transfer prices (from credit risk allocation system) Mispricing: gaps target prices versus effective prices

Reporting to general management

The ALCO Typical Agenda and Issues Without discussing further some obvious items of the list in Table 19.1, we detail some representative issues that need ALCO attention. Historical Analysis

Typical questions include the following: • How do volumes, margins and fees behave? • How do previous projections differ from realizations, in terms of interest margins, fees and volumes? • What explains the actual–projected volume gaps, based on actual and projected interest rate gaps, and the product mix changes?

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A review of commercial actions and their results is necessary to see which corrective actions or new actions are required. These involve promotions and incentives for customers, or all development policies in terms of product mix. Without this historical, or ex post, analysis, it remains difficult to look forward. On-balance Sheet Actions

There is an important distinction between existing portfolios, which generate the bulk of revenues, and new business for future developments. Actions on the volume–product mix and pricing have an impact on new business. All future actions need analysis both in business terms and in risk–return terms. New development policies require projections of risk exposure and revenue levels. Conversely, interest rate and liquidity projections help define new business policies. This is a two-way interaction. Interest rate risk and liquidity risk are not the only factors influencing these decisions. However, the new transactions typically represent only a small fraction of the total balance sheet. Other actions on the existing portfolio might be more effective. They include commercial actions, such as incentives for customers to convert some products into others, for example fixed to floating rate loans and vice versa, or increasing the service ‘intensity’, i.e. the number of services, per customer. Off-balance Sheet Actions

Off-balance sheet actions are, mainly, hedging policies through derivatives and setting up a hedging programme for both the short and long-term. Hedges crystallize in the revenues the current market conditions, so that there are no substitutes to ‘on-balance sheet’ actions. In low interest rate environments, forward hedges (swaps) embed current low rates in the future revenues. In high interest rate environments, high interest rate volatility becomes the issue, rather than the interest rate level. The hedging policy arbitrages between revenues level, interest rate risk volatility and the costs of hedging (both opportunity costs and direct costs). The financial policy relates to funding, hedging and investing. However, it depends on business policy guidelines. Defining and revising the hedging programme depends on expectations with respect to interest rates, on the trade-off between hedging risk versus level of revenues, and on the ‘risk appetite’ of the bank, some willing to neutralize risk and others betting on favourable market movements to various degrees.

OVERVIEW OF THE ALM REPORTING SYSTEM The ALM reporting system moves from the ALM risk data warehouse down to final reports, some of them being purely financial and others cross-tabulating risk and return with business dimensions. The business perspective requires in fact any type of breakdown of the bank aggregates, by product, market segment and business unit. In addition, reporting the sources of gaps, along any of these relevant dimensions, raises the issue of what contributes to risk and return. Therefore, the system should provide functionalities to ‘drill down’ into aggregates to zoom in on the detailed data. Moreover, front-ends should include ‘what if’ and simulation capabilities, notably for decision-making purposes (Figure 19.1).

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Front-ends and Reporting

ALM Data Warehouse Products & Markets

Aid to Decisionmaking: Pricing Gaps, Simulations & ALM Reports

Volumes Balance Amortization Renewal & expected usage Implicit options Currency ...

Actual / Projected Variance Analysis Mispricing to Benchmarks

Reference rate Rate level, guaranteed historical rate, historical target rate Nature of rate (calculation mode). Interest rate profile Sensitivity to reference rates Options characteristics ...

FIGURE 19.1

OLAP Analysis

Rates

Business Business Reports: Reports: Markets & Markets & Products: Products: 'Risk−Return' "Risk - Return" ...

ALM data warehouse, front-ends and reporting

BREAKING DOWN GAPS BY SEGMENTS In this section, we illustrate the breakdown of gaps and ‘all-in revenues’ (interest plus fees) along the product and market dimensions. We show gaps and volumes first, and gaps and volumes in conjunction with ‘all-in margins’ in a second set of figures. All sample charts focus on risks measured by gaps. Since gaps are differences in volumes of assets and liabilities, there is no difficulty in splitting these aggregates into specific product or market segment contributions measured by their outstanding balances of assets and liabilities. The time profile of existing assets and liabilities is the static gap profile. Figure 19.2 shows the declining outstanding balances of assets and liabilities. The static gap, measured by assets minus liabilities, tends to be positive in the first periods, which implies a need for new funding. The picture is very different from the dynamic profiles because of the volume of new transactions during each of the first three periods. Figure 19.3 shows total asset and liability values, together with the liquidity gaps and the variable interest rate gaps. Since the total volumes of assets and liabilities are used, these gaps are dynamic. The positive liquidity gaps represent a permanent need for new funds at each period, and the negative variable interest rate gaps show that the bank interest income is adversely influenced by rising interest rates. The dynamic time profile of gaps increases up to period 3 and then declines because we assume that no projections are feasible beyond

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237

1500 Existing liabilities (total) 1000

Existing assets (total)

Amounts

500 0 1

2

3

4

5

6

−500 −1000 Time

FIGURE 19.2

Static gaps time profile 2000

Amounts & Gaps

1500 1000 500 0 −500 −1000 1

2

3

4

5

6

−1500 Time Total Assets

FIGURE 19.3

Total Liabilities

Interest Rate Gap

Assets and liabilities and the liquidity and variable interest rate gaps

period 3. After period 3, the time profile becomes static and total assets and liabilities decline.

Breakdown of Dynamic Gaps by Segments Figures 19.4 and 19.5 show the dynamic time profiles of assets and liabilities and their breakdown into six hypothetical segments, at each date. Bars represent volumes of assets or liabilities. The differences are the gaps. Each bar is broken down into the six segments of interest. The bar charts show which assets and liabilities contribute more or less to the gap. This type of reporting links the gap and asset and liability time profiles to the commercial policy. For instance, we see that the top segment of each time bar is among the highest volumes of assets contributing to the positive liquidity gap, or the need for

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2000 1500 Assets

Amount

1000 500 0

Time

−500 −1000

1

2

−1500

FIGURE 19.4

3 Liabilities

4

5

6 Bars broken down by product−market segments

Breakdown of the liquidity gaps by segments 2000 Bars broken down by product−market segments

Assets

1500

Amount

1000 500 0 −500 −1000 −1500 −2000

1 2

3

4

5

6

Liabilities

Time Liquidity Gap New Liabilities (cum.) Variable Rate Existing Liabilities Fixed Rate Existing Liabilities New Assets (cum.) Variable Rate Existing Assets Fixed Rate Existing Assets

FIGURE 19.5

Breakdown of interest rate gaps and volumes by segments

new funds. Some assets and liabilities are variable rate and others fixed rate. Figure 19.5 shows a similar breakdown using the type of rate, fixed or variable. These figures are not sufficient, however, since we have only the contributions to gaps without the levels of margins.

BREAKDOWN OF REVENUES AND GAPS, AND THE PRODUCTS–MARKETS MIX The revenues might be: • All-in margins, meaning that they combine various sources of revenues.

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• Pure interest rate spreads, such as customers’ rate minus any internal transfer rate used as the reference for defining commercial margins. • Pre-operating expenses, and in some cases, post-direct operating expenses.

Breakdown of Interest Income by Segments Figure 19.6 shows the contributions to the ‘all-in margin’ (interest margin and fees) of the different types of products or markets across time. The vertical axis shows the all-in revenue divided by the asset volume, and the horizontal axis shows time. The profitability measure is a ‘Return On Assets’ (ROA). This is not a risk-adjusted measure, since that would require capital allocation. In this case, the total cumulated margin, as a percentage, seems more or less constant at 10%. Hence, the interest incomes in dollars are increasing with volume since the percentage remains constant while volumes grow in the first 3 years. 12%

All-in Margin

10% 8% 6% 4%

Bars broken down by product−market segments

2% 0% 1

2

3

4

5

6

Time

FIGURE 19.6

Breakdown of percentages of ‘all-in margins’ by segments

This chart shows only interest income contributions, without comparing them with contributions to gaps. Therefore, it is incomplete.

Contributions of Segments to Revenues and Gaps Figure 19.7 compares both interest income and volumes. The chart brings together contributions to gaps and margins in percentages (of volumes) at a given date. The six segments of the example correspond to the six bars along the horizontal axis. The comparison shows contrasting combinations. Low volume and low margin for the fourth segment make it a loser (relatively), high volume and high margin for the first segment make it a winner (relatively).

‘Gap Risk’–Return Profiles A common financial view plots volumes (gap risk) against incomes (in value) in the same chart. The traditional business view suggests that more volume with high income is

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3.5%

Volumes ($)

300

3.0%

250 Volume & Margin

Margins (%)

2.5%

200

2.0% 150 1.5% 100

1.0%

50

0.5%

0

0.0% Segments

FIGURE 19.7

Comparing margins and volumes by products and market lines

better, and conversely. Under this view, the volume–income space is split into four basic quadrants, and the game is to try to reach the area above the diagonal making volume at least in line with margins: more volume with higher margins and less volume with lower incomes. An alternative view highlights volumes as contributions to gaps. It turns out that this is the same as the risk–income view when volume measures risk, since volume is the contribution to gaps. Hence, the two views are consistent. In Figure 19.8, each dot represents one of the six segments. Some contribute little to gaps and have higher income, 3.5% 3.0%

Higher income

100

Return On Asset

2.5% 2.0%

200

1.5%

200

100

1.0%

150

Lower 'risk'

250

0.5% 0.0% 0

50

100

150

200

250

300

Volume (Value)

FIGURE 19.8

Margins versus volumes (contribution to liquidity and interest rate risks)

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241

some contribute to both with similar magnitude, and others contribute to gaps and little to income. ‘Losers’ are the segments combining high volumes with low incomes. The low volume–high income segments are dilemmas, since it might be worth taking on more risk for these. The first diagonal dots are acceptable, with the ‘winners’ being the high volume–high income combinations. Another general comment is that the view on risks is truncated. There is not yet any adjustment for credit risk in incomes since gap risk is not credit risk. Hence, credit risk versus income, or the view of Risk-adjusted Return on Capital (RaRoC), provides a global view on the complete risk–return profiles. The conclusion is that the reports of this chapter provide useful views on the sources of gap risk, but they do not serve the purpose of optimizing the portfolio considering all risks. This issue relates more to the credit risk and to the related RaRoC of segments or transactions.

ANALYSIS OF THE VARIATIONS OF INTEREST INCOME Often, it is necessary to explain the changes in the interest margin between two dates. In addition, with projections and budgets, there are deviations between actual interest income and projected values from the previous projections. The sources of variations of interest income are: the change in portfolio structure by level of interest rate; the new business of the elapsed period; the changes in the yield curve. • The amortization of the existing portfolio creates variations of interest income because assets and liabilities earn or pay different interest rates. When they amortize the corresponding revenues or costs disappear. This effect is unrelated to the variations of interest rates. Tracking this variation between any two dates requires having all information on flows resulting from amortization and on the level of rates of each individual asset and liability. To isolate this effect, it is necessary to track the portfolio structure by interest rate level. • The new business creates variations as well, both because of the pure volume effect on income and costs and because of the new rates of these assets and liabilities. New assets and liabilities substitute progressively for those that amortize, but they earn or pay different rates. • The changes in the yield curve create variations of the interest income independent of volume variations of assets and liabilities. Note that the gap provides only the change due to a parallel shift of the yield curve, while commercial banks are sensitive to the slope of the yield curve because of the maturity gaps between interest rates of liabilities and assets. Moreover, the level of interest income depends on the portfolio structure by products and market segments since each of these generates different levels of spread. It becomes necessary to trace back such deviations to the actual versus projected differences between all interest margin drivers, volumes, interest rates and commercial percentage margins. This

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allows us to allocate the predicted–actual margin variance to errors in past interest expectations and to the actual–predicted variances between volumes and percentage margins. This analysis is required to understand the changes in interest margins between two ALM committees. Corrective actions depend on which drivers explain the variations of the interest margin. Using gaps to explain the changes in interest income is not sufficient in general. First, gaps show the sensitivity of the interest income due to a parallel shift of the interest rate used as reference. If we use a unique gap, we ignore non-parallel shifts of the yield curve. With as many gaps as there are interest rates, it is feasible to track the change in interest income due to changes of all interest rates used as references. If we assume no portfolio structure change between two dates, implying that the interest margin as a percentage is independent of volume, the change in interest income between any two dates depends on the shifts of interest rates, the change in volumes and the interaction effect of volume–interest rate. This is the usual problem of explaining variances of costs or revenues, resulting from variances of prices, volumes and product mix. In the case of interest income, there is variation due to the pure volume change, the gap effect due to the variations of the interest rates between the two dates, and an interaction effect. The interaction effect when both price (interest rate) and volume vary results from the simplified equation using a single interest rate: IM = [i × (A − L)] = (A − L) × i + i × (A − L) + i × (A − L) In the equation, IM is the interest margin, (A − L) is the difference between interest rate-sensitive assets and liabilities, or the variable rate gap, and i is the unique interest rate driver. The interaction term results from the change in product between the gap and the interest rate. Table 19.2 shows how these effects combine when there is no portfolio structure effect. TABLE 19.2 The analysis of volume and price effects on the variation of interest margin Actual–predicted variance IM = IM1 − IM0 IM due to interest rate variations at constant margins and volume IM due to volume variations at constant interest rate and margins IM due to price variations (interest rate and margins) combined with volume variations (interaction effect) IM

2.80 −0.90 3.00 0.70

2.80

Note that the calculations depend on the numerical values of the start and end values of rates and volumes. In this example, the 2.80 increase of margin is due to: 1. A decrease of −0.90 due to a negative variable rate interest gap at initial date combined with an interest rate increase. 2. An increase of +3.00 due to the expansion of assets and liabilities between initial and final dates.

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243

3. An increase of +0.70 due to the increased customer rates of new assets and liabilities, resulting from the increase in interest rates and from any variation of commercial margins. In general, there is an additional effect portfolio structure, depending on the amortization of existing assets and liabilities plus the new business. For instance, there are only two assets and a single debt. The two assets earn the rates 10% and 12%, while the debt pays 8%. Total assets are 100, divided into two assets of 50 each, and the debt is 100. At the end of period 1 (between dates 0 and 1), the asset earning 12% amortizes and the bank originates a new loan earning 11%. During period 1, the revenue is 50 × 10% + 50 × 12% − 100 × 8% = 0.050 + 0.060 − 0.080 = 0.030. Even if there is no change in interest rate and overall volume, the interest income of the next period 2 changes due to the substitution of the old loan by the new loan. The revenue becomes 50 × 10% + 50 × 11% − 100 × 8% = 0.050 + 0.055 − 0.080 = 0.025. This change is due to the change in portfolio structure by level of interest rates. This change might be independent of interest rate change if, for instance, the new loan earns less than the first one due to increased competition between lenders.

SECTION 7 Options and Convexity Risk in Banking

20 Implicit Options Risk

This chapter explains the nature of embedded, or implicit, options and details their payoff, in the event of exercise for the individual borrower. Implicit options exist essentially in retail banking and for individuals. A typical example of an implicit option is the facility of prepaying, or renegotiating, a fixed rate loan when rates decline. The borrower’s benefit is the interest costs saving, increasing with the gap between current rates and the loan fixed rate, and with the residual maturity of the loan. The cost for the bank is the mirror image of the borrower’s benefit. The bank’s margin over the fixed rate of debt matching such loans narrows, or eventually turns out negative. Other banking options relate to variable rate loans with caps on interest rates, or to transfers of resources from demand deposits to interest-earning products, when interest rates increase, raising the financial cost to the bank. Embedded options in banking balance sheets raise a number of issues: What is the cost for the lender and the pricing mark-up that would compensate this cost? What is the risk–return profile of implicit options for the lender? What is the portfolio downside risk for options, or downside ‘convexity risk’? How can banks hedge such risks? Because the outstanding balances of loans with embedded options are quite large, several models based on historical data help in assessing the prepayment rates of borrowers. Prepayment models make the option exercise a function of the differential between the rate of loans and the prevailing rates, and of demographic and behavioural factors, which also influence the attitude of individuals. From such prepayments, the payoffs for borrowers and the costs for lenders follow. Payoffs under exercise differ from the value of the option. The value of the option is higher than the payoff as long as there is a chance that future payoffs increase beyond their current value if rates drift further away. This chapter assesses the benefit to the borrower, calculated as the payoff of the option under various interest rate levels. For a long-term fixed rate loan, the option payoff is the time profile of the differential savings (annuities) after and before exercising the

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renegotiation option. Alternatively, it is the present value of these cash savings at the date of renegotiation. The payoff increases with the differential between the initial and the new interest rates and the residual maturity of the loan. The next chapter addresses the valuation of options based on the simulation of representative interest rate scenarios. The first section details the nature of options embedded in various banking and insurance contracts. Subsequent sections illustrate prepayment risk. The second section refers briefly to prepayment models. The last section calculates the payoff of the option under various levels of interest rates, using a simple fixed rate loan as an example.

OPTIONAL RISK: TWO EXAMPLES The most well known options are the prepayment options embedded in mortgage loans. These represent a large amount of the outstanding balances, and their maturity is usually very long. The prepayment option is quite valuable for fixed rate loans because it usually has a very long horizon and significant chances of being in-the-money during the life of the loan. There are other examples of embedded options. Guaranteed rate insurance contracts are similar for insurance companies, except that they are liabilities instead of assets. The prepayment option has more value (for a fixed rate loan) when the interest rate decreases. Prepaying the loan and contracting a new loan at a lower rate, eventually with the same bank (renegotiation), might be valuable to the customer. The prepayment usually generates a cost to the borrower, but that cost can be lower than the expected gain from exercise. For instance, a prepayment penalty of 3% of the outstanding balance might be imposed on the borrower. Nevertheless, if the rate declines substantially, this cost does not offset the gain of borrowing at a lower rate. Prepayments are an issue for the lender if fixed rate borrowings back the existing loans. Initially, there is no interest rate risk and the loan margin remains the original one. However, if the rates decline, customers might substitute a new loan for the old one at a lower price, while the debt stays at the same rate, thereby deteriorating the margin. For insurance contracts, the issue is symmetrical. The insurance companies provide guaranteed return contracts. In these contracts, the customer benefits from a minimum rate of return over the life of the contract, plus the benefit of a higher return if interest rates increase. In order to have that return, the insurance company invests at a fixed rate to match the interest cost of the contract. If the rate increases, the beneficiary can renew the contract in order to obtain a higher rate. Nevertheless, the insurance company still earns the fixed rate investment in its balance sheet. Hence, the margin declines. The obvious solution to prepayments of fixed rate loans is to make variable rate loans. However, these do not offer much protection against a rise in interest rates for the borrower, unless covenants are included to cap the rate. Therefore, fixed rate loans or variable rate loans with a cap on the borrower’s rate are still common in many countries. A theoretical way of limiting losses would be to make the customer pay for the option. This implies valuing the option and that competition allows pricing it to customers. Another way would be to hedge the option risk, and pay the cost of hedging. Caps and floors are adapted for such protections. The lender can hedge the risk of a decrease in the rate of the loan by using a floor that guarantees a minimum fixed return, even though the rate of the new loan is lower. An insurance company can use a cap to set a maximum value on the rate guaranteed to customers in the case of a rise in interest rates. However,

IMPLICIT OPTIONS RISK

249

such hedges have a cost for banks. The underlying issue is to determine the value that customers should pay to compensate the additional cost of options for banks.

MODELLING PREPAYMENTS The modelling of prepayments is necessary to project the gap profiles. There is a significant difference between contractual flows and effective maturity. The purpose of models is to relate prepayment rates to those factors that influence them1 . Prepayment models can also help to value the option. If such models capture the relationship between prepayments and interest rate, it becomes easier to value options by simulating a large number of prepayment scenarios. When prepayments are considered, the expected return of a loan will differ from the original return due to the rollover of the loan at a lower rate at future dates. Models can help to determine the amount subject to renewal, and the actual cost of these loans, as a function of interest scenarios. If multiple scenarios are used, the expected return of the loan is the average over many possible outcomes. The valuation of options follows such lines, except that models are not always available or accurate enough. Models specify how the prepayment rate, the ratio of prepayment to outstanding balances, changes over time, based on several factors. The simplest model is the Constant Prepayment Rate (CPR) model that simply states that the prepayment at any date is the product of a prepayment rate with the outstanding balances of loans (Figure 20.1). The prepayment rate usually depends on the age of the loans. During the early stages of the loans, the prepayment rate increases. Then, it tends to have a more or less stable value for mature loans. Modifying the model parameters to fit the specifics of a loan portfolio is feasible. The simple prepayment rate model captures the basic features of the renegotiation behaviour: renegotiation does not occur in the early stage of a loan because the rates do not drift suddenly away and because the rate negotiation is still recent. When the loan gets closer to maturity, the renegotiation has fewer benefits for the borrower because savings occur only in the residual time to maturity. Therefore, renegotiation rates increase steadily, reach a level stage and then stay there or eventually level off at periods close to maturity. Hence, the models highlight the role of ageing of loans, as a major input of Prepayment rate

First period

FIGURE 20.1 1 See,

Time

‘Constant Prepayment Rate’ models

for example, Bartlett (1989) for a review of prepayment issues and models serving for securitizations of residential mortgages.

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prepayment models, in addition to the rate differentials. This makes it necessary to isolate cohorts of loans, defined by period of origination, to capture the ‘seasoning effect’, or ‘ageing effect’, as well as the rate differential, which varies across cohorts. Other models use such factors as the level of interest rate, the time path of interest rates, the ageing of the loan, the economic conditions and the specifics of customers. These factors serve to fit the modelled prepayment rates to the observed ones. However, such models require a sufficient amount of historical data. An additional complexity is that the time path of interest rates is relevant for options. If the rates decline substantially at the early stage of a loan, they trigger an early prepayment. If the decline occurs later, the prepayment will also occur later. Therefore, the current level of interest rates is not sufficient to capture timing effects. The entire path of interest rates is relevant. This is one reason why the prepayment rate does not relate easily to the current interest level, as intuition might suggest. The valuation of the option uses models of the time path of rates and of the optimum prepayment or renegotiation behaviour (see Chapter 21).

GAINS AND LOSSES FROM THE PREPAYMENT OPTION In this section, an example serves to make explicit the gains for the borrower and the losses for the lender. The gain is valued under the assumption of immediate exercise once the option is in-the-money. In reality, the option is not exercised as soon as it becomes in-the-money. The exercise behaviour depends on future expected gains, which can be higher or lower than the current gain from immediate exercise. This is the time value of the option. The next chapter captures this feature using a valuation methodology of options. The example used is that of a fixed rate and amortizing loan, repaid with constant annuities including interest and capital. The customer renews the loan at a new rate if he exercises the prepayment option. The loan might be representative of a generation of loan portfolios. In actual portfolios, the behaviour of different generations of loans (differing by age) will differ. The gain from prepayment results from the characteristics of the new loan. The new loan has a maturity equal to the residual maturity of the original loan at the exercise date. The decision rule is to prepay and renew the debt as soon as the combined operation ‘prepayment plus new loan’ becomes profitable for the borrower. If the borrower repays the loan, he might have penalty costs, for instance 3% of the outstanding balance. The gain from prepayment results from the difference between the annuities of the original and the new loan plus the penalty cost. It spreads over the entire time profile of differential annuities between the old and the new loan until maturity. Either we measure these differences or we calculate their present value. The original loan has a 12% fixed rate, with an amount of 1000, and with a maturity of 8 years. Table 20.1 details the repayment schedule of the loan. The repayment or renegotiation occurs at date 4, when the interest rate decreases to 10%. The new annuities result from the equality between their present value, calculated as of date 4, and the amount borrowed. The amount of the new loan is the outstanding balance at date 4 of the old loan plus the penalty, which is 3% of this principal balance. It is identical to the present value of the new annuities calculated at the date of renewal at the 10% rate until maturity: Present value of future annuities at t = 4, at 10% = outstanding balance at date 4 × (1 + 3%)

IMPLICIT OPTIONS RISK

TABLE 20.1

251

Repayment schedule of the original loan

Amount 1000 Date

0

Maturity 8 years 1

Discount rate 12% 3 4 5

2

6

Annuity 201.30 7

8

a

PV at 12% 1000 Annuity (rounded) 201.30 201.30 201.30 201.30 201.30 201.30 201.30 201.30 Principal repayment 81.30 91.06 101.99 114.22 127.93 143.28 160.48 179.73 Outstanding balance 918.70 827.64 725.65 611.43 483.50 340.21 179.73 0.00 Discount rateb 12.00% a PV

is the present value of future annuities. discount rate is the rate that makes the present value of future annuities equal to the amount borrowed. It is equal in this case to the nominal rate (12%) used to calculate interest payments. b The

The gain for the borrower is the present value, as of date 4, of the difference between the old and new annuities. The present value of the old annuities at the new rate is the market value of the outstanding debt at date 4. The value of the old debt becomes higher than the outstanding balance if the rate decreases. This change is a gain for the lender and a loss for the borrower. The economic gain from prepayment, for the borrower, as of the rate reset date T , is equal to the difference between the present values of the old and the new debts: PVT ,new rate (old debt) − PVT ,new rate (new debt) = PVT ,new rate (old debt) − outstanding balance × (1 + 3%) PVT ,new rate is the present value at the prepayment date (T = 4) and at the new rate (10%). The calculation of both market values of debt and annuities is shown in Table 20.2. The new annuity is 198.75, lower than the former annuity of 201.30, in spite of the additional TABLE 20.2

Gains of prepayment for the borrower

Amount Initial rate Original maturity in years Annuity Schedule Date

1000 12.00% 8 201 1

2

Principal repayment 81.30 91.06 Outstanding balance 918.70 827.64 Penalty (3%) 0 0 New debt 0 0 Old annuity 201.30 201.30 New annuity 0 0 Annuity profile 201.30 201.30 Differential flows (new–old) 0 0 Gain for the borrower Present value at date 4 at the 10% rate PV old debt PV new date Present value of gain at 4

Penalty Date of prepayment New rate

3.00% 4 10.00%

3

4

5

6

7

8

101.99 725.65 0 0 201.30 0 201.30 0

114.22 611.43 18.34 629.77 201.30 0 201.30 0

127.93 483.50 0 0 201.30 198.75 198.75 2.63

143.28 340.21 0 0 201.30 198.75 198.75 2.63

160.48 179.73 0 0 201.30 198.75 198.75 2.63

179.73 0.00 0 0 201.30 198.75 198.75 2.63

638.10 629.77 8.33

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cost due to the 3% penalty. The annual gain in value is 2.63. The present value of this gain, at date 4, and at the new rate (10%), is 8.33. The borrower gain increases when the new rate decreases. Above 12%, there is no gain. Below 12%, the gain depends on the difference between the penalty in case of prepayment and the differential gain between the old and the new loans. The gain can be valued at various levels of the new interest rate (Table 20.3). Table 20.3 gives the value of the old debt at the current new rate. The outstanding balance at date 4 is 611.43. The market value of the debt is equal to 611.43 only when the rate is 12%. The exercise price of the option is equal to the prepayment amount, which is the outstanding balance plus the 3% penalty, or 611.43(1 + 3%) = 629.77, rounded to 630. This price is determined by the prepayment date since the amortization schedule is contractual. The present value of gains, at date 4, is the difference between the market value of the old debt and 630, which is the present value of the new debt. The exercise value of the option is the maximum of 0 and the market value of the old debt minus 630. It is given in the last column of Table 20.3. TABLE 20.3 New rate

Borrower gains from prepayment at date 4 PV(old debt) at 4 12% (a)

PV(new debt) at 4 (b)

698 682 667 652 638 625 611 599 587

630 630 630 630 630 630 630 630 630

6% 7% 8% 9% 10% 11% 12% 13% 14%

PV of gains at 4 (c = b − a)

Exercise value of the option at 4 (d = max[0, c])

68 52 37 22 8 −5 −18 −31 −43

68 52 37 22 8 0 0 0 0

Gain from the option 80 60 40 20 0 6%

FIGURE 20.2

7%

8%

9%

10% 11% 12% 13% 14% Interest rate

Payoff profile of the prepayment option

The borrower gains are higher when the interest rate decreases. If the rate increases, the option has a zero exercise value. The profile of payoffs as a function of interest rate has the usual shape of the payoff profile of an option. Positive gains appear when the decline in interest rate below 12% is sufficient to offset the penalty (Figure 20.2).

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The same profile is the cost of the exercise of the option for the lender. This cost becomes very important when the decline in rates is significant. If the rate moves down to 8%, the present value of the loss for the bank reaches 37. This is 3.7% of the original capital, or 37/667 = 5.5% of the outstanding capital at date 4. This example is simplified, but it demonstrates the importance of valuing options. The next chapter provides an example of the valuation of a prepayment option.

21 The Value of Implicit Options

The value of an option combines its liquidation value, or the payoff under exercise, plus the value of waiting further for larger payoffs. Valuing an option requires simulating all future outcomes for interest rates, at various periods, to determine when the option is in-the-money and what are the gains under exercise. The option value discounts the expected future gains, using the simulated rates as discount rates. Various interest rate models potentially apply to the valuation of options1 . The purpose of this chapter is to illustrate how they apply for valuing prepayment options. The value of options depends on the entire time path of the interest rates from origination to maturity, because interest rate changes can trigger early or late exercise of the option. This is a significant departure from classical balance sheet Asset–Liability Management (ALM) models, which use in general a small number of scenarios. The valuation of options necessitates simulating the future time paths of interest rates and deriving all future corresponding payoffs. The interest simulation technique applies because it generates the entire spectrum of payoffs of the option up to a certain horizon, including all intermediate values. The valuation is an average of all future payoffs weighted by probabilities of occurrences. In this chapter, we use the simple technique based on ‘binomial trees’. Other examples of time path simulations are given in Section 11 of this book. From a risk perspective, the issue is to find the expected and worst-case values of the option from the distribution of its future values at various time points. From a pricing perspective, the option is an asset for the borrower and it generates an expected cost for the bank, which the bank should charge to borrowers. The worst-case value makes sense in a ‘Value at Risk’ (VaR) framework, when considering worst-case losses for determining 1 The

bibliography is rather exhaustive and technical. A good introduction to the valuation of options is Hull (2000).

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economic capital. The sections on Net Present Value (NPV) and optional risk develop this view further. The expected value of options relates to pricing. Pricing optional risk to borrowers requires converting a single present value as of today into a percentage mark-up. The mark-up reflects the difference in values between a straight fixed rate debt, that the borrower cannot repay2 , and a ‘callable debt’, which is a straight debt combined with the prepayment option. The callable loan is an asset for the lender that has a lower value than the straight debt for the bank, the difference being the option value. Conversely, the callable loan is a liability for the borrower that has a lower value for him because he can sell it back to the lender (he has a put option). The percentage mark-up is the ‘Option-Adjusted Spread’ (OAS). The OAS is the spread added to the loan rate making the value of the ‘callable’ loan identical to the value of a straight loan. The callable debt, viewed as an asset held by the bank, has a lower value than the straight debt for the lender. To bring the value of the straight debt in line with the value of the debt minus the value of the option, it is necessary to increase the discount rates applied to the future cash flows of the debt. The additional spread is the OAS. For pricing the option, banks add the OAS to the customer rate, if competition allows. The first section summarizes some basic principles and findings on option valuation. The second section uses the binomial tree technique to simulate the time path of interest rates. The third section derives the value of the prepayment option from the simulated ‘tree’ of interest rate values. The last section calculates the corresponding OAS.

RISK AND PRICING ISSUES WITH IMPLICIT OPTIONS The valuation process of implicit options addresses both issues of pricing and of risk simultaneously. From a pricing perspective, the expected value of the option should be a cost transferred to customers. From a risk perspective, the issue is to find the worst-case values of options. In fact, the same technique applies to both issues, since valuation requires simulation of the entire time paths of interest rates to find all possible payoffs. The valuation issue shows the limitations of the discrete yield curve scenarios used in ALM. The ALCO would like to concentrate on a few major assumptions, while the option risk requires considering a comprehensive set of interest rate time paths. Pricing an option as the expected value of all outcomes requires finding all its potential values under a full range of variations of interest rates. This is why the two blocks of ‘ALCO simulations’ and ‘options valuation’ often do not rely on the same technique. The value of an option is an expected value over all future outcomes. The interest rate risk is that of an increase in that value, since it is a loss for the bank, measured as the loss percentile at a forward horizon. Simulating all interest rate scenarios up to the forward horizon and after provides both. 2 The

terminology is confusing. In fact the ‘callable loan’ is a straight fixed rate loan plus a put option to sell it at a lower than market value when interest rates decline. The value of the loan for the bank (lender) is that of the straight debt minus the put value, which increases when rates decline. It is more like a ‘putable loan’.

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The value of an interest rate option depends on several parameters. They include the interest rate volatility, the maturity of the option and the exercise price, the risk-free rate. There are many option valuation models. Black’s (1976) model allows pricing interest rate options. The model applies to European options, and assumes that the underlying asset has a lognormal distribution at maturity. In the case of loan prepayments, the borrower has the right to sell to the bank the loan for its face value, which is a put option on a bond. Black’s model assumes that the bond price is lognormal. This allows us to derive formulas for the option price. In this chapter, we concentrate on numerical techniques, which allow us to show the entire process of simulations, rather than using the closed-form formula. For long-term options, such as mortgage options, the Heath, Jarrow and Morton model (Heath et al., 1992) allows more freedom than previous models in customizing the volatility term structure, and addressing the need to periodically revalue long-term options such as those of mortgage loans. They require using Monte Carlo simulations. However, such advanced models are more difficult to implement. Simple techniques provide reasonable estimates, as shown in the example. We concentrate on ‘American’ options allowing exercise at any time between current date and maturity. As soon as the option provides a positive payoff, the option holder faces a dilemma between waiting for a higher payoff and the risk of losing the current option positive liquidation value because of future adverse deviations. The choice of early or late exercise depends on interest rate expectations. The risk drivers of options are the entire time paths of interest rates up to maturity. The basic valuation process includes several steps, which are identical to those used for measuring market risk VaR, because this implies revaluing a portfolio of market instruments at a forward date. In the current case, we need both the current price for valuing the additional cost charged to borrowers and the distribution of future values for valuing downside risk. The four main steps addressing both valuation as of today and forward downside risk are: • Simulate the stochastic process of interest rates, through models or Monte Carlo simulations or an intermediate technique, such as ‘binomial trees’, explained below. • Revalue the option at each time point between current date and horizon. • Derive the ‘as of date value’ from the entire spectrum of future values and their probabilities. The current valuation addresses the pricing issue. • When the range of values is at a forward date, the process allows us to find potential deviations of values at this horizon. This technique serves for measuring the downside risk rather than the price. Intermediate techniques, such as binomial trees, are convenient and simple for valuing options3 . To illustrate the technique, the following section uses the simplest version of the common ‘binomial’ methodology. The technique of interest rate trees is a discrete time representation of the stochastic process of a short-term rate, such as those mentioned in Chapter 30. 3 There

are variations around the basic original binomial model. Hull and White (1994, 1996) provided techniques for building ‘general trees’ which comply with various modelling constraints on interest rates for incorporating mean-reversion. Mean-reversion designates the fact that interest rates tend to revert to their long-term mean when they drift along time. Some additional details on interest rate models are given in Chapter 31.

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A SIMPLE ‘BINOMIAL TREE’ TECHNIQUE APPLIED TO INTEREST RATES We use the simplest possible model of interest rates4 for illustration purposes. We limit the modelling of interest rates to the simple and attractive ‘binomial’ model. The ‘binomial’ name of the technique refers to the fact that the interest rate can take only two values at each step, starting from an initial value.

The Binomial Process Each ‘step’ is a small interval of time between two consecutive dates t and t + 15 . Given a current value of the rate at date t, there are only two possible values at t + 1. The rate can only move up or down by a fixed amount. After the first step (period), we repeat the process over all periods dividing the horizon of the simulations. The shorter this period, the higher the number of steps6 . Starting from a given initial value, the interest rate moves up or down at each step. The magnitudes of these movements are u and d. They are percentage coefficients applied to the starting value of the rate to obtain the final values after one step. If it is the interest rate at date t: it+1 = u × it

or

d × it

It is convenient to choose d = 1/u. Since u × d = 1, an up step followed by a down step results in a previous value, minimizing the number of nodes. The rate at t + 1 is u × rate(t) or d × rate(t). With several steps, the tree looks like the chart in Figure 21.1. In addition to u and d, we also need to specify what is the probability of an up and of a down move.

i(0) u x u i(0) u i(0) u x d

i(0) i(0) d

i(0) d x d

FIGURE 21.1 4 See

The binomial tree of rates

Cornyn and Mays (1997) for a review of interest rate models and their applications to financial institutions. literature uses a small interval t, which tends towards zero. 6 The binomial approach to options is expanded in Cox et al. (1979) and Cox and Rubinstein (1985).

5 The

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Natural and ‘Risk-neutral’ Probabilities The ‘binomial tree’ of rates requires defining the values of the ‘up’ and ‘down’ moves, u and d, and of their probabilities consistent with the market. Finding the appropriate values requires referring to risk-neutrality. This allows us to derive u and d such that the return and volatility are in line with the real world behaviour of interest rates. The financial theory makes a distinction between actual, or ‘natural’, probabilities and the so-called ‘risk-neutral’ probabilities. Risk-neutrality

Risk-neutrality implies indifference between the expected value of an uncertain gain and a certain gain equal to this expected value. Risk aversion implies that the value of the uncertain gain is lower than its expected value. Risk-neutrality implies identity between the expected outcome and the certain outcome (see Chapter 8 for brief definitions). The implication is that, under risk-neutrality, all securities should have the same risk-free return. There exists a set of risk-neutral probabilities such that the expected value equals the value under risk aversion. Intuitively, the risk-neutral probabilities are higher for downside deviations than actual probabilities. For instance, if an investor values a bet providing 150 and 50 with the same probability, his expected gain is 100. Under risk aversion, the value might fall to 90. This value under risk aversion is the expected value under riskneutral probabilities. This implies that the risk-neutral probability of the down move is higher than the actual natural downside probability, 0.5. The risk-neutral probability of the down move is p ∗ such that p ∗ × 50 + (1 − p ∗ ) × 150 = 90, or p ∗ = 0.6. The Parameters of the Binomial Model

Risk-neutrality implies that there exists a set of risk-neutral probabilities, p ∗ and 1 − p ∗ , of up and down moves such that: • The return is the risk-free rate r, or the expectation of the value after one period is 1 + r. • The volatility of distribution of the ‘up’ value and the ‘down’ value is identical to the √ market volatility σ t for a small period t. The corresponding equations are: p ∗ u + (1 − p ∗ )d = 1 + r

Resulting in:

[p ∗ u2 + (1 − p ∗ )d 2 ] − [p ∗ u + (1 − p ∗ )d]2 = σ 2 t √ u = exp[σ t]

and

√ d = exp[−σ t]

The derivation of these equations is given in the appendix to this chapter.

Using Numerical Data In the example used in this section, the yearly volatility is 15% and the steps correspond to a 1-year period. The numerical values selected for u and d are:

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259

u = exp(15%) = 1.1618

and

d = exp(15%) = 0.8607

With the above numerical values, the sequence of rates is that of Table 21.1. The riskneutral probabilities are simply 50% and 50% to make calculations more tractable. This implies that the real world probability of a down jump in interest rate is lower than 50% since the actual risky return should be higher than the risk-free rate in the real market. TABLE 21.1 values

The tree of numerical

Dates

0

1

2

Rates

10.00%

11.62% 8.61%

13.50% 10.00% 7.41%

After two up and down steps, the value at date 2 is the initial value, 10%, since u = 1/d.

VALUING DEBT AND OPTIONS WITH SIMULATED RATES In this section, we apply the binomial tree technique to value a risky debt. This implies ensuring that the binomial tree is consistent with market prices. To meet this specification, we need to proceed with various practical steps.

Value of Debt under Uncertainty Any asset generates future flows and its current value is the discounted value of those flows. The simulated rates serve for discounting. For instance, a zero-coupon bond generates a terminal flow of 100 at date 2. Its current value is equal to the discounted value of this flow of 100 using the rates generated by the simulation. There are as many possible values as there are interest rate paths to arrive at date 2. In a no-uncertainty world, the time path of rates is unique. When rates are stochastic, the volatility is positive and there are numerous interest rate paths diverging from each other. Each sequence of rates is a realization of uncertain paths of rates over time. For each sequence of rates, there is a discounted value of this stream of flows. With all sequences, there are as many discounted values as there are time paths of rate values. The value of the asset is the average of these values assuming that all time paths of rates are equally probable7 . The simple zero-coupon bond above illustrates the calculation process. The terminal flow of 100 is certain. Its present value as of date 1 uses the rate at period 2. Its present value as of date 0 uses the rate of period 1, which is the current rate. There are only two possible sequences of rates in this example, and hence two possible values of the same asset: 100/[(1 + 10.00%)(1 + 11.62%)] = 81.445 100/[(1 + 10.00%)(1 + 8.61%)] = 83.702 7 The maximum number of time paths is 2n where n is the number of periods, since at each date there are twice as many nodes as at the preceding date.

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The average value is (81.445 + 83.702)/2 = 82.574, which is the expected value given the uncertainty on interest rates (Table 21.2). TABLE 21.2 Values of the bond at various dates Dates

0

1

2

Values

82.574

89.590 92.073

100 100 100

Another technique for calculating this value serves later. At date 2, the value of the flow is 100 for all time paths of rates. However, this flow has two possible values at date 1 that correspond to the two possible values of the rates: 100/(1 + 11.62%) = 89.590 and 100/(1 + 8.61%) = 92.073. The current value results from discounting the average value at date 1, using the current rate, which gives [(89.590 + 92.073)/2]/(1 + 10.00%) = 82.574. The result is the same as above, which is obvious in this case. The benefit of this second method lies in calculating a current value using a recursive process starting from the end. From final values, we derive values at preceding dates until the current date. This second technique illustrates the mark-to-future process at date 1. In this case, there are two possible states only at date 1. From this simple distribution, we derive the expected value and the worst-case value. The expected value at date 1 is the average, or (89.590 + 92.073)/2 = 91.012. The worst-case value is 89.590. The binomial model is general. The only difference between a straight debt and an option is that the discounted flows of the option change with rates, instead of being independent of interest rate scenarios. Since it is possible to use flows defined conditionally upon the value of the interest rates, the model applies.

The Global Calibration of the Binomial Tree The value of listed assets, calculated with all the time paths of interest rates, should replicate those observed on the market. In order to link observed prices and calculated prices, the rates of each node of the tree need an adjustment, which is another piece of the calibration process. The binomial tree above does not yet include all available information. The rate volatility sets the upward and downward moves at each step. However, this unique adjustment does not capture the trend of rates, any liquidity premium or credit spread observed in markets, and the risk aversion making expected values less than actual values. All factors should show up in simulated rates as they do in the market. Instead of including progressively all factors that influence rates, it is easier to make a global calibration. This captures all missing factors in the model. In the above example, the zero-coupon value is 82.574. Its actual price might diverge from this. For instance, let us assume that its observed price is 81.5. The simulated rates need adjustment to reconcile the two values. In this example, the simulated rates should be above those used in the example, so that the simulated price decreases until it reaches the observed price. There is no need to change the volatility, since the up and down moves already replicate the market volatility for all rates individually. In the case of zero-coupon maturity at date 2, the initial rate value is 10%, and the rates at period 2 do not count.

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261

Hence, only rates as of date 1 need adjustment. The easiest way to modify them is to add a common constant to both rates at date 1. The constant has an empirical value such that it matches the calculated and observed prices. This constant is the ‘drift’. In the example, the drift should be positive to increase the rate and decrease the discounted value. Numerical calculation shows that the required drift is around 1.40% to obtain a value of 81.5. The calibrated rates at 1 become: i1u = 11.62% + 1.40% = 13.02% and i1d = 8.61% + 1.40% = 10.01%. Once the rates at date 1 are adjusted, the calibration extends to subsequent periods using the prices of listed assets having longer maturity. It is necessary to repeat this process to extend the adjustment of the whole tree. This technique enforces the ‘external’ consistency with market data.

The Valuation of Options In order to obtain an option value, we start from its terminal values given the interest rate values simulated at this date. Once all possible terminal values are determined from the terminal values of rates, the recursive process serves to derive the values at other dates. For American options, whose exercise is feasible at any time between now and maturity, there is a choice at any date when the option is in-the-money. Exercising the option depends on whether the borrower is willing to wait for opportunities that are more profitable or not. An American option has two values at each node (date t) of the tree: a value under no exercise and the exercise value. The value when waiting, rather than exercising immediately, is simply the average of the two possible values at the next date t + 1, just as for any other asset. Under immediate exercise, the value is the difference between the strike price and the gain. The optimum rule is to use the maximum of these two values. With this additional rule, the value of the options results from the same backward process along the tree, starting from terminal values.

THE CURRENT VALUATION OF THE PREPAYMENT OPTION This methodology applies to a loan amortized with constant annuities, whose maturity is 5 years and fixed rate 10%. The current rate is also 10%. The yearly volatility of rates is 15%. The borrower considers a prepayment when the decline in rates generates future savings whose present value exceeds the penalty of 3%. However, being in-the-money does not necessarily trigger exercise, since immediate prepayment may be less profitable than deferred prepayment. The borrower makes the optimal decision by comparing the immediate gain with the expected gains of later periods. The expected gain is the expected value of all discounted gains one period later, while the immediate gain is the exercise value. The process first simulates the rate values at all dates, then revalues the debt accordingly, calculates the strike price as 1.03 times the outstanding principal, and calculates the payoff under immediate exercise. The last step is to calculate the optimum payoff at each node and value the present value of these optimum payoffs to get the option value.

Simulating Rates Equal yearly periods divide the future. The binomial tree starts at 10%. However, there is a need to adjust the rates with a uniform drift of −0.10% applied to all rates to find the exact value 1000 of the straight debt under interest rate uncertainty. The values of u

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TABLE 21.3

The binomial tree of rates (5 years, 1-year rate, initial rate 10%)

Dates (Eoy) Ratesa

0

1

2

3

4

5

9.90%

11.52% 8.51%

13.40% 9.90% 7.31%

15.59% 11.52% 8.51% 6.28%

18.12% 13.40% 9.90% 7.31% 5.39%

21.07% 15.59% 11.52% 8.51% 6.28% 4.63%

a

In order to find exactly the value of 1000 when moving along the ‘tree’ of rates, it is necessary to adjust all rates by deducting a constant drift of −0.10%. Using the original rates starting at 10% results in a value of the debt at date 0 higher than 1000 because we average all debt values which are not a linear function of rates. Instead of starting at 10% we start at 10% − 0.10% = 9.90%. The same drift applies to all rates of the tree. This uniform drift results from a numerical calculation.

and d correspond to a yearly volatility of 15%. The binomial tree is the same as above, and extends over five periods. Dates are at ‘end of year’ (Eoy), so that date 1 is the end of period 1 (Table 21.3).

The Original Loan Since dates refer to the end of each year, we also need to specify when exactly the annuity flow occurs. At any date, end of period, the borrower pays immediately the annuity after the date. Hence, the outstanding debt includes this annuity without discounting. The date 0 is the date of origination. The terminal date 5 is the end of the last period, the date of the last annuity, 263.80, completing the amortization. Hence, the last flow, 263.80, occurs at date 5. The value of existing debt is exactly 263.80 before the annuity is paid. Table 21.4 shows the original repayment schedule of the loan. The constant annuity is 263.80 with an original rate of 10%. TABLE 21.4

Repayment schedule of loan

Dates (Eoy) Loan Annuities Capital reimbursement Outstanding balance

0

1

2

3

4

5

263.80 163.80 836.20

263.80 180.18 656.03

263.80 198.19 457.83

263.80 218.01 239.82

263.80 239.82 0.00

1000

The present value of all annuities at a discount rate of 10% is exactly 10008 .

The Market Value of a Loan, the Exercise Price and the Payoff of the Option Given the above strike price, the borrower can repurchase the debt and contract a new debt at the new rate. The market value of the old debt varies with interest rates. This market 8 The

1000 values discount all annuities of 263.80, starting at Eoy 1, so that: 1000 =

5

t=1 263.80/(1

+ i)t .

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263

value is the underlying of the prepayment option. It is the present value of all subsequent annuities, given the market rate. The rate values of each node serve to calculate this time profile of the present value. The gain from prepayment is the difference between the current value of the debt and the current strike price (Table 21.5). TABLE 21.5

Market value of loan, exercise price and payoff from immediate exercise

Dates

0

1

Value at origination

1000.00

1071.72 1126.34

Principal + 3%

1030.00

836.20

Gains

0.00

2

3

Valuation: 881.73 689.25 920.29 712.25 951.60 730.78 745.41

Exercise price: 656.03 457.83

Payoff from immediate exercise:a 0.00 0.00 0.00 26.34 0.47 0.00 31.78 9.15 23.79

4

5

487.12 496.42 503.82 509.62 514.10

263.80 263.80 263.80 263.80 263.80 263.80

239.82

0.00

0.00 0.00 0.21 6.01 10.49

0.00 0.00 0.00 0.00 0.00

a Value

= max(market value of debt − strike, 0). The strike is the outstanding balance plus a penalty. The above values are rounded at the second digit: the payoff 26.34 at date 1 is 1126.34 − 263.80 − 836.20 = 26.34.

As an example, the following calculations are detailed at date 1, when the rate is 8.51% (line 2 and column 1 of the binomial tree and of the debt value table): the value of the debt; the exercise price; the payoff from exercise; the value of the option. Value of Debt

The recursive process starting from the terminal values of the debt applies to the calculation of the debt value along any time path of rates. At date 4, the debt value discounts this 263.80 flow with the interest rates of the binomial tree. The same process applies to all previous dates until date 0. At any intermediate date, the value of the debt includes the annuity of the period, without discounting, plus the average discounted value of the debt at date 2 resulting from previous calculations. This value is: 1126.34 = 263.80 + 0.5(920.29 + 951.60)/(1 + 8.51%) The Payoff from Exercise of the Option

The outstanding balance is the basis for calculating the penalty, set at 3% at any date. The strike price of the option is 1.03 times the outstanding balance at all dates before maturity. The gain, when exercising the prepayment option, is the difference between the

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market value of existing debt and the exercise price, which is the value of new debt plus penalty. The time profile of the exercise price starts at 1030 and declines until maturity. This difference between market value of debt and strike price is a gain. Otherwise, the exercise value of the option is zero. A first calculation provides the gain with immediate exercise, without considering further opportunities of exercise. For instance, at date 1 the value of debt, after payment of the annuity, is 1126.34 − 263.80 = 862.54. The exercise price is 836.20. The liquidation value of the option is 862.54 − 836.20 = 26.34. In many instances, there is no gain, the option is ‘out-of-the-money’ and the gain is zero.

Valuation of the Option under Deferred Exercise The immediate gain is not the optimum gain for the borrower since future rate deviations might generate even bigger gains. The optimum behaviour rule applies. The expected value of deferred gains at the next period is the average of their present values. Finally, the value of the option at t is the maximum between immediate exercise and the present value of expected gains of the next period, or zero. This behavioural rule changes the time profile of gains. The rounded present value of the option as of today is 12.85 (Table 21.6). TABLE 21.6 Dates Value of option

a Option

The expected gains from the prepayment optiona 0

1

2

3

4

5

12.85

1.90 26.34

0.04 4.21 31.78

0.00 0.09 9.15 23.79

0.00 0.00 0.21 6.01 10.49

0.00 0.00 0.00 0.00 0.00 0.00

value = max(exercise value at t, expected exercise values at t + 1 discounted to t, 0).

The value of the prepayment option is 12.85, for a loan of 1000, or 1.285% of the original value of the loan.

CURRENT VALUE AND RISK-BASED PRICING Given the value of the option, pricing requires the determination of an additional markup corresponding to the expected value to the borrower. This mark-up, equivalent to the current value of the option, is the OAS. The original loan has a value of 1000 at 10%, equal to its face value when considering a straight debt without the prepayment option. The value of the debt with the prepayment option is equal to the value of the straight debt less the value of the option given away, or 1000 − 12.85 = 987.15. For the lender, the loan is an asset whose value is less than that of a straight debt of 1000 at 10%, the difference being the value of the prepayment option. For the borrower, the loan is a liability whose value is also lower than the 1000 at 10%. The lender gave away a right whose value is 12.85.

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265

This cost is 1.285% of the current loan principal. With a current rate of 10%, the ‘debt plus option’ package has a value of 987.15, or only 98.715% of the value of the debt without option. This decline in loan value by the amount of the option value should appear under ‘fair value’ in the balance sheet. It represents the economic value of the debt, lower than the book value by that of the option given away by the bank. In this case, fair value is a better image than book value, because it makes explicit the losses due to prepayments rather than accruing losses later on when borrowers actually repay or renegotiate their loan rate. For practical purposes, banks lend 1000 when customer rates are 10%. In fact, the customer rate should be above this value since the 10% does not account for the value of the option. Pricing this prepayment risk necessitates a mark-up over the straight debt price. This periodical mark-up is an additional margin equivalent to the value of the option. The spread required to make the value of the debt alone identical to that of the debt with the option is the OAS. To determine the OAS, we need to find the loan rate making the loan value less the option value equal to 1000. The straight loan value is 1000 + 12.85 = 1012.85. In order to decrease this value to 1000, we need to increase the yield of the loan. This yield is 10.23%. At this new yield, the annuity is 265.32 instead of 263.80. This is a proxy calculation. The OAS is 23 basis points (Table 21.7). TABLE 21.7

Debt 1000 Current rate 10.00%

Summary of characteristics of the loan A: Loan without prepayment option Rate 10.00% B: Loan with prepayment option Value of option Value of debt without option 12.85 1000 Maturity 5

Annuity 263.80 Value of debt with option 1012.85

OAS 0.23%

The methodology used to determine the OAS is similar to that of calibration. However, the goal differs. Calibration serves to link the model to the market. OAS serves to determine the cost of the option embedded in a loan and the required mark-up for pricing it to the borrower.

THE RISK PROFILE OF THE OPTION To determine the risk profile of the option at a given horizon, we need its distribution. The simulation of interest rate paths provides this distribution. In this simple case, the up and down probabilities are 0.5 each. From these up and down probabilities, we can derive the probabilities of reaching each node of the tree. At any future horizon, the losses are simply the payoffs to the option holder, under exercise. Using the end of period 3 as a horizon, the probabilities of each payoff depend on how many paths end up with the same value of the debt, remembering that several paths can end up at the same value, because each ‘up and down’ combination results in the same final value as a ‘down and up’ combination. At date 3, there are four values of the option payoff. There are more than four time paths leading to these four different values of rates at date 3. The probability of a single time path for three consecutive periods is

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0.53 = 12.5% because we use a constant up and down probability of 0.5. When three time paths lead to the same node, the probability is 3 × 12.5% = 37.5%. At date 3, we have one single path leading to each extreme node and three time paths leading to each of the two intermediate nodes. The probability distribution of reaching each of the four nodes follows. For each node, we have the payoff of the option, which equates the loss for the bank. The distribution of the payoffs, or the bank losses, follows (Table 21.8). TABLE 21.8 option

Payoff, or bank’s cost, distribution from an

Date

Probabilities 0.53 3 × 0.53 3 × 0.53 0.53

Total

Bank’s cost = option payoff

12.5% 37.5% 37.5% 12.5%

0.00 0.09 9.15 23.79

100%

The lowest value is 23.79 and the worst-case value at the 75% confidence interval is 9.15. Detailed simulations of interest rates would lead to a more continuous distribution. The loss distribution is highly skewed to the left, as for any option, with significant losses for two nodes (9.15 and 23.79) (Figure 21.2).

Probability

40% 30% 20% 10% 0%

Loss 0

FIGURE 21.2

10

20

30

Standalone risk of the implicit option

Extending the simulation to the entire banking portfolio implicit options would provide the VaR due to option risk. The NPV simulations derive such an ALM VaR for optional risk, as illustrated in Chapter 25.

APPENDIX: THE BINOMIAL MODEL The natural probability p is that of an up move, while 1 − p is the probability of a down move: p = 1 − p = 50%. Similar definitions apply to p∗ and 1 − p ∗ in a risk-neutral world. There is a set of risk-neutral probabilities p ∗ such that the expectation becomes equal to 90: p ∗ × 150 + (1 − p∗ ) × 50 = 90 or: p ∗ = (90 − 50)/(150 − 50) = 40%

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The risk-neutral probability p ∗ differs from the natural probability p = 50% of the real world because of risk aversion. Generalizing, the expected value of any security using risk-neutral probabilities is E ∗ (V ), where E is the expectation operator and ∗ stands for the risk-neutral probability in calculating E ∗ (V ). This risk-neutral expected value is such that it provides the risk-free return by definition. Since we know the risk-free return r, we have a first equation specifying that u and d are such that E ∗ (V ) = 1 + r for a unit value. The expectation of the value after one time interval is: p × u + (1 − p) × d = 1 + r. This is not enough to specify u and d. We need a second equation. A second specification of the problem is that the volatility should mimic the actual volatility of rates. The volatility in a risk-neutral world is identical to the volatility in the real world. Given natural or risk-neutral probabilities, the variance over a small time interval t of the values is9 : [p ∗ u2 + (1 − p ∗ )d 2 ] − [p ∗ u + (1 − p ∗ )d]2 = σ 2 t This equality holds with both p and p ∗ . Under risk-neutrality, the expected return is the risk-free return r. Under risk aversion, it has the higher expected value µ. Therefore, p and p ∗ are such that: p = (eµt − d)/(u − d)

p ∗ = (ert − d)/(u − d)

Using p ∗ × u + (1 − p ∗ ) × d = 1 + r and p ∗ = (ert − d)/(u − d), we find the values of u and d: √ √ u = exp[σ t] and d = exp[−σ t] The yearly volatility is σ and t is the unit time for one step, say in years. Hence, u and d are such that the volatility is the same in both real and risk-neutral worlds. However, the return has to be the risk-free return in the risk-neutral world and the expected return, above r, in the real world.

9 The

variance of a random variable X is V (X) = E(X2 ) − [E(X)]2 .

SECTION 8 Mark-to-Market Management in Banking

22 Market Value and NPV of the Balance Sheet

The interest income is a popular target of interest rate risk management because it is a simple measure of current profitability that shows up directly in the accounting reports. It has, however, several drawbacks. Interest income characterizes only specific periods, ignoring any income beyond the horizon. Measuring short-term and long-term profitability provides a more comprehensive view of the future. The present values of assets and liabilities capture all future flows generated by assets and liabilities, from today up to the longest maturity. The mark-to-market value of the balance sheet discounts all future flows using market discount rates. It is the Net Present Value (NPV) of the balance sheet, since it nets liability values from asset values. The calculation of an NPV does not imply any assumption about liquidity of loans. It is a ‘fair’ or ‘economic’ value calculation, not the calculation of a price at which assets could trade: • It differs from the ‘fair values’ of assets by the value of credit risk of borrowers, because it uses market rates, without differentiating them according to the risk of individual borrowers. • It also differs from the value of equity because it considers the balance sheet as a portfolio of fixed income assets, long in loans and short in the bank’s liabilities, exposed to interest rate risk only, while the stock price depends on equity market parameters and is subject to all bank risks. To put it more simply, NPV can be negative, while stock prices cannot. • The NPV at market rates represents the summation of all future interest income of assets and liabilities, plus the present value of equity positioned as of the latest date of calculations.

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This last property provides a major link between the accounting income measures and the NPV, under some specific assumptions. It makes NPV a relevant long-term target of ALM policy, since controlling the risk of the NPV is equivalent to controlling the risk of the entire stream of interest incomes up to the latest maturity. The presentation of the NPV comprises two steps. This chapter discusses the interpretation of the NPV and its relationship with net interest margins and accounting profitability. The first section discusses alternative views of NPV. The second section demonstrates, through examples, the link between NPV at market rates and the discounted value of the future stream of periodical interest incomes up to the longest maturity, plus an adjustment for equity. Chapter 23 deals with the issue of controlling the interest rate risk using the NPV, rather than interest income, as the target variable of ALM. Duration gaps substitute the classical interest rate gaps for this purpose.

VIEWS ON THE NPV The NPV of the balance sheet is the present value of assets minus the present value of liabilities, excluding equity. This present value can use different discount rates. When using the market rates, we have an ‘ALM’ NPV that has the properties described below. ‘Fair value’ calculation would use the yields required by the market dependent on credit spreads of individual assets. This fair value differs from the value of assets under ‘ALM’ NPV, which is a mark-to-model calculation. There are various possible views on NPV. We review some of them, focusing on what they imply and eventual inconsistencies. Possible views of NPV are: a mark-to-market value of a portfolio; a measure of performance defined by the gaps between the asset and liability yields and the market rates; a measure of the future stream of interest incomes up to the latest maturity in the banking portfolio.

Definition The present value of the stream of future flows Ft is: Ft /(1 + yt )t V = t

The market rates are the zero-coupon rates yt derived from yield curves, using the cost of the debts of the bank. The formula provides a value of any asset that generates contractual flows. The NPV of the balance sheet is the value of assets minus that of debts. Both result from the Discounted Cash Flow (DCF) model. The calculation of NPV uses market rates with, eventually, the credit spread of the bank compensating its credit risk. The NPV using market rates, with or without the bank’s credit spread, differs from the ‘fair value’ of the balance sheet. Hence, NPV is more a ‘mark-to-model’ value whose calculation rules derive from the interpretations of its calculation, as explained below.

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273

NPV as a Portfolio The NPV is the theoretical value of a portfolio long in assets and short in liabilities. The NPV can be either positive or negative, since the NPV is the net value of assets and liabilities. The change in NPV depends entirely on the sensitivities of these two bonds. A subsequent chapter (Chapter 23) discusses the behaviour of NPV when market rates change. The NPV ‘looks like’ a proxy for equity value since it nets the values of assets and debts, so that equity is the initial investment of an investor who buys the assets and borrows the liabilities. However, NPV and market value of equity are not at all identical. First, NPV can be negative whereas equity value is always positive. Second, this interpretation assimilates equity to a bond, which it is not. Using market rates as discount rates is equivalent to considering the investor as a ‘net’ lender, instead of an equity holder. Since equity investors take on all risks of the bank, the required rate of return should be above the cost of debt. Therefore, it does not make sense to consider the equity investor as a ‘pure lender’. In addition, the NPV at market rates represents the market value of the balance sheet even though assets and liabilities are not actually liquid. If assets and liabilities were marketable, it would be true that an investor could replicate the bank’s portfolio with a net bond. In such a case, the market value of equity would be identical to the NPV at market rates because the investor would not be an equity investor. He would actually hold a ‘leveraged’ portfolio of bonds. The problem is that the market assets do not truly replicate operating assets and liabilities because they are neither liquid nor negotiable. Hence, the market value of equity is actually different from the theoretical NPV at market rates. The market value of equity is the discounted value of the flows that compensate equity, with a risk-adjusted discount rate. These flows to equity are dividends and capital gains or losses. In addition, the relevant discount rate is not the interest rate, but the required return on equity, given the risk of equity.

NPV and Assets and Liabilities Yields The NPV at market rates is an economic measure of performance. If the value of assets is above face value, it means that their return is above the market rates, a normal situation if loans generate a margin above market rates. If the bank pays depositors a rate lower than market rates, deposits cost less than market rates and the market value of liabilities is below the face value. This results in a positive NPV, even if there were no equity, because assets are valued above face value and liabilities below face value. Accordingly, when the NPV increases, margins between customer rates and market rates widen, the economic performance improves, and vice versa. When the bank lends at 10% for 1 year, and borrows in the market at 8%, it gets 1100 as proceeds from the loan in 1 year and repays 1080 for what it borrows. The difference is the margin, 20. In addition, the present value of the asset is 1100/(1 + 8%) = 1018.52, above 1000. The gap of 18.52 over face value is the present value of the margin at the

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prevailing market rate. If asset values at this rate are higher than book values, they provide an excess spread over the bank’s cost of debt. If not, they provide a spread that does not compensate the bank’s cost of debt. If assets yield exactly the cost of debt for the bank, their value is at par. The same process applies to bank deposits. When using the market rates for the bank’s deposits, their value is under face value. If the bank collects deposits of 1000 for 1 year at 5% when the market rate is 10%, the value of the deposits is 1050/(1 + 8%) = 972.22, lower than the face value. The difference is the present value of the margin at the market rate. The interpretation is that lending deposits costing 5% in the market at 10% would provide a margin of 1000 × 5% = 50. This simple reasoning is the foundation of the interpretation of the NPV as the present value of future periodical interest incomes of the bank.

NPV and the Future Stream of Interest Incomes Since we include all flows in discounting, the NPV should relate to the margins of the various future periods calculated using market rates as references. We show in the next section that the NPV is equal to the discounted value of the future stream of interest incomes, with an adjustment for equity. Since this NPV represents the discounted value of a stream of future interest incomes, all sensitivity measures for the NPV apply to this discounted stream of future margins. This is an important conclusion since the major shortcoming of the interest margin as a target variable is the limited horizon over which it is calculated. Using NPV instead as a measure of all future margins, the NPV sensitivity applies to the entire stream of margins up to the longest horizon. The next chapter expands the NPV sensitivity as a function of the durations of assets and liabilities.

NPV AND INTEREST INCOME FOR A BANK WITHOUT CAPITAL The relationship between NPV and interest income provides the link between market and accounting measures. It is easier to discuss this relation in two steps: the case of a bank without equity, which is a pure portfolio of bonds; a bank with equity capital. This section considers only the zero equity case and the next the case where equity differs from zero. The discussion uses an example to make explicit the calculations and assumptions. The discount rate used is a flat market rate of 10% applying to the bank, which can fluctuate.

Sample Bank Balance Sheet The bank funding is a 1-year debt at the current rate of 9%. This rate is fixed. The asset has a longer maturity than debt. It is a single bullet loan, with 3-year maturity, having a contractual fixed rate of 11%. Hence, the bank charges to customers the current market rate plus a 1% margin. The current market rate is 10%. This margin is immune to interest

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275

TABLE 22.1 Example of a simplified balance sheet

Amount Fixed rate Maturity

Assets

Liabilities

1000 11% 3 years

1000 9% 1 year

rate changes for the first year only, since the debt rate reset occurs only after 1 year. The balance sheet is given in Table 22.1. The cash flows are not identical to the accounting margins (Table 22.2). The differences are the principal repayments. For instance, at year 1, two cash flows occur: the interest revenue of 110 and the repayment of the bank’s debt plus interest cost, or 1110. However, the accounting margin depends only on interest revenues and costs. It is equal to 110 − 90 = 20. For subsequent years, the projection of margins requires assumptions, since the debt for years 2 and 3 has to be renewed. TABLE 22.2 The stream of cash flows generated by assets and liabilities Dates Assets Liabilities

0

1

2

3

1000 1000

110 1090

110

1110

The profile of cash flows remains unchanged even though the interest rate varies. This is not so for margins, after the first period, because they depend on the cost of debt, which can change after the first year. Therefore, projecting margins requires assumptions with respect to the renewal of debt. We assume that the debt rolls over yearly at the market rate prevailing at renewal dates. The horizon is the longest maturity, or 3 years. The cost of debt is 80 if the market rate does not change. If the market rate changes, so does the cost of debt beyond 1 year. The rollover assumption implies that the cash flows change, since there is no repayment at the end of period 1, and until the end of period 3, where the debt ceases to roll over. If the market rate increases by 1%, up to 11%, the cost of debt follows and rises to 10%. The stream of cash flows for the rollover debt becomes (90, 90, 1090) and the margin decreases starting from the second year (Table 22.3). TABLE 22.3 The future stream of cash flows and margins generated by assets and liabilities Dates

0

1

2

3

Assets 1000 110 110 1110 Liabilities 1000 90 100 1100 Interest revenues and costs when debt is rolled over at the current rate Revenues — 110 110 110 Costs — 90 100 100 Margins — 20 10 10

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NPV and Projected Interest Income with Constant Market Rate The values of assets and liabilities, using the original rate, and the NPV are: PV(asset) = 110/(1 + 10%) + 110/(1 + 10%)2 + 1110/(1 + 10%)3 = 1024.869 PV(debt) = 90/(1 + 10%) + 90/(1 + 10%)2 + 1090/(1 + 10%)3 = 975.131 The NPV is the difference: NPV = PV(asset) − PV(debt) = 49.737 Next, we calculate the present value of all future interest incomes at the prevailing market rate of 10%: Present value of interest income = 20/(1 + 10%) + 20/(1 + 10%)2 + 20/(1 + 10%)3 = 49.737 These two calculations provide evidence that the NPV from cash flows is exactly equal to the present value of interest incomes at the same rate: NPV = present value of margins = 49.737

NPV and Projected Interest Income with Market Rate Changes When the market rate changes, the cash flows do not change, but the projected margins change since the costs of debt are subject to resets at the market rates. We assume now parallel shifts of the yield curve. The flat market rate takes the new value of 9% from the end of year 1 and up to maturity. The cost of debt changes accordingly to 8% assuming a constant margin between the cost of debt and the market rate. The flows are as given in Table 22.4. TABLE 22.4 years Periods

Flows generated over three 0

1

2

3

Assets 1000 110 110 1110 Liabilities 1000 1090 Rollover of debt at 8%, market rate at 9% Revenues — 110 110 110 Costs — 90 80 80 Margins — 20 30 30

The NPV calculation remains based on the same flows, but the discount rate changes. For simplicity, we assume that the rate changes the very first day of the first period. The margins also change with the rate, and the present value discounts them with the new market rate. With a 9% market rate, the margins increase after the first year. The calculations of the NPV and the discounted value of margins are: PV(asset) = 110/(1 + 9%) + 110/(1 + 9%)2 + 1110/(1 + 9%)3 = 1050.626 PV(debt) = 90/(1 + 9%) + 90/(1 + 9%)2 + 1090/(1 + 9%)3 = 983.861

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277

The NPV is the difference: NPV = PV(asset) − PV(debt) = 66.765 The discounted value of periodical interest incomes is: Discounted margins = 20/(1 + 9%) + 30/(1 + 9%)2 + 30/(1 + 9%)3 = 66.765 When the rate changes, the equality still holds. Using the 9% flat rate, or any other rate, shows that this equality holds for any level of market rates.

The Interpretation of the Relationship between Discounted Margins and NPV For the above equality to hold, we need to reset both the discount rate and the margin values according to rate changes. Taking an extreme example with a sudden large jump of interest rate, we see what a negative NPV means. The negative NPV results from projected margins becoming negative after the first year. Both asset and liability values decline. With a 15% market rate, the projected margins beyond year 1 become 110 − 140 = −30. The two calculations, discounted flows and discounted margins, provide the same value: NPV = discounted interest incomes = 20/(1 + 15%) − 30/(1 + 15%)2 − 30/(1 + 15%)3 = −25.018 The change from a positive NPV to a negative NPV is the image of the future negative margins given the rising cost of debt. This results from the discrepancies of the asset and liability maturities and the resulting mismatch between interest rate resets. The short-term debt rolls over sooner than assets, which generate a constant 11% return. The variable interest rate gap is negative, calculated as the difference between interest-sensitive assets and liabilities. This implies that the discounted margins decrease when the rate increases. In the example, discounted margins change from 49.737 to −25.018. The change in NPV is the difference between variations of the asset and liability values. When rates increase, both decrease. Nevertheless, the asset value decreases more than the debt value. This is why the NPV declines and becomes negative. The sensitivity of the asset value is higher than the sensitivity of the debt value. The following chapter discusses the sensitivity of the mark-to-market value, which is proportional to duration. In the current example, the duration of assets is above that of debt. Accordingly, assets are more sensitive to rate changes than debt. The above results are obtained for a balance sheet without equity. Similar results are obtained when equity differs from zero, but they require an adjustment for the equity term.

DISCOUNTED MARGINS AND NPV WITH CAPITAL With capital, the debt is lower than assets and the interest income increases because of the smaller volume of debt. In the example below, equity is 100, debt is 900 and assets are 1000. All other assumptions are unchanged. At the 10% market rate, the first year

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cost of debt becomes 9% × 900 = 81 instead of 90 because the amount of debt is lower (Table 22.5). TABLE 22.5 Example of a simplified balance sheet Assets Loans at 11% Capital Debt at 9%

Liabilities

1000 100 900

We can duplicate the same calculations as above with the new set of data. Cash flows and projected margins correspond to the new level of debt: PV(asset) = 110/(1 + 10%) + 110/(1 + 10%)2 + 1110/(1 + 10%)3 = 1024.869 PV(debt) = 81/(1 + 10%) + 81/(1 + 10%)2 + 1081/(1 + 10%)3 = 877.618 The NPV is the difference: NPV = PV(asset) − PV(debt) = 147.250 The stream of future margins becomes 29 for all 3 years. Its discounted value at the market rate of 10% becomes: Discounted margins = 29/(1 + 10%) + 29/(1 + 10%)2 + 29/(1 + 10%)3 = 72.119 The present value of margins becomes lower than the balance sheet NPV. The gap results from equity. The difference between the NPV and the discounted margins is 147.250 − 72.119 = 75.131. We see that this difference is the present value of the equity, at the market rate, positioned at the latest date of the calculation, at date 3: 100/(1 + 10%)3 = 75.131 Therefore, the NPV is still equivalent to the discounted value of future interest income plus the present value of equity positioned at the latest date of the calculation. Table 22.6 illustrates what happens when the market rate varies, under the same assumptions as above, with a constant margin of −1% for the rollover debt. The discount rate being equal to the market rate, it changes as well. TABLE 22.6 Market rate 9% 10% 11%

Discounted values of cash flows and margins at market rates Present value of margins

Present value of 100 at date 3

Present value of margins + capital

NPV

87.932 72.119 56.982

77.218 75.131 73.119

165.151 147.250 130.102

165.151 147.250 130.102

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279

Conclusion All the above calculations use a very simple balance sheet. However, since all assets and liabilities can be broken down into simple zero-coupon items, the calculations still hold for real balance sheets. In the end, the assumptions required for the equivalence of discounted margins plus capital and NPV are the following: • Projections of interest income should extend up to the longest maturity of balance sheet items. • Both the stream of projected margins and the discount rate need adjustment when rates change: future margin resets occur when the current market rate changes until maturity1 . • The discount rates are the market rates. • The NPV at market rates, or at the bank’s market rates, differs from the ‘fair value’ which uses rates embedding the credit spreads of the borrower. • The difference between NPV and discounted margins at market rates is the present value of equity positioned at a date that is the longest maturity of all assets and liabilities.

1 Rate

deviations from 8% require adjusting both margins and discount rate.

23 NPV and Interest Rate Risk

The Net Present Value (NPV) is an alternative target variable to interest income. The sensitivity of interest rate assets and liabilities is the ‘duration’. Since the NPV is the difference between mark-to-market values of loans and debts, its sensitivity to rate changes depends on their durations. The duration is the percentage change of a market value for a unit ‘parallel’ shift of the yield curve (up and down moves of all rates). The duration of assets and liabilities is readily available from their time profile of cash flows and the current rates. Intuitively, the interest rate sensitivity of the NPV depends on mismatches between the duration of assets and the duration of liabilities. Such mismatches are ‘duration gaps’. Controlling duration gaps is similar to controlling interest rate gaps. Derivatives and futures contracts alter the gap. Simple duration formulas allow us to define which gaps are relevant for controlling the sensitivity of several NPV derived target variables. These include the NPV of the balance sheet, the leverage ratio of debt to asset in mark-tomarket values, or the duration of ‘equity’ as the net portfolio of loans minus bank debts. Maintaining adequate duration gaps within bounds through derivatives or futures allows us to control the interest rate sensitivities of mark-to-market target variables. Duration-based sensitivities do not apply when the yield curve shape changes, or when the interest shock is not small. Durations are, notably, poor proxies of changes with embedded options (for further details see Chapter 24). This chapter focuses on duration gaps between assets and liabilities, and how to use them. The first section describes duration and its properties. The second section lists NPV derived target variables of the Asset–Liability Management (ALM) policy, and defines the relevant duration gaps to which they are sensitive. Controlling these duration gaps allows us to monitor the risk on such target variables. The last section explains why derivatives and futures modify durations and help to hedge or control the risk of the NPV.

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281

THE SENSITIVITY OF MARKET VALUES AND DURATION The sensitivity is the change in market value generated by a parallel shift of the yield curve. In technical terms, it is the derivative of the value of the asset with respect to interest rate. The sensitivity is the modified duration of an asset. The duration is the ratio of the present value of future flows, weighted by dates, to the market value of the asset. The modified duration applies when the yield curve is flat, since it is the ratio of the duration to (1 + y), where y is the flat rate. However, it is always possible to derive a sensitivity formula when the yield curve is not flat, by taking the first derivative with respect to a parallel shift of the entire yield curve. In this case, we consider that a common shock y applies to all rates yt and take the derivative with respect to y rather that y. The appendix to this chapter uses this technique to find the condition of immunization of the interest income over a given period when flows do not occur at the same date within the period. The cash flows are Ft for the different dates t, and yt are the market rates. Then the duration is:

N N tFt /(1 + yt )t Duration = Ft /(1 + yt )t t=1

t=1

The duration formula seems complex. In fact, it represents the average maturity of future flows, using the ratio of the present value of each flow to the present value of all flows as weights for the different dates1 . For a zero-coupon, the duration is identical to maturity. In this case, all intermediate flows are zero. The formula for the duration shows that D = M, where M is the zero-coupon maturity. The duration has several important properties that are explained in specialized texts2 . Subsequent sections list the properties of duration relevant for ALM. The duration is a ‘local’ measure. When the rate or the dates change, the duration drifts as well. This makes it less convenient to use with important changes because duration ceases to be constant.

Sensitivity of Market Values to Changes in Interest Rates The modified duration is the duration multiplied by 1/(1 + y). The modified duration is the sensitivity to interest rate changes of the market value and applies when the yield curve is flat. The general formula of sensitivity is: V /V = −[D/(1 + y)] × y The relative change in value, as a percentage, is equal to the modified duration times the absolute percentage interest rate change. This formula results from the differentiation of the value with respect to the interest rate. The sensitivity of the value is the change in value of the asset generated by a unit change in interest rate. This value is equal to the market value of the asset multiplied by the modified duration and by the change in interest rate: V = −[D/(1 + y)] × V × y 1 This

is different from the simple time weighted average of flows because of discounting. (1987) book is entirely dedicated to duration definitions and properties.

2 Bierwag’s

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The duration of a portfolio is the average of the durations of assets weighted by their market values. This property is convenient to derive the durations of a portfolio as a simple function of the durations of its individual components.

Duration and Return Immunization The return of an asset calculated over a horizon equal to its duration is immune to any interest rate variation. With fixed rate assets, obtaining the yield to maturity requires holding the asset until maturity. The yield to maturity is the discount rate making the present value of the future cash flow identical to its price. When selling the asset before maturity, the return is uncertain because the price at the date of sale is unknown. This price depends on the prevailing interest rate at this date. The Discounted Cash Flow (DCF) model, using the current market rates as discount rates, provides its value. If the rates increase, the prices decrease, and conversely. The holding period return combines the current yield (the interest paid) and the capital gain or loss at the end of the period. The total return from holding an asset depends on the usage of intermediate interest flows. Investors might reinvest intermediate interest payments at the prevailing market rate. The return, for a given horizon, results from the capital gain or loss, plus the interest payments, and plus the proceeds of the reinvestments of the intermediate flows up to this horizon. If the interest rate increases during the holding period, there is a capital loss due to the decline in price. Simultaneously, intermediate flows benefit from a higher reinvestment rate up to the horizon at a higher rate. If the interest rate decreases, there is a capital gain at the horizon. At the same time, all intermediate reinvestment rates get lower. These two effects tend to offset each other. There is some horizon such that the net effect on the future value of the proceeds from holding the asset, the reinvestment of the intermediate flows, plus the capital gain or loss cancel out. When this happens, the future value at the horizon is immune to interest rate changes. This horizon is the duration of the asset3 .

Duration and Maturity The duration increases with maturity, but less than proportionately. It is a ‘convex’ function of maturity. The change in duration with maturity is the slope of the tangent to the curve that relates the duration to the maturity for a given asset (Figure 23.1). When time passes, from t to t + 1, the residual life of the asset decreases by 1. Nevertheless, the duration decreases by less than 1 because of convexity. If the duration is 2 years in January 1999, it will be more than 1 year after 1 year, because it diminishes less than residual maturity. This phenomenon is the ‘duration drift’ over time. Due to duration drift, any constraint on duration that holds at a given date does not after a while. For instance, an investor who wants to lock in a return over a horizon of 5 years will set the duration of the portfolio to 5 at the start date. After 1 year, the residual life decreases by 1 year, but the duration decreases by less than 1 year. The residual time to the horizon is 4 years, but the duration will be somewhere between 5 and 4 years. The portfolio duration needs readjustment to make it equal to the residual time of 4 years. This adjustment is continuous, or frequent, because of duration drift. 3 This

is demonstrated in dedicated texts and discussed extensively by Bierwag (1987).

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Duration

t

FIGURE 23.1

t+1

Maturity

The relationship between duration and maturity

THE DURATION GAP AND THE TARGETS OF INTEREST RATE POLICY The target variables of ALM policies include: the interest income over specified periods, the NPV of the balance sheet; the leverage ratio expressed with market values; the market return of the portfolio of assets and liabilities. The immunization conditions for all variables hold under the assumption of parallel shifts in the yield curve.

Immunization of the Interest Margin With the gap model, all flows within periods are supposed to occur at the same date. This generates errors due to reinvestments or borrowings within the period. The accurate condition of immunization of the interest margin over a given horizon depends on the duration of the flows of assets and liabilities over the period. The condition that makes the margin immune is: VA(1 − DA ) = VL(1 − DL ) where VA and VL are the market values of assets and liabilities, and DA and DL are the durations of assets and liabilities. The demonstration of this formula is in the appendix to this chapter. The formula summarizes the streams of flows for both assets and liabilities with two parameters: their market values and their durations. It says that the streams of flows generated by both assets and liabilities match when their durations, weighted by their market values, are equal. Figure 23.2 illustrates the immunization condition that makes market values of assets and liabilities, weighted by duration, equal.

NPV of the Balance Sheet To neutralize the sensitivity of the market value, the variations in values of assets and liabilities should be identical. This condition implies a relationship between the market values and the duration of assets and liabilities. With a parallel shift of the yield curve equal to i, the condition is: NPV/i = (VA − VL)/i

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VA 1−DA

DL DA

VL

0

FIGURE 23.2

1−DL

1

Summarizing a stream of flows into a single ‘equivalent flow’

The changes in values of assets and liabilities result from their durations, DA and DL . They are (−DA × VA × i) and (−DL × VL × i). If we divide by i, we obtain: NPV/i = [1/(1 + i)][−DA × VA + DL × VL] The immunization condition is: VA × DA = VL × DL This condition states that the changes in the market value of assets and liabilities are equal. When considering the net portfolio of assets minus liabilities, the condition stipulates that the change in asset values matches that of liabilities. Note that this duration gap is in value, not in years. When adjusting durations, it is more convenient to manipulate durations in years. The immunization condition is: DA /DL = VL/VA The formula stipulates that the ratio of the duration and liabilities should be equal to the ratio of market values of liabilities to assets. The NPV sensitivity is the sensitivity of the net portfolio of assets minus liabilities. Its duration is a linear function of these, using market value weights: NPV/NPV = [1/(1 + i)]{[−DA × VA + DL × VL]/NPV} × i The sensitivity of the NPV has a duration equal to the term in brackets. An alternative designation for the duration of NPV is the ‘duration of equity’. It implies dividing the duration gap in value by the NPV. This formula expresses the duration of NPV in years rather than in value. However, it leverages the unweighted duration gap (DA − DL ) by dividing by the NPV, a lower value than those of assets and liabilities. For instance, using the orders of magnitudes of book values, we can consider approximately that VL = 96% × VA since equity is 4% of assets (weighted according to regulatory forfeits). Assuming that the NPV is at par with the book value, it represents around 4% of the balance sheet. If we have a simple duration gap DA − DL = 1, with DA = 2 while DL = 1, the weighted duration gap is around: −2 × 100% + 1 × 96% = −1.04 The ‘equity’ or NPV duration is −1.04/4% = −26, which is much higher than the unweighted duration gap because of the leverage effect of equity on debt. On the other

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hand, the ratio of the durations of assets and liabilities is around DA /DL = 0.96/1, or 96%. These conditions are extremely simple. The durations of assets and liabilities are averages of durations of individual lines weighted with their market values. Therefore, the duration of any portfolio is easy to obtain from the durations of the assets. This makes it easy to find out which assets or liabilities create the major imbalances in the portfolio.

Leverage The leverage is the ratio of debt to equity. With market values, the change of market leverage results from the durations of assets and liabilities. The condition of immunization of the market leverage is very simple, once calculations are done4 : DA = DL This condition says that the durations of assets and liabilities should match, or that the duration gap should be set at zero. By contrast, immunization of the NPV would imply different durations of assets and liabilities, proportional to their relative values, which necessarily differ in general because of equity. The intuition behind this condition is simple. When durations match, any change in rates generates the same percentage change of asset and debt values. Since the change of the debt to asset ratio is the difference between these two percentage changes, it becomes zero. If the debt to asset ratio is constant, the debt to equity ratio is also constant. Hence, leverage is immune to changes in interest rates.

The Market Return on Equity for a Given Horizon The market return on equity is the return on the net value of assets and liabilities. Its duration is that of the NPV. Setting the duration to the given horizon locks in the return of the portfolio over that horizon: DE = DNPV = H

CONTROLLING DURATION WITH DERIVATIVES The adjustment of duration necessitates changing the weights of durations of the various items in the balance sheet. Unfortunately, customers determine what they want, which 4 The

change in asset to equity ratio is: [VA/VL]/i = [−1/E 2 ]{E × VA/i − VA × E/i} = −[1/E 2 ]{[−E × VA × DA /(1 + i)] + [VA × E × DE /(1 + i)]}

This can be simplified to: [VA/E]/i = [−VA × E/E 2 ][1/(1 + i)](DA − DL ) The sensitivity of the ratio is zero when DA = DL .

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sets the duration value. Hence, modifying directly the durations of assets and liabilities is not the easiest way to adjust the portfolio duration. An alternative method is to use derivatives. Hedges, such as interest rate swaps, modify duration to the extent that they modify the interest flows. When converting a fixed rate asset into a variable rate asset through an interest rate swap, the value becomes insensitive to interest rates, except for the discounted value of the excess margin over the reference rate. In the case where the excess margin is zero, we have a ‘floater’. The value of a floater is constant because the discounted value of future cash flows, both variable interest revenues and capital repayments, remains equal to face value. Its duration is zero. When the floater provides an excess margin over the discount rates, the value becomes sensitive to rate because its value is that of a floater plus the discounted value of the constant margin. But its duration becomes much lower than a fixed rate asset with the same face value and the same amortizing profile because the sensitivity relates to the margin amount only, which is much smaller than the face value of the asset. Future contracts have durations identical to the underlying asset. Any transaction on the futures market is therefore equivalent to lending or borrowing the underlying asset. Hence, futures provide a flexible way to adjust durations through off-balance sheet transactions rather than on-balance sheet adjustments.

APPENDIX: THE IMMUNIZATION OF THE NET MARGIN OVER A PERIOD The horizon is 1 year. The rate-sensitive assets within this period are Aj for date j and the rate-sensitive liabilities are Lk for date k. Before assets and liabilities mature, it is assumed that the assets generate a fixed return rj until date j and that the liabilities generate a fixed cost ij until date j . When they roll over, the rates take the new values prevailing at dates j and k (Figure 23.3). Aj 1−tj

tj 0

1

tk tj

1−tk

Lk

FIGURE 23.3

Time profile of cash flows within a period

The reinvestment of the inflows Aj occurs over the residual period 1 − tj . The refunding of the outflows Lk occurs during the residual period 1 − tk . Until the revision dates, the

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return rj and the costs rk are fixed. At the time of renewal, rate resets occur according to market conditions. The reinvestment, or the new funding, uses the rates ij or ik , assumed constant until the end of the period. The Interest Margin (IM) of the period results from both historical rates and the new rates used for rolling over the flows. The interests and costs accrue from the dates of the flows, j or k, up to 1 year. The residual periods are 1 − tj and 1 − tk . The net margin is: {Aj [(1 + rj )tj (1 + ij )1−tj − (1 + rj )tj ]} IM = j

−

k

{Lk [(1 + rk )tk (1 + ik )1−tk − (1 + rk )tk ]}

This expression combines revenues and costs calculated at the historical rates with those calculated after the rollover of transactions at the new rates. For instance, an inflow of 100 occurs at tj = 30, with a historical rate of 10% before renewal. The rate jumps to 12% after reinvestment. The total interest flow up to the end of the period is: 100(1 + 10%)30/360 [(1 + 12%)330/360 − 1] The amount 100(1 + 10%)30/360 is the future value of the capital at the renewal date. We reinvest this amount up to 1 year at 12%. We deduct the amount of capital from the future value to obtain the interest flows only. To find out when the net margin is immune to change in interest rates, we calculate the derivative with respect to a change in interest rate and make it equal to zero. The change in rate is a parallel shift λ of the yield curve: ij −→ ij + λ ik −→ ik + λ The derivative of the margin with respect to λ is: [Aj (1 + rj )tj (1 + ij ) − tj (1 − tj )] δIM/δλ = j

−

k

[Lk (1 + rk )tk (1 + ik ) − tk (1 − tk )]

The market values of all assets and liabilities using the historical rates prevailing at the beginning of the period are: VAj = Aj (1 + rj )tj /(1 + ij )tj

and VLk = Lk (1 + rk )tk /(1 + ij )tk

Using these market values, the expression of the derivative simplifies and becomes: VLk (1 − tk ) VAj (1 − tj ) − δIM/δλ = j

k

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This condition becomes, using VA and VL for the market values of all assets and all liabilities: VA × 1− tk VLk /VL = 0 1− tj VAj /VA − VL × j

k

VA and VL are weighted by 1 minus the present values of all assets and liabilities times the reset dates. The weights are equal to 1 minus the duration of assets and liabilities DA and DL . This formula is in the text.

24 NPV and Convexity Risks

Using NPV as target variable, its expected value measures profitability, while risk results from the distribution of NPV around its mean resulting from interest rate variations. Market values and interest rates vary inversely because market values discount the future flows with interest rates. The NPV–interest rate profile is an intermediate step visualizing how interest rate deviations alter the NPV. It is convenient for making explicit ‘convexity risk’. The NPV risk results from its probability distribution, which can serve for calculating an Asset–Liability Management (ALM) ‘Value at Risk’ (VaR). Convexity risk arises from the shape of the relationship between asset and liability values to interest rates. Both values vary inversely with rates, but the slopes as well as the curvatures of the asset–rate curve and the liability–rate curve generally differ. When slopes only differ, the NPV is sensitive to interest rate deviations because there is a mismatch between asset value changes and liability value changes. Controlling the duration gap between assets and liabilities allows us to maintain the overall sensitivities of the NPV within desired bounds. When interest rate drifts get wider, the difference in curvatures, or ‘convexities’, of the asset–rate and the liability–rate profiles becomes significant, resulting in significant variations of the NPV even when slopes are similar at the current rate. This is ‘convexity risk’, which becomes relevant in times of high interest rate volatility. Options magnify convexity risk. ‘Gapping’ techniques ignore implicit options turning a fixed rate loan into a variable rate one because of renegotiation, or a variable rate loan into a fixed rate one because of a contractual cap on the variable rate. With such options, the NPV variation ceases to be approximately proportional to interest rate deviations, because the NPV durations are not constant when rates vary widely and when options change duration. ‘Convexity risk’ becomes relevant in the presence of interest rate options even when interest rate volatility remains limited. Finally, NPV sensitivity results from both duration and convexity mismatches of assets and liabilities. Convexity risk remains hidden unless interest rates vary significantly.

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NPV–interest rate profiles visualize this risk. Interest rate volatility and implicit options make a strong case for measuring the downside risk of the balance sheet. NPV is an adequate target variable for revealing such risk because it captures long-term profitability when interest rate shifts become more important. Even though there is no capital requirement for ALM risk, VaR techniques applied to NPV capture this downside risk and value it. This chapter focuses on the specifics of ‘convexity’ risk for the banking portfolio starting from the ‘market value–interest rate’ profiles of assets and liabilities. The first section details the relationship between the market value of the balance sheet and the interest rate. The second section visualizes duration and convexity effects. The third section illustrates how various asset and liability duration and convexity mismatches alter the shape of the NPV–interest rate profile, eventually leading to strong adverse deviations of NPV when interest rates deviate both upwards and downwards. The last section addresses hedging issues in the context of low and high volatility of interest rates. Chapter 25 discusses the valuation of downside NPV risk with VaR techniques.

THE MARKET VALUE OF THE BALANCE SHEET AND THE INTEREST RATE The sensitivity to interest rate variations of the NPV is interesting because the NPV captures the entire stream of future flows of margins. The easiest way to visualize the interest risk exposure of the NPV is to use the ‘market value–interest rate’ profile. For any asset, the value varies inversely with interest rates because the value discounts future cash flows with a discount rate that varies. The relationship is not linear (Figure 24.1). The curvature of the shape looks upwards. This profile is a complete representation of the sensitivity of the market value. Market value

V

i

FIGURE 24.1

Rate

The market value–interest rate profile

For the NPV, the profile results from the difference of two profiles: the asset profile and the liability profile. If the NPV calculation uses the costs of liabilities (wacc), it is equal to the present value of all streams of margins up to the longest maturity, except for equity. The shape of the NPV profile results from those of the asset and liability profiles, as shown in Figure 24.2.

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Value

Assets

Liabilities

Rate NPV

Rate

FIGURE 24.2

The NPV–interest rate profile

The two profiles, for assets and liabilities, have the usual shape with an upward looking convexity. However, the difference, the NPV profile, has a shape that depends on the relative curvatures of the profiles of assets and liabilities. Since the market values of assets and liabilities can vary over a large range, and since their curvatures are generally different, the NPV profile can have very different shapes. In Figure 24.2, the NPV is positive, and the curvature of liabilities exceeds that of assets. Nevertheless, this is only an example among many possibilities. In this example, the NPV is almost insensitive to interest rate variations. The sensitivity analysis of NPV leads to duration gaps and suggests matching the durations of assets and liabilities to make the NPV immune to variations of interest rates. However, this is a ‘local’ (context-dependent) rule. Sensitivities of both assets and liabilities and, therefore, the sensitivity of the NPV depend on the level of interest rates. When the variation of interest rates becomes important, the overall profile provides a better picture of interest rate risk. Neutralizing the duration gap is acceptable when the variations of interest rates are small. If they get larger, the NPV again becomes sensitive to market movements. The next section discusses duration and convexity.

DURATION AND CONVEXITY The duration is a good criterion whenever interest rate variations remain small and when the convexity of asset and liability profiles with rates remains small. If one of these two conditions does not hold, matching durations do not make the NPV immune to rate changes and the duration gaps cease to be a good measure of sensitivity. Since the duration is the first derivative of value with respect to interest rate, the second derivative shows how it changes with rates, and there are techniques to overcome the duration limitations and provide a better proxy for the real ‘value–rate’ profile.

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Convexity The sensitivity of a financial asset is the variation of its value when the market rate moves. Graphically, the slope of the ‘market value–interest rate’ profile relates to duration. When the interest rate moves, so does the duration. Graphically, the curvature of the profile shows how the slope changes. The curvature means that the sensitivity to downward moves of the interest rate is higher than the sensitivity to upward moves. The effect of a decrease in rates, from 9% to 8%, is bigger than the change generated by a move from 4% to 3%. ‘Convexity’ measures the change in duration when the rates move. Because of convexity, the market value changes are not linear. For small variations of rates, the duration does capture the change in asset value. Nevertheless, for bigger variations, the convexity alters the sensitivity significantly (Figure 24.3). Market values

Sensitivity

Rate

FIGURE 24.3

Sensitivity and the level of interest rates

Visually, convexity is the change in slope that measures duration at various levels of interest rates.

The Sources of Convexity The first source of convexity is that the relationship between market value and discount rates is not linear. This is obvious since the calculation of a present value uses discount factors, such as (1 + i)t . A fraction of convexity comes from this mathematical formula. Because of discounting, convexity depends on the dates of flows. The sensitivity is proportional to duration, which represents a market value weighted average maturity of future flows. Convexity, on the other hand, is more a function of the dispersion of flows around that average. Financially speaking, modifying convexity is equivalent to changing the time dispersion of flows. A third source of convexity is the existence of options. The relationship between the market value of an option and the underlying parameter looks like a broken line. When the option is out-of-the-money, the sensitivity is very low. When the option is in-themoney, the sensitivity is close to 1. Therefore, the ‘market value–interest rate’ profile is quite different for an option. It shows a very strong curvature when the options are at-the-money. Figure 24.4 shows the profile of a floor that gains value when the interest rate decreases.

NPV AND CONVEXITY RISKS

Value

293

Floor in-the-money, high sensitivity Floor out-of-the-money, low sensitivity

Rate Strike = Guaranteed rate

FIGURE 24.4

Market value–interest rate profile: example of a floor

The implication is that, with options embedded in the balance sheet of banks, the convexity effects increase greatly.

The Measure of Convexity Controlling convexity implies measuring convexity. Mathematically, convexity results, like duration, from the formula that gives the price as the discounted value of future cash flows. The duration is the first-order derivative with respect to the interest rate. The convexity is the second-order derivative. This formula of convexity allows us to calculate it as a function of the cash flows and the interest rate. For options, the convexity results from the equation that relates the price to all parameters that influence its value. Just as for duration, convexity is also a local measure. It depends on the level of interest rate. Therefore, building the ‘market value–interest rate’ profile requires changing the yield curve and revaluing the portfolio of assets and liabilities for all values of the interest rates. Simulations allow us to obtain a complete view of the entire profile. The convexity has value in the market because it provides an additional return when the volatility of interest rates becomes significant. The higher the volatility of interest rates, the higher the value of convexity. The implication is that it is better to have a higher convexity of assets when the interest rate volatility increases, because this would increase the NPV. This intuition results from Figure 24.5. The convexity of assets makes Market value of assets

V

i

FIGURE 24.5

The value of convexity

Rate

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the average value of assets between two distant interest rates higher than the current value. Hence, interest rate volatility increases the expected value of assets. The converse is also true. When convexity is negative, a high volatility of rates results in a lower value of assets, everything else held constant. The value of convexity alters the NPV–rate profile when large deviations of rates occur. The appendix to this chapter expands the visualization with both assets and liabilities having a common duration but different convexities.

THE SENSITIVITY OF NPV The value–interest profiles of NPV have various shapes depending on the relative durations and relative convexities of the assets and liabilities. Some are desirable profiles and others are unfavourable. What follows describes various situations.

Duration Mismatch A duration mismatch—weighted by asset and liability values—between assets and liabilities makes the NPV sensitive to rates. The change in NPV is proportional to the change in interest rates since the values of assets and liabilities are approximately linear functions of the rate change. The NPV changes if the durations do not match. Figure 24.6 shows that there is a value of interest rates such that the NPV is zero, reached when rates decrease. When the interest rate increases, the NPV increases. Value

Liabilities

Assets

Rate NPV

Rate

FIGURE 24.6

Value–rate profile and NPV–duration mismatch

By modifying and neutralizing the duration gap, it is possible to have a more favourable case. In Figure 24.7, the convexities are not very important, and the durations of assets

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Small variations of rates

Value

Liabilities Assets

NPV

Rate

Rate

FIGURE 24.7

Variations in NPV with matched durations and different convexities

and liabilities match at the level where the NPV reaches its maximum. With significant changes in interest rates, the NPV remains positive. However, the different convexities weaken the protection of matching durations when interest rates move significantly away from the current value. With options embedded in the balance sheet, the NPV becomes unstable, because options increase convexity effects. In such cases, the simulation of the entire profile of the NPV becomes important.

The Influence of Implicit Options Options are the major source of convexity in balance sheets, and the main source of errors when using local measures of sensitivity to determine risk. They cap the values of assets and impose floors on the values of liabilities. For instance, the value of a loan with a prepayment option cannot go above some maximum value since the borrower can repay the loan at a fixed price (outstanding capital plus penalty). Even though the borrower might not react immediately, he will do so beyond some downward variation of the interest rate. In this case, a decline in interest rate generates an increase in the value of the asset, until we reach the cap when renegotiation occurs. The value of a straight debt (without prepayment option) would increase beyond that cap. The difference between the cap and the value of the straight debt is the value of the option. On the liability side, options generate a floor for the value of some liabilities when the interest rates rise. This can happen with a deposit that earns no return at all, or provides a fixed rate to depositors. Beyond some upper level of market rates, the value of the deposit hits a floor instead of decreasing with the rise in interest rates. This is when depositors shift their funds towards interest-bearing assets. In such a case, even if the resources stay within the bank, they keep from now on a constant value when rates increase since they become variable rate assets.

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Everything happens as if fixed rate assets and liabilities behaved as variable rate assets and liabilities beyond some variations of interest rates. The shape of the ‘value–interest rate profile’ flattens and becomes horizontal when hitting such caps and floors, as shown in Figure 24.8. Profile of the asset without option

Value Value of the option

Loan with prepayment Deposits

Rate

FIGURE 24.8

Values of assets and liabilities under large interest rate shifts

NPV and Optional Risks The options generate a worst-case situation for banks. The combination of cap values of assets with floor values of liabilities generates a ‘scissor effect’ on the NPV (Figure 24.9). Value Capped value of assets

Floor value of liabilities

Rate NPV Rate

NPV < 0

FIGURE 24.9

NPV > 0 NPV > 0

The ‘scissor effect’ on NPV due to embedded options

This is because the convexity of the assets is the opposite of the convexity of the liabilities beyond some variations. When the interest rate is in the normal range, the NPV remains

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positive. However, if it deviates beyond upper and lower bounds, the options gain value. The change in rate generates a negative NPV beyond those values that trigger the caps or floors. This happens for both upward and downward variations of the interest rates. Since there are multiple embedded options with varying exercise prices, the change is progressive.

CONTROLLING AND OPTIMIZING THE NPV RISK–RETURN PROFILE The goal of interest rate policies is to control the risk of the NPV through both duration and convexity gaps. The common practice is to limit the NPV sensitivity by closing down the duration gap. However, this is not sufficient to narrow the convexity gap. Theoretically, it is, however, possible to eliminate the risk of negative NPV and turn the NPV convexity upside down.

Duration and Convexity Gaps To immunize the NPV to changes in interest rates, the asset-weighted duration and the liability-weighted duration should match. This policy works for limited variations of rates. In order to keep the NPV immune to interest rates for larger variations, the convexities of assets and liabilities should also match. There is a better solution. When the convexity of assets is higher than the convexity of liabilities, any variation of interest rate increases the NPV as shown in Figure 24.10. The NPV then has a higher expected value when the volatility of rates increases. Value

Rate NPV

Rate

FIGURE 24.10

Convexities and NPV sensitivity to interest rate optimization

Controlling Convexity Risk The theoretical answer to protect the NPV against duration and convexity risks is to match duration and increase the convexity of assets. Optional hedges allow us to modify

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the NPV–interest rate profile, to obtain the desired shape. Theoretically, floors protect the lender against a decrease in return from loans. Caps protect the bank against a rise in cost of resources. Floor values will decrease significantly when interest rates decrease significantly. Cap values will increase the NPV when interest rates increase significantly. Both caps and floors correct convexity effects and flatten the asset and liability profiles when rates change. Since optional hedges are expensive, we face the usual trade-off between the cost of hedging and the gain from hedging. Any technique to reduce the cost of such hedges helps. The simplest is to set up an early hedge out-of-the-money. Doing otherwise might imply a prohibitive decrease in NPV.

APPENDIX: WHERE DOES CONVEXITY VALUE COME FROM? The value of convexity appears visually when considering the ‘market value–interest rate’ profile. To make the comparison between two assets of different convexities visible, it is preferable to start with assets of equal value and equal duration, but with different convexities (Figure 24.11).

High convexity

Low convexity

Value High convexity

Max = 17 Max = 15 Mean = 14

Low convexity

Mean = 12.5 Min = 11

Min = 10 Min = 6%

Value of convexity = 14 − 12.5

FIGURE 24.11

Max = 10%

Rate

Expected rate = 8%

Expected values of assets of different convexities

It is not obvious how to define assets with the same value and the same duration. One way is to use a zero-coupon and a composite asset called a ‘dumbbell’. The zero-coupon duration is equal to its maturity. The dumbbell combines two zero-coupons of different durations. The dumbbell duration is the weighted average of the duration of the two zerocoupons. If the average duration of this portfolio matches that of the zero-coupon, we have portfolios of identical market value and identical duration. A property of zero-coupons is to have the lowest convexity, everything else being equal1 . The ‘market value–interest rate’ profiles of these assets, the single zero-coupon and the dumbbell, look like those of Figure 24.11. They intersect at the current price and have 1 What

drives convexity is not always intuitive. This property holds only when everything else is equal. Otherwise, it might not be true.

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the same tangent at this price, since they have the same sensitivity. However, when the interest rate moves away from its original value, the market values diverge because of the differences in convexity. Under certainty, the two assets would have the same value. Under uncertainty, they have an expected value. Uncertainty begins with two possible values of interest rate. Two interest rates (6% and 10%) with equal probabilities result in a common expected value of interest rate identical to the current value (8%). Even though the volatility of interest rates increases when the variations from the mean grow bigger, the average value of the interest rate stays the same. However, the expected value of the asset changes when the rate volatility increases. The reason is that the increase in value when the interest rate declines is higher than the decrease in value when the interest rate moves upwards by the same amount. This asymmetry in the deviation of market values is higher when the convexity (the curvature of the profile) is higher. The expected value of the assets under the two equal probability rate scenarios is the average of the corresponding two values of the assets. The least convex has a value of (15 + 10)/2 = 12.5, and the most convex has an expected value of (17 + 11)/2 = 14. Therefore, the expected value of the most convex asset (14) is higher than the expected value of the least convex asset (12.5). This difference is the value of convexity (1.5). The value of convexity starts at zero, under certainty, and increases with the volatility of rates. Convexity explains the behaviour of the NPV when the volatility of rates increases. With the same variations, it depends essentially on the duration gap. With larger deviations of rates, it depends also on the convexity gap.

25 NPV Distribution and VaR

Multiple simulations serve to optimize hedges and determine interest income at risk, or ‘Earnings at Risk’ (EaR) for interest rate risk as the worst-case deviation with a preset confidence level of interest income. The same technique transposes readily to Net Present Value (NPV) and values downside risk when both duration and convexity mismatches magnify the NPV risk. The current regulations do not impose capital requirements for Asset–Liability Management (ALM), making the ALM ‘Value at Risk’ (VaR) of lower priority than market or credit risk VaR. However, the important adverse effects of convexity risk suggest that it is worthwhile, if not necessary, to work out NPV VaR when considering long-term horizons. The construction of a distribution of NPV at future time points follows the same basic steps as other VaR calculations. The specific of ALM VaR is that interest rate variations beyond some upper and lower bounds trigger adverse effects whatever the direction of interest rate moves, because of convexity risk. Such adverse effects make it necessary both to value this risk and to find these bounds. The same remarks apply for interest income as well, except that NPV crystallizes in a single figure the long-term risk, which short-term simulations of interest income might not reveal or underestimate. The first section discusses the specifics of NPV downside risk. The second section shows that modelling the NPV VaR uses similar techniques as market risk VaR. The third section illustrates the techniques with a simplified example, in two steps: using the Delta VaR technique designed for market risk (see Section 11 of this book); showing how convexity risk alters the shape of the NPV distribution and the NPV VaR.

OVERVIEW AND SPECIFICS OF NPV RISK AND DOWNSIDE RISK VaR applies to NPV as it does for other risks, but there are a number of basic differences. They result in an NPV distribution whose profile is highly skewed when the balance sheet embeds a significant convexity risk.

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301

Interest Income and NPV Distributions The interest income VaR approach looks similar to the EaR approach. In fact, it differs because the interest income results from explicit calculations, whereas EaR concentrates on observed earnings rather than modelling them. In addition, EaR does not use simulations, whereas ALM models do. When transposing the technique to NPV, it is sufficient to identify the main risk drivers and their distributions. These are a small number of correlated interest rates. Characterizing the NPV risk traditionally uses the NPV sensitivity. The higher the sensitivity, the higher the risk. In addition, setting limits implies bounding this sensitivity. To get downside risk requires going one step further. This is not current practice, but its value is to demonstrate that the same technique applies to various risks and to demonstrate the value of the NPV downside risk. The next step consists of calculating a distribution of NPV values at future time points. The time points need to fit the needs of ALM to have medium-term visibility. A 1 to 2-year horizon is adequate, as for credit risk. Determining a forward distribution of NPV at the 1-year time point requires considering all cash flows accruing from assets and liabilities up to maturity. ALM models consider business risk over some short to medium-term horizon. Since we need probability distributions, we ignore business risk here, thereby getting closer to the ‘crystallized’ portfolio approach of market and credit risks. The usual steps are generating correlated distributions of the main interest rate drivers and revaluing the NPV for each scenario. In what follows, we perform the exercise using only a flat yield curve and a single rate. Using correlated interest rates would require the techniques to be expanded for market and credit risks. Because of the relatively high correlations of interest rates for a single currency, parallel shifts of yield curve assuming perfect correlations are acceptable as a first proxy. They do not suffice, however, because a fraction of the interest income results from the maturity gap between assets and liabilities, making both interest income and NPV sensitive to the steepness of the yield curve. Going further would entail using correlations or factor models of the yield curve. For illustration purposes, there is no need to do so here. Deriving the NPV value from its current sensitivities is not adequate for a medium-term horizon and in the presence of options. Full simulations of the NPV values, requiring revaluations of all assets and liabilities, and of options, are necessary to capture the extreme deviations when discussing VaR. Without option risk, the NPV distribution would be similar to that of interest rates, since interest margin variations remain more or less proportional to interest rate deviations. With options, the NPV distribution drifts away from the interest rate distribution because options break down this simple relationship and introduce abrupt changes in the sensitivity to interest rates. Because of convexity risk, the reason for fat tails in the NPV distributions lies with the adverse effect of large deviations of interest rates on both sides of current interest rates. The fat tails of the NPV distribution result from convexity risk. Once the NPV distribution is identified, it becomes easy to derive the expected value and the worst-case deviations at preset confidence levels. Since the NPV is usually positive, losses occur once deviations exceed the expected NPV value. The VaR is this adverse deviation minus any positive expected NPV.

Downside Risk and Convexity Downside risk appears when interest rates deviate both upwards and downwards. From a VaR standpoint, this is a specific situation. It stems from the nature of implicit options,

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some of them contributing to the decrease in NPV when rates increase, and others when rates decrease. This ‘double adverse exposure’ results from the implicit options in loans and deposits. As a result, downside risk appears in two worst-case scenarios of rising and decreasing rates. Since it is possible to model interest rate deviations, it is easy to assign probabilities to such deviations and to assign probabilities to the NPV adverse changes. Note, however, that the ‘convexity’ risk is not mechanically interest risk-driven. It depends also on behavioural patterns of customers, and demographic or economic factors for prepayments or renegotiations of loans, or on legal and tax factors (for deposits). In addition, we ignore business risk here. We could use business scenarios with assigned probabilities. This would result in several NPV distributions that we should combine into one. For such reasons, the probability distribution of adverse variations of NPV is not purely interest rate-driven. Figure 25.1 shows the VaR of the NPV at a preset confidence level. Since the NPV is positive, losses occur only when the NPV decreases beyond the positive expected value. Probability Interest rate distribution

NPV

Interest rate

NPV distribution Probability + 0 NPV −

FIGURE 25.1

Interest rate distribution NPV at given confidence level

NPV, implicit options and VaR

NPV VAR In order to derive the NPV distribution from interest rate distributions, ignoring other risk factors, a first technique mimics the technique for market risk and, notably, Delta VaR as expanded in Section 11 of this book. It relies on the sensitivity, which is the NPV duration. Current durations do not account for changes in sensitivity due to time drift of duration and convexity risk. Forward calculations of duration correct the first drawback. They imply positioning the portfolio at the future time points and recalculating the durations for each value of the future interest rate. We simplify somewhat the technique and assume them given. High interest rate volatility is dealt with through convexity later on. The alternative technique uses multiple simulations of risk drivers plus revaluation of the NPV for each set of trial values of rates, to obtain the NPV distribution, similarly to the simulation technique used for market risk. We summarize here the main steps for both techniques. The process replicates the techniques expanded in Chapter 29 and serves as a transition. The sensitivity of the NPV is the net duration of assets and liabilities, weighted by the mark-to-market value of assets and liabilities. This allows us to convert any drift of rates into a drift of NPVs. Proceeding along such lines makes the NPV random, as the interest rate is. We use bold characters in what follows for random variables. As usual,

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303

VaR requires looking for the maximum variations of the NPV resulting from the shift of interest rates. Let Si be the sensitivity of the NPV to interest rate i, this sensitivity resulting from the above duration formula. The volatility of the NPV is: σ (NPV) = Si × σ (i) With a given interest rate volatility, the maximum change, at a given confidence level, is a multiple of the volatility. The multiple results from the shape of the distribution of interest rates. With a normal curve, as a proxy for simplification, a multiple of 2.33 provides the maximum change at 1% confidence level: VaR = 2.33 × Si × σ (i) When there are several interest rates to consider, and when a common shift is not an acceptable assumption, the proxy for the variation of the NPV is a linear combination of the variations due to a change in each interest rate selected as a risk driver: NPV = S1 × i1 + S2 × i2 + S3 × i3 + · · · The indices 1, 2, 3 refer to different interest rates. Since all interest rate changes are uncertain, the volatility of the NPV is the volatility of a sum of random variables. Deriving the volatility of this sum requires assumptions on the correlations between interest rates. The issue is the same for market risk except that interest rates correlate more than other market parameters do. This simple formula is the basic relationship of the simple Delta VaR model for market risk. The distribution of the NPV changes is the sum of the random changes of the mark-to-market values of the individual transactions, which depend on those of underlying rates. Using the normal approximation makes the linear NPV changes normal as well. Figure 25.2 represents the distribution and VaR for NPV at a single time point in the future. V NP

Probabilities

Lower bound of NPV at a given confidence level

e

Tim R

Va

FIGURE 25.2

Projecting NPV at a future horizon (no convexity risk)

In fact, convexity risk plus the long-term horizon makes this technique inappropriate. When time passes, duration drifts. In addition, interest rate volatility grows and activates options. The Delta VaR relationship ceases to apply. Multiple revaluations of NPV for various sets of rates are necessary to capture the changes in durations and NPV. In order

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to illustrate the effect of convexity risk, we develop in the next section a simplified example.

MULTIPLE SIMULATIONS AND NPV VAR This section provides simulations of NPV with varying rates and uses Monte Carlo simulation, with a flat yield curve, to calculate the NPV VaR. In order to proceed through the various steps, we need to simulate random values of a single risk driver, which is a flat market rate. The forward revaluation of the NPV is straightforward with only two assets, the loan portfolio and the debt. Assets and debt are both fixed rate in a first stage. We introduce convexity risk later on in the example using a variable rate loan with a cap on the interest rate of the loan. Using ‘with and without’ cap simulations illustrates how the cap changes the NPV distribution. Revaluation depends on the interest rates of assets and liabilities. The simulation without cap uses a fixed rate, while the simulation with cap on the variable rate loan necessitates that the loan be variable up to the cap. The first subsection provides a set of simplified data. The next two subsections provide the two simulations.

The Bank’s Balance Sheet Data The bank sample portfolio is as given in Table 25.1. TABLE 25.1

Market Asset Liability Date Asset Liability

Sample bank portfolio Rate 8.0% Rate

Amount

9.0% 8.0% 0

1000.00 960.00 1 90.00 1036.80

2

3

4

5

90.00

90.00

90.00

1090.00

The bank NPV at the flat yield curve, with interest rates at 8%, is the present value at 8% of all asset and liability cash flows, as in Table 25.2. TABLE 25.2 Sample bank portfolio NPV Present value Asset Liability NPV

1123.12 1036.80 86.32

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305

When the interest rate increases, both values of assets and liabilities decrease. There are two cases: without significant convexity risk and with convexity risk for assets due to a prepayment option.

NPV VaR with Fixed Rate Asset and Liability In the simpler case, without convexity, both asset and debt have fixed rates, the loan yielding a 9% rate and the liability rate being 8%. The NPV calculation is a forward calculation at date 1. Since the liability matures at date 1, its value remains constant whatever the rate value because there is no discounting at this date. Both asset and liability rates are fixed rates. The asset and liability value–rate profiles and the NPV profile when the market rate changes are given in Figure 25.3. 1600 Asset value

1400 1200 Values & NPV

1000 800

Debt value

600 NPV

400 200 0 −200 0%

2%

4%

6%

8%

10% 12% 14% 16%

−400 Market Rate

FIGURE 25.3

Asset, debt and NPV values when the market rate changes

The NPV decreases continuously with rates because the asset value decreases, while the debt value remains constant. To calculate the NPV VaR, we vary the interest rate randomly, using a normal curve for the flat rate, with an expected value of 8% and a yearly volatility of 2%1 . With 1000 simulations, we generate the interest rate values and the corresponding NPV values. With the distribution of NPV values, we derive the NPV at various confidence levels and the VaR at these confidence levels. The break-even rate making the NPV negative is around 11% (Figure 25.4). The NPV reaches negative values, triggering losses. The VaR requires setting confidence levels. The VaR at various NPV percentiles results from the distribution (Figure 25.5). The graph shows that the NPV VaR at 1% confidence level is close to −60. The NPV expected value is 86.32, and the NPV (1%) shows that the loss from the expected value is close to 146 (86 + 60). 1 We

need to ensure that the interest rate does not become negative. We could also use a lognormal curve for the market rate. In this simulation, the rate remains well above 0.

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12% 10%

NPV

8% 6% 4% 2% 0% −20

−10

0

0

0

100

200

300

400

Frequency (%)

FIGURE 25.4

NPV distribution with market rate changes 20

NPV

0 −20 0%

2%

4%

6%

8%

10%

−40 −60 −80 Confidence Level

FIGURE 25.5

NPV values and confidence levels

NPV VaR with Convexity Risk in Assets In the second case, we simulate convexity for the asset by introducing a formula that caps the asset rate: asset rate = max(market rate + 1%, 9%). The asset rate is either the market rate plus a margin of 1% above the market rate, or 9%. It is a floater when the market rate is lower than 8%. When the market rate hits 8%, the loan has a fixed rate equal to 9%. This simulates a prepayment option triggered at 9% by borrowers. The asset value varies when the market rate changes, with a change of slope at 8%. The increase with rate when rate values are lower than 8% results from the discounting flows indexed to rates, and when the asset rate hits the 9% cap the asset value starts decreasing. The liability value remains constant because we have a unique flow at date 1, which is the date for the forward valuation. Consequently, the NPV shows a bump at 8%, and becomes negative when the market rate exceeds some upper bound close to 11% (Figure 25.6). Figure 25.7 shows the NPV bump. When proceeding with similar simulations of market rate as above, we obtain the NPV distribution and the NPV percentiles for deriving the VaR (Figure 25.8).

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307

Asset value

1200 1000

Liability value Values & NPV

800 600 400

NPV

200 0 −200 0%

2%

4%

6%

8%

10%

12%

14%

16%

Market Rate

FIGURE 25.6

Asset, debt and NPV values when the market rate changes 100 50

NPV

0 −50

0%

2%

4%

6%

8%

10%

12%

−100 −150 Market Rate

FIGURE 25.7

NPV ‘bump’ when the market rate changes 45% 40% 35% Frequency

30% 25% 20% 15% 10% 5% 0% −150

−100

−50

0

50

NPV

FIGURE 25.8

Distribution of NPV with market rate changes

14%

16%

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20

NPV

0 −20 0%

2%

4%

6%

8%

10%

12%

−40 −60 −80 Confidence Level

FIGURE 25.9

NPV values and confidence levels

The NPV distribution becomes highly skewed, illustrating convexity effects. The bump effectively flattens the NPV–rate profile, with a concentration of value around the bump which results in a much higher frequency around the maximum value. On the other hand, both increases and decreases in market rate have an adverse effect due to convexity risk. Hence, there are higher frequencies of adverse deviations and negative values (Figure 25.9).

SECTION 9 Funds Transfer Pricing

26 FTP Systems

The two main tools for integrating global risk management with decision-making are the Funds Transfer Pricing (FTP) system and the capital allocation system. As a reminder, FTP serves to allocate interest income, while the capital allocation system serves to allocate risks. Transfer prices serve as reference rates for calculating interest income of transactions, product lines, market segments and business units. They also transfer the liquidity and the interest rate risks from the ‘business sphere’ to Asset–Liability Management (ALM). The capital allocation system is the complement to the FTP system, since it allocates capital, a necessary step for the calculation of risk-based pricing and risk-based performance (ex post). Chapters 51 and 52, discussing ‘risk contributions’, address the capital allocation issue. The current chapter addresses three major FTP issues for commercial banking: • The goals of the transfer pricing system. • The transfer of funds across business units and with the ALM units. • The measurement of performance given the transfer price. It does not address the issue of defining economic transfer prices, deferred to the next chapter, and assumes them given at this first stage. The FTP system specifications are: • Transferring funds between units. • Breaking down interest income by transaction or client, or any subportfolio such as business units, product families or market segments. • Setting target profitability for business units. • Transferring interest rate risk, which is beyond the control of business units, to ALM. ALM missions are to maintain interest rate risk within limits while minimizing the cost of funding or maximizing the return of investments.

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• Pricing funds to business units with economic benchmarks, using economic transfer prices. • Eventually combining economic prices with commercial incentives. An internal system exchanges capital between units. Transfers go through ALM. The ALM unit is the central pole buying all resources from business lines, collecting them through deposits and selling funds to the lending business lines. Transfer prices allow the calculation of interest income based on transfer prices, in such a way that the algebraic addition of interest income of all units, including ALM, sums up to the accounting interest income of the overall banking portfolio (because all internal sales and purchases of funds compensate). The first section introduces the specifications of the transfer pricing system. The second section discusses the netting mechanisms for excesses and deficits of funds within the bank. The third section details the calculation of performance, for business lines as well as for the entire bank, and its breakdown by business units. The fourth section provides simple calculations of interest income, and shows how the interest incomes allocated to business units or individual transactions add up exactly to the bank’s overall margin. The fifth section addresses the choice of target profitability and risk limits for ALM and business lines. The last section contrasts the commercial and financial views of transfer prices, and raises the issue of defining ‘economic’ transfer prices, providing a transition to the next chapter.

THE GOALS OF THE TRANSFER PRICING SYSTEM The FTP system serves several major purposes, to: • Allocate funds within the banks. • Calculate the performance margins of a transaction or any subportfolio of transactions and its contributions to the overall margin of the bank (revenues allocation). • Define economic benchmarks for pricing and performance measurement purposes. This implies choosing the right reference for economic transfer prices. The ‘all-in’ cost of funds to the bank provides this reference. • Define pricing policies: risk-based pricing is the pricing that would compensate the risks of the bank, independent of whether this pricing is effective or not, because of competition, in line with the overall profitability target of the banks. • Provide incentives or penalties, differentiating the transfer prices to bring them in line with the commercial policy, which may or may not be in line with the target risk-based prices. • Provide mispricing reports, making explicit the differences between the effective prices and what they should be, that is the target risk-based pricing. • Transfer liquidity and interest rate risk to the ALM unit, making the performance of business lines independent of market movements that are beyond their control. The list demonstrates that the FTP system is a strategic tool, and that it is the main interface between the commercial sphere and the financial sphere of the bank. Any malfunctioning or inconsistency in the system interferes with commercial and financial management, and

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313

might create a gap between global policies and operations management. Moreover, it is not feasible to implement a risk management system without setting up a consistent and comprehensive FTP system. All business units of a financial institution share a common resource: liquidity. The first function of FTP is to exchange funds between business units with ALM, since business units do not have balanced uses and resources. The FTP nets the balances of sources and uses of funds within the bank. The exchange of funds between units with ALM requires a pricing system. The FTP system serves the other major purpose of setting up internal transfer prices. Transfer prices serve to calculate revenues as spreads between customers’ prices and internal references. Without them, there is no way to calculate internal margins of transactions, product lines, customer or market segments, and business units. Thus, transfer prices provide a major link between the global bank earnings and individual subportfolios or individual transactions. The presentation arranges the above issues in two main groups (Figure 26.1): Allocate funds Measure performance

FTP system

Define economic benchmarks Pricing

Economic transfer prices

Transfer risks to ALM

FIGURE 26.1

The ‘Funds Transfer Pricing’ system and its applications

• The organization of the FTP. • The definition of economic transfer prices.

THE INTERNAL MANAGEMENT OF FUNDS AND NETTING Since uses and resources of funds are generally unbalanced for business units, the FTP system allows netting the differences and allocating funds to those having liquidity deficits, or purchasing excesses where they appear. There are several solutions for organizing the system. An important choice is to decide which balances are ‘netted’ and how.

ALM, Treasury and Management Control The organization varies across banks although, putting together all interested parties, it should allow us to perform the necessary functions of the FTP. Various entities are

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potentially interested in an FTP scheme. Since the Treasury is the unit which, in the end, raises debts or invests excesses in the market, it is obviously interested in the netting of cash flows. Since internal prices also serve to monitor commercial margins, management control is also involved. Since ALM is the unit in charge of managing the liquidity and interest rate exposures of the bank, the internal prices should be consistent with the choices of ALM. These various units participate in transfer pricing from various angles. They might have overlapping missions. The organization might change according to specific management choices. The definition of the scope of each unit should avoid any overlapping. In this chapter, we consider that ALM is in charge of the system, and that other management units are end-users of the system.

Internal Pools of Funds Pools of funds are virtual locations where all funds, excesses and deficits, are centralized. The concept of pools of funds is more analytical than real. It allocates funds and defines the prices of these transfers. Netting

The FTP nets the excesses of some business units with the deficits of others. This necessitates a central pool of resources to group excesses and deficits and net them. The central pool lends to deficit units and purchases the excesses of others. Starting from scratch, the first thought is to use this simplest system. The simplest solution is to use a unique price for such transfers. The system is ‘passive’, since it simply records excesses and deficits and nets them. Some systems use several pools of funds, for instance grouping them according to maturity and setting prices by maturity. Moreover, at first sight, it sounds simple to exchange only the net balances of business units (Figure 26.2). Market

Sale of resources to A

Purchase of net excess of B

Central pooling of net balances Business unit A Deficit of funds

FIGURE 26.2

Business unit B Excess of funds

Transfers of net balances only

Since assets and liabilities are netted before transfer to the central pool, all assets and liabilities, generated by the operations of A and B, do not transit through the system.

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315

Therefore, the transfer prices apply only to net balances. Their impact is limited to netted balances. The system is simple. The flows are consequences of the operations of business units and the fund transfer system does not influence them. A more active management requires a different system. Pricing all Outstanding Balances

Instead of exchanging only the net excesses or deficits, an alternative is that the ALM purchases all resources and sets a price for all uses of funds of business units, without prior local netting of assets and liabilities. The full amounts of assets and liabilities transit through the central pool. This central pool is the ALM unit. This scheme creates a full internal capital market with internal prices. Post-transfers, the ALM either invests any global excess or funds any global deficit in the capital markets. This system is an active management tool. It does more than ‘record’ the balances resulting from the decisions of business units. The major difference with the system exchanging net local balances only is that the internal prices hit all assets and liabilities of each business unit. By setting transfer prices, the ALM has a powerful leverage on business units. The decision-making process, customer pricing and commercial policy of these units become highly dependent on transfer prices. Transfer prices might also serve as incentives to promote some product lines or market segments through lower prices, or to discourage, through higher prices, the development of others. Such decisions depend on the bank’s business policy. Using such a system in a very active manner remains a management decision. In any case, this scheme makes the FTP a very powerful tool to influence the commercial policies of all business units (Figure 26.3). Market

Sale of all uses of funds

Purchase of all resources

Central pool of all assets and liabilities

Purchase of all resources Sale of all uses of funds

Business unit A

FIGURE 26.3

Business unit B

The central pool of all assets and liabilities

This system serves as a reference in what follows because it ‘hits’ all assets and liabilities. For instance, it was impossible to calculate a margin of an individual transaction in the previous system, since there was no way to ‘hit’ that transaction. Now, the FTP system makes it possible, since all transactions transfer through the central pool. This is a critical property to develop the full functions of the FTP system. Given the powerful potential of this solution, the issue of defining relevant economic prices is critical.

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MEASURING PERFORMANCE Any transfer pricing system separates margins into business, or commercial, margins and financial margins. The commercial margin is the spread between customer prices and internal prices. The financial margin is that of ALM, which results from the volumes exchanged plus the spreads between internal prices and the market prices used to borrow or invest. In addition, calculating the spread is feasible for any individual transactions as well as for any subportfolio. This allows the FTP to isolate the contribution to the overall bank margin of all business units and of ALM. In the FTP, the banking portfolio is the mirror image of the ALM portfolio since ALM buys all liabilities and sells all assets. The sum of the margins generated by the business units and those generated by the ALM balance sheet should be equal to the actual interest margin of the bank (Figure 26.4). Business balance Business sheet balance sheet

Commercial margins Customer prices Transfer prices

ALM balance sheet ALM balance sheet

ALM margins on internal prices Revenues and costs from investing or borrowing in the market

Accounting margin of the bank Accounting margin of the bank

FIGURE 26.4

The bank’s balance sheet and the ALM balance sheet

In general, the summation of all business line internal margins calculated over the transfer prices will differ from the accounting margin since it ignores the ALM margin. Without the latter, there is a missing link between the commercial margins and the accounting earnings of the bank. On the other hand, calculating all margins properly, including the ALM margin, makes them sum the accounting margin of the bank because internal transfers neutralize over the entire bank. The FTP implies an internal analytical recording of all revenues to allow reconciliation with the bank’s income statement. The next paragraph illustrates this reconciliation process. The calculations use a simple example. The commercial margin calculation scope is the entire balance sheet of all business units.

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317

• For the bank, we simply sum all revenues and costs from lending and borrowing, including any funding on the market. • For the business units, revenues result from customer prices minus the cost of any internal purchase of resources by the central unit (ALM). Costs result from the interest paid to customers (depositors) and the revenues of selling these resources to the ALM at transfer prices charged by the ALM. • For the ALM unit, revenues result from charging the lending units the cost of their funds, and the costs result from purchasing from them their collected resources. In addition, ALM gets a market cost when funding a deficit and a market revenue from investing any global excess of funds.

THE FTP SYSTEM AND MARGIN CALCULATIONS OF BUSINESS UNITS The margin calculations use a simplified balance sheet. After calculating the bank’s accounting margin, we proceed by breaking it down across business units and ALM using the transfer prices. The system accommodates cost allocations for completing the analytical income statements, and any differentiation of multiple transfer prices, down to individual transactions. As long as the ALM records properly internal transactions, the overall accounting profitability reconciles with the interest income breakdown across business lines and transactions.

The Accounting Margin: Example The business balance sheet generates a deficit funded by ALM. The average customer price for borrowers is 12% and the average customer rate paid to depositors is 6%. The ALM borrows in the market at the 9% current market rate. There is a unique transfer price. It applies to both assets sold by ALM to business units and resources purchased by ALM from business units. This unique value is 9.20%. Note that this value differs from the market rate deliberately. There could be several transfer prices, but the principle would be the same. Table 26.1 shows the balance sheet and the averaged customer rates. TABLE 26.1

Assets Loans Total Resources Deposits Funding Total

Bank’s balance sheet Volume

Rate

2000 2000

12.00%

1200 800 2000

6.00% 9.00%

The full cost of resources is the financial cost plus the operating cost, either direct or indirect. The analytical profitability statements, using both transfer prices and cost allocations, follows. The transfer price mechanism ensures the consistency of all analytical

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income statements of the business units. If transfer prices change, they simply transfer margins from one unit to another, but they do not change the overall bank margin. Cost allocations add up to the overall costs as well. To illustrate the process, we subdivide the bank into three business units: • Collecting resources. • Originating loans. • ALM, managing financial risks. Table 26.1 shows the aggregated balance sheet. The volumes of loans, deposits and external funding are under the control of each of the three business units. The direct calculation of the accounting margin is straightforward since it depends only on customer rates and the funding cost by the ALM: 2000 × 12% − 1200 × 6% − 800 × 9% = 96

Breaking Down the Bank Margin into Contributions of Business Units From the average customer rates, the transfer price set by the ALM unit and the market rate (assuming a flat yield curve), we calculate the margins for each commercial unit purchasing or selling funds to the ALM. The ALM unit buys and sells funds internally and funds externally the bank’s deficit. The corresponding margins are determined using the usual set of calculations detailed in Table 26.2. TABLE 26.2 Market rate Transfer price Margin

The calculation of margins 9.00% 9.20% Calculation

Direct calculation of margin Accounting margin 2000 × 12.00% − 1200 × 6.00% − 800 × 9.00% Commercial margins Loans 2000 × (12.00% − 9.20%) Deposits 1200 × (9.20% − 6.00%) Total commercial margin 2000 × (12.00% − 9.20%) + 1200 × (9.20% − 6.00%) ALM margin 2000 × 9.20% − 1200 × 9.20% − 800 × 9.00% Bank margin Commercial margin + ALM margin

Value 96.0 56.0 38.4 94.4 1.6 96.0

With a unique internal price set at 9.20%, the commercial margin as a percentage is 12% − 9.20% = 2.80% for uses of funds and 9.20% − 6% = 2.80% for resources. ALM charges the lending activities at the transfer price and purchases the resources collected at this same price. The total commercial margin is 94.4, and is lower than the bank’s accounting margin. This total commercial margin adds the contributions of the lending activity and the collection of resources. The lending activity generates a margin of 56.0, while the collection of resources generates 38.4. These two activities could be different business units. The system allocates the business margin to the different business units.

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The ALM has symmetrical revenues and costs, plus the cost of funding the deficit of 800 in the market. Its margin is 1.6. The commercial margin is 94.4, but the accounting margin of the bank is 96.0. This is due to the positive ALM margin, because ALM actually overcharges the market rate to the business units with a transfer price of 9.20%, higher than the market rate (9%). If we add the contributions of business units and the ALM margin, the result is 96.0, the accounting margin. The mechanism reconciles the contribution calculations with the bank’s accounting margin. It is easy to verify that, if the ALM margin is zero, the entire bank’s margin becomes equal to the commercial margin. This is a reference case: ALM is neutral and generates neither profit nor loss. Then the entire bank’s margin is ‘in’ the commercial units, as in the calculations below. Note that we keep the customers’ rates constant, so that the internal spreads change between customers’ rates and transfer prices. They become 9% − 8% = 1% for lending and 8% − 3% = 5% for collecting deposits. Evidently, since all customers’ rates are constant as well as the cost of market funds, the accounting margin remains at 4. Nevertheless, the allocation of this same accounting margin between business units and ALM differs (Table 26.3). TABLE 26.3

The calculation of margins

Market rate Transfer price Margin

9.00% 9.00% Calculation

Direct calculation of margin Accounting margin 2000 × 12.00% − 1200 × 6.00% − 800 × 9.00% Commercial margins Loans 2000 × (12.00% − 9.00%) Deposits 1200 × (9.00% − 6.00%) Total commercial margin 2000 × (12.00% − 9.00%) + 1200 × (9.00% − 6.00%) ALM margin 2000 × 9.20% − 1200 × 9.20% − 800 × 9.00% Bank margin Commercial margin + ALM margin

Value 96.0 60.0 36.0 96.0 0.0 96.0

The above examples show that: • Transfer prices allocate the bank’s margin between the business units and ALM. • The overall net interest margin is always equal to the sum of the commercial margin and the ALM margin.

Analytical Income Statements The cost allocation is extremely simplified. We use three different percentages for lending, collecting resources and the ALM units. The percentages apply to volumes (Table 26.4). Combining the cost allocations with the interest margins, we check that all interest margins add up to the overall bank margin and that the aggregated operating income is the bank operating income (Table 26.5).

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TABLE 26.4 units

Operating costs allocated to business

Operating cost

Volume

Loans Collection of resources ALM Total operating cost

2.00% 1.80% 0.08% 3.88%

Cost 2000 1200 3200

40.00 21.60 2.56 64.00

TABLE 26.5 Business unit income statements and bank consolidated income statements Income statement Interest margin Operating cost (2%) Operating income

‘Loans’

‘Deposits’

‘ALM’

‘Bank’

56.0 40.0 16.0

38.4 26.7 11.7

1.6 2.6 −1.0

96.0 69.3 26.7

Differentiation of Transfer Prices The above calculations use a unique transfer price. In practice, it is possible to differentiate the transfer prices by product lines, market segments or individual transactions. If we use a different transfer price for lending and collecting deposits, the customer rates being 12% for lending and 6% for deposits, the margin is: Commercial margin = assets(12% − TPassets ) + liabilities(TPliabilities − 6%) Commercial margin = assets × assets contribution (%) + resources × resources contribution (%) The commercial margin depends directly on the transfer prices and it changes if we use different ones. But whatever transfer prices we use, the ALM uses the same. When consolidating business transaction margins and ALM internal transactions, we end up with the accounting margin of the bank. This provides degrees of freedom in setting transfer prices. It is possible to use economic transfer prices differentiated by transaction, as recommended in the next chapter, and still reconcile all transaction margins with the bank’s accounting margin through the ALM margin. The facility of differentiating prices according to activities, product lines or markets, while maintaining reconciliation through the ALM margin, allows us to choose whatever criteria we want for transfer prices. The next chapter shows that economic prices reflecting the ‘true’ cost of funding are the best ones. However, it might be legitimate to also use transfer prices as commercial signals for developing some markets and products while restricting business on others. As long as we record the transfer prices and their gaps with economic prices, we know the cost of such a commercial policy. Transfer of revenues and costs neutralizes within the bank, whether or not there is a unique transfer price or several. In fact, no matter what the transfer prices are, they allocate the revenues. Setting economic benchmarks for these prices is an issue addressed in the next chapter.

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ALM AND BUSINESS UNIT PROFITABILITY GOALS The overall profitability target should translate into consistent objectives across business units and the ALM unit. We discuss setting objectives for the ALM unit and move on to the definition of business line target profitability levels, from an overall profitability goal. The overall profitability relates to the overall risk of the bank, as target profitability on economic capital. We assume it given, and examine how to break it down.

ALM Profitability and Risks ALM is a special business unit since it can have various missions. If its target profit is set to zero, it behaves as a cost centre whose responsibility is to minimize the cost of funding and hedge the bank against interest rate risk. If the bank has excess funds, the mission of ALM is still to keep the global interest rate risk within limits, while maximizing the return on excess funds. This is consistent with a strategy that makes commercial banking the core business, and where the ALM policy is to hedge liquidity and interest rate risks with cost-minimizing, or revenue-maximizing, objectives. However, minimizing the cost of funding could be an incentive for ALM to maintain some exposure to interest rate risk depending on expectations with respect to interest rates. In general, banking portfolios generate either deficits or excesses of funds. Nevertheless, a policy of hedging systematically all exposures, for all future periods, has a cost. This is the opportunity cost of neutralizing any potential gain resulting from market movements. However, without a full hedge, there is an interest rate risk. The ALM unit should be responsible for this risk and benefit from any saving in the cost of funds obtained thanks to maintaining risk exposure. This saving is the difference between the funding cost under full hedging policy and the effective funding cost of the actual policy conducted by ALM. This cost saving is its Profit and Loss (P&L). Since it is not reasonable to speculate on rates, interest rate risk limits should apply to ALM exposures. Giving such flexibility to ALM turns it into a profit centre, which is in charge of optimizing the funding policy within specified limits on gaps, earnings volatility or ‘Value at Risk’ (VaR). Making such an organization viable requires several conditions. All liquidity and interest rate risks should actually be under ALM control, and the commercial margins should not have any exposure to interest rate risk. This imposes on the FTP system specifications other than above, developed subsequently. In addition, making the ALM a profit centre subject to limits requires proper monitoring of what the funding costs would be under full hedging over a period. Otherwise, it is not possible to compare the effective funding costs with a benchmark (Figure 26.5).

Setting Target Commercial Margins Banks have an overall profitability goal. They need to send signals to business units and allocate target contributions to the various business units consistent with this overall goal. These contributions are the spreads over transfer prices times the volume. Assuming transfer prices given, the issue is to define commercial contributions of the various units such that they sum exactly to the target accounting net margin.

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Risks Risks

ALM ALM P&L P&L Maintain risk within limits

FIGURE 26.5

Return Return

Minimize funding cost Maximize investment return

Policies and profitability of ALM

The accounting net interest margin is the summation of commercial margins, or contributions, and of the ALM margin. If ALM is a pure hedging entity, business units generate the entire margin. In such a case, the sum of commercial margins becomes identical to the target accounting margin. If ALM is a profit centre, any projected profit or loss of ALM plus the commercial margins is necessarily equal to the consolidated margin. Taking the example of lending only, we have only one business unit. The commercial margin, as a percentage of assets, is the spread between the unknown customer rate and the transfer price, or X − 8%. The value of the margin is the product of this spread with asset volume. The accounting margin is, by definition, the total of the commercial margin of business units and of the ALM margin: Mbank = Mcommercial + MALM Therefore, the commercial margin is the difference between the accounting margin and the internal ALM margin. The process requires defining what the ALM margin is. The ALM margin might be zero, making the sum of commercial contributions to the margin identical to the accounting margin. This would be the benchmark case, since any P&L of the ALM is a ‘speculative’ cost saving ‘on top’ of the net margin of the banking book. The simple example below shows how to move from a target overall accounting interest margin towards the commercial contributions (margins) of business units. Once these margins are defined, the target customer prices follow, since they are equal to the transfer prices plus the percentage margin on average. We use the following data. The volume of assets is 100, the cost of borrowing in the market is 10% and the internal transfer price is 8%. There is a difference between the market rate and the internal transfer rate to make the argument general. The volume of equity is 4% of assets, or 40. The target net interest margin is 25% of equity, or 10. The outstanding debt is 1000 − 40 = 960. The cost of debt is 8% × 960 = 76.8. Table 26.6 summarizes the data. The issue is to find the appropriate target commercial margin over transfer price. The ALM margin is: MALM = 8% × 1000 − 10% × 960 = −16 It is negative because the ALM prices internal funds at a rate lower than the market rate and subsidizes the commercial units. The target commercial margin is the difference

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TABLE 26.6

Setting a target economic margin

Target accounting margin Cost of debt Transfer price Volume of assets Equity Debt Target commercial margin over transfer price

10 10% 8% 1000 40 960 ?

between the targeted accounting margin and that of the ALM: Target IMcommercial = target IMoverall − IMALM = 10 − (−16) = 26 Given the volume of assets, it is easy to derive the average target customer rate X from this target margin. The margin value 26 represents 2.6% of outstanding assets. The required spread as a percentage of assets is: X − 8% = 2.6% The average rate X charged to customers should be 10.6%. It is sufficient to set a target commercial margin at 26, setting some minimum customer rate to avoid selling below cost, so that commercial entities might have flexibility in combining the percentage margin and the volume from one transaction to another as long as they reach the overall 26 goal. We check that the calculation ensures that the net interest margin is 10. It is the total revenue less the cost of debt: 10.6% × 1000 − 96 = 10. This equation allows us to calculate the average customer rate directly since the target accounting margin, the cost of debt and the size of assets are given. It simply says: (X − 8%) × 1000 − 96 = 10. The same overall summations hold with multiple transfer prices. Therefore, transfer prices allow us to define the global commercial margin above transfer prices consistent with a given target net accounting margin. Setting the actual business line target contributions to this consolidated margin implies adjusting the profitability to the risk originated by business units. For lending, credit risk is the main one. Within an integrated system, the modulation of target income depends on the risk contributions of business lines to the overall risk of the portfolio (as defined in Chapters 51 and 52).

THE FINANCIAL AND COMMERCIAL RATIONALE OF TRANSFER PRICES The FTP system is the interface between the commercial universe and the financial universe. In order to serve this purpose, the transfer prices should be in line with both commercial and financial constraints. From a financial standpoint, the intuition is that transfer prices should reflect market conditions. From a commercial standpoint, customer prices should follow business policy guidelines subject to constraints from competition. In other words, transfer prices should be consistent with two different types of benchmarks: those derived from the financial universe and those derived from the commercial policy. Transfer prices should also be consistent with market rates. The next chapter develops this rationale further. However, the intuition is simple. If transfer prices lead to much higher customer rates than the market offers, it would be impossible to sustain competition

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and customers would arbitrage them with the market. For buying resources, a similar rationale applies. The alternative uses of resources are lending or investing. If lending provides much less than the market, the bank is better off investing. If the opposite holds, resources priced internally at more than market prices would hurt the lending margin. Hence, the market provides a benchmark. On the other hand, ignoring competitors’ prices does not make much sense. This leads to the usual trade-off between profitability and market share. Banks are reluctant to give up market share, because it would drive them out of business. Since competition and market rates are not easy to reconcile, there has to be some mispricing. Mispricing is the difference between ‘economic prices’ and effective pricing. Mispricing is not an error since it is business-driven. Nevertheless, it deserves monitoring for profitability and business management, and preventing mispricing from being too important to be sustainable. Monitoring mispricing implies keeping track of target prices and effective prices, to report any discrepancy between the two. At this stage, the mispricing concept applies to economic transfer prices limited to such benchmarks as the cost of funds for lending and benchmarks of investment return for collecting resources. It extends to riskbased pricing, when target prices charge the credit risk to customers. This has to do with capital allocation. A logical conclusion is that using two sets of internal prices makes sense. One set of transfer prices should refer to economic benchmarks, such as market rates. The other set of transfer prices serves as commercial signals. Any discrepancy between the two prices is the cost of the commercial policy. These discrepancies are the penalties (mark-up) or subsidies (mark-down) consistent with the commercial policy. This scheme reconciles diverging functions of the transfer prices and makes explicit the cost of enforcing commercial policies that are not in line with market interest rates.

27 Economic Transfer Prices

This chapter focuses on economic benchmarks for transfer prices that comply with all specifications of the transfer pricing scheme. The basic principle for defining such ‘economic’ transfer prices is to refer to market prices because of arbitrage opportunities between bank rates and market rates whenever discrepancies appear. Economic benchmarks derive from market prices. Mark-ups or mark-downs over the economic benchmarks serve for pricing. There are two types of such add-ons. Some serve for defining risk-based pricing. Others are commercial incentives or penalties resulting from the business policy, driving the product–market segments mix. Note that implementing this second set of commercial mark-ups or mark-downs requires tracking the discrepancies with economic prices which, once aggregated, represent the cost of the commercial policy. Economic benchmarks for transfer prices are ‘all-in’ costs of funds. The ‘all-in’ cost of funds applies to lending activities and represents the cost of obtaining these funds, while complying with all banks’ balance sheet constraints, such as liquidity ratio. It is a market-based cost with add-ons for liquidity risk and other constraints. It is the cost of a ‘notional’ debt mimicking the loans. It is notional because no rule imposes that Asset–Liability Management (ALM) continuously raises such debts. The rationale of this scheme is to make sure that lending provides at least the return the bank could have on the market, making it worthwhile to lend. The target price adds up the cost of funding with mark-ups and mark-downs, plus a target margin in line with the overall profitability goal. To ensure effective transfer of interest rate risk from business units to ALM, economic prices of individual transactions are historical, calculated from prevailing market rates at origination, and guaranteed over the life of the transaction. The guaranteed prices make the interest incomes of transactions and business lines insensitive to interest rates over subsequent periods until maturity of the transaction. The cost of funds view applies to lending activities. For excess resources, market rates are the obvious references. The difference is that investments generate interest rate risk

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and exposure to the credit risk of issuers, while lending generates exposure to the credit risk of borrowers, ALM taking care of interest rate risk. Investments of excess funds raise the issue of defining meaningful risk and return benchmarks. These result, in general, from investment portfolios serving as ‘notional’ references. The first section expands the arbitrage argument to define transfer prices referring to market rates. The second section provides an overview of the transfer pricing scheme with all inputs, from the cost of funds to the final customer price, using a sample set of calculations. The third section defines the economic cost of funds, the foundation on which all additional mark-ups and mark-downs pile up. The fourth section makes explicit the conditions for ensuring effective transfer of interest rate risk to the ALM unit.

COMMERCIAL MARGIN AND MATURITY SPREAD Discrepancies of banks’ prices with market prices lead to arbitrage by customers. The drawback of a unique transfer price, equal to the average cost of funds of the bank, is that it would serve as a reference for both long and short loans. If the term structure of market rates is upward sloping, customer prices are below market rates on the long end of the curve and above market rates in the short maturity range (Figure 27.1). The unique transfer subsidizes long-term borrowers and penalizes short-term borrowers. A first conclusion is that transfer prices should differ according to maturities. Rate Rate for borrowers Commercial margins

Transfer price Rate for depositors Maturity

FIGURE 27.1

Drawbacks of a single transfer price

It is sensible to remove the maturity spread from the commercial margins, since this spread is beyond the control of the business lines. The maturity spread depends on market conditions. On the other hand, it is under the control of ALM, which can swap longterm rates against short-term rates, and the spread is in the accounting margin of the bank. Most commercial banks ride the yield curve by lending long and borrowing short, benefiting from this spread. On the other hand, riding the yield curve implies the risk of ‘twists’ in the curve. The risk of shifts and twists should be under the responsibility of ALM, rather than affecting commercial margins. Otherwise, these would embed interest rate risk. Business lines would appear responsible for financial conditions beyond their control. Making commercial margins subject to shifts and twists of the yield curve would ultimately lead to closing and opening offices according to what happens to the yield curve. The implication is that ‘commercial’ margins do not include the contribution of the market spread between long and short rates, although it contributes to the accounting margin, and this contribution should be under the control of ALM.

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In order to ensure that the bank makes a positive margin whatever the rates, and to isolate the maturity spread from the commercial margin, it is necessary to relate transfer prices to maturity. Bullet loan prices should be above market rates of the same maturity. Term deposit rates should be below the market rate of the same maturity. Demand deposit rates should be below the short-term rate. When lending above the market rate, the bank can borrow the money on the market and have a positive margin whatever the maturity. When collecting resources below market rates, the bank can invest them in the market and make a profit. Figure 27.2 illustrates this pricing scheme. Customer rate Rate Market spread

Commercial spread Customer rate

Time

FIGURE 27.2

Commercial and market spreads

This suggests that market rates are the relevant benchmarks. It also illustrates the arbitrage argument making discrepancies irrelevant for both customers and the bank. This basic pricing scheme is the foundation of economic transfer prices, although the relevant benchmarks require further specifications.

PRICING SCHEMES Transfer prices differ for lending and calculating margins on resources. For assets, transfer prices include all financial costs: they are ‘all-in’ costs of funds, with all factors influencing this cost. For deposits, the transfer prices should reflect the market rates on investing opportunities concurrent with lending.

Lending Activities There are several pricing rationales in practice, mixing economic criteria and commercial criteria. It is possible to combine them if we can fully trace the components of pricing and combine them consistently for reporting purposes. • Risk-based pricing is the benchmark, and should be purely economic. It implies two basic ingredients: the cost of funds plus mark-ups, notably for credit risk in lending. The mark-ups for risk result from the risk allocation system, and derive from capital allocations. We consider them as given here and concentrate on the economic costs of funds.

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• Commercial pricing refers to mark-ups and mark-downs over economic benchmarks to drive the business policies through incentives and penalties differentiated by product and market. Such mark-ups and mark-downs are purely business-driven. A comprehensive pricing scheme might include risk-based references plus such commercial mark-ups and mark-downs. ‘Effective pricing’ refers to actual prices used by banks. They might differ from the target risk-based prices simply because pricing is subject to competitive pressures. Mispricing is the difference between effective prices and target prices. The ‘all-in’ cost of funds serves as the foundation for transfer prices. To get the economic transfer price, other economic costs should add up. They include the expected loss on credit risk. Risk-based pricing implies a mark-up related to the credit standing of the borrower. This all-in cost of funding, plus these economic mark-ups, is the foundation of the transfer price. Note that, at this level, the bank does not earn any profit yet since it simply balances the overall cost of funding and risk. Conventions are necessary at this stage. If the cost of allocated risk includes the target profitability, the bank actually defines a transfer price that provides adequate profitability. An alternative presentation would be to add a margin, which is the target profitability on allocated capital. Such a target margin, or contribution, should be in line with the overall profitability goal of the bank, and absorb the allocated operating costs. We adopt this second presentation below, because it makes a distinction between the economic transfer price and the risk-based price, the difference being precisely the target return on allocated capital. Table 27.1 summarizes the format of the economic income statement. TABLE 27.1 incentives

Risk-based pricing and commercial

Component

%

Cost of funding +Cost of liquidity =‘All-in’ cost of funding −Expected loss from credit risk =Economic transfer price +Operating allocated costs +Risk-based margin for compensating credit risk capital =Target risk-based price +Business mark-ups or mark-downs =Customer price

Such transfer prices are before any business mark-ups or mark-downs. Additional mark-ups or mark-downs, which result from deliberate commercial policies of providing incentives and penalties, could also affect the prices. Our convention is to consider only the economic transfer price plus a profitability contribution. The rationale is that we need to track these to isolate the cost of the business policy, while other mark-ups or mark-downs are purely business-driven. Moreover, we also need to track mispricing or the gap between effective prices and target risk-based prices for reporting purposes, and take corrective action. Figure 27.3 summarizes all schemes and mispricing.

ECONOMIC TRANSFER PRICES

'Pure' risk "Pure" based Risk Based pricing Pricing

All-in cost of funds

329

Expected loss

Mark-up for risk: capital

Risk Risk based based price price

Mispricing Effective Effective pricing pricing

FIGURE 27.3

All-in cost of funds

Effective mark-up/ down

Effective Effective pricing pricing

Pricing schemes and mispricing

At this stage, we have not yet defined the ‘all-in’ cost of funds. The next section does this.

Transaction versus Client Revenues and Pricing Risk-based pricing might not be competitive at the individual transaction level simply because market spreads are not high enough to price all costs to a large corporate which has access to markets directly. This is a normal situation, because there is no reason why local markets should price credit risk, as seen in credit spreads, in line with the target of banks based on capital allocation for credit risk. Section 16 of this book addresses again the ‘consistency’ issue between internal pricing and external pricing of risk. However, this does not imply that the overall client revenue cannot meet the bank’s target profitability. Banks provide products and services and obtain as compensation interest spreads and fees. The overall client revenue is the relevant measure for calculating profitability, because it groups all forms of revenues from all transactions, plus all services resulting from the bank’s relationship with its clients. Loans and services are a bundle. Using risk-based pricing at the transaction level might simply drive away business that would be profitable enough at the client level. The client is a better base for assessing profitability than standalone transactions. This is a strong argument for developing economic income statements at the client level.

Target Risk-based Pricing Calculations Economic pricing for loans includes the cost of funds plus any mark-up for risk plus a target margin in line with the overall profitability goal. The cost of funding depends on rules used to define the funding associated with various loans. The pure economic benchmark is the one notional economic cost of funds, described below, which is the cost of funding that exactly mirrors the loan. We assume here that this cost of funds is 7%. The ‘all-in’ funding cost of the loan adds up the cost of maintaining the liquidity ratio and the balance sheet structure at their required level. We use here an add-on of 0.2% to obtain an ‘all-in’ cost of funds of 7.2%. Other relevant items are expected loss and allocated cost. Note that the overall profitability of the bank should also consider an additional contribution for non-allocated operating costs1 . 1 Chapters

53 and 54 provide details on full-blown economic statements pre- or post-operating costs.

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The required margin results from the required capital and the target return on that capital. For an outstanding balance of 1000 that requires no capital, the price needs to absorb only the all-in cost of funding. When there is credit risk, the rationale for defining the target required margin based on allocated capital is as follows. If capital replaces a fraction of debt because the loan is risky, there is an additional cost because capital costs more than debt. The additional cost is the differential cost between equity and debt weighted by the amount of capital required. The same calculations apply to regulatory capital or economic capital. The pre-tax cost of equity is 25% in our example. The additional cost is the amount of equity times the differential between the cost of equity and the cost of debt: (25% − cost of debt) × equity. In order to transform this value into a percentage of the loan outstanding balance, we divide this cost in value by the loan size, assuming that the ratio of equity to assets is 4% for example. This gives the general formula for the risk premium: (25% − cost of debt) × 4%. In this example, the cost of capital is (25% − 7%) × 4% = 0.72%. All components of transfer prices are given in Table 27.2. To obtain a pure economic price, we set commercial incentives to zero. TABLE 27.2 prices

Components of transfer

Component Cost of debt +Cost of liquidity +Expected losses +Operating costs =Transfer price +Risk-based margin =Target risk-based price +Commercial incentives =Customer rate

% 7.00 0.20 0.50 0.50 8.20 0.72 8.92 0 8.92

The cost of debt, set to 7%, is the most important item. The next section discusses the all-in cost of funds.

THE COST OF FUNDS FOR LOANS The transfer price depends on the definition of the funds backing the assets. There are two views on the issue of defining the transaction that best matches the assets or the liabilities. The first is to refer to existing assets and resources. The underlying assumption is that existing resources fund assets. The second view is to define a ‘best’ funding solution and use the cost of this funding solution as the pure cost of funding.

The Cost of Existing Resources Using the existing resources sounds intuitively appealing. These resources actually fund assets, so their cost has to be relevant. However, this solution raises conceptual inconsistencies. There are several types of assets and liabilities with different characteristics.

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Each type of resource has a different cost. Therefore, we use either the ‘weighted average cost of capital’ (wacc), the average cost of resources or several costs. We pointed out earlier the shortcomings of a single transfer price. Another solution could be to match assets and resources based on their similar characteristics. For instance, we can try to match long-term resources with long-term assets, and so on for other maturities. This is the rationale for multiple pools of funds based on maturity buckets. This principle raises several issues: • Since the volumes of assets and resources of a given category, for instance long-term items, are in general different, the match cannot be exact. If long-term assets have a larger volume than long-term resources, other resources have to fill up the deficit. Rules are required to define which other resources will fund the mismatch, and the resulting cost of funds becomes dependent on such conventions. • For business units, there is a similar problem. Generally, the balance sheet under the control of a business unit is not balanced. Any deficit needs matching by resources of other business units. However, it does not sound right to assign the cost of funds of another business unit, for instance a unit collecting a lot of ‘cheap’ deposits, to one that generates a deficit. The collection of cheap resources should rather increase the profitability of the unit getting such resources. • Any cheap resource, such as deposits, subsidizes the profitability of assets. Matching a long-term loan with the core fraction of demand deposits might be acceptable in terms of maturity. However, this matching actually transfers the low cost of deposits to the margin allocated to loans. Economically, this assumes that new deposits fund each new dollar of loans, which is unrealistic. This view is inconsistent from both organizational and economic standpoints. It implies a transfer of income from demand deposits to lending. In general, transferring the low cost of some resources to lending does not leave any compensation for collecting such resources. Low cost resources subsidize lending and lose their margin in the process! Given inconsistencies and unrealistic assumptions, the matching of assets with existing resources is not economic. Transfer prices require other benchmarks.

The ‘Notional’ Funding of Assets The unique funding solution that neutralizes both liquidity and interest rate risk of a specific asset is the funding that ‘mirrors’ the time profile of flows and the interest rate nature. For instance, with a fixed rate term loan, the funding should replicate exactly the amortization profile and carry a fixed rate. Such funding is more ‘notional’ than real. It does not depend on the existing resources. It does not imply either that ALM actually decides to implement full cash flow matching. It serves as a benchmark for determining the cost of funds backing any given asset. In some cases, the replication is obvious and the cost of funds also. A bullet debt matches a bullet loan. The relevant rate is the market rate corresponding to the maturity of the transaction. In other cases, the notional funding is different. For an amortizing loan, the outstanding balance varies over time until maturity. Therefore, the market rate of this maturity is not adequate. Using it would assume that the loan does not amortize over

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time. The funding that actually replicates the time profile of the loan is a combination of debts of various maturities. In the example below, a loan of 100 amortizes in 2 years, the repayments of capital being 40 and 60. Figure 27.4 shows the profile of the reference debt. The funding solution combines two spot bullet debts, of 1 and 2-year maturity, contracted at the market rates2 . This example demonstrates that it is possible to find a unique notional funding for any fixed rate asset whose amortizing schedule is known. Outstanding balance

40

60

Debt spot 1 year

Debt spot 2 years

1 year

FIGURE 27.4

2 years

Time

A 2-year amortizing loan

The solution is even simpler with floating rates than with fixed rates. For floating rate transactions, the replicating funding is a floating rate debt with reset dates matching those of the assets. The amortization profile should mirror that of the floating rate asset. However, for those assets that have no maturity, conventions are necessary.

The Cost of Funds For a fixed rate loan, the funding solution used as reference has a cost that depends on the combinations of volumes borrowed and maturities. The cost is not a single market rate, but a combination of market rates. In the above example, there are two layers of debt: 60 for 2 years and 40 for 1 year. The relevant rates are the spot rates for these maturities. However, there should be a unique transfer price for a given loan. It is the average cost of funds of the two debts. Its exact definition is that of an actuarial rate. This rate is the discount rate making the present value of the future flows generated by the two debts equal to the amount borrowed. The future outflows are the capital repayments and interest. The interest flows cumulate those of the 1-year and 2-year market rates. If the debts are zero-coupon, the interest payments are at maturity. The flows are 40(1 + r1 ) in 1 year and 60(1 + r2 )2 in 2 years. 2 There

is another solution for backing the loan. For instance, a spot debt for 1 year could be contracted for the full amount of the loan, that is 100 for 1 year at the spot market rate, followed by another 1-year debt, for an amount of 60, starting 1 year from now. Nevertheless, such funding does not replicate the loan, since a fraction of the debt needs renewal for another year at a rate unknown today.

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The cost r of this composite funding is a discount rate such that: 100 = 40 × (1 + r1 )/(1 + r) + 60 × (1 + r2 )2 /(1 + r)2 The discount rate is somewhere between the two market rates. An approximate solution uses a linear approximation of the exact formula: 100 = 40(1 + r1 − r) + 60(1 + 2r2 − 2r) r = (40 × r1 + 60 × 2 × r2 )/(40 + 2 × 60) The rate r is the weighted average of the spot rates for 1 and 2 years, using weights combining the size of each debt and its maturity. For instance, r1 counts roughly twice and r2 only once. With rates r1 and r2 equal to 8% and 9%, r = 8.75%. The rate is closer to 9%, because the 2-year debt is the one whose amount and maturity are the highest. In practice, transfer price tables provide the composite market rates immediately, given the time profile of loans and the current market rates.

The Benefits of Notional Funding Using the cost of funds of a debt that fully replicates the assets offers numerous economic benefits. Perfect matching implies that: • The margin of the asset is immune to interest rate movements. • There is no need for conventions to assign existing resources to usages of funds. • There is no transfer of income generated by collecting resources to the income of lending activities. • The calculation of a transfer price is mechanical and easy. This objective choice avoids possible debates generated by conventions about transfer prices. In addition, this reference actually separates the income of the banking portfolio from those of ALM. It also separates business risk from financial risk and locks in commercial margins at origination of the loan, if we use as transfer price the cost of this notional funding mimicking the loan. The ALM unit does not have to use a funding policy that actually immunizes the interest margin of the bank. The debt replicating the asset characteristics is a ‘notional debt’. The ALM policy generally deviates from perfect matching of the flows of individual assets depending on interest rate views for example. On the other hand, a perfect match is feasible at the consolidated level of the bank for neutralizing both liquidity and interest rate risk. Doing so at the individual transaction level would result in over-hedging because it would ignore the natural offsets between assets and liabilities, resulting in gap profiles. Comparing the perfect matching cost with the effective cost resulting from the ALM policy determines the performance of the ALM. If ALM policy results in a cost saving compared to the cost of a global perfect match, this saving is the profit of ALM. If the effective policy results in a higher cost, ex post, the cost differential over the benchmark is a loss. This requires keeping track of the cost of funding under perfect matching and the effective cost of funding for the same period. The differential is the ALM contribution.

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Transfer Prices for Resources Assigning transfer prices to resources should follow the same principles, except that some resources have no tangible maturity. The arbitrage rationale suffices to define economic transfer prices to ALM. It is feasible to invest resources with a contractual maturity up to that maturity, so that the corresponding market rate serves as reference. For resources without maturity, such as the volatile fraction of demand deposits, the maturity is conventional, such as a 1-year maturity, or modelled to some extent according to the effective time profile of resources.

TRANSFERRING LIQUIDITY AND INTEREST RATE RISK TO ALM Two factors help to fully separate commercial and financial risks. First, the commercial margins become independent of the market maturity spread of interest rates and of the ‘twist’ of the yield curve. ALM is responsible for managing the spread risk. Second, referring to a debt replicating the asset removes the liquidity and the market risks from the commercial margin. This takes care of the transfer of interest rate risk to ALM. However, guaranteeing this risk transfer requires another piece of the Funds Transfer Pricing (FTP) to actually protect business margins against financial risks. For instance, if a fixed rate asset generates 11% over a transfer price of 9%, the margin is 2%. Recalculating the margin of the same asset with a new rate at the next period would result in a change, unless the calculation follows a rule consistent with the system. For instance, resetting the transfer price after one period at 10%, because of an upward shift of 1% in the yield curve, results in a decrease of the margin to 1%. This decrease is not acceptable, because it is not under the control of business lines. The philosophy of the system is to draw a clear-cut frontier between commercial and financial risks. An additional component of the FTP scheme allows us to achieve this goal. The calculation of transfer price as the cost of the mirror debt occurs at the origination date of the asset. This rules out transfer price resets because of market rate changes. The transfer price assigned to a transaction over successive periods is the historical transfer price. Doing otherwise puts the commercial margin at risk, instead of locking it over the life of the transaction. Margins calculated over reset transfer prices are apparent margins. They match current market conditions and change with them. The FTP system should ensure that transfer prices are assigned to individual transactions for the life of the transactions. These prices are historical prices ‘guaranteed’ for transactions and business units. Using such historical prices for each transaction puts an additional burden on the information system. Nevertheless, without doing so, the margins lose economic meaning. In addition, the consistency between what business units control by and their performance measures breaks down. Table 27.3 shows the time profile of both apparent and locked in margins calculated, respectively, over reset prices and guaranteed historical transfer prices. The pricing remains in line with a target margin of 2% over the transfer price. Meeting this goal at origination does not lock in the apparent margin over time. Only the margin over guaranteed prices remains constant. This last addition makes the transfer pricing system comprehensive and consistent.

ECONOMIC TRANSFER PRICES

TABLE 27.3

335

Margins on apparent transfer prices and historical transfer prices

Transactions

Period 1

Period 2

Period 3

A B C

100

100 100

100 100 100

Current rate Target % margin Customer rate

10% 2% 12%

11% 2% 13%

12% 2% 14%

Apparent margin over current rate Apparent margin (value) Apparent margin (%)

100 × (12 − 10)

100 × (12 − 11) +100 × (13 − 11)

100 × (12 − 12) + 100 ×(13 − 12) + 100 × (14 − 12)

2

3

3

2/100 = 2%

3/200 = 1.5%

3/300 = 1%

Margin over guaranteed historical price Margin over guaranteed price (value) Margin over guaranteed price (%)

100 × (12 − 10)

100 × (12 − 10) +100 × (13 − 11)

100 × (12 − 10) + 100 ×(13 − 11) + 100 × (14 − 12)

2

4

6

2%

2%

2%

BENCHMARKS FOR EXCESS RESOURCES Some banks have structurally excess resources or dedicate deliberately some resources to an investment portfolio, more or less independently of the lending opportunities. Under a global view, the bank considers global management of both loan and investment portfolios. The management of invested funds integrates with ALM policy. Alternatively, it is common to set up an investment policy independently of the loan portfolio, because there are excess funds structurally, or because the bank wants to have a permanent investment portfolio, independent of lending opportunities. This policy applies for example when investing equity into a segregated risk-free portfolio rather than putting these funds at risk with loans. The transfer price for the portfolio becomes irrelevant because the issue is to maximize the portfolio return under risk limits, and to evaluate performance in relation to a benchmark return. The benchmark return depends on the risk. The bank needs to define guidelines for investing funds. When minimizing risk is the goal, a common practice consists of smoothing out market rate variations by structuring adequately the investment portfolio. It implies breaking down the investment into several fractions invested over several maturities and rolling them over, like a ‘tractor’, when each tranche matures. This is the ‘ladder’ policy. It avoids crystallizing the current yield curve in large blocks of investments. The ‘tractor’ averages the time series of all yield curves up to the management horizon. Segregation of assets in different subportfolios creates potential conflicts with the global ALM view of the balance sheet. Any segmentation of the portfolio of assets might

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result in overall undesired mismatches not in line with global guidelines. For instance requiring a long-term maturity for an investment portfolio, while the loan maturities are business-driven, could increase the gap between the average reset date of assets and that of liabilities. Hedging this gap with derivatives is inconsistent with the long-term goal for the portfolio, because it is equivalent to closing the gap by synthetically creating a short-term rate exposure rather than the long-term one. In general, segmentation of the balance sheet creates mismatch and does not comply with the basic ALM philosophy. If controlling the overall remains the goal, gaps over all compartments should add up. If they do not, the bank maintains an undesired global exposure.

SECTION 10 Portfolio Analysis: Correlations

28 Correlations and Portfolio Effects

Combining risks does not follow the usual arithmetic rules, unlike income. The summation of two risks, each equal to 1, is not 2. It is usually lower because of diversification. Quantification based on correlations shows that the sum is in the range of 0 to 2. This is the essence of diversification and correlations. Diversification reduces the volatility of aggregated portfolio income, or Profit and Loss (P&L) of market values, because these go up for certain transactions and down for others, thereby compensating each other to some extent. Moreover, credit risk losses do not occur simultaneously. In what follows, we simply consider changes of values of transactions, whatever the nature of risk that triggers them. Portfolio risk results from individual risks. The random future portfolio values, from which potential losses derive, are the algebraic sum of individual random future transaction values or returns. Returns are the relative changes in values between the current date and the horizon selected for measuring risk. The distribution of the portfolio values is the distribution of this sum. The sum of random individual transaction values or returns follows a distribution dependent on the interdependencies, or correlations, between these individual random values. The expected value, or loss, does not depend on correlations. The dispersion around the mean of the portfolio value, or volatility, and the downside risk do. Individual risks and their correlations drive the portfolio risk and the extent of the diversification effect, which is the difference between the (arithmetic) sum of these individual transaction risks and the risk of the sum. Because portfolio risk depends so much on interdependencies between individual transaction risks, they play a key role in portfolio risk models. Various techniques serve for capturing the correlation, or diversification, effect:

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• Correlations quantify interdependencies in statistical terms, and are critical parameters for valuing the random changes in portfolio values. Basic statistical rules allow us to understand how to deal with sums of correlated random individual values of facilities. • When dealing with discrete default events, the credit standing of individual obligors depend on each other. Conditional probabilities are an alternative view of correlations between individual risks that facilitates the modelling of some random events. • When looking forward, the ‘portfolio risk building block’ of models relies on simulations of random future changes of value, or ‘returns’, of transactions, triggered by risk events. Forward-looking simulations require generating random values of ‘risk factors’ in order to explore all future scenarios and determine the forward distribution of the portfolio values. Such random scenarios should comply with variance and correlation constraints between risk factors. Special techniques serve to construct such simulations. Several examples use them later on for constructing portfolio loss distributions. The first section explains why correlations are critical parameters for determining portfolio risk. The second section defines correlations and summarizes the essential formulas applying to sums of correlated random variables. The third section introduces conditional probabilities. The fourth section presents some simple basic techniques, relying on factor models, for generating correlated values of risk drivers.

WHY ARE CORRELATIONS IMPORTANT? Correlations are measures of the association between changes in any pair of random variables. In the field of risk management, the random variables are individual values of facilities that the random portfolio value aggregates. The subsequent sections expand the essentials for dealing with correlated value changes.

Correlations and Interdependencies between Individual Losses within a Portfolio For the market portfolio, the P&L is the variation of market values. Market values are market-driven. The issue for measuring downside risk is to quantify the worst-case adverse value deviations of a portfolio of instruments whose individual values are market-driven. It does not make sense to assume that all individual values deviate simultaneously on the ‘wrong’ side because they depend on market parameter movements, which are interdependent. To measure market risk realistically, we need to assess the magnitudes of the deviations of the market parameters and how they relate to each other. The interdependencies, or correlations, between parameters, plus their volatilities, drive the individual values. This is the major feature of market Value at Risk (VaR) calculations. The modelling relates individual value deviations, or returns, to the market parameter co-movements. For credit risk, the same phenomenon occurs. It does not make sense to assume that all obligors are going to default at the same time. The principle for modelling simultaneous changes in credit states, including the migrations to the default state, is to relate the credit standing of firms to factors, such as economic–industry conditions, that influence all of them. If these get worse, defaults and credit deteriorations increase, if they improve,

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the reverse happens. Such dependence creates some association between credit states of all obligors and their possible defaults. This implies that we need to model correlations between credit events, both default and migration events. In all cases, correlations are key parameters for combining risks within portfolios of transactions, both for market and credit risk. This chapter has the limited goal of summarizing the basics of correlations and the main formulas often used for measuring portfolio statistics, notably volatility of losses. Such definitions are a prerequisite whenever risk aggregation or risk allocation are considered. Throughout the chapter, we use bold characters for random variables and italics or regular characters for their particular values1 .

The Portfolio Values as the Aggregation of Individual Facility Values In order to proceed, we need the basic definitions, notations and statistical formulas applying to the summation of random variables. As a preliminary step, we show that the portfolio value is the summation of all individual transaction values. The presentation does not depend on whether we deal with market risk or credit risk. In both cases, we have changes of facility values between now (date 0) and a future horizon H . For market risk, the horizon is the liquidation period. For credit risk, it is the horizon necessary to manage the portfolio risk or the capital, typically 1 year in credit portfolio models. For market risk, we link instrument values to the underlying market parameters that drive them. For credit risk, the value changes depend on credit risk migrations, eventually related to factors common to all borrowers such as the economic–industry conditions. The value change of any transaction i between current date 0 and horizon H is: Vi = (Vi , H − Vi,0 ) = Vi,0 × yi The unit value change yi is a random variable independent of the size of exposure i. It is simply the discrete time return of asset i. A loss is a negative variation of value, whether due to default or a downside risk migration. For market risk, the changes in values of each individual transaction are either gains or losses. The algebraic summation of all gains and losses is the portfolio loss. Since the individual value changes are random, so is the algebraic summation. The distribution of negative changes, or adverse changes, is the loss distribution. The random change portfolio value Vp is the sum of the random individual losses Vi , and the change in the portfolio value Vp is the sum of the changes in all individual transactions Vi : Vp = Vi i

Vp = 1 For

Vi

i

instance X is a random variable, and X or X designates a particular numerical value.

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The variations of values are Vi = (Vi ,H − Vi,0 ) = Vi,0 × yi so that the portfolio return yp is: wi,0 × yi Vi,0 × yi /Vp,0 = Vi /Vp,0 = yp = Vp /Vp,0 = i

i

i

In this formula, the portfolio return is the weighted sum of the individual asset returns, the weights wi,0 being the ratios of initial exposures to the total portfolio value. By definition, they sum to 1. The same definitions apply for credit risk, except that credit risk results from credit risk migrations, including the migration to the default state. Whenever there is a risk migration, there is a change in value. Whenever the migration ends in the default state, the value moves down to the loss given default, that is the loss net of recoveries. Models ignoring migrations are default only. In default mode, all changes in values are negative, or losses, and there are only two credit states at the horizon.

PORTFOLIO RISK AS A FUNCTION OF INDIVIDUAL RISKS This section uses the classical concepts of expectations and variance applied to the portfolio value or loss viewed as the algebraic sum of individual value changes. We provide here only a reminder of basic, but essential, definitions.

Expectation of Portfolio Value or Loss The expectation of the sum of random variables is the sum of their expectations. This relation holds for any number of random variables. In general, using compact notations for the summation: ai E(Xi ) E ai Xi = i

i

The immediate application is that the expectation of the change in value of a portfolio is simply the sum of all individual value changes of each individual instrument within the portfolio: E(Vp ) = i Vi . This result does not depend on any correlation between the individual values or losses.

Correlations The value or loss volatility of the portfolio measures the dispersion of the distribution of values. The volatility of value is identical to that of value variations because the initial value is certain. The loss volatility, although distinct from loss percentiles, is an intermediate proxy measure to the unexpected loss concept2 . The portfolio value or loss volatility depends on the correlation between individual values or losses and on their variance–covariance matrix. The correlation measures the extent to which random variables change together or not, in the same direction or in opposite directions. Two statistics characterize this association: the 2 Loss

percentiles are often expressed as a multiple of loss volatility. See Chapters 7 and 50 for specific details.

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correlation coefficient and the covariance. A correlation and a covariance characterize pairs of random variables. The covariance is the weighted sum of the products of the deviations from the mean of two variables X and Y, the weights being the joint probabilities of occurrence of each pair of values. The coefficient of correlation is simpler to interpret because it is in the range −1 to +1. It is calculated as the ratio of the covariance by the product of the volatilities (which are the square roots of variances) of X and Y. The value +1 means that the two variables change together. The correlation −1 means that they always vary in opposite directions. The zero correlation means that they are independent. The formulas are as follows: σXY = ρXY σX σY where σXY is the covariance between variables X and Y; ρXY is the correlation between variables X and Y; σX and σY are the standard deviations, or volatilities, of variables X and Y.

Correlation and Volatility of a Sum of Random Variables The volatility of a sum depends on the correlations between variables. It is the square root of the variance. The variance of a sum is not, in general, the sum of the variances. It is the sum of the variances of each random variable plus all covariance terms for each pair of variables. Therefore, we start with the simple case of a pair of random variables and proceed towards the extension to any number of random variables. The general formula applies for determining the portfolio loss volatility and the risk contributions to the portfolio volatility of each individual exposure. Hence, it is most important. Two Variables

For two variables, the formulas are as follows: V (X + Y) = σ 2 (X + Y) = V (X) + V (Y) + 2 Cov(X, Y)

V (X + Y) = σ 2 (X) + σ 2 (Y) + 2ρXY σ (X)σ (Y)

The covariance between X and Y is Cov(X, Y), V (X) is the variance of X, σ (X) is the standard deviation and ρXY the correlation coefficient. Since the covariance is a function of the correlation coefficient, the two above formulas are identical. If the covariances are not zero, the variance of the sum differs from the sum of variances. The correlation term drops out of the formulas when the two variables are independent: V (X + Y) = σ 2 (X) + σ 2 (Y) σ (X + Y) = σ 2 (X) + σ 2 (Y)

The variance of the sum becomes the sum of variances only when all covariances are equal to zero. This result is valid only when all variables move independently of each other. The volatility is the square root of the variance. Since the variance of the sum is the sum of variances and the volatility is less than the sum of volatilities. It is the square root of the sum of the squared values of volatilities.

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As an example, the volatilities of two equity returns are 2.601% and 1.826%, and the correlation between returns is 27.901%. The sum of volatilities is 2.601% + 1.826% = 4.427%. The variance of the sum is: V (X + Y) = 2.601%2 + 1.826%2 + 2 × 37.901% × 1.826% × 2.601% = 0.137% The volatility is: σ (X + Y) =

√

0.137% = 3.7014% < 4.427%

This volatility is lower than the sum of volatilities. This is the essence of the diversification effect, since it shows that deviations of X and Y compensate to some extent when they vary. Combining risks implies that the risk of the sum, measured by volatility, is less than the sum of risks. Risks do not add arithmetically, except in the extreme case where correlation is perfect (value 1). Extension to Any Number of Variables

This formula extends the calculation of the variance and of the volatility to any number N of variables Zi . This is the single most important formula for calculating the portfolio return or value volatility. The portfolio volatility formula also serves for calculating the risk contribution of individual facilities to the total portfolio loss volatility because it adds the contributions of each facility to this volatility. The formula also illustrates why the sum of individual risks is not the risk of the sum, due to the correlation effects embedded in the covariance terms. The general formula is: N N N 2 σ Zi = σij = σi2 + ρij σi σi σi2 + i=1

i =j

i=j =1

i=j =1

i =j

In this formula, σi2 is the variance of parameter i, equal to the square of the standard deviation. σi and σij are the covariance between variables Zi and Zj . The summation corresponds to a generalized summation over all pairs of random variables Zi and Zj . It collapses to the above simple formulas for two variables. The summation i =j is equiva lent to the more explicit formula i =j with i and j =1,N , a summation over all values of i and j , being allowed to vary from 1 to N , but with i not equal to j since, in this case, we have the first variance terms in the above formula. A similar, and more compact, notation writes that the variance of the random variables summing N random variables is the summation of all σij using the convention that, whenever i = j , we obtain the variance σi2 = σj2 : N 2 σ Zi = σij i,j with i and j =1,N

i=1

Reverting to the portfolio return, we need to include the weights of each individual exposure in the formulas. If we use the portfolio return as the random variable characterizing the relative change in portfolio value, using the simple properties of variances and covariances, we find that: N N N 2 σ ρij wi wj σi σi wi wj σij = wi2 σi2 + wi2 σi2 + Zi = i=1

i=j =1

i =j

i=j =1

i =j

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The variance σi and the covariance σij now refer to the individual asset returns. The weights are constant in the formula, which implies a crystallized portfolio, with constant asset structure. This is the basic formula providing the relative portfolio value over a short period. It applies as long as the weights are approximately constant.

Visual Representation of the Diversification Effect The diversification effect is the gap between the sum of volatilities and the volatility of a sum. In practice, the random variables are the individual values of returns of assets in the portfolio. The diversification effect is the gap between the volatility of the portfolio and the sum of the volatilities of each individual loss. A simple image visualizes the formula of the standard deviation of a sum of two variables. The visualization shows the impact of correlation on the volatility of a sum. A vector whose length is the volatility represents each variable. The angle between the vectors varies inversely with correlation. The vectors are parallel whenever the correlation is zero and opposed when the correlation is −1. With such conventions, the vector equal to the geometric summation of the two vectors representing each variable represents the overall risk. The length of this vector is identical to the volatility of the sum of the two variables3 . Visually, its length is less than the sum of the lengths of the two vectors representing volatilities, except when the correlation is equal to +1. The geometric visualization shows how the volatility of a sum changes when the correlation changes (Figure 28.1). Variable 2

1+2

Angle = correlation −0 if ρ = +1 −90° if ρ = 0 −180° if ρ = −1 Variable 1 Length of vectors = volatility

FIGURE 28.1 variables

Geometric representation of the volatility of the sum of two random

Figure 28.2 groups different cases. The volatilities of the two variables are set to 1. The only change is that of their correlation changes. The volatility of the sum is the length of the geometric summation of vectors 1 and 2. It varies between 0 when the correlation is −1, up to 2 when the correlation is +1. The intermediate case, when correlation is zero, √ shows that the volatility is 2. 3 This

result uses the formula for the variance of a sum, expressing the length of the diagonal as a function of the sides of the square of which it is a diagonal.

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Correlation +1

σ1+2 = 2

FIGURE 28.2

Correlation > 0

Independence Correlation 0

Correlation < 0 Correlation − 1

σ1+2 = √2

σ1+2 = 0

The change in volatility of a sum when the correlation changes

CONDITIONAL AND UNCONDITIONAL PROBABILITIES Conditional probabilities provide an alternative view to correlations that serves in several instances. The unconditional probabilities are the ‘normal’ probabilities, when we have no information on any event that influences the random events, such as defaults. Conditional probabilities, on the other hand, embed the effects of information correlated with the random event.

Conditional Probabilities and Correlations In the market universe, we have time series of prices and market parameter values, making it easy to see whether they vary together, in the same direction, in opposite directions, or when there is no association at all. The universe of credit risk is less visible. We have information on the assessment of the risk of obligors, but the scarcity of data makes measures of correlations from historical data hazardous. In addition, we cannot observe the correlation between default events of two different firms in general because the joint default of two firms is a very rare event. Correlation still exists. Typically, the default probabilities vary across economic cycles. Since the credit standing of all firms depends on the state of the economy, the variations of economic conditions create correlations between default events. If economic or industry conditions worsen, all default probabilities tend to increase and vice versa. Therefore, the same default probability should differ when there is no information on future economic conditions and if it is possible to assess how likely the future states of the economy will be, better or worse than average. Probabilities depending on random economic scenarios are conditional probabilities on the state of the economy. In general, when common factors influence various risks, the risks are conditional on the values of these factors. Correlations influence the probability that two obligors default together. Default probabilities depending on the default events of the other firm are conditional probabilities on the status, default or non-default, of the other firm. The purpose of the next sections is to explain the properties of joint and conditional probabilities, and to illustrate how they relate to correlations. Applications include the definitions of joint default or migration events for a pair of obligors, the derivation of default probabilities conditional on ‘scoring models’ providing information on the credit standing of borrowers, or making risk events dependent on some common factor, such as the state of the economy, that influences them.

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Conditional Probabilities Conditional probabilities are probabilities conditional on some information that influences this probability. If X is a random event whose probability of occurrence depends on some random scenario S, then the probability of X occurring given S is the conditional probability P (X|S). We characterize default events by a random variable X taking only two values, default or non-default, or 1 and 0 respectively for these two states. The unconditional default probability of X taking the default value 1 is P (X). This is an unconditional default probability that applies when we have no particular information on whatever influences this probability. The random state of the economy S influences the likelihood of X values. The probability of X defaulting once S is known, or P (X|S), is the conditional default probability of default of X given S. The conditional probability depends on the correlation between the two events. For example, if economic conditions worsen, the default probability of all firms increases. In such a case, there is a correlation between the default event X and the occurrence of worsening economic conditions, a value of S. If there is no relation, the conditional probability P (X|S) collapses to the unconditional probability P (X). If there is a positive relation, the conditional probability increases above the unconditional probability, and if there is a negative correlation, it decreases. For instance, if the unconditional default probability is 1% for a firm, its conditional probability on S is lower than 1% if the economy improves and higher when it deteriorates.

Joint Probabilities A joint probability is the probability that two events occur simultaneously. It applies to both continuous and discrete variables. The two random events are X and Y. The relationship between the joint probability that both X and Y occur and the conditional probabilities of X given occurrence of Y, or Y given occurrence of X, is: P (X, Y) = P (X) × P (Y|X) = P (Y) × P (X|Y) A first example relates the default probability to the state of the economy. A firm has an ‘unconditional’ default probability of 1%. The random value F, taking the values ‘default’ and ‘non-default’, characterizes the status of the firm. The firm’s unconditional default probability P (F = default) = 1%. The unconditional probability represents the average across all possible states of the economy. Let us now consider three states of the economy: base, optimistic and worst-case. The probability of observing the worst-case state is 20%. Let us assume that the corresponding default probability increases to 2%. This probability is conditional on S. The formula is: P (F = default|S = worst-case) = 2%. The joint probability of having simultaneously a default of the firm and a worst-case state of the economy is, by definition: P (F = default, S = worst-case) = P (S = worst-case) × P (F = default|S = worst-case) = 20% × 2% = 0.4% As a summary: P (F = default) = 1%

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P (F = default|S = worst-case) = 2% P (F = default, S = worst-case) = 0.4% When X and Y are independent, the joint probability collapses to the product of their standard probabilities, and the conditional probabilities are equal to the standard probabilities: P (X, Y) = P (X) × P (Y) P (X|Y) = P (X)

and P (Y|X) = P (Y)

If the state of the economy does not influence the credit standing of the firm, the conditional probability of default becomes equal to the unconditional default probability, or 1%. This implies that P (F = default|S = worst-case) = P (F = default) = 1%. As a consequence, the joint probability of having a worst-case situation plus default drops to P (F = default, S = worst-case) = 1% × 20% = 0.2% < 0.4%. The equality of conditional to unconditional probabilities, under independence, is consistent with a joint probability equal to the product of the unconditional probabilities: P (X, Y) = P (X) × P (Y). With correlated variables X and Y, the joint probability depends on the correlation. If the correlation is positive, the joint probability increases above the product of the unconditional probabilities. This implies that P (X|Y) > P (X) and P (Y|X) > P (Y). With positive correlation between X and Y, conditional probabilities are higher than unconditional probabilities. This is consistent with P (X, Y) being higher than the product of the unconditional probabilities P (X) and P (Y) since: P (Y|X) = P (X, Y)/P (X) and P (X|Y) = P (X, Y)/P (Y). In the universe of credit risk, another typical joint probability of interest is the joint default of two obligors X and Y, or P (X, Y). If the chance that X defaults increases when Y does, there is a correlation between the two defaults. This correlation increases the joint probability of default, the conditional probability of X defaulting given Y defaults, or P (X|Y), and P (Y|X) as well. If X and Y are positively correlated events, the occurrence of Y increases the probability of X occurring. This is equivalent to increasing the joint default probabilities of X and Y. Correlation relates to conditional probabilities. It is equivalent to say that there is a correlation between events A and B, and to consider that the probability that B occurs increases above B’s ‘unconditional’ probability if A occurs. For instance, the unconditional probabilities of default of A and B are 1% and 2% respectively. If the default events A and B are independent, the Joint Default Probability (JDP) is 2% × 1% = 0.02%. However, if the probability that B defaults when A does becomes 10%, the conditional probability of B given A’s default P (B|A) is 10%, much higher than the unconditional probability of B’s default, which is P (B) = 2%. This implies a strong correlation between A and B default events. Then, the joint default probability becomes: JDP(A, B) = P (A|B) × P (B) = 10% × 1% = 0.1% This is much higher than 2% × 1% = 0.02%, the joint default probability if the defaults were independent.

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Averaging Conditional Probabilities and Bayes’ Rule When inferring probabilities, the rule for averaging conditional probabilities is helpful. Let’s assume there are three (or more) possible values for S, representing the state of the economy. The default probabilities of firms vary with the state of the economy. If it worsens, the default probabilities increase and vice versa. Let’s assume we have only three possible states of the economy, with unconditional probabilities P (S = S1), P (S = S2) and P (S = S3) respectively equal to 50%, 30% and 20%, summing up to 1 since there is no other possible state. The third case is the worst-case scenario of the above. The averaging rule stipulates that the default probability of a firm F is such that: P (F) = P (F|S1) × P (S = S1) + P (F|S2) × P (S = S2) + P (F|S3) × P (S = S3) The probability of X defaulting under a particular state of the economy varies with the scenario since we assume a correlation between the two events, default and state of the economy. Taking 0.5%, 1% and 1.5% as conditional default probabilities for the three states respectively, the unconditional default probability has to be: P (F) = 0.5% × 50% + 1% × 30% + 1.5% × 20% = 0.85% The unconditional (or standard) default probability is the weighted average of the conditional default probabilities under various states of the economy, the weights being the probabilities of occurrence of each state. Its interpretation is that of an average over several years where all states of the economy occurred. The unconditional probability is the weighted average of all conditional probabilities over all possible states of the economy: K P (F) = P (F|Si) × P (Si) i=1

If S is a continuous variable, we transform the summation symbol into an integration symbol over all values of S using P (S) as the probability of S being in the small interval dS: P (X) = P (X|S) × P (S) dS

Conditioning and Correlation Correlation implies that conditional probabilities differ from unconditional probabilities, and the converse is true. Correlation and conditional probabilities relate to each other, but they are not identical. Conditional probabilities are within the (0, 1) range, while correlations are within the (−1, +1) range. Therefore, it is not simple to define the exact relationship between conditionality and correlation. As an illustration of the relation, the joint probability of default of two obligors depends on the correlation and the unconditional default probabilities of the obligors. The relation results from a simple analysis, developed in Chapter 46, explaining the loss distribution for

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a portfolio of two obligors X and Y, whose random defaults are variables X and Y taking the value 1 for default and 0 for non-default. Such ‘binary’ variables are called Bernoulli variables. For discrete Bernoulli variables, the relationship between the unconditional probabilities and correlation is: P (Y, Z) = P (Y) × P (Z) + ρYZ × P (Y)[1 − P (Y)] × P (Z)[1 − P (Z)]

The joint default probability increases with a positive correlation between default events. However, there are boundary values for correlations and for unconditional default probabilities. Intuitively, a perfect correlation of +1 would imply that X and Y default together. This is possible only to the extent that the default probabilities are identical. Different unconditional probabilities are inconsistent with such perfect correlation.

Conditioning, Expectation and Variance Some additional formulas serve in applications. The conditional probability distribution of X subject to Y is the probability distribution of X given a value of Y. The conditional expectation of X results from setting Y first, then cumulating (‘integrating’) over all values of X: E(X) = E[E(X|Y)]

E(X|Y) = E(X) implies that X does not depend on Y. The expectation of a variable can be calculated by taking the expectation of X when Y is set, then taking the expectation of all expected values of X given Y when Y changes. E(X|Y) = E(X) implies that X does not depend on Y. Conditional expectations are additive, like unconditional expectations. These formulas serve to define the expectation when a loss distribution of a portfolio is conditional on the value of some external factors, such as the state of the economy. The variance of a random event X depending on another event Y results from both the variance of X given Y and the variance of the conditioning factor Y: Var(X) = Var[E(X|Y)] + E[Var(X|Y)]

It is convenient to use this formula in conjunction with: Var(X) = E(X2 ) − [E(X)]2 . The variance of X is the expectation of the square of X minus the square of the expectation. This allows decomposition of the variance into the fraction of variance due to the conditioning factor Y and the variance due to the event X.

Applications of Joint and Conditional Probabilities for Credit Events Joint probabilities apply to a wide spectrum of applications. They serve in credit risk analysis when dealing with credit events. Such applications include the following. The joint default probability of a pair of obligors depends on the correlation between their default events, characterized by probabilities of default conditional on the default or non-default of the second obligor. The two-obligor portfolio is the simplest of all portfolios. Increasing the default probability of an obligor conditional on default of the other increases correlation between defaults. There is a one-to-one correspondence between default correlation and conditional probability, once the unconditional default probabilities are defined.

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The same two-obligor portfolio characterizes the borrower–guarantor relation, since default occurs only when both default together. This is the ‘double’ default event when there are third-party guarantees. The technique allows us to value third-party guarantees in terms of a lower joint default probability. Modelling default probability from scoring models relies on conditional probabilities given the value of the score, because the score value provides information on the credit standing of a borrower. The score serves for estimating ‘posterior’ default probabilities, differing from ‘prior’ probabilities, which are the unconditional probabilities, applying when we do not have the score information. When defaults are dependent on some common factors, they become conditional on the factor values. This is a simple way to determine the shape of the distribution of correlated portfolio defaults by varying the common factors that influence the credit standing of all obligors within the portfolio.

GENERATING CORRELATED RISK DRIVERS WITH SINGLE FACTOR MODELS This section shows how to generate, through multiple simulations, two correlated variables, with a given correlation coefficient. The correlation coefficient ρij is given. The variables can be credit risk drivers such as asset values of firms or economic factors influencing the default rates of segments. We first show how to generate two random normal variables having a preset correlation coefficient. Then, we proceed by showing how to generate N random variables having a uniform correlation coefficient. This is a specific extension of the preset pair correlation case. It applies when simulating values of credit risk drivers for a portfolio of obligors with known average correlation. We implement this technique to demonstrate the sensitivity of the loss distribution to the average correlation in Chapters 45–49. Next, we drop the uniform correlation and discuss the case of multiple random variables with different pair correlations. To generate random values complying with a variance–covariance structure, Cholevsky’s decomposition is convenient. This technique serves to generate the distribution of correlated variables when conducting Monte Carlo simulations.

Generating Two Correlated Variables Each random variable Zi is a function of a common factor Z and a specific factor εi . The specific factor is independent of the common factor. All cross-correlations between Z and εi or εj are 0. All random variables are normal standardized, meaning that they have zero mean and variance equal to 1. The correlation is between each Zi variable and the common factor is ρi , while the correlation between the two random variables Zi and Zj ρij . We use the equations for two different variables Zi and Zj : Zi = ρ × Z + 1 − ρij2 × εi Zj = ρ × Z + 1 − ρij2 × εj

The variance of Zi is:

Var(Zi ) = ρi2 Var(Z) + (1 − ρi2 ) Var(εi ) + 2ρi 1 − ρi2 × Cov(Zi , εi )

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Given that Var(Zi ) = Var(Z) = Var(εi ) = 1 and Cov(Zi , εi ) = 0, we have: Var(Zi ) = ρi2 + (1 − ρi2 ) + 0 = 1

Risk models make a distinction between the general risk, related to a common factor influencing the values of all transactions, and the specific risk, which depends on the transaction only. If Zi represents the transaction ‘i’ and if Z represents a factor influencing its risk and that of other transactions, the general risk is the variance of Zi due to the factor Z. From above, the general risk is simply equal to the correlation coefficient ρi2 and the specific risk to its complement to 1, or (1 − ρi2 ). In this very simple case, we have a straightforward decomposition of total risk. Another correlation of interest is the pair correlations between any pair of variables, Zi and Zj . The covariance and their correlation between these variables derive from their common dependence on the factor Z. All cross-covariances between the factor Z and the residual terms εi and εj are zero by definition. Therefore: Cov(Zi , Zj ) = Cov(ρi × Z + 1 − ρij2 × εi , ρj × Z + 1 − ρij2 × εj ) Cov(Zi , Zj ) = Cov(ρi Z, ρj Z) = ρi ρj Var(Z) = ρi ρj

Hence, all pairs such as Zi and Zj have a predetermined correlation ρij = ρi ρj Var(Z). This technique shows how to generate two correlated variables having a predetermined correlation coefficient ρij by generating independent normal variables: • Generate a standardized normal variable εi . • Generate a standardized normal variable εj . • Generate a standardized normal variable Z. Then Zi and Zj , calculated with the above linear function using ρij as the correlation coefficient, have a correlation equal to this ρij . The technique applies for generating N correlated variables with various pair correlations. To use the above technique, we generate independently K random draws of the common factor Z and K random draws of each of the specific risks εi (N variables) following standardized normal distributions. K is the number of trials or runs. Combining the random

values of Z and εi with the above equation Zi = ρ × Z + 1 − ρi2 × εi , we get K values of the N Zi . The K values of all Zi are such that all cross-correlations are equal to ρ. To generate K = 1000 trials for N variables, we need to generate 1000 × (N + 1) values since there are N residuals εi plus the unique common factor. In the case of a uniform correlation, all pair correlations and covariances of Zi and Zj are equal to a common ρ. The technique generates random values with standardized normal distributions and all pair correlations equal to a given value ρ. The technique serves for generating samples of N uniformly correlated normal variables. This procedure is implemented in Chapter 48.

Generating Random Values Complying with a Correlation Structure The correlation structure results from multi-factor modelling of credit drivers and credit events, or any proxy such as the equity return correlations of Credit Metrics. Simulation

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requires generating risk drivers, such as asset values, complying with this correlation structure. Of course, in such a case, variances and covariances differ across pairs of variables. Cholevsky’s decomposition allows us to impose directly a variance–covariance structure on N random normal variables4 . It allows us to derive from the generation of N independent variables N other random variables that comply with the given correlation structure, and that are linear functions of the independent variables. The difference with the previous technique is that we start from a full variance–covariance matrix. First, we generate K independent samples of values of N variables Xi . It is convenient to use normal standardized variables. K is the number of simulation runs. Then, we convert the N Xi variables into another set of N Zj variables. The Zj variables are linear combinations of the independent Xi . The Cholevsky decomposition provides the coefficients of these linear combinations. The coefficients are such that the new Zj variables comply with the imposed variance–covariance matrix. The Decomposition Technique

We use the simple case of two random variables. We start from X1 , a normal standardized variable. Once we have X1 , we derive the α12 and α22 for X2 : Z1 = α11 X1 Z2 = α12 X1 + α22 X2 These numbers are such that: 2 Var(Z1 ) = α11 =1

This requires α11 = 1, or Z1 = X1 :

2 2 Var(Z2 ) = α12 + α22 =1

Cov(Z1 , Z2 ) = Cov(α11 X1 , α12 X1 + α22 X2 ) = α11 × α12 = ρ12 Since α11 = 1, we have two equations with two unknown values α12 and α22 : 2 2 α12 + α22 =1

α12 = ρ12 2 α22 = 1 − ρ12 Matrix Format with Two Random Variables

Therefore, the two variables Z1 , Z2 are linear functions of standardized independent normal variables X1 , X2 according to: Z 1 = X1 Z2 = ρX2 + 4 If

1 − ρ 2 X2

all variables are normal standardized, the variance–covariance matrix collapses to the correlation matrix, with correlation coefficients varying for each pair of variables and all diagonal terms equal to 1.

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The above equations show that ZT = LXT using L as the lower triangular matrix of the αij coefficients, where XT is the column vector of Xi variables, and ZT is also a column vector of Zj variables. Using the matrix format of Cholevsky’s decomposition, the process starts from the variance–covariance matrix of two normal standardized variables, which is extremely simple. If Zk are standardized normal variables, they have covariance matrix: 1 ρ

= ρ 1 The Cholevsky decomposition of is: 1 1 0 1 ρ ×

= = ρ 1 0 ρ 1 − ρ2

ρ 1 − ρ2

Therefore, the variables Z1 and Z2 having correlation ρ are linear functions of independent standardized normal variables X1 and X2 according to the relation: 1 0 X1 Z1 × = 2 X2 Z2 ρ 1−ρ

Then:

Z1 Z2

=

X 1 ρX2 + 1 − ρ 2 X2

These are the relations used in preceding sections for generating pairs of correlated variables. The appendix to this chapter provides general formulas with N variables.

APPENDIX: CHOLEVSKY DECOMPOSITION This appendix shows how to generate any number of correlated normal variables, starting with a set of independent variables and using Cholevsky’s distribution to obtain linear combinations of these independent variables complying with a given correlation structure.

The Matrix Format for N Variables Generalizing to N variables requires an N -dimensional squared variance–covariance matrix . In general, L and U stand respectively for ‘lower’ triangular matrix and ‘upper’ triangular matrix. The U matrix is the transpose of the L matrix: U = LT . By construction of L = ZT (XT ) − 1, LU = LLT is the original variance–covariance matrix . The above equations show that ZT = LXT using L as the lower triangular matrix of the αij coefficients, where XT is the column vector of Xi variables, and ZT is also a column vector of Zj variables. Solving the matrix equation ZT = LXT determines L: Z j = L × Xi α11 Z1 Z2 α12 = .. .. . .

0 α22 .. .

X1 ... X2 ... × .. .. . .

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The solution uses an iterative process as indicated below for determining the coefficients, as illustrated with the two-variable case.

Determining the Set of Coefficients of the Cholevsky Decomposition The Cholevsky decomposition determines the set of coefficients αim of the linear combinations of the Xj such that the Zj comply with the given variance–covariance structure: Zj =

j

αim Xi

m=1

This set of coefficients transforms the independent variables Xi into correlated variables Zj . The coefficients result from a set of equations. The variance of Zj is the sum of the variances of the Xi weighted by the squares of αij , for i varying from 1 to j . It is equal to 1. This results in a first set of equations, one for each j , with j varying from 1 to N : j j 2 αim =1 Var(Zj ) = Var αim Xi = m=1

m=1

The index m varies from 1 to j , j being the index of Zj varying from 1 to N because there are as many correlated variables as independent variables Xi . Then, for all i different from j , we impose the correlation structure ρij . All variances are 1 and the covariances are equal to the correlation coefficients. The covariance of Zj and Zk is: j k αj m Xi , Cov(Zj , Zk ) = Cov αj k Xi = ρij m=1

m=1

This imposes a constraint on the coefficients: αj m αj k = ρij j =m

The problem is to find the set of αj m for each value of j and m, both varying from 1 to N , transforming the independent variables Xi into correlated Zj that are linear functions of the Xi . These values are the solution of the above set of equations for all αim . Determining these values follows an iterative process.

Example of Cholevsky’s Decomposition The original variance–covariance matrix is . The Cholevsky decomposition of a symmetrical matrix is such that = LU. The following is an example of such a decomposition using a variance–covariance matrix:

4 2 14

=L×U 2 14 2 0 0 2 17 5 = 1 4 0 × 0 5 83 7 3 5 0

1 7 4 3 0 5

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If ZT is the column vector of the correlated normal Zj , and XT is the column vector of the independent normal Xi variables, ZT derives from the XT vector of independent variables through: ZT = LXT If we proceed for all Zi , we find k values for each Zi , and the values comply with the correlation structure. Table 28.1 makes two sample calculations of two sets of three values for the three variables Zi starting from two sets of three values for each of the three independent variables Xm , of which the Zi are linear combinations. The construction of the coefficients of the L matrix ensures that the values of the Zi comply with the original variance–covariance matrix. TABLE 28.1 The Z values from the values of the independent variable using L L

X1

Column vector X: X2 X3

1 2 ··· ···

3 1 ··· ···

2 3 ··· ···

2 0 0 1 4 0 7 3 5 Column vector Z = XL: Z1 Z2 Z3 19 26 ··· ···

18 13 ··· ···

10 15 ··· ···

The first row of sample values 19, 18, 10 for Z1 , Z2 , Z3 results from the first row of X values and the matrix L coefficients: 1 × 2 + 3 × 1 + 2 × 7 = 19 1 × 0 + 3 × 4 + 2 × 3 = 18 1 × 0 + 3 × 0 + 2 × 5 = 10 We can generate as many values as we need from random trials of normally independent X. The above Z values comply with the variance–covariance matrix = LLT = LU.

SECTION 11 Market Risk

29 Market Risk Building Blocks

Market risk is the potential downside deviation of the market value of transactions and of the trading portfolio during the liquidation period. Therefore, market risk focuses on market value deviations and market parameters are the main risk drivers. Figure 29.1 summarizes the specifics of the market risk building blocks using the common structure followed throughout the book for all risks. We focus on the first two main blocks: standalone risk of individual transactions and portfolio risk. The capital allocation and risk–return building blocks are technically identical to that of credit risk.

BLOCK I: STANDALONE RISK Market risk results from the distribution of the value variations between current date and horizon of individual assets. In percentage terms, these deviations are the random asset returns. Consequently, all we need for measuring risk are the distributions of the returns between now and the future short-term horizon. The risk drivers are all market parameters to which mark-to-market values are sensitive. They include all interest rates by currency, equity indexes and foreign exchange rates. The exposures are mark-to-market values. Exposure values map to different market parameters, depending on the type of transaction. Simple instruments map to few main parameters, such as equity indexes for stocks. Derivatives are sensitive to several market parameters, such as underlying asset value, underlying asset volatility, interest rate and time to maturity. Mapping exposures to risk drivers is a prerequisite for determining the distribution of random future asset returns. Valuing risk implies defining the downside deviations between current value and the random market value at the horizon of the liquidation period. These variations of

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Building Blocks: Market Risk Building Blocks: Market Risk I.

I.

II. II.

1

Risk drivers

Market parameters (interest rates, equity indexes, foreign exchange rates)

2

Risk exposures

Mark-to-market values. Mapped to market parameters

3

Standalone risk

Valuation of adverse deviations of market returns over liquidation period

Correlations

Correlations between selected market parameters mapped to individual exposures

4

5

6

FIGURE 29.1

Portfolio risk

Capital

Loss distributions aggregate algebraically the individual asset returns. The 'Delta− normal' or Monte Carlo simulation techniques serve for deriving the individual and correlated asset returns VaR is a loss percentile

Major building blocks of market risk

values result directly from the distribution of the future asset returns. Financial models traditionally view small period returns as following specific random processes. Using such processes, it is possible to generate a random set of time paths of market parameter changes or of individual asset returns to obtain their final values at horizon. Then, we can revalue each asset for each final outcome. These are the basics of the full revaluation technique, which is resource intensive. The Delta ‘Value at Risk’ (VaR) technique uses several simplifications to bypass full revaluations: mapping individual asset returns to market parameter returns; constant sensitivities of asset returns to underlying value drivers; linear relationship between asset returns and market parameter returns; normal distributions of market parameter returns as a proxy for actual distributions over short horizons. This solution is evidently more economical than full revaluation because there are many fewer market parameters than individual assets. Alternatively, when it is not possible to ignore the changes in sensitivity with market movements, full revaluations at horizon under various scenarios serve for improving the accuracy of the distribution of the simulated values (Figure 29.2). All loss statistics and loss percentiles derive from the distribution of horizon returns of each individual exposure. The distribution of returns characterizes the standalone risk. When moving on to portfolios, diversification eliminates a large fraction of the sum of individual risks. The portfolio risk building block needs to model correlations and use them to model the distribution of the portfolio return.

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361

Distribution of forward market values

Loss

Current value

FIGURE 29.2

Forward valuation and loss valuation

BLOCK II: PORTFOLIO RISK Portfolio risk results from the distributions of all individual and correlated asset returns. In practice, it is more economical to derive asset returns from the common factors that influence them, using sensitivities. These factors, or risk drivers, are the market parameters. Correlations between the returns of individual exposures are derived from those of market parameters. Sometimes, it is sufficient to observe directly the market parameters for measuring historical correlations. Sometimes, it is necessary to use ‘factor models’ of market parameters for simulating possible scenarios. Factor models make the market parameters dependent on a common set of factors, thereby correlating them. The most well known single-factor model relates individual equity returns to the single equity index return. Some interest rate models and yield curve models can also serve to capture the distributions of the various rates. Because it is complex to model all rates, other models serve for modelling the random yield curve scenarios from factors that represent the main changes in shape of the yield curve. This is a two-stage modelling process, from factors to market parameters, and from these risk drivers to individual market values of exposures. Since factors correlate, risk drivers also do, as well as the returns of individual exposures (Figure 29.3).

FIGURE 29.3

Factors

Market parameters

Correlated factors

Correlated market parameters

Correlated asset values & returns

Modelling risk driver correlations

Portfolio risk results from the forward revaluation of portfolio returns, as the algebraic summation of all individual asset value changes, under various market parameter scenarios. VaR is the loss percentile at the preset confidence level. Generating the various

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Variance covariance structure

Market risk drivers

Simulate new 'risk driver' values

−

Loss Distribution VaR

Revalue portfolio

New 'trials'

FIGURE 29.4 distribution

Current MTM values

Gain / loss

0

+

The multiple simulation process for generating the market risk value

scenarios requires techniques for obtaining sets of market parameter values complying with the observed volatilities and correlations. The Delta VaR methodology, also known as the ‘linear–normal model’, relates individual transaction returns to market parameter changes through sensitivities. It suffers from limitations due to the assumption of constant sensitivities. Under these simplifying assumptions, the distribution of the portfolio return between current date and horizon is a normal distribution. Loss statistics and loss percentiles, which measure VaR, are obtained from analytical formulas. Nevertheless, the sensitivity-based approach is not appropriate with optional instruments because their sensitivities are not stable. The multiple simulation methodology (Monte Carlo simulation) remedies the drawback of the attractive, but too simple, linear model. It reverts to the direct modelling of asset values from the underlying random and correlated market parameters. The process implies generating correlated random values of market parameters and deriving, for each set, the values of individual assets. The random risk drivers need to comply with the actual variance–covariance structure observed in the market. Multiplying the number of trials results in a distribution of market parameter values. A distribution of the portfolio returns and value changes is then derived from these runs. This allows us to determine the VaR. The drawback of the Monte Carlo simulation is that it is calculation intensive, so that banks prefer the Delta VaR technique or intermediate techniques whenever possible (Figure 29.4).

30 Standalone Market Risk

This chapter reviews the techniques for measuring the market risk of individual transactions. Market risk results from the distribution of the value variations between current date and horizon of individual assets. In percentage terms, these deviations are the random asset returns. Consequently, all we need for measuring risk are the distributions of the returns of individual assets between the current date and the short-term liquidation horizon. Losses are downside moves from the current value, when ignoring the expected return for the short liquidation period. The main drivers influencing the returns of market instruments are the market parameters. Individual asset values also serve for options, such as stock options, of which values depend on that of the underlying stock. The literature designates them as ‘risk factors’: they are ‘risk drivers’ as well as ‘value drivers’ because risk materializes as changes of values or of market parameters. The risk measures are loss statistics, such as value volatility and downside variations of values at a preset confidence level. All statistics result from the distribution of the random asset values at the liquidation horizon. Pricing models in finance consider current prices as the present value of all future outcomes for individual assets. The VaR perspective is the reverse of the pricing perspective. We start from current prices and need to derive all potential adverse deviations of asset values or returns until a future time point. There are two basic techniques for doing so. Full revaluation determines the distribution of all asset values and returns after having modelled the full time path until horizon of those parameters that influence each individual asset price. Partial revaluation relies only on asset returns over the small time period until the liquidation horizon and uses shortcuts to relate these variations directly to market parameter returns over the period through sensitivities. Full revaluation at a future date for VaR purposes is intensive in terms of resources and often impractical at the full scale of the entire portfolio. Pricing models provide

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closed-form formulas facilitating revaluation at the horizon once we have the distribution of the parameters that drive values and returns of individual assets. The theoretical process comprises three steps: choosing intermediate time points as ‘vertices’ for intermediate calculations; generating random time paths of asset and market parameter returns from now to the horizon; revaluing from these time paths the assets at the liquidation horizon. The Delta VaR technique uses several simplifications for bypassing full revaluations: mapping individual asset returns to market parameter returns; constant sensitivities of asset returns to underlying value drivers; linear relationships between asset returns and market parameter returns; normal distributions of market parameter returns as a proxy for actual distributions over short horizons. This solution is more economical than full revaluation because there are many fewer market parameters than individual assets. The Delta–normal VaR technique that provides an analytically tractable distribution of asset returns and values at the horizon makes it simple to derive value statistics, volatility and value percentiles for a single period. When complex assets, such as exotic options, make this process excessively simplistic, we need to revert to full revaluation. However, the full power of the Delta–normal technique appears when dealing with portfolios of assets, rather than a single asset, because the portfolio return depends in a linear way on all market parameter returns mapped to the various assets in the portfolio. The values of all instruments depend on a common set of value drivers that comply with the variance–covariance structure. Therefore, we need to mimic this correlation–volatility structure before revaluation. The portfolio return, or final value, sums algebraically all individual asset returns and depends on offsetting effects between positive and negative variations and on correlations between individual asset returns, as subsequent chapters explain. The current chapter shows the process in the case of a single asset, as an intermediate step. The first section provides an overview of VaR and modelling of returns, and discusses the full valuation technique and the Delta VaR technique. The second section describes well-known stochastic processes for returns, stock prices and interest rates. The third section addresses the volatility measurement issue, which is a major ingredient for modelling variations of returns. The fourth section explains how to derive standalone risk measures from the normal and lognormal distribution of prices. The last section addresses the mapping issues and describes some common sensitivities, which are the foundations of the Delta VaR technique.

VAR AND FUTURE ASSET RETURNS The market risk loss is the adverse deviation of value due to market movements. This variation is Vi = (Vi ,H − Vi,0 ) = Vi,0 × ri . It is the deviation from current value or, equivalently, the return times the current value. The return measures the unit deviation of value. Hence, measuring value deviations over short time periods is equivalent to measuring the random returns from the current value in percentage terms. Therefore, to measure market risk VaR, it is sufficient to model random returns between the current date and the horizon. In what follows, we use both value and cumulative returns equivalently since they map on a one-to-one basis at the horizon. The valuation building block of market risk models relies on two basic techniques:

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365

• Full revaluation at a future horizon, set as the liquidation period. • Partial revaluation of individual asset returns from market movements, also known as the Delta VaR technique, which applies to both single assets and portfolios.

Full Revaluation The finance literature relies extensively on modelling returns for pricing models. Theoretical finance derives prices from modelled time paths of returns. Returns follow stochastic processes. Stochastic processes specify the distribution of asset returns, or of market parameters, over each time interval, as a function of their expected values and volatilities. Implementing these stochastic processes allows us to model future outcomes, and to find all future asset payoffs, notably when these payoffs are contingent on outcomes, such as for options. For common assets, including standard options, modelling the stochastic processes of returns results in closed-form pricing formulas. For complex options, the modelling of return processes allows us to conduct simulations of all future outcomes and derive the current price as the expected present value of all option payoffs over this range. Derivative pricing models derive the return of derivatives from the return process of the underlying asset. The most well known example is the pricing of European stock options. The Black–Scholes equation prices a European equity option by modelling the return of the option as a function of the underlying stock return using Itˆo’s lemma. The principle applies to all derivatives1 . Accordingly, the literature on stochastic processes expanded considerably. Stochastic processes serve for modelling all asset returns and prices. They apply to stock prices, interest rates and derivatives, whose returns depend on the underlying asset return process2 . The VaR perspective is a sort of ‘reverse’ pricing perspective. Instead of finding the current price from the present value of all possible future outcomes, we derive all possible asset values for all future outcomes from the current data. There is a duality between cumulative returns and values. The time paths of returns map on a one-to-one basis to future values. This is obvious using the single discrete time formula for the random return y(0, t) = (Vt − V0 )/V0 , with Vt and V0 being random values at date t and the current certain value at date 0. However, discrete time returns have drawbacks, as illustrated in the section on returns below, making it necessary to use small time intervals and construct the time path of returns, over a sequence of small periods. The random cumulative return links final value to initial value. Continuous finance makes extensive use of instantaneous ‘logarithmic’ returns over small time intervals, rather than arithmetic returns, because they are additive, as demonstrated in the appendix to this chapter. There are important variations across asset classes for modelling future outcomes. For stocks, we need to model the time path of stock returns and derive the distribution of future prices, given the expected return and its volatility as measured from historical observations. We find, as shown below, that stock prices follow a lognormal distribution whose parameters depend on the expected return and its volatility. For bonds, we need 1 See

Hull (2000) and Merton (1990) for comprehensive reviews. Neftci (1996) for an introduction to the mathematics of stochastic processes, Smith (1976) and Cox and Ross (1976) for option pricing.

2 See

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to derive all future interest scenarios at the future date from their stochastic processes to revalue the bond for each of these scenarios by discounting at appropriate market rates the future contractual cash flows. For options, we need to derive all future outcomes for all their value drivers, which include the underlying prices, such as stock prices for equity options, and revalue the options at the future date. For complex derivatives, the process is the same except that we do not have closedform pricing formulas. Therefore, we need to revert to simulations of future option payoffs. The difference is that the price as of a given date results from all possible future outcomes. A forward revaluation requires determining all possible outcomes as of this future date. Unfortunately, there are many starting points at future dates, so we need to replicate the process for each of them. This mechanism is a ‘simulation within a simulation’. The first simulation provides all outcomes at the horizon. A second set of simulations replicates the process for all dates beyond the horizon starting from each horizon state. Evidently, such a process consumes many resources, making it unpractical. Derivatives looking backward, such as barrier options and ‘look back’3 options, raise different obstacles. The values of these options depend on all intermediate outcomes between the current date and a future date. Revaluation at a future date depends on these intermediate states. The ‘full revaluation’ technique requires all outcomes for all ingredients of future prices and recalculates future prices for each scenario. The distribution of future cumulative returns between today and the horizon, provides all that is needed for VaR calculations. Accordingly, the next section details some basic stochastic processes, those applying to stock prices and those applying to interest rates, to show the implications of using the full revaluation technique. Stochastic processes specify the distribution of asset returns, or of market parameters, over each time interval, as a function of the expected value of the return and its volatility. For stocks, the process allows stock prices to drift away from initial value without imposing any bound on prices. However, the time paths of interest rates follow specific stochastic processes because, unlike stock prices, they tend to revert towards a long-term average value. The next section provides an overview of these basic stochastic processes. Once stochastic processes of value–risk drivers are specified, the technique implies: • Choosing intermediate time points as ‘vertices’ for intermediate calculations. • Generating random time paths of assets and market parameter returns from now to each time point. • Revaluing from these time paths the asset at the liquidation horizon.

Partial Revaluation and Delta VaR It is obvious from the above that the process is resource intensive and impracticable in most cases. Additional shortcuts help resolve such complexities. Partial revaluation uses much simpler techniques. It starts from the observation that all we need for VaR purposes 3A

‘look back’ option has a payoff linking to past values of the underlying. For instance, the payoff of a look back equity option could be the maximum of the stock price between inception and the horizon. Barrier options disappear when the underlying hits a barrier, and also depend on the time path of past values.

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are variations of values or returns for each individual asset, and that value drivers for assets are market parameters. To be more specific, the drivers of stock returns are the equity indexes. The drivers of interest-bearing assets are the interest rates along the yield curve. The drivers of option values are these market parameters plus the underlying asset prices, for instance an interest rate or a stock price. There are many fewer market parameters than individual assets in a portfolio. Therefore, starting from market parameters to obtain asset returns is evidently more economical than modelling each asset return. The first prerequisite is to map asset returns to market parameters. Stock returns map to equity indexes, interest-bearing assets map to interest rates, stock options map to the market parameters of interest rates, stock return volatilities and stock prices, and so on for other options. The next step is to relate asset returns to the selected underlying market parameters. Sensitivities provide a simple proxy relationship as long as the variations of market parameters are not too important. This is acceptable in many cases for short-term horizons, except for options when market movements trigger abrupt shifts in sensitivities. Using constant sensitivities makes the relationship between market parameter changes and asset returns linear. This is a major simplification compared to using the pricing formulas of individual assets. The next step is to model the distribution of the market parameter returns. A normal distribution has very attractive properties: the linear combination of random normal variables is also a normal variable. Combining these shortcuts results in a simple technique. The asset returns follow normal distributions derived from a linear combination of random normal parameter returns. The derivation of all loss statistics and VaR becomes quite easy since we know all percentiles of the normal distribution. For single assets, a single value driver is sufficient. For options, there are several value drivers, but the option returns remain a linear combination of these. For a portfolio, various assets depend on different market parameters, so that we always have a linear function of all of them. These are the foundations of the ‘Delta–normal VaR’ technique. It provides normal distributions of asset returns from which VaR derives directly. The Delta VaR technique faces some complexities with interest rates. Modelling future outcomes of the yield curve is not an easy task. Simple interest rate models might suffice, for instance when there are only parallel shifts in the yield curve. In general, however, interest-bearing asset values depend on the entire yield curve. It is not practical to use full-blown yield curve models. One technique consists of selecting only some interest rates along the yield curve for references. An alternative technique consists of using factor models of the yield curve. These make the yield curve a function of a few factors that behave randomly, with specific volatilities, subject to prior fitting to historical data. They allow us to simulate random shocks on the yield curve, including those that alter its shape (level, slope and ‘bump’). Converting these factors into yield curves provides the required distribution of selected interest rates. The partial revaluation process requires the following steps, that substitute for full revaluation: • • • •

Mapping asset returns to value–risk drivers. Simple modelling of the market parameter returns. Measuring the sensitivities of asset returns to market parameters. Deriving the asset return distribution as a linear function of these distributions.

Limitations of the Delta–normal technique are:

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Constant sensitivities. Proxy mapping to market parameters, generating basis risk. Proxy of actual distributions of value–risk drivers. Ignoring ‘fat tails’ of actual distributions.

Notably, the Delta VaR technique collapses when sensitivities cannot be considered constant, as for options and look back derivatives. Moreover, only on sensitivities for stocks ignores risk unrelated to market parameters, or the specific risk of stocks. The diagonal model of stock returns remedies this drawback, as explained in the next chapter. Nevertheless, the Delta VaR technique shines for portfolios. Portfolios of assets make the overall return dependent on the entire set of market parameters. Their random movements depend on their correlations. Since the Delta VaR technique makes individual asset returns a linear function of random parameters, it also applies to the portfolio return. The difference when dealing with multiple market parameters resides in correlations affecting their co-movements. Since it is a relatively simple matter to model correlated random variables, using techniques detailed in Chapter 28, the Delta VaR technique allows us to handle correlations without difficulty.

STOCHASTIC PROCESSES OF RETURNS To fully revaluate asset values at the horizon, the common technique consists of modelling the individual asset returns to derive the value at the horizon by cumulating intermediate returns. This section explains the basic mechanisms of common stochastic processes and how time paths of returns relate to final values. Pricing models view returns as following ‘stochastic’ processes across successive short time intervals. Measuring risk requires the distribution of cumulative returns to the horizon given the current value. The first subsection illustrates the drawbacks of a single-period discrete return. The next subsections discuss two types of processes serving to model asset returns: the Wiener process for stock prices and processes applicable to interest rates. The last subsection discusses volatility measures, which are critical inputs to risk measures and stochastic models of returns.

Single-period Return The return over a discrete period is the relative variation of value between two dates. For a short time interval, the single-period return over that period is sufficient to obtain the final value because there is a one-to-one correspondence between the random return value and the final value, according to the above equation. Random returns between dates 0 and H are y(0, H ), values indexed by date are V0 for the current (certain) value and VH for the random value at horizon H . The return y, between any two dates, and the value at the horizon, are: y(0, H ) = (VH − V0 )/V0 VH = V0 (1 + y) With a single-period return, the value distribution at H derives from that of the return because VH = V0 (1 + y). The obvious limitation of this technique for modelling returns

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and final values is that a normal distribution allows the random return to hit values lower than −100%, resulting in non-acceptable negative values of the asset. For a sufficiently short horizon, this formulation is convenient because normal distributions are easier to handle. When the period gets very small, the return becomes y = dV/V , where dV is the small change in value. This makes it more acceptable to use normal distributions. On the other hand, it makes it necessary to use cumulative returns, which are not normally distributed, between discrete dates, current date and horizon.

Stock Return Processes Under the efficient markets hypothesis, all stock values reflect all past information available. Consequently, variations of values in successive periods are independent from one period to another because the flow of new information, or ‘innovations’, is random. In addition, small period returns follow approximately normal distributions with a positive mean. This assumption applies notably to stock prices, but it is also a proxy for short-term interest rates. The mean is the expected return, such as the risk-free return for a risk-free asset and the risky expected market yield for risky assets. Over very small intervals, returns follow stochastic processes. The Wiener process is a common stochastic process, adequate for stock prices: dVt = µVt dt + σ Vt dzt This equation stipulates that the small change in value of a stock price Vt is the sum of a term µVt dt, which reflects the expected return µ per unit time, times the initial stock value, and proportional to the time interval dt, plus a random term. This first term is the expected value of the relative increase dVt /Vt of the asset price, because the expected value of the random term dzt is zero. The random component of the return is σ Vt dzt , proportional to the return volatility σ , the price Vt and a random ‘noise’, dzt , following a standardized normal variable (mean 0 and volatility 1). This simple process models the behaviour of the price through time. dVt follows a ‘generalized Wiener process’, with drift µ and volatility σ . The process makes the instantaneous return dVt /Vt a direct function of its mean and volatility σ : y = dVt /Vt = µdt + σ dzt This equation shows that the return follows a normal distribution around the time drift defined by the expected return µ dt. Since dx/x is the derivative of ln(x), this equation shows that the logarithm of stock price at date t follows a normal distribution4 . Therefore, the stock price at date t follows a lognormal distribution whose equation is5 : Vt = V0 exp[(µ − 21 σ 2 )t + σ zt ] 4 This

necessitates integration from date 0 to t of the small value increments over each dt. lognormal distribution is the distribution of a variable whose logarithm follows a normal distribution. The summation of all dzt over time results in a normal variable for zt . The equation of the final value of the stock results from expressing the logarithm of the stock price as following the process d[ln(S)] = [(µ − 12 σ 2 )dt + σ dz]. 5A

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For stocks, the random process is consistent with the efficient markets hypothesis stating that, if all prices fully reflect the available information, they fluctuate randomly with the flow of new information. Therefore, new prices are dependent only on ‘innovations’ dzt , not on past prices. The dzt are uncorrelated through time and follow the same distribution at all dates. Under this assumption, the process is ‘stationary’, meaning that it remains the same through time. Innovations are identically independently distributed (iid). Because the innovations dzt are independent variables of variance (σ dt)2 , their sum zt has a variance equal to the sum of variances. Using dt = 1, and t time√intervals, the variance of the cumulated innovations is t × σ 2 , and the volatility is σ t. It increases as the square root of time. This is the ‘square root of time’ rule for the volatility. It results in an ever-increasing volatility of the random price. The formula makes it easy to generate time paths of the St by generating normal standardized dzt and cumulating them until date t, with preset values of the instantaneous expected return µ and its volatility σ . This simple process is very popular. It applies to asset prices and market parameters whose variance increases over time.

Generating Time Paths of Stock Values: Sample Simulation The simulation of time paths of value drivers allows us to model the forward values of any asset. We illustrate here the mechanism with the simple Wiener process, which results in a lognormal distribution. The technique comprises three steps: • Choosing intermediate time points as ‘vertices’ for intermediate calculations of the returns. • Generating random time paths of market parameter returns from now to the horizon based on the specific parameters characterizing the asset process. The final asset value distribution at the horizon results from cumulative returns along each time path. To generate a discrete path, we use a unit value for t = t − (t − 1) and the standard N(0, 1) normal variable for the innovations z. With several time intervals, we obtain the intermediate values of the price. The process uses the following inputs: • • • • •

Initial price V0 = 10. Time interval t = 0.1 year. Expected return 10% yearly, or 10%/10 = 1% per 1/10 of a year. Yearly volatility of the random component 20%, or 20%/10 = 2% per 1/10 of a year. The innovation at each time interval z as a standardized normal distribution.

The price after one step is: Vt − Vt−1 = Vt−1 (µ + z × t) = Vt−1 (1% × 0.1 + z × 0.2 ×

√

0.1)

Simulating a time path over 1 year consists of generating a time series of 10 random draws of 10 innovation values. Simulating several time paths consists of repeating the process as many times as necessary. This allows us to calculate any final value that depends on past values, such as the value of a look back option on the stock. Figure 30.1 shows various time paths and the final price at date 10.

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13 12 11

Value

10 9 8 7 6 5 4 0

3

4

5 6 Time

7

8

9

10

22.5

18.8

15

11.3

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20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% 0

FIGURE 30.2

2

Time paths of value

Probability

FIGURE 30.1

1

Distribution of final value

Figure 30.2 shows the simulation of the final prices after 1000 time path simulations, which is a lognormal distribution. The appendix to this chapter shows why logarithmic returns provide good proxies of instantaneous returns and facilitate handling cumulative returns when the number of subperiods makes them small enough. The formulas provide useful shortcuts when modelling cumulative returns.

Interest Rate Processes Interest rate processes are more complex than the basic stock price processes for two reasons: • There are several interest rates, so we need to model the behaviour of several parameters rather than only one. • Interest rates tend to revert to some long-term average over long periods. To capture the behaviour of interest rates, we need to model the entire time structure of rates, with models using ‘mean reverting’ stochastic processes. Interest rate models view

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interest rates as following a stochastic process through time. A stochastic process specifies the change in the rate as a function of the time interval and a random ‘noise’ mimicking the volatility of rates. Several stochastic processes apply to interest rates. Mean reversion implies that rates revert to some long-term average and avoid extreme variations of the interest rate to mimic the actual behaviour of rates. Mean reverting processes prevent rates from drifting too far away from a central tendency materialized by a long-term rate. This contrasts with stock prices, which can drift away from initial values without bounds. A ‘term structure model’ describes the evolution of the yield curve through time. Models generate the entire yield curve using one or two factors. When using models to generate interest rate scenarios, the technique consists of generating random values of the factor(s) driving the rate and the error term. Random interest rate scenarios follow. To find all possible values of a bond at horizon, we need to simulate yield curve scenarios between now and the horizon. For each scenario, there is a unique value of the bond price resulting from its discounting contractual cash flows at maturity with rates of simulated yield curves. Since the same yield curve applies to all interest-driven assets, ignoring the fluctuation of credit spreads, we use the same yield curve scenarios for all bonds. For derivatives, the final values result from the underlying asset values, as for equity derivatives. One-factor models involve only one source of uncertainty, usually the short-term rate. They assume that all rates move in the same direction. However, the magnitude of the change is not the same for all rates. The following equation represents a one-factor model of short-term rates rt . Using such a model implies that all rates fully correlate with this short-term rate. The random dzt term follows a standardized normal distribution and makes the rate changes stochastic: drt = κ(θ − rt )dt + σ (rt )γ dzt The parameter κ < 1 determines the speed of mean reversion to a long-term mean θ . When the rate is high, the term in parentheses becomes negative, and contributes to a downward move of the rate. When the parameter γ = 0, the change of rate is normally distributed as the random term. When γ = 1, the rate rt follows a lognormal distribution, as the stock prices do with the standard Wiener process. When γ = 0.5, the variance of the rate is proportional to its level, implying that low interest rates are less volatile than high interest rates. One-factor models use the following processes: Rendleman and Bartter: dr = κrdt + σ rdz Vasicek’s model: Cox, Ingersoll and Ross model:

dr = κ(θ − r)dt + σ dz √ dr = κ(θ − r)dt + σ r dz

Two-factor models assume two sources of uncertainty (Brennan and Schwartz, 1979; Cox et al., 1985; Longstaff and Schwarz, 1992). The ‘no-arbitrage’ models start by fitting the current term structure of rates, which provides the benefit of starting from actual rates that can serve for pricing purposes or defining interest rate scenarios from the current state. These models comply with the principle of no risk-free arbitrage across maturities of the spot yield curve. However, this constraint implies that the current shape of the yield curve drives to a certain extent the future short rates. The Ho and Lee (1986) model uses a binomial tree of bond prices. Hull

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and White extended the original technique6 . Advanced models allow long-term modelling of the yield curves (Heath et al., 1992). For VaR purposes, it is not practical to use elaborate interest rate models. Simpler techniques are necessary. Selecting some interest rates as references and allowing them to vary, while complying with given variance–covariance structure, is the simplest technique. Principal component analyses use historical data on market parameters to define components that explain the movements7 . The application of this technique to term structure modelling shows that the most important factors represent the parallel shifts, twists (slope changes) and curvature of the term structure of yield. Principal component analysis has many attractive properties. The rates of different maturities are a linear function of the factors. The technique provides the sensitivities of rates to each factor and the factor volatilities. Since factors are independent, it is easy to derive the volatility of a rate given sensitivities and factor volatilities. For simulation, allowing factors to vary randomly, as well as the random error, generates a full distribution of yield curves.

MEASURING VOLATILITIES The volatility is a basic ingredient of all processes leading to the horizon value distribution. There are several techniques to obtain volatilities. The first is to derive them from historical observations of prices and market parameters. More elaborate approaches try to capture the instability of volatility over time. They use moving average, with or without assigning higher weights to the most recent observations, or ARCH–GARCH models of the time behaviour of observed volatilities. A third technique is to use the implicit volatilities embedded in options prices, by reverse engineering pricing models of options to derive from the price the volatility parameter. Implicit volatilities look forward by definition, as market prices do, but they might be very unstable.

Square Root of Time Rule Historical volatility measures require us to define the horizon of the period of observations and the frequency of observations. The frequency stipulates whether we measure a daily, monthly or yearly volatility. It is easier to use daily measures because there is more information. Then, we move to other periods using the ‘square root of time rule’ for volatilities. The square root of time rule applies only when there are no bounds to cumulative returns, as mentioned above for stock prices, and if the underlying process is stationary. The rule stems from the independence assumption across time of consecutive value changes. For instance, the change in values between dates 0 and 2, V (0, 2), is the sum of the change between dates 0 and 1 and between dates 1 and 2 (end of period dates), V (0, 1) + V (1, 2). The issue is to find the volatility of the change σ02 of V (0, 2) between 0 and 2 from the volatilities between 0 and 1, σ01 , and between 1 and 2, σ12 , which are supposed identical, σ01 = σ12 = σ1 , the unit period volatility. A simple rule of summation indicates the volatility of the sum is the square 6 See 7 See

Hull (2000) for a review of all models and details on the Hull and White extensions. Frye (1997) for factor models of interest rates, also Litterman and Scheinkman (1988).

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√ root of the sum of the squared volatilities, or σ02 = σ1 2. The rule extends easily to any number of units of time, if variations across time periods remain independent and have a constant √ volatility. Using the rule, with a 1% daily volatility, the yearly volatility is σ1year = σ1day 250 = 1% × 15.811 = 15.811%. This simple rule allows us to convert market volatilities over various lengths of time. The relationship between the volatility σt √ over t periods and the volatility over a unit σ1 period is σt = σ1 t (Figure 30.3). σ

σ √t

Time t

FIGURE 30.3

Volatility time profile of a market parameter over various horizons

Practical rules are helpful, even though reality does not comply with the underlying assumptions. For long-term horizons, the above volatility would become infinite, which is unacceptable. Many phenomena are ‘mean reverting’, such as interest rates. The above formula provides a practical way to infer short-term volatilities only. In addition, real phenomena are not ‘stationary’, implying that volatilities are not stable over time, and the assumption does not hold.

Non-stationarity Various techniques deal with non-stationarity, which designates the fact that the random process from which we sample the volatility observations is not stable over time. Since the VaR is highly sensitive to volatilities of market parameters, it is of major importance to use relevant and conservative values. They extend from simple conservative rules to elaborate models of the time behaviour of volatility, ARCH–GARCH models (Engle, 1993; Nelson, 1990a). Volatilities are highly unstable. Depending on the period, they can be high or very low. There are a number of techniques for dealing with this issue8 . Some are very simple, such as taking the highest between a short-term volatility (3 months) and a long-term volatility (say 2 years). If volatility increases, the most recent one serves as a reference. If there are past periods of high volatility, the longer period volatility serves as a reference. Other techniques consist of using moving averages of volatilities. A moving average is the average of a determined number k of successive observations. The average moves with time since the sample of the last k observations moves when time passes. For example, we could have a time series of values such as: 2, 3, 5, 6, 4, 3. A three-period moving average would calculate the average over three consecutive observations. At the third date it is 8 For

updating volatilities in simulations, see Hull (2000).

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(2 + 3 + 5)/3, then it becomes (3 + 5 + 6)/3 at the fourth date, (5 + 6 + 4)/3 at the fifth date, and finally (6 + 4 + 3)/3 at the last date. Moving averages minimize deviations from the long-term mean. Arithmetic moving averages place equal weights on all observations, ignoring the fact that the latest ones convey more information. Exponential smoothing places higher weights on the most recent observations and mimics better the most recent changes. The Exponentially Weighted Moving Average (EWMA) model uses weights decreasing exponentially when moving back in time. Risk Metrics uses a variation of these ‘averaging’ techniques9 . The GARCH family of models aims to model the time behaviour of volatility. Volatility seems to follow a mean reverting process, so that high values tend to smooth out after a while. The models attempt to capture these patterns assuming that the variance calculated over a given period as of t depends on the variance as of t − 1, over the same horizon, plus the latest available observation. The model equation, using ht as the variance and rt as the return, is: 2 ht = α0 + α1 rt−1 + α2 ht−1 The variance ht is a conditional variance, dependent on past observations. By contrast, the long-term unconditional mean is h, obtained by setting h = ht = ht−1 and observing that E(r2t−1 ) = h : h = α0 + α1 h + α2 h implies h = α0 /(1 − α1 − α2 ). The model results in a series of variance shocks followed by smoothed values of variance.

VALUE DISTRIBUTIONS AT THE HORIZON AND STANDALONE MARKET RISK The measures of standalone market risk of single instruments are loss statistics, loss volatility and loss percentiles, all resulting from the distribution of final values at horizon. Applied to a single asset, we obtain the standalone VaR at a preset confidence level. This section uses the normal and lognormal distributions to derive these measures. The normal distribution has the drawbacks mentioned above, but serves for the Delta–normal technique for a short-term horizon. When reverting to continuous stochastic processes of the time paths of asset returns, we obtain lognormal distributions at the horizon for stock prices. The value percentiles derive from this lognormal distribution.

Normal Values Distribution at Horizon Return yt and value Vt are such that Vt = V0 (1 + yt ) and both follow a normal distribution. The confidence level for the value percentiles is such that: Prob[Vt ≤ V (α)] = α Converting this equation into a condition on the random return between 0 and t: Prob[V0 (1 + yt ) ≤ V (α)] = α Prob{yt ≤ [V (α) − V0 ]/V0 } = α 9 J.

P. Morgan, Risk Metrics Monitor, 1995.

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yt follows N(µ, σ ) and zt = (yt − µ)/σ follows N(0, 1), the normal standardized distribution. Since yt = σ zt + µ, the preceding inequality becomes: Prob{zt ≤ [V (α)/V0 ] − (1 + µ)/σ } = α By definition, Prob[zt ≤ z(α)] = α. The probability of a standardized normal variable being lower than a given threshold z(α) is by definition (α), where is the cumulative standard distribution. By definition, z(α) = −1 (α), with −1 being the standardized normal inverse. This inequality defines z(α) and the value percentile V (α) derives from the linear relationship between z(α) and V (α): z(α) = −1 (α) = [V (α)/V0 − (1 + µ)]/σ V (α)/V0 = 1 + µ + −1 (α)σ In this formula, −1 (α) is negative, so that the final value can be lower than the initial value as long as the downside deviation exceeds the upside variation due to the asset return. As an example, let’s assume that the expected yearly return µ is 10%, the yearly return volatility σ is 15%, the horizon t is 1 year and the value at date 0, V0 , is 1. This simplifies the formula. If the confidence level is 1%, Prob[zt ≤ z(α)] = α implies that z(α) = −1 (α) = −2.3263. Hence: V (α)/V0 = 1 + 10% − 2.3263 × 15% = 75.11% The downside deviation of the value at the preset confidence level of 1% is 1 − 75.11% = 24.89% in absolute value.

Lognormal Values Distribution at Horizon When using normal distributions of continuous returns, the distribution at the horizon is a lognormal distribution. The lognormal distribution applies notably to stock prices. Defining the distribution requires the return volatility. Deriving value percentiles from a lognormal distribution is less straightforward than for a normal distribution. If the return follows a Wiener process, the value at date t is: √ Vt = V0 exp[(µ − 21 σ 2 )t + σ t zt ] The random component zt follows a standardized normal distribution N(0, 1). The parameters µ and σ are the mean and volatility of the instantaneous rate of return of the firm. The value percentile V (α) at the preset confidence level α is such that: Prob[Vt ≤ V (α)] = α From the distribution of the value at t, we derive the value percentile V (α) using the inequality: √ Prob{exp[(µ − 12 σ 2 )t + σ t zt ] ≤ V (α)/V0 } = α This inequality requires that zt be such that: √ Prob{(µ − 21 σ 2 )t + σ t zt ≤ ln[V (α)/V0 ]} = α √ Prob{zt ≤ ln[V (α)/V0 ] − (µ − 21 σ 2 )t/σ t} = α

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The probability of a standardized normal variable being lower than a given threshold z(α) is by definition (α), where is the cumulative standard distribution, and z(α) = −1 (α), with −1 being the standardized normal inverse. The equality Prob[zt ≤ z(α)] = α defines z(α) and, then, the value percentile V (α). This requires determining z(α) first, and moving from there to V (α): √ z(α) = −1 (α) = {ln[V (α)/V0 ] − (µ − 12 σ 2 )t}/σ t √ ln[V (α)/V0 ] = −1 (α)σ t + (µ − 12 σ 2 )t √ V (α)/V0 = exp[−1 (α)σ t + (µ − 21 σ 2 )t] In this formula, −1 (α) is negative, so that the final value is lower than the initial value as long as the downside deviation exceeds the upside variation due to the asset return adjusted for the volatility term. As an example, we assume that the expected yearly return µ is 10%, the yearly return volatility σ is 15%, the horizon t is 1 year and the value at date 0 is 1, so that we have the final value as a percentage of the initial value. This simplifies the formula. If the confidence level is 1%, z(α) = −1 (α) = −2.3263. The argument of the exponential above is: √ −1 (α)σ t + (µ − 12 σ 2 )t = −2.3263 × 15% + (10% − 12 × 15%2 ) = −0.260195 The exponential of this term is 77.09%. The downside deviation of the value at the preset confidence level of 1% is 1 − 77.09% = 22.91% in absolute value. We observe that this downside deviation is lower than under the normal approximation, because the normal approximation has a ‘fatter’ downside tail than the lognormal distribution.

MAPPING INSTRUMENTS TO RISK FACTORS Modelling all individual asset returns within a portfolio is overly complex to handle. Since the number of market indexes is much smaller than that of individual assets, it is more efficient to derive asset returns from the time paths of market parameters that influence their values. On the other hand, it necessitates modelling the relation between asset returns and market parameters. The process requires two steps: identifying those market parameters which influence values, a process called ‘mapping’; modelling the sensitivity of individual asset values to market parameters. Mapping and using sensitivities to underlying market parameters is the foundation of the Delta VaR technique. The first subsection describes the mapping process, the second one explains how to derive standalone market risk under the Delta VaR technique with a single asset. The next subsections specify the sensitivities of various types of assets.

The Mapping Process The mapping process results from pricing models in order to identify the main value drivers of asset values. Mapping individual asset returns to market parameters often uses only a subset of all market parameters for simplification purposes. The subset of interest

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rates corresponds to selected maturities. For interest rate references that do not correspond to these maturities, it is possible to interpolate a proxy of the corresponding rate using the references. The exclusion of some value drivers results in ‘basis risk’, since the modelled values do not track exactly the actual asset prices any more. The variations of value of any market instrument result from its sensitivities to market parameters and the volatility of these parameters. The mapping of individual asset values to market parameters (also designated as risk drivers or risk factors) results from pricing models, showing which parameters influence values. Most relations between asset values and value drivers have a linear approximation. In some cases, the relationship is statistical, and implies always a significant error, such as for stocks. In such cases, it becomes necessary to account for the error in estimating the risk. In other cases, the relation results from a closed-form formula such as for bonds or simple derivatives. When dealing with derivatives, such as interest rate swaps or foreign exchange swaps, or forward rate agreements, it is necessary to break down the derivative into simpler components to reduce complexity and find direct relations with market parameters driving their values. For example, a forward transaction is a combination of long and short transactions, such as a forward interest rate (lend long until final date and borrow short until start date). An interest rate swap is a combination of fixed and variable flows indexed to the term structure of rates. A forward swap is a combination of a long swap and a short swap. Sensitivities are unstable because they are local measures. This is a major limitation, notably for options. Their sensitivity with respect to the underlying parameter is the delta of the option. An ‘out-of-the-money’ option has a relatively stable and low delta. When the option is ‘at-the-money’, its delta changes significantly with the underlying price. When it is ‘in-the-money’, the value increases almost proportionally with the market parameter and the delta tends progressively towards 1. With out-of-the-money options, the portfolio behaves as if no options exist. When options are in-the-money, the sensitivity gets close to 1. In other words, the portfolio behaves as if its sensitivity structure changes with variations in the underlying assets. This is the convexity risk, encountered with implicit options in Asset–Liability Management (ALM) models. Whenever convexity is significant within a portfolio, the simple Delta VaR model becomes unreliable10 . Extensions, as described in the next chapter, partially correct such deficiencies. Sensitivities are ‘local’ measures that change with the context. Accordingly, calculating the value change of an instrument as a linear function of market parameter changes is only an approximation. When a model relates the price of an asset to several parameters, the ‘Taylor’ expansion11 of the formula is a proxy relationship of the value change resulting from a change in value drivers. It relates the value change to changes in all parameters using all derivatives of the actual function. A Taylor series makes the change of the value of instruments as a function of the first, second, third-order derivatives, and so on. For large changes, it is preferable to consider additional terms of Taylor expansion of the equation beyond 10 Another

case in point is the measure of credit risk for options. Since the horizon gets much longer than for market risk, the assumption of constant sensitivities collapses. 11 See the Taylor expansion formula in Chapter 32.

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the first one. Additional terms better proxy the curvature of the relationship between the asset returns and their drivers.

Standalone Risk from Market Returns The return sensitivity change in asset i is, in general, a function of several market parameters k: K sik × mk /mk0 yi = Vi /Vi0 = 1

The index k refers to any of the K relevant market parameters, with k varying from 1 to K. The index i refers to any one of N transactions, with i varying from 1 to N . Since one transaction often depends on only one market parameter, and in general on only a small number of them, most of the sensitivities are zero. The standalone risk of an instrument results directly from the sensitivities and volatilities of market parameters. The equity index return has a volatility of 20%. The volatility of the stock return results from the above relationship: σ (yi ) = σ (Vi /Vi0 ) = sik × σ (mk /mk0 ) σ (Vi ) = σ (yi ) × Vi0 = sik × σ (mk /mk0 ) × Vi0 Let’s assume that the stock return has a sensitivity of 2 to the equity index. Then: σ (yi ) = σ (Vi )/Vi0 = sik × σ (mk /mk0 ) = 2 × 20% = 40% If the initial asset value is 1000, the resulting value volatility is 400. If the value follows approximately a normal distribution, the loss percentile at the 1% confidence level is 2.33 × 400 in value, or 932. Calculations are very simple because the linear approximation implies that the product of a random normal return by a constant follows also a normal distribution.

Stocks For stocks, the ‘beta’ β relates the change in stock price to the change in market index for the same period. It is the sensitivity of the equity return (a percentage change) to the index return (a percentage change). β is the risk measure in the Capital Asset Pricing Model (CAPM), the well-known model12 that provides the theoretical price of a stock and the required return by the market as a function of risk. It is the coefficient of the regression of the historical equity periodical returns against the similar index return providing β as the coefficient of the regression. The return is the ratio of the variation in stock price, or 12 The

ments.

original presentation is in Sharpe (1964). See also Sharpe and Alexander (1990) for subsequent develop-

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index value, to the original value, for any period. There are as many return observations as periods (daily, weekly, monthly, and so on). β is higher or lower than 1, and it is 1 by definition for the equity index. The return of the stock i and of the index Im are: ri = Pi /Pi rm = Im /Im β results from a regression on historical returns fitting the equation below to time series of observations: r i = βi r m + α + ε i In the case of stocks, there is no deterministic relationship between the equity index and the stock return. Rather, there is a general risk resulting from general market index changes plus an error term. Ignoring the error term: ri = βi rm β relates the relative price changes of a stock and an equity index, which is the market parameter in this case. If a stock has a β of 1.2, a change in index return rm of 1% results, on average, in a 1.2% increase of the price return ri . Hence ri = Pi /Pi increases by ri = 1.2%. Considering the initial stock price as given, the variation Pi is 1.2% × Pi when the variation of the index Im is 1% × Im . Hence, β relates the relative and the absolute changes in stock price and equity index. Using statistical fits implies an error term. The stock return depends on the equity index return plus a random ‘innovation’ term, which is independent of the equity index return by definition. The error term is the fraction of return of the stock unexplained by general index variations. This fraction is the specific risk of the stocks, as opposed to the general risk shared by all stocks related to the equity index. The variance of stock return is: σ 2 (ri ) = βi σ 2 (rm ) + σ 2 (εi ). It sums the variance due to general risk and specific risk. In practice, since error terms offset to a significant extent, simplified techniques ignore the sum of the specific risks for all stock prices, which is the variance of a sum of independent variables. For portfolio models, the ‘diagonal model’ of stock returns allows us to consider the specific risk from the innovation term for all stocks.

Bonds and Loans The sensitivity of bond prices to interest rate shocks, shocks being parallel shifts of the entire spectrum of rates, is the ‘duration’. If the duration of a bond is 5, it means the bond value will change by 5% when all rates deviate by 1%. A common measure of sensitivity for bonds is the basis point measure. A basis point (bp) is 1% of 1%, or 0.0001. It is the deviation, expressed in basis points, for a unit basis point change of interest rates. In the previous example, the basis point change of the bond value is 5 bp for a 1 bp change in all interest rates. This is the ‘DV01’ measure of bond sensitivity.

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Sensitivities also apply to credit risk, because bond prices depend on ‘credit spreads’, the difference between the market rate applying to a risky bond and the risk-free market rate of government bonds. Since bond values are sensitive to credit spread, any widening of spreads reduces the value. The sensitivity of bonds with respect to credit spreads is similar to that of interest rates. For non-tradable assets, such as banking loans, sensitivities to interest rates also apply to their mark-to-market values and are also durations. These sensitivities serve for measuring the risks of the mark-to-market value of the balance sheet, or Net Present Value (NPV). Durations help when considering only parallel shifts of the yield curves. They do not capture changes in the shape of the curve. Considering all changes in interest rates over all maturities requires the sensitivities to all market rates to obtain proxy asset returns. For the NPV calculations, for ALM, assets and liabilities are fully revalued for each yield curve scenario, without relying on sensitivities. Technically, the change of a single rate along the curve implies a change of the discount factor relative to that rate. For VaR purposes, it is necessary to simulate yield curve changes, either from correlations across rates or from principal components factor models.

Foreign Exchange Exposures The sensitivity of the dollar value of any exposure labelled in foreign currency to the exchange rate is the variation of this dollar value due to a unit variation of the exchange rate. For instance, the dollar value of 1000 euros, with an exchange rate of 1 EUR/USD, is 1000 USD. If the exchange rate becomes 8 EUR/USD, the dollar value becomes 800 USD. The variation is −20%, which is in value −1000 × 20%. The relative change in dollar value of exposure and that of the market parameter value are identical to −20%. The value sensitivity of the exposure in USD with respect to the exchange rate is simply 1.

Options Options are the right, but not the obligation, to buy or sell some underlying parameter. Derivative, and option, sensitivities result from models, or ‘pricers’, which relate their values to their drivers. The value of options depends on a number of parameters, as shown originally in the Black–Scholes model: the value of the underlying, the horizon to maturity, the volatility of the underlying, the risk-free interest rate. The model is in line with intuition. The option value increases with the underlying asset value, its volatility and maturity, and decreases with interest rate. Option sensitivities are known as the ‘Greek letters’. The sensitivity with respect to the underlying asset is the ‘delta’ δ. Intuitively, the δ is low if the option is ‘out-of-the-money’ (asset price below strike) because we do not get any money by exercising unless the asset value changes significantly. However, when exercising provides a positive payoff, the δ gets closer to 1: if strike is 100 and asset price is 120, the payoff is 120 − 100 = 20; if the asset price increases by 1 to 121, the payoff increases by 1. The sensitivity can be anywhere in the entire range between 0 and 1, when

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the sensitivity increases from near zero values to values close to one when the option is in-the-money. The variation of δ is the ‘convexity’ of the option. Gamma (γ ) is the change in δ when the underlying changes. It is the change in slope of the curve representing the option value as a function of the underlying. The longer the horizon, the higher the chances that the stock moves above the strike price. Hence, the option is sensitive to the time to maturity, the sensitivity being theta (θ ). The shorter the time to maturity, the lower is the value of the option. This value change when time passes is the ‘time decay’ of the option value. The higher the volatility of the underlying asset, the higher the chances that the value moves above the strike during a given period. Hence, the option has a positive sensitivity to the underlying asset volatility, which is the ‘vega’ (ν). Since any payoff appears only in the future, its value today requires discounting, and so that it varies inversely with the level of interest rates. Rho (ρ) is the change due to a variation of the risk-free rate. The option has several sensitivities, one for each relevant parameter that influences its value.

APPENDIX 1: CUMULATIVE RETURNS OVER TIME PERIODS When using a single terminal point, we have a straightforward correspondence between the discrete one-period return and the final value. This is not so for cumulative returns over a large number of intervals. Terminal values result from cumulated returns over intermediate periods. In practice, we simulate returns for each set of subperiods, for instance 10 single-period returns, as in the example below, by drawing randomly 10 values from the distribution fitting the stochastic process of returns. Each of the 10 endof-period values results from the value at the start date times the random percentage return plus 1 (Vt /Vt−1 = 1 + y). Calculating the 10 end-of-period values directly provides the final one at horizon. The example above illustrates the process. There are shortcuts for obtaining cumulative returns if returns are ‘logarithmic’ and when using small intervals. The arithmetic return is yt = (Vt − Vt−1 )/Vt−1 . The logarithm of the price ratio, ln(Vt /Vt−1 ), is identical to ln[1 + y(t)]13 . It is approximately equal to the arithmetic return when the return is small because ln(1 + y) is approximately identical to y. From the logarithmic definition, the value after a time interval t is such that: Vt+t = Vt exp(yt). With arithmetic return yt = (Vt − Vt−1 )/Vt−1 , the final value Vt becomes negative for values of y lower than −100%. With logarithmic returns, the final value cannot be negative even with negative returns. This makes the normal distribution for the return y acceptable. If the logarithm of Vt /V0 follows a normal distribution of y, Vt follows a lognormal distribution, by definition. Finally, logarithmic returns are additive across periods. When combining returns across consecutive periods, the return between dates 0 and t is y(0, t) = ln(Vt /V0 ). The logarithmic returns are additive across periods. With two consecutive periods, we have: y(0, 2) = ln(V2 /V0 ) = ln[(V2 /V1 ) × (V1 /V0 )] y(0, 2) = ln(V2 /V1 ) + ln(V1 /V0 ) = y(0, 1) + y(1, 2) If, for instance, V1 /V0 = 120% and V2 /V1 = 90%, with V0 = 1, V1 = 1.2 and V2 = 90% × 1.2 = 1.08, the overall return is 8%, which differs from the arithmetic summation 20% − 10% = 10% because the percentages apply to different initial values. In the 13 The

return is yt = (Vt − Vt−1 )/Vt−1 and 1 + yt = 1 + (Vt − Vt−1 )/Vt−1 = 1 + (Vt /Vt−1 − 1) = Vt /Vt−1 .

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example of Table 30.1, the exact cumulative return between initial and final values is 8% from the actual values. This is very close to the logarithm cumulative return of 7.696%, which is the exact summation of single-period logarithmic returns. The arithmetic summation of the single-period arithmetic returns is 10%, significantly above the actual 8%. TABLE 30.1

Cumulating arithmetic and logarithm returns

Vt V0 V1 V2

1.00 1.20 1.08

Cumulative return

Vt /Vt−1 = 1 + y

ln(Vt /Vt−1 )

Arithmetic return yi

120.00% 90.00% V2 /V0 − 1

18.232% −10.536% ln(V2 /V0 )

20.00% −10.00% y1 + y2

8.00%

7.696%

10.00%

Since the sum of random normal variables is also a normal variable, so is y(0, t) for any horizon t. Since y(0, t) = ln(Vt /V0 ) follows a normal distribution, the final value follows a lognormal distribution.

31 Modelling Correlations and Multi-factor Models for Market Risk

To model correlations, the common principle for all portfolio models is to relate the individual risks of each transaction to a set of common factors. For market risk, the market values of individual transactions are sensitive to risk drivers that alter their values. When all risk factors vary, the dependence of the individual asset returns on this set of common factors generates correlations between them. Because common factors influence simultaneously all transaction risks, they create a ‘general’ risk, defined as the risk common to all assets. The fraction of individual risk unrelated to common factors is ‘specific’ risk. Factor models are two-sided, depending on the initial purpose. The first purpose is to model the correlations between risk drivers on which individual risk correlations depend. The second purpose is to construct loss distributions for portfolios. The next chapter explains how to proceed with factor models. We address only the first issue in this chapter. It is necessary to capture the interdependencies between market parameters to correlate future values of market parameters and assets. Individual asset value variations depend on these correlations, some of them being positive and others negative. They sum algebraically to obtain portfolio values. The diversification effect results from offsetting effects between individual variations. For market risk, correlations derive directly from observed market prices or market parameters. Pricing models of derivatives or bonds help because they relate in a deterministic way the prices of assets to the market parameters that derive them. In such a

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case, it is sufficient to correlate the market parameters to obtain correlated individual asset returns (or values) using the pricing models with the correlated market parameter values as inputs. When the relationship to market parameters is stochastic, for example between stock returns and equity indexes, it includes an error term, measuring the random component of individual asset returns unrelated to market parameters. When using factor models, asset return variations result from the volatility of the factors plus that of the error term, measuring the volatility of the asset return unrelated to factors. The error term contributes to the risk. The volatility of the common factors is the ‘general risk’, and the volatility of the error term is the ‘specific risk’. Specific risk is important for stock returns with general risk driven by equity indexes and specific risk related to each individual stock. These principles underlie the ‘diagonal’ model for stocks. Fitting factor models to returns allows us to determine both the return variances plus all their covariance terms from the coefficients of the factor models. In addition, the contribution of common factors to the return volatility is the general risk, while the volatility of the error term is the specific risk. The one-factor model of stocks illustrates these properties. Multi-factor models provide the same information. ‘Orthogonal’ factor models are the simplest because the factors are independent of each other. The first section briefly summarizes why correlations between risk drivers are key inputs for capturing diversification effects through variance–covariance matrices. The second section describes the specifics of the ‘correlation building block’ of market risk. The third section explains how to derive correlations, volatilities and general plus specific risk from factor models of individual asset returns. The appendix describes various types of factor models.

WHY IS THE VARIANCE–COVARIANCE MATRIX NECESSARY FOR MODELLING PORTFOLIO LOSSES? The risk over a portfolio is the change in all mark-to-market values of individual instruments. The loss of the portfolio is the algebraic sum of all gains and losses for all individual positions. Some exposures gain values, others lose values. These changes are market parameter-driven. Since these correlate, all individual returns correlate as well. When they relate deterministically to risk drivers, the entire value variations result from sensitivities. When there exists only a statistical relationship, the risk drivers generate general risk, to which it is necessary to add specific risk, the fraction of risk not related to the common factors. A simplistic view would use simultaneous adverse changes, captured as a function of both the risk parameter volatility and the instrument sensitivities. This is a most conservative rule because it is not possible that all market parameters will change simultaneously in such a way that they trigger losses simultaneously for all individual positions. Such arithmetic addition of all risks would greatly over-estimate the portfolio risk. Sensitivities do not add. Adding them is equivalent to assuming that all parameters change simultaneously in adverse directions, which is unrealistic. The key to measuring portfolio risk lies in capturing the market parameter interdependencies. It is easy to observe that some vary together and others inversely. Sometimes the relationship is strong and sometimes it is loose. The structure of these relationships results in a variance–covariance matrix and a correlation matrix between all market parameters

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to which the instrument values map. The process for measuring portfolio risk comprises two steps: • Imposing this variance–covariance structure on market parameter deviations. • Modelling the portfolio return distribution when all market parameters vary in compliance with such variance–covariance structure. The correlation methodology aims to replicate realistic simultaneous changes in all market parameters, in line with observed changes. This is a prerequisite for revaluing the portfolio at the horizon. When the correlation structure is available, the correlations between individual asset values result from those of the stochastic processes driving the asset values. Several assets depend on the correlated market parameters, such as equity indexes. It is possible to correlate market parameters and prices, assuming a normal distribution of their values as an approximation. When implementing full valuation techniques, the issue is to correlate random time paths of returns for different assets. To generate the paths of N correlated assets, indexed i, we use several correlated random processes as follows. The stochastic process driving the market parameter or the individual asset return includes a random term, such as the process applying to stock prices: yi = dVit /Vti = µi dt + σ i dzit , where i is the index specific to an individual asset. This process results in the value distribution at√a forward date t lognormal distribution of stock prices: Sti = S0i exp[(µi − 21 σi2 )t + σi t]. When considering a pair of stocks, we apply a correlation on the random innovations dzit of the process. In order to simulate correlated normal innovations, standard procedures apply, such as the Cholevsky decomposition, explained below. The final prices of the pair of stocks correlate even though they follow lognormal distributions. The techniques for portfolio revaluation at the horizon vary from using sensitivities to value small changes of instrument values to full revaluation of all instruments given multiple correlated scenarios of time paths of the market parameters. The linear relationship between values and underlying market parameters is the foundation of the Delta VaR model. It ends up as an analytical model because the weighted summation of random normal variables is also a normal variable. Hence, only the mean and standard deviation of the portfolio value suffice to define the entire distribution. Monte Carlo simulations allow us to bypass the restrictive assumption that individual returns are linear functions of market parameters. In both cases, the prerequisite is to have the correlations and the variance–covariance matrix of all relevant market parameters.

MODELLING CORRELATIONS BETWEEN INDIVIDUAL ASSET RETURNS AND MARKET PARAMETER RETURNS Correlations and variances of individual asset returns and market parameters are observable. Therefore, the first technique for obtaining the matrix is to measure it through direct observations, usually on an historical basis. This raises several difficulties. First, the high number of assets makes it unpractical to create a matrix with all pair covariances between individual asset returns. Second, the observed variances and covariances might not comply

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with the basic property of variance–covariance matrices, that of being semi-definite positive. This happens because some observed parameters correlate strongly1 . Because of the large number of asset returns, it is more efficient to derive correlations from the correlation of the ‘factors’ driving these returns. This option requires a prior mapping of asset returns to market parameters. Risk Metrics provides correlations between the main market parameters. Risk drivers are interdependent because they depend on a common set of factors. For instance, all stock returns relate to equity index returns through statistical fitting. The technique extends to multiple factors. One example of a multiple-factor model is the Arbitrage Pricing Theory (APT) model (see Ross, 1976), which considers that more than one factor influences the equity returns. Other factor models model interest rates, making them correlated. Principal Component Analysis (PCA) uses orthogonal factors influencing all rates for generating yield curve changes. Correlations between instrument returns within each block result from their common dependence on such factors. For market risk, the risk factors are market parameters, or observable parameters that drive the risk. Factors and risk drivers are not necessarily identical however. For example, it is possible to identify a number of factors that influence the equity index returns or the interest rates. Equity indexes and interest rates are the direct drivers of asset returns. On the other hand, factors influence risk drivers without interacting directly with returns as drivers do. They serve to correlate the distributions of risk drivers as an alternative technique to direct measures of correlations. The common dependence of risk drivers, such as interest rates, on a set of factors makes them correlate. Hence, we need to distinguish three levels: factors, risk drivers and market values (Figure 31.1). From Risk Factors to Risks Market Risk

FIGURE 31.1

Factors

Risk factors

Risk drivers

Market parameters

Individual risks

Market value changes

From risk factors to correlation of individual risks

In many instances, however, risk factors are identical to the risk drivers for market risk. They are the market parameters, under the broad sense of yield curves, foreign exchange rates, equity indexes, plus their volatilities. Since market parameters are directly observable, there is no need to model them. Historical direct observations are feasible. Interest rate models allow us to generate yield curves, such that all interest rates remain consistent with arbitrage relationships. Factor models offer a convenient alternative technique, notably in two cases: • For equity returns, the Capital Asset Pricing Model (CAPM) is a single-factor model serving to calculate all variances and covariances of individual stocks, plus the specific 1 When

two variables have a correlation of 1, the matrix does not have the desired properties. Measuring errors of highly correlated variables might give a similar result.

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risk of each stock. The APT model uses ‘orthogonal’ factors to achieve the same purpose, using more than one factor to model equity returns. • For yield curves, multi-factor models help summarize in a convenient way the basic transformation of interest rates, parallel shifts, slope variations and bumps of actual yield curves. Figure 31.2 provides an overview of the correlation modelling building block of market risk models. The purpose of the ‘revaluation’ block of models is to link risk drivers to asset returns and values. As such, this revaluation block does not trigger correlations. Correlation between risks results rather from the correlations between the risk drivers. Since returns result from risk drivers through the revaluation process, correlated risk drivers generate return correlations.

Factors

Revaluation Revaluation building block building block

Factors are either observed explicit market parameters or principal components factors ...

Sensitivities, revaluations

Market risk driver correlations

Market risk drivers are market parameters driving transaction values

Market risk

Asset return correlations

Correlated MTM changes

Asset A

Asset B

Asset ...

Market risk: Correlations and volatility Market risk: Correlations and volatility

FIGURE 31.2

Individual value changes correlate due to the dependence on correlated risk drivers

From market factors to correlated transaction market risk

IMPLEMENTING FACTOR MODELS Factor models serve, notably, to simplify the modelling of correlations between individual stocks and that of interest rates. In this section we show how to derive correlations from factor models. The main example deals with stock prices. For stocks, it is important both to model correlations and to isolate specific risk, the risk unrelated to common factors. The section addresses the one-factor model of stock returns and multiple-factor models, such as those of PCA.

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Deriving the Variance–Covariance for Stocks The direct measures of the variances–covariances of stock returns would imply measuring N 2 terms, for a portfolio of N stocks. The one-factor model allows us to measure the covariances using a linear relationship between the stock return and the market index return. This is a direct application of a one-factor model, whose formulas are given in the appendix to this chapter. The market model is: ri = αi + βi rm + εi The relationships simply indicate that all stock returns co-vary with the index, although their sensitivities βi vary with each stock2 . The statistical relationship is a one-factor model, with rm being the explicit factor influencing the individual returns. For equity returns, we illustrate first the attractive property of using one factor only and proceed to describe the general ‘diagonal’ model of stock return correlations. Pairs of Stocks

By definition of the regression model, the random equity index return is independent of the residual εi . As a result, the covariance between any pair of equity returns depends only on the variance of the single factor and the coefficients β1 and β2 . The covariance and the correlation between any pair of equity returns r1 and r2 is simply3 : Cov(r1 , r2 ) = β1 β2 σ 2 (rm ) ρij = Cov(ri , rj )/σ (ri )σ (rj ) = βi βj σ 2 (rm )/σ (ri )σ (rj ) Moreover, the volatility of any individual asset return is: [σ (ri )]2 = βi2 [σ (rm )]2 + [σ (εi )]2 The return ri volatility is the sum of the systematic variance of the market, the general risk generated by rm weighted by the squared sensitivity βi2 , and of the specific, or idiosyncratic, risk of an individual stock. Let’s assume that all returns are ri standardized normal variables, with unit variance, for the index return as well as for the stocks picked. The coefficients are βi = 1 and βj = 1.5. The ri represent the equity returns of two obligors. The model fit sets the volatility of the residual, measuring specific risk: σ 2 (r1 ) = (1)2 1 + 1.5 = 2.5 = 1.5812 σ 2 (r2 ) = (1.2)2 1 + 1 = 2.94 = 1.7152 Cov(r1 , r2 ) = 1 × 1.2 × 1 = 1.2 ρ12 = 1.2/(1.581 × 1.715) = 44.25% 2 The CAPM models this empirical finding and shows that the return on any asset i is the risk-free rate r plus f a risk premium equal to the differential of the random market return rm and the risk-free rate, times the βi of the asset: ri = rf + βi (rm − rf ). 3 Cov(r , r ) = Cov(α + β r + ε , α + β r + ε ) = β β Cov(r , r ) because all cross-covariance 1 2 1 1 m 1 2 2 m 2 1 2 m m terms, Cov(r1 , ε1 ), Cov(r1 , ε2 ), Cov(ε1 , ε2 ), as well as the covariances between r2 and the residuals, are zero. In addition, Cov(rm , rm ) is the variance of the equity index return.

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In the case of the first obligor, the error variance is 1.5 and the variance of the factor is 1. Total variance is 2.5. The R 2 is the ratio of 1/2.5 = 40%. In the second case, total variance is 2.94, and the explained variance is 1.44, hence R 2 is 1.44/2.94 = 48.98%. The R 2 provides a direct estimate of the general versus specific risk. When no fit is available, for private firms for example, it is necessary to specify the R 2 using for instance the average over all firms, which is in the range of 20% to 30% for listed companies in the main stock exchanges. Deriving the Variance–Covariance for the Entire Stock Portfolio

The covariance between asset i and asset j returns depends only on βi and βj , and on the factor variance. With N assets, there are N variances and N × (N − 1) covariance terms, a total of N 2 terms in the variance–covariance matrix. Using the one-factor model allows us to calculate the N × (N − 1) covariance terms from N coefficients βi plus the factor volatility, or N + 1 terms compared to N × (N − 1). Hence, we summarize N 2 terms of the variance–covariance matrix using only N + 1. In matrix format, the variance–covariance matrix becomes: σ (ε1 )2 0 0 ··· β 1 β 1 β1 β2 β1 β 3 · · · β 2 β 1 β2 β2 β2 β 3 · · · 0 0 ··· σ (ε2 )2 2

= σm2 β3 β1 β3 β2 β3 β3 · · · + 0 0 σ (ε3 ) · · · .. .. .. .. .. .. .. .. . . . . . . . .

All the off-diagonal terms result from the βi , plus the market index volatility. All the diagonal terms depend on the βi , plus the market index volatility, plus the specific risk term. This provides the entire matrix. The first term of the matrix is simply ββ T × σm2 , where β T stands for the transpose vector of the sensitivities. This is the ‘diagonal’ model of stock returns variances and covariances.

The APT Model for Stock Returns A well-known example of multi-factor modelling is the APT model of equity returns. Contrasting with the CAPM, or the simpler ‘market model’, the APT model makes the equity returns dependent on several independent factors rather than a single one. Multiple factors generate some additional complexity. Common factors are the source of general credit risk, the risk to which each obligor is subject. The error term is the specific or idiosyncratic risk of asset i. By construction, it is independent of all common factors. The analytical form of risk results from the variance of the Yi : σ 2 (Y1 ) = (βi1 )2 Var(X1 ) + (βi2 )2 Var(X2 ) + (βi3 )2 Var(X3 ) + Var(εi ) σ 2 (Y1 ) =

K i=1

(βik )2 Var(Xk ) + Var(εi )

The general risk is the sum of the variances of the factors weighted by the model coefficients, and the specific risk is the residual variance. The appendix to this chapter explains how to obtain these formulas.

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Orthogonal Multiple-factor Models A practical model for changing randomly the shape of the yield curve is PCA. In this case, factors are linear functions of observed variables and are independent of each other. In such a case, it is easy to derive the variances and covariances, or correlations, from the factor variances and the model coefficients. We use here a simple example of such calculations, with a two-factor model of two variables. The general formulas are given in the appendix to this chapter. The models for two correlated indexes and two factors are: Y1 = β10 + 1 × X1 + 1.2 × X2 + ε1 Y2 = β20 + 0.8 × X1 + 0.5 × X2 + ε2 X1 and X2 are standardized normal independent factors (zero mean and unit variance). The variances of the residuals Var(ε1 ) and Var(ε2 ) are respectively 1.5 and 1. The crosscovariance between X1 and X2 is zero. The volatility of the residuals measures the specific risk and results from the model fit. The variance adds the general factor risk plus the specific risks Var(ε1 ) and Var(ε2 ). The variances of the indexes Y1 and Y2 follow: σ 2 (Y1 ) = (1)2 1 + (1.2)2 1 + 1.5 = 3.94 and σ (Y1 ) = 1.9852 σ 2 (Y2 ) = (0.8)2 1 + (0.5)2 1 + 1 = 0.64 + 0.25 + 1 = 1.89 and σ (Y2 ) = 1.3752 The R 2 of the regression of Y1 and Y2 (Yi ) on X1 and X2 (Xk ) is the ratio of explained variance by all factors to the error variance or, equivalently, the ratio of general to total risk. In the case of the first asset, the error variance is 1.5 and that of factors X1 and X2 is 2.44. Total variance is 3.94. The R 2 is 2.44/3.94 = 61.93%. In the second case, the total variance is 1.89 and the explained variance is 0.89, hence R 2 is 0.89/1.89 = 47.09%. The covariance between Y1 and Y2 is: Cov(Y1 , Y2 ) = 1 × 1.2 × 1 + 0.8 × 0.5 × 1 = 1.600 The corresponding correlation coefficient is: ρ12 = 1.600/(1.985 × 1.375) = 58.63%

General versus Specific Risk When relating individual returns to factors, the factors generate general risk, the risk common to all assets, and the residual risk is the specific risk. Specific risk appears only whenever there is no deterministic relationship between the factors and the risk drivers, or between the risk drivers and the returns (such as the closed-form formulas of pricing models). This is the case for stocks, as illustrated above. Specific risk also appears when modelling interest rates from underlying factors. The specific risk is the variance of the error terms in these models. When using the Delta VaR technique, ignoring specific risk underestimates the overall risk. There is always a fraction of the portfolio volatility that does not relate to common factors or risk drivers. The diagonal model described above makes explicit the specific risk component.

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APPENDIX: IMPLEMENTING FACTOR MODELS FOR MODELLING CORRELATIONS Factor models serve to model correlations between risk drivers generating correlated distributions of credit events within a portfolio. To model correlations, we face the issue of constructing a variance–covariance matrix for credit risk drivers, such as asset values or factors that drive the credit standing. From the matrix, we can infer any pair correlation. A distinct, but related, issue is that such a matrix is information intensive for a portfolio. A variance–covariance matrix is squared, and has N 2 terms for a portfolio of N obligors. Therefore, we need to reduce the information to manageable dimensions. Factor models address these issues. We provide examples of the single-factor model, or orthogonal-factor (independent factors) models obtained with PCA, and of multiple-factor models with correlated factors.

Measuring and Modelling Correlations with Single-factor Models In the case of a single factor, the decomposition is obvious because the residual term is independent, by definition of the single factor. The Single-factor Model

We start with this simple case. The one-factor model form is: Yi = βi0 + βi X1 + εi The variance of Yi is simply the sum of the variance due to the single factor and to that of the residual: σ 2 (Yi ) = (βi )2 Var(X1 ) + Var(εi ) The covariance between any two Yi and Yj is: Cov(Yi , Yj ) = Cov(βi0 + βi X1 + εi , βj 0 + βj X1 + εj ) Cov(Yi , Yj ) = βi βj Var(X1 ) This formula simplifies because all cross-correlations between factors and residuals are zero by construction of the model. All residuals εi are independent of X1 . The correlation between Yi and Yj is: ρij = Cov(Yi , Yj )/σ (Yi )σ (Yj ) = βi βj Var(X1 )/σ (Yi )σ (Yj ) When the single factor explains a large fraction of the risk driver volatility, the systematic risk is relatively high, and the specific risk is low. The opposite holds when the single factor explains only a small fraction of the volatility. Note that the R 2 of the regression of Yi on Xi is, by definition, the ratio of variance explained by the factor to the total variance or, equivalently, the ratio of general to total risk. For N assets, there are N variances plus covariances, resulting in N 2 terms. Using the single-factor model, we need only the N β i , plus the N residual variances, plus the

MODELLING CORRELATIONS AND MULTI-FACTOR MODELS

393

single factor variance, or 2N + 1 items of information. This is the diagonal model of correlations, as shown when modelling equity risk.

Multi-factor Models for Measuring Correlations The first subsection extends the definitions to a multi-factor setting. The second subsection shows how to derive variances, covariances and correlations from models using orthogonal factors, and the third subsection is a brief extension to correlated, rather than independent, multiple factors. General and Specific Risk in Multi-factor Models

In general, a multi-factor model relates some random variable Y to common factors Xk : Yi = βi0 + βi1 X1 + βi2 X2 + βi3 X3 + · · · + εi The index i refers to asset returns, while the index k refers to the factor. In the equity universe, the variable explained by factors is the equity return of stocks. They are random just as the factors and the residual are, but they are all sensitive to each of the common factors. Multiple-factor Models with Orthogonal Factors

To illustrate the general formulas, we use a two-factor model. The two factors are independent, or ‘orthogonal’. Yi is the random asset return of asset i. The two-factor model is: Yi = βi0 + βi1 X1 + βi2 X2 + εi The general formula is: Yi =

k

βi1 Xk + εi

The index i refers to the assets, while the index k refers to the factor. The R 2 of the regression of Yi on Xk is the ratio of explained variance by all factors to the total variance. From this general formula, it is easy to derive the variance of Yi , which simplifies because we have a linear combination of independent variables. For pairs of assets, we derive both covariances and correlations. Again, the formulas simplify because of zero cross-correlations between factors and residuals. All Xk are standardized normal orthogonal factors, the variances of the residuals σ 2 (εi ) are respectively 1.5 and 1 for obligors 1 and 2. The cross-covariances between all Xk are zero. The volatility of the residuals measures the specific risk and results from the model fit. The general formula for the variance of the variable is: σ 2 (Y1 ) = Cov(β11 X1 + β12 X2 + ε1 , β11 X1 + β12 X2 + ε1 ) σ 2 (Y1 ) = (β11 )2 Var(X1 ) + (β12 )2 Var(X2 ) + Var(ε1 )

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The extension to any number K of factors is straightforward. The variance is the summation of general risk variances plus the specific risk variance of the residual. The variance is: (βi1 )2 Var(Xk ) + Var(εi ) Var(Yi ) = k

With orthogonal factors, all covariances between factors and residuals are zero and the other covariances are: Cov(Y1 , Y2 ) = Cov(β11 X1 + β12 X2 + ε1 , β21 X1 + β22 X2 + ε2 ) = β11 β21 Var(X1 ) + β12 β22 Var(X2 ) The covariance between Yi and Yj collapses to: Cov(Yi , Yj ) = βik βj k Var(Xk ) k

The correlation between Yi and Yj is: ρij = Cov(Yi , Yj )/σ (Yi )σ (Yj ) For two assets i and j , the corresponding correlation coefficient is: ρij = [β1i β1j Var(X1 ) + β2i β2j Var(X2 )]/σ (Yi )σ (Yj )

General Multi-factor Models The models are similar except that there is no simplification due to zero cross-correlations. All formulas for variances, covariances and correlations depend on all cross-correlations, making them more complex. Generally, relating asset returns to market parameters is a multi-factor setting where factors are market parameters. Therefore, we need to account for the observed cross-correlations of market parameters. When factors are not orthogonal, all cross-covariances between factors contribute to both variances and covariances of the Yi . Covariances of Risk Drivers

The covariance of two indexes Y1 and Y2 is: Cov(Y1 , Y2 ) = Cov(β10 + β11 X1 + β12 X2 + ε1 , β20 + β21 X1 + β22 X2 + ε2 ) All the covariances between constant and any random variables are zero. The covariances between any factor and the residuals are zero because the factors extract the correlations between the random asset values leaving only an uncorrelated residual. We expand all formulas to see the details: Cov(Y1 , Y2 ) = Cov(β11 X1 + β12 X2 + ε1 , β21 X1 + β22 X2 + ε2 ) Simplifying: Cov(Y1 , Y2 ) = β11 β21 Var(X1 ) + β12 β22 Var(X2 ) + β11 β22 Cov(X1 , X2 ) + β12 β21 Cov(X2 , X1 ) + β13 β21 Cov(X3 , X1 )

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395

All cross-covariances of factors apply for calculating the asset covariance, but all crosscovariances between factors and residuals are zero by construction of the model. Variances of Risk Drivers

The variance of an asset value results from setting Y1 = Y2 : σ 2 (Y1 ) = Cov(β11 X1 + β12 X2 + ε1 , β11 X1 + β12 X2 + ε1 ) Simplifying4 : σ 2 (Y1 ) = (β11 )2 Var(X1 ) + (β12 )2 Var(X2 ) + 2β12 β11 Cov(X1 , X2 ) + Var(ε1 ) For a single obligor, there are two factor covariance terms, plus two factor variance terms, plus one variance of the error term. The extension to K factors is straightforward. The variance depends on the variances of each factor, plus the covariance terms, plus the variance of the error term representing the specific risk. In general, for each obligor or segment, there are K factor variance terms, plus K × (K − 1) covariance terms, plus one specific risk term. Aggregating over the N obligors or segments generates N times these terms. There are N × K variance terms, plus N specific risk terms, plus N × (N − 1) × K × (K − 1) covariance terms. The N × K variance terms count less than the covariance terms. This mechanism makes the specific risk much lower than general risk when the number of obligors increases.

4 Var(Y

1)

= (β11 )2 Var(X1 ) + β11 β12 Cov(X1 , X2 ) + β12 β11 Cov(X2 , X1 ) + (β12 )2 Var(X2 ) + Var(ε1 ).

32 Portfolio Market Risk

The goal of modelling portfolio risk is to obtain the distribution of the portfolio returns at the horizon, set at the liquidation period. This implies a forward revaluation at the horizon date of all instruments once market parameters change. Since these are the value drivers of individual assets within the portfolio, the prerequisite is to model the market parameter random deviations complying with their correlation structure. The next step is to derive individual asset return distributions from market parameters to get all possible portfolio returns. Loss statistics and loss percentiles providing the market risk ‘Value at Risk’ (VaR) derive from the portfolio return distribution. To achieve this ultimate goal, techniques range from the Delta VaR technique to full-blown simulations of market parameters and portfolio values. Portfolios benefit from diversification. The portfolio return volatility decreases with the number of assets, down to a floor resulting from general risk. However, the value of risk relates to portfolio value rather than return. The portfolio value volatility increases with the number of assets, and the incremental volatility for a new asset increases with the average correlation of the portfolio. When dealing with single assets, there is no need to worry about standalone risk. For portfolios of assets, we need to include the effect of the correlation between individual asset returns and between risk drivers, or market parameters, influencing these individual returns. The Delta VaR model relates linearly asset returns to market parameter returns using instrument sensitivities. The essentials are that the portfolio return is a linear combination of random normals. Therefore it follows a normal distribution. The volatility of the portfolio return applies to any set of linear combinations of random variables. Since we can calculate the volatility of the portfolio return, and since we know that it is normally distributed, we have all that we need to measure VaR. When the assumptions get unrealistic, we need to extend the simple Delta VaR technique or rely on full revaluation at horizon.

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397

Full-blown simulations consist of generating risk driver returns complying with the variance–covariance structure observed in the markets, and revaluating each individual transaction. Revaluation uses pricing models, or simulation techniques for complex derivatives. The portfolio return distribution results from the full revaluation for all trials. Forward looking simulations generate random market parameter values complying with market volatilities and correlations. Intermediate techniques use sensitivities to save the time intensive calculations. Historical simulations use past values of all risk drivers, which effectively embed existing correlations. Other intermediate techniques include ‘Delta–Gamma’ techniques, or grid simulations. Because of model risk, modelled returns deviate from actual returns. This necessitates back testing and stress testing modelled VaR to ensure that tracking errors remain within acceptable bounds. The first section shows how the portfolio return and the portfolio value volatility vary with the number of assets in a simple portfolio. The second section summarizes the Delta VaR technique. The third section expands fully the calculation of the Delta–normal VaR technique using the simple example of a two-asset portfolio. The fourth section reviews intermediate techniques. The fifth section expands the simulation technique and details some intermediate stages before moving to full-blown Monte Carlo simulations. The last section addresses back and stress testing of market risk VaR.

THE EFFECT OF DIVERSIFICATION AND CORRELATION ON THE PORTFOLIO VOLATILITY The standard representation of the diversification effect applies to portfolio and asset returns. The principles date from Markowitz principles (see Markowitz, 1952). The portfolio return varies with common factors and because of specific ri

RISK MANAGEMENT IN BANKING ¨ Bessis Joel

Copyright 2002 by

John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex, PO19 1UD, England National 01243 779777 International (+44) 1243 779777 e-mail (for orders and customer service enquiries): [email protected] Visit our Home Page on http://www.wileyeurope.com or http://www.wiley.com

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London, UK W1P 9HE, without the permission in writing of the publisher. Jo¨el Bessis has asserted his right under the Copyright, Designs and Patents Act 1988, to be identified as the author of this work. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA WILEY-VCH Verlag GmbH, Pappelallee 3, D-69469 Weinheim, Germany John Wiley & Sons (Australia) Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W 1L1, Canada Library of Congress Cataloguing-in-Publication Data Bessis, Jo¨el. [Gestion des risques et gestion actif-passif des banques. English] Risk management in banking/Jo¨el Bessis.—2nd ed. p. cm. Includes bibliographical references and index. ISBN 0-471-49977-3 (cloth)—ISBN 0-471-89336-6 (pbk.) 1. Bank management. 2. Risk management. 3. Asset-liability management. I. Title. HG1615 .B45713 2001 332.1′ 068′ 1—dc21 2001045562 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-471-49977-3 (cloth) ISBN 0-471-89336-6 (paper) Typeset in 10/12pt Times Roman by Laserwords Private Limited, Chennai, India. Printed and bound in Great Britain by TJ International Ltd, Padstow, England. This book is printed on acid-free paper responsibly manufactured from sustainable forestation, for which at least two trees are planted for each one used for paper production.

Contents

Introduction

ix

SECTION 1 Banking Risks

1

1 2

Banking Business Lines Banking Risks

SECTION 2 Risk Regulations 3

Banking Regulations

SECTION 3 Risk Management Processes 4 5

Risk Management Processes Risk Management Organization

SECTION 4 Risk Models 6 7 8 9

Risk Measures VaR and Capital Valuation Risk Model Building Blocks

3 11 23 25 51 53 67 75 77 87 98 113

SECTION 5 Asset–Liability Management

129

10 11 12 13 14

131 136 151 164 180

ALM Overview Liquidity Gaps The Term Structure of Interest Rates Interest Rate Gaps Hedging and Derivatives

vi

CONTENTS

SECTION 6 Asset–Liability Management Models

191

15 16 17 18 19

193 201 210 224 233

Overview of ALM Models Hedging Issues ALM Simulations ALM and Business Risk ALM ‘Risk and Return’ Reporting and Policy

SECTION 7 Options and Convexity Risk in Banking

245

20 Implicit Options Risk 21 The Value of Implicit Options

247 254

SECTION 8 Mark-to-Market Management in Banking

269

22 23 24 25

271 280 289 300

Market Value and NPV of the Balance Sheet NPV and Interest Rate Risk NPV and Convexity Risks NPV Distribution and VaR

SECTION 9 Funds Transfer Pricing

309

26 FTP Systems 27 Economic Transfer Prices

311 325

SECTION 10 Portfolio Analysis: Correlations

337

28 Correlations and Portfolio Effects

339

SECTION 11 Market Risk

357

29 30 31 32

359 363 384 396

Market Risk Building Blocks Standalone Market Risk Modelling Correlations and Multi-factor Models for Market Risk Portfolio Market Risk

SECTION 12 Credit Risk Models

417

33 Overview of Credit Risk Models

419

SECTION 13 Credit Risk: ‘Standalone Risk’

433

34 35 36 37 38

435 443 451 459 479

Credit Risk Drivers Rating Systems Credit Risk: Historical Data Statistical and Econometric Models of Credit Risk The Option Approach to Defaults and Migrations

CONTENTS

39 40 41 42 43

Credit Risk Exposure From Guarantees to Structures Modelling Recoveries Credit Risk Valuation and Credit Spreads Standalone Credit Risk Distributions

vii

495 508 521 538 554

SECTION 14 Credit Risk: ‘Portfolio Risk’

563

44 45 46 47 48 49 50

565 580 586 595 608 622 627

Modelling Credit Risk Correlations Generating Loss Distributions: Overview Portfolio Loss Distributions: Example Analytical Loss Distributions Loss Distributions: Monte Carlo Simulations Loss Distribution and Transition Matrices Capital and Credit Risk VaR

SECTION 15 Capital Allocation

637

51 52

639 655

Capital Allocation and Risk Contributions Marginal Risk Contributions

SECTION 16 Risk-adjusted Performance

667

53 54

669 679

Risk-adjusted Performance Risk-adjusted Performance Implementation

SECTION 17 Portfolio and Capital Management (Credit Risk)

689

55 56 57 58 59 60

691 701 714 721 733 744

Portfolio Reporting (1) Portfolio Reporting (2) Portfolio Applications Credit Derivatives: Definitions Applications of Credit Derivatives Securitization and Capital Management

Bibliography

762

Index

781

Introduction

Risk management in banking designates the entire set of risk management processes and models allowing banks to implement risk-based policies and practices. They cover all techniques and management tools required for measuring, monitoring and controlling risks. The spectrum of models and processes extends to all risks: credit risk, market risk, interest rate risk, liquidity risk and operational risk, to mention only major areas. Broadly speaking, risk designates any uncertainty that might trigger losses. Risk-based policies and practices have a common goal: enhancing the risk–return profile of the bank portfolio. The innovation in this area is the gradual extension of new quantified risk measures to all categories of risks, providing new views on risks, in addition to qualitative indicators of risks. Current risks are tomorrow’s potential losses. Still, they are not as visible as tangible revenues and costs are. Risk measurement is a conceptual and a practical challenge, which probably explains why risk management suffered from a lack of credible measures. The recent period has seen the emergence of a number of models and of ‘risk management tools’ for quantifying and monitoring risks. Such tools enhance considerably the views on risks and provide the ability to control them. This book essentially presents the risk management ‘toolbox’, focusing on the underlying concepts and models, plus their practical implementation. The move towards risk-based practices accelerated in recent years and now extends to the entire banking industry. The basic underlying reasons are: banks have major incentives to move rapidly in that direction; regulations developed guidelines for risk measurement and for defining risk-based capital (equity); the risk management ‘toolbox’ of models enriched considerably, for all types of risks, providing tools making risk measures instrumental and their integration into bank processes feasible.

THE RATIONALE FOR RISK-BASED PRACTICES Why are visibility and sensitivity to risks so important for bank management? Certainly because banks are ‘risk machines’: they take risks, they transform them, and they embed

x

INTRODUCTION

them in banking products and services. Risk-based practices designate those practices using quantified risk measures. Their scope evidently extends to risk-taking decisions, under an ‘ex ante’ perspective, and risk monitoring, under an ‘ex post’ perspective, once risk decisions are made. There are powerful motives to implement risk-based practices: to provide a balanced view of risk and return from a management point of view; to develop competitive advantages, to comply with increasingly stringent regulations. A representative example of ‘new’ risk-based practices is the implementation of riskadjusted performance measures. In the financial universe, risk and return are two sides of the same coin. It is easy to lend and to obtain attractive revenues from risky borrowers. The price to pay is a risk that is higher than the prudent bank’s risk. The prudent bank limits risks and, therefore, both future losses and expected revenues, by restricting business volume and screening out risky borrowers. The prudent bank avoids losses but it might suffer from lower market share and lower revenues. However, after a while, the risk-taker might find out that higher losses materialize, and obtain an ex post performance lower than the prudent lender performance. Who performs best? Unless assigning some measure of risk to income, it is impossible to compare policies driven by different risk appetites. Comparing performances without risk adjustment is like comparing apples and oranges. The rationale of risk adjustment is in making comparable different performances attached to different risk levels, and in general making comparable the risk–return profiles of transactions and portfolios. Under a competitive perspective, screening borrowers and differentiating the prices accordingly, given the borrowers’ standing and their contributions to the bank’s portfolio risk–return profile, are key issues. Not doing so results in adverse economics for banks. Banks who do not differentiate risks lend to borrowers rejected by banks who better screen and differentiate risks. By overpricing good risks, they discourage good borrowers. By underpricing risks to risky customers, they attract them. By discouraging the relatively good ones and attracting the relatively bad ones, the less advanced banks face the risk of becoming riskier and poorer than banks adopting sound risk-based practices at an earlier stage. Those banking institutions that actively manage their risks have a competitive advantage. They take risks more consciously, they anticipate adverse changes, they protect themselves from unexpected events and they gain the expertise to price risks. The competitors who lack such abilities may gain business in the short-term. Nevertheless, they will lose ground with time, when those risks materialize into losses. Under a management perspective, without a balanced view of expected return and risk, banks have a ‘myopic’ view of the consequences of their business policies in terms of future losses, because it is easier to measure income than to capture the underlying risks. Even though risks remain a critical factor to all banks, they suffer from the limitations of traditional risk indicators. The underlying major issue is to assign a value to risks in order to make them commensurable with income and fully address the risk–return trade-off. Regulation guidelines and requirements have become more stringent on the development of risk measures. This single motive suffices for developing quantified risk-based practices. However, it is not the only incentive for structuring the risk management tools and processes. The above motivations inspired some banks who became pioneers in this field many years before the regulations set up guidelines that led the entire industry towards more ‘risk-sensitive’ practices. Both motivations and regulations make risk measurement a core building block of valuable risk-based practices. However, both face the same highly challenging risk measuring issue.

INTRODUCTION

xi

RISK QUANTIFICATION IS A MAJOR CHALLENGE Since risks are so important in banking, it is surprising that risk quantification remained limited until recently. Quantitative finance addresses extensively risk in the capital markets. However, the extension to the various risks of financial institutions remained a challenge for multiple reasons. Risks are less tangible and visible than income. Academic models provided foundations for risk modelling, but did not provide instrumental tools helping decision-makers. Indeed, a large fraction of this book addresses the gap between conceptual models and banking risk management issues. Moreover, the regulators’ focus on risks is still relatively recent. It dates from the early stages of the reregulation phase, when the Cooke ratio imposed a charge in terms of capital for any credit risk exposure. Risk-based practices suffered from real challenges: simple solutions do not help; risk measures require models; models not instrumental; quantitative finance aimed at financial markets more than at financial institutions. For such reasons, the prerequisites for making instrumental risk quantifications remained out of reach.

Visibility on Losses is Not Visibility on Risks Risks remain intangible and invisible until they materialize into losses. Simple solutions do not really help to capture risks. For instance, a credit risk exposure from a loan is not the risk. The risk depends on the likelihood of losses and the magnitude of recoveries in addition to the size of the amount at risk. Observing and recording losses and their frequencies could help. Unfortunately, loss histories are insufficient. It is not simple to link observable losses and earning declines with specific sources of risks. Tracking credit losses does not tell whether they result from inadequate limits, underestimating credit risk, inadequate guarantees, or excessive risk concentration. Recording the fluctuations of the interest income is easy, but tracing back such changes to interest rates is less obvious. Without links to instrumental risk controls, earning and loss histories are of limited interest because they do not help in taking forward looking corrective actions. Visibility on losses is not visibility on risks.

Tracking Risks for Management Purposes Requires Models Tracking risks for management purposes requires models for better capturing risks and relating them to instrumental controls. Intuitively, the only way to quantify invisible risks is to model them. Moreover, multiple risk indicators are not substitutes for quantified measures. Surveillance of risk typically includes such various items as exposure size, watch lists for credit risk, or value changes triggered by market movements for market instruments. These indicators capture the multiple dimensions of risk, but they do not add them up into a quantified measure. Finally, missing links between future losses from current risks and risk drivers, which are instrumental for controlling risk, make it unfeasible to timely monitor risks. The contribution of models addresses such issues. They provide quantified measures of risk or, in other words, they value the risk of banks. Moreover, they do so in a way that allows tracing back risks to management controls over risk exposures of financial institutions. Without such links, risk measures would ‘float in the air’, without providing management tools.

xii

INTRODUCTION

Financial Markets versus Financial Institutions The abundance of models in quantitative finance did not address the issues that financial institutions face until recently, except in certain specific areas such as asset portfolio management. They undermined the foundations of risk management, without bridging the gap between models and the needs of financial institutions. Quantitative finance became a huge field that took off long ago, with plenty of pioneering contributions, many of them making their authors Nobel Prize winners. In the market place, quantification is ‘natural’ because of the continuous observation of prices and market parameters (interest rates, equity indexes, etc.). For interest rate risk, modelling the term structure of interest rates is a classical field in market finance. The pioneering work of Sharpe linked stock prices to equity risk in the stock market. The Black–Scholes option model is the foundation for pricing derivative instruments, options and futures, which today are standard instruments for managing risks. The scientific literature also addressed credit risk a long time ago. The major contribution of Robert Merton on modelling default within an option framework, a pillar of current credit risk modelling, dates from 1974. These contributions fostered major innovations, from pricing market instruments and derivatives (options) that serve for investing and hedging risks, to defining benchmarks and guidelines for the portfolios management of market instruments (stocks and bonds). They also helped financial institutions to develop their business through ever-changing product innovations. Innovation made it feasible to customize products for matching investors’ needs with specific risk–return bundles. It also allowed both financial and corporate entities to hedge their risks with derivatives. The need for investors to take exposures and, for those taking exposures, to hedge them provided business for both risk-takers and risk-hedgers. However, these developments fell short of directly addressing the basic prerequisites of a risk management system in financial institutions.

Prerequisites for Risk Management in Financial Institutions The basic prerequisites for deploying risk management in banks are: • Risks measuring and valuation. • Tracing risks back to risk drivers under the management control. Jumping to market instruments for managing risks without prior knowledge of exposures to the various risks is evidently meaningless unless we know the magnitude of the risks to keep under control, and what they actually mean in terms of potential value lost. The risk valuation issue is not simple. It is much easier to address in the market universe. However, interest rate risk requires other management models and tools. All banking business lines generate exposures to interest rate risks. However, linking interest income and rates requires modelling the balance sheet behaviour. Since the balance sheet generates both interest revenues and interest costs, they offset each other to a certain extent, depending on matches and mismatches between sizes of assets and liabilities and interest rate references. Capturing the extent of offsetting effects between assets and liabilities also requires dedicated models.

INTRODUCTION

xiii

Credit risk remained a challenge until recently, even though it is the oldest of all banking risks. Bank practices rely on traditional indicators, such as credit risk exposures measured by outstanding balances of loans at risk with borrowers, or amounts at risk, and internal ratings measuring the ‘quality’ of risk. Banking institutions have always monitored credit risk actively, through a number of systems such as limits, delegations, internal ratings and watch lists. Ratings agencies monitor credit risk of public debt issues. However, credit risk assessment remained judgmental, a characteristic of the ‘credit culture’, focusing on ‘fundamentals’: all qualitative variables that drive the credit worthiness of a borrower. The ‘fundamental’ view on credit risk still prevails, and it will obviously remain relevant. Credit risk benefits from diversification effects that limit the size of credit losses of a portfolio. Credit risk focus is more on transactions. When moving to the global portfolio view, we know that a large fraction of the risk of individual transactions is diversified away. A very simple question is: By how much? This question remained unanswered until portfolio models, specifically designed for that purpose, emerged in the nineties. It is easy to understand why. Credit risk is largely invisible. The simultaneous default of two large corporate firms, for whom the likelihood of default is small, is probably an unobservable event. Still, this is the issue underlying credit risk diversification. Because of the scarcity of data available, the diversification issue for credit risk remained beyond reach until new modelling techniques appeared. Portfolio models, which appeared only in the nineties, turned around the difficulty by modelling the likelihood of modelled defaults, rather than actual defaults. This is where modelling risks contributes. It pushes further away the frontier between measurable risks and invisible–intangible risks and, moreover, it links risks to the sources of uncertainty that generate them.

PHASES OF DEVELOPMENT OF THE REGULATORY GUIDELINES Banks have plenty of motives for developing risk-based practices and risk models. In addition, regulators made this development a major priority for the banking industry, because they focus on ‘systemic risk’, the risk of the entire banking industry made up of financial institutions whose fates are intertwined by the density of relationships within the financial system. The risk environment has changed drastically. Banking failures have been numerous in the past. In recent periods their number has tended to decrease in most, although not all, of the Organization for Economic Coordination and Development (OECD) countries, but they became spectacular. Banking failures make risks material and convey the impression that the banking industry is never far away from major problems. Mutual lending–borrowing and trading create strong interdependencies between banks. An individual failure of a large bank might trigger the ‘contagion’ effect, through which other banks suffer unsustainable losses and eventually fail. From an industry perspective, ‘systemic risk’, the risk of a collapse of the entire industry because of dense mutual relations, is always in the background. Regulators have been very active in promoting pre-emptive policies for avoiding individual bank failures and for helping the industry absorb the shock of failures when they happen. To achieve these results, regulators have totally renovated the regulatory framework. They promoted and enforced new guidelines for measuring and controlling the risks of individual players.

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INTRODUCTION

Originally, regulations were traditional conservative rules, requiring ‘prudence’ from each player. The regulatory scheme was passive and tended to differentiate prudent rules for each major banking business line. Differentiated regulations segmented the market and limited competition because some players could do what others could not. Obvious examples of segmentation of the banking industry were commercial versus investment banking, or commercial banks versus savings institutions. Innovation made rules obsolete, because players found ways to bypass them and to compete directly with other segments of the banking industry. Obsolete barriers between the business lines of banks, plus failures, triggered a gradual deregulation wave, allowing players to move from their original business field to the entire spectrum of business lines of the financial industry. The corollary of deregulation is an increased competition between unequally experienced players, and the implication is increased risks. Failures followed, making the need for reregulation obvious. Reregulation gave birth to the current regulatory scheme, still evolving with new guidelines, the latest being the New Basel Accord of January 2001. Under the new regulatory scheme, initiated with the Cooke ratio in 1988, ‘risk-based capital’ or, equivalently, ‘capital adequacy’ is a central concept. The philosophy of ‘capital adequacy’ is that capital should be capable of sustaining the future losses arising from current risks. Such a sound and simple principle is hardly debatable. The philosophy provides an elegant and simple solution to the difficult issue of setting up a ‘pre-emptive’, ‘ex ante’ regulatory policy. By contrast, older regulatory policies focused more on corrective actions, or ‘after-the-fact’ actions, once banks failed. Such corrective actions remain necessary. They were prompt when spectacular failures took place in the financial industry (LTCM, Baring Brothers). Nevertheless, avoiding ‘contagion’ when bank failures occur is not a substitute for pre-emptive actions aimed at avoiding them. The practicality of doing so remains subject to adequate modelling. The trend towards more internal and external assessment on risks and returns emerged and took momentum in several areas. Through successive accords, regulators promoted the building up of information on all inputs necessary for risk quantification. Accounting standards evolved as well. The ‘fair value’ concept gained ground, raising hot debates on what is the ‘right’ value of bank assets and how to accrue earnings in traditional commercial banking activities. It implies that a loan providing a return not in line with its risk and cost of funding should appear at lower than face value. The last New Basel Accord promotes the ‘three pillars’ foundation of supervision: new capital requirements for credit risk and operational risks; supervisory processes; disclosure of risk information by banks. Together, the three pillars allow external supervisors to audit the quality of the information, a basic condition for assessing the quality and reliability of risk measures in order to gain more autonomy in the assessment of capital requirements. Regulatory requirements for market, credit and operational risk, plus the closer supervision of interest rate risk, pave the way for a comprehensive modelling of banking risks, and a tight integration with risk management processes, leading to bank-wide risk management across all business lines and all major risks.

FROM RISK MODELS TO RISK MANAGEMENT Risk models have two major contributions: measuring risks and relating these measures to management controls over risks. Banking risk models address both issues by embedding the specifics of each major risk. As a direct consequence, there is a wide spectrum of

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modelling building blocks, differing across and within risks. They share the risk-based capital and the ‘Value at Risk’ (VaR) concepts that are the basic foundations of the new views on risk modelling, risk controlling and risk regulations. Risk management requires an entire set of models and tools for linking risk management (business) issues with financial views on risks and profitability. Together, they make up the risk management toolbox, which provides the necessary inputs that feed and enrich the risk process, to finally close the gap between models and management processes.

Risk Models and Risks Managing the banking exposure to interest rate risk and trading interest rate risk are different businesses. Both commercial activities and trading activities use up liquidity that financial institutions need to fund in the market. Risk management, in this case, relates to the structural posture that banks take because of asset and liability mismatches of volumes, maturity and interest rate references. Asset–Liability Management (ALM) is in charge of managing this exposure. ALM models developed gradually until they became standard references for managing the liquidity and interest rate risk of the banking portfolio. For market risk, there is a large overlap between modelling market prices and measuring market risk exposures of financial institutions. This overlap covers most of the needs, except one: modelling the potential losses from trading activities. Market risk models appeared soon after the Basel guidelines started to address the issues of market risk. They appeared sufficiently reliable to allow internal usage by banks, under supervision of regulators, for defining their capital requirements. For credit risk, the foundations exist for deploying instrumental tools fitting banks’ requirements and, potentially, regulators’ requirements. Scarce information on credit events remains a major obstacle. Nevertheless, the need for quantification increased over time, necessitating measuring the size of risk, the likelihood of losses, the magnitude of losses under default and the magnitude of diversification within banks’ portfolios. Modelling the qualitative assessment of risk based on the fundamentals of borrowers has a long track record of statistical research, which rebounds today because of the regulators’ emphasis on extending the credit risk data. Since the early nineties, portfolio models proposed measures of credit risk diversification within portfolios, offering new paths for quantifying risks and defining the capital capable of sustaining the various levels of portfolio losses. Whether the banks should go along this path, however, is no longer a question since the New Basel Accord of January 2001 set up guidelines for credit risk-sensitive measures, therefore preparing the foundations for the full-blown modelling of the credit risk of banks’ portfolios. Other major risks appeared when progressing in the knowledge of risks. Operational risk became a major priority, since January 2001, when the regulatory authorities formally announced the need to charge bank capital against this risk.

Capital and VaR It has become impossible to discuss risk models without referring to economic capital and VaR. The ‘capital adequacy’ principle states that the bank’s capital should match risks. Since capital is the most scarce and costly resource, the focus of risk monitoring and

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INTRODUCTION

risk measurement follows. The central role of risk-based capital in regulations is a major incentive to the development of new tools and management techniques. Undoubtedly a most important innovation of recent years in terms of the modelling ‘toolbox’ is the VaR concept for assessing capital requirements. The VaR concept is a foundation of risk-based capital or, equivalently, ‘economic capital’. The VaR methodology aims at valuing potential losses resulting from current risks and relies on simple facts and principles. VaR recognizes that the loss over a portfolio of transactions could extend to the entire portfolio, but this is an event that has a zero probability given the effective portfolio diversification of banks. Therefore, measuring potential losses requires some rule for defining their magnitude for a diversified portfolio. VaR is the upper bound of losses that should not be exceeded in more than a small fraction of all future outcomes. Management and regulators define benchmarks for this small preset fraction, called the ‘confidence level’, measuring the appetite for risk of banks. Economic capital is VaRbased and crystallizes the quantified present value of potential future losses for making sure that banks have enough capital to sustain worst-case losses. Such risk valuation potentially extends to all main risks. Regulators made the concept instrumental for VaR-based market risk models in 1996. Moreover, even though the New Accord of 2001 falls short of allowing usage of credit models for measuring credit risk capital, it ensures the development of reliable inputs for such models.

The Risk Management Toolbox Risk-based practices require the deployment of multiple tools, or models, to meet the specifications of risk management within financial institutions. Risk models value risks and link them to their drivers and to the business universe. By performing these tasks, risk models contribute directly to risk processes. The goal of risk management is to enhance the risk–return profiles of transactions, of business lines’ portfolios of transactions and of the entire bank’s portfolio. Risk models provide these risk–return profiles. The risk management toolbox also addresses other major specifications. Since two risks of 1 add up to less than 2, unlike income and costs, we do not know how to divide a global risk into risk allocations for individual transactions, product families, market segments and business lines, unless we have some dedicated tools for performing this function. The Funds Transfer Pricing (FTP) system allocates income and the capital allocation system allocates risks. These tools provide a double link: • The top-down/bottom-up link for risks and income. • The transversal business-to-financial sphere linkage. Without such links, between the financial and the business spheres and between global risks and individual transaction profiles, there would be no way to move back and forth from a business perspective to a financial perspective and along the chain from individual transactions to the entire bank’s global portfolio.

Risk Management Processes Risk management processes are evolving with the gradual emergence of new risk measures. Innovations relate to:

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• The recognition of the need for quantification to develop risk-based practices and meet risk-based capital requirements. • The willingness of bankers to adopt a more proactive view on risks. • The gradual development of regulator guidelines for imposing risk-based techniques, enhanced disclosures on risks and ensuring a sounder and safer level playing field for the financial system. • The emergence of new techniques of managing risks (credit derivatives, new securitizations that off-load credit risk from the banks’ balance sheets) serving to reshape the risk–return profile of banks. • The emergence of new organizational processes for better integrating these advances, such as loan portfolio management. Without risk models, such innovations would remain limited. By valuing risks, models contribute to a more balanced view of income and risks and to a better control of risk drivers, upstream, before they materialize into losses. By linking the business and the risk views, the risk management ‘toolbox’ makes models instrumental for management. By feeding risk processes with adequate risk–return measures, they contribute to enriching them and leveraging them to new levels. Figure 1 shows how models contribute to the ‘vertical’ top-down and bottom-up processes, and how they contribute as well to the ‘horizontal’ links between the risk and return views of the business dimensions (transactions, markets and products, business lines). Risk Models & Tools

Credit Risk

FTP Capital Allocation

Market Risk ALM Others ...

Risk Processes Global Policy

Business Policy

Risk−Return Policy

Reporting Risk & Capital

Earnings

Business Lines

FIGURE 1

Comprehensive and consistent set of models for bank-wide risk management

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INTRODUCTION

THE STRUCTURE OF THE BOOK The structure of the book divides each topic into single modular pieces addressing the various issues above. The first section develops general issues, focusing on risks, risk measuring and risk management processes. The next major section addresses sequentially ALM, market risk and credit risk.

Book Outline The structure of the book is in 17 sections, each divided into several chapters. This structure provides a very distinct division across topics, each chapter dedicated to a major single topic. The benefit is that it is possible to move across chapters without necessarily following a sequential process throughout the book. The drawback is that only a few chapters provide an overview of interrelated topics. These chapters provide a synthesis of subsequent specific topics, allowing the reader to get both a summary and an overview of a series of interrelated topics. Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Banking Risks Risk Regulations Risk Management Processes Risk Models Asset–Liability Management Asset–Liability Management Models Options and Convexity Risk in Banking Mark-to-Market Management in Banking Funds Transfer Pricing Portfolio Analysis: Correlations Market Risk Credit Risk Models Credit Risk: ‘Standalone Risk’ Credit Risk: ‘Portfolio Risk’ Capital Allocation Risk-adjusted Performance Portfolio and Capital Management (Credit Risk)

Building Block Structure The structure of the book follows from several choices. The starting point is a wide array of risk models and tools that complement each other and use sometimes similar, sometimes different techniques for achieving the same goals. This raises major structuring issues for ensuring a consistent coverage of the risk management toolbox. The book relies on a building block structure shared by models; some of them extending across many blocks, some belonging to one major block. The structuring by building blocks of models and tools remedies the drawbacks of a sequential presentation of industry models; these bundle modelling techniques within each building block according to the model designers’ assembling choices. By contrast,

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a building block structure lists issues separately, and allows us to discuss explicitly the various modelling options. To facilitate the understanding of vendors’ models, some chapters provide an overview of all existing models, while detailed presentations provide an overview of all techniques applying to a single basic building block. Moreover, since model differentiation across risks is strong, there is a need to organize the structure by nature of risk. The sections of the book dealing directly with risk modelling cross-tabulate the main risks with the main building blocks of models. The main risks are interest rate risk, market risk and credit risk. The basic structure, within each risk, addresses four major modules as shown in Figure 2.

FIGURE 2

I

Risk drivers and transaction risk (standalone)

II

Portfolio risk

III

Top-down & bottom-up tools

IV

Risk & return measures

The building block structure of risk models

The basic blocks, I and II, are dedicated by source of risk. The two other blocks, III and IV, are transversal to all risks. The structure of the book follows from these principles.

Focus The focus is on risk management issues for financial institutions rather than risk modelling applied to financial markets. There is an abundant literature on financial markets and financial derivatives. A basic understanding of what derivatives achieve in terms of hedging or structuring transactions is a prerequisite. However, sections on instruments are limited to the essentials of what hedging instruments are and their applications. Readers can obtain details on derivatives and pricing from other sources in the abundant literature. We found that textbooks rarely address risk management in banking and in financial institutions in a comprehensive manner. Some focus on the technical aspects. Others focus on pure implementation issues to the detriment of technical substance. In other cases, the scope is unbalanced, with plenty of details on some risks (market risk notably) and fewer on others. We have tried to maintain a balance across the main risks without sacrificing scope. The text focuses on the essential concepts underlying the risk analytics of existing models. It does detail the analytics without attempting to provide a comprehensive coverage of each existing model. This results in a more balanced view of all techniques for modelling banking risk. In addition, model vendors’ documentation is available directly from sites dedicated to risk management modelling. There is simply no need to replicate such documents. When developing the analytics, we considered that providing a

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INTRODUCTION

universal framework, allowing the contrast of various techniques based on their essential characteristics, was of greater help than replicating public documentation. Accordingly, we skipped some details available elsewhere and located some technicalities in appendices. However, readers will find the essentials, the basics for understanding the technicalities, and the examples for grasping their practical value added. In addition, the text develops many numerical examples, while restricting the analytics to the essential ingredients. A first motive is that simple examples help illustrate the essentials better than a detailed description. Of course, simple examples are no substitute for full-blown models. The text is almost self-contained. It details the prerequisites for understanding fullblown models. It summarizes the technicalities of full-blown models and it substitutes examples and applications for further details. Still, it gathers enough substance to provide an analytical framework, developed sufficiently to make it easy to grasp details not expanded here and map them to the major building blocks of risk modelling. Finally, there is a balance to strike between technicalities and applications. The goal of risk management is to use risk models and tools for instrumental purposes, for developing risk-based practices and enriching risk processes. Such an instrumental orientation strongly inspired this text. *

*

*

The first edition of this book presented details on ALM and introduced major advances in market risk and credit risk modelling. This second edition expands considerably on credit risk and market risk models. In addition, it does so within a unified framework for capturing all major risks and deploying bank-wide risk management tools and processes. Accordingly, the volume has roughly doubled in size. This is illustrative of the fast and continuous development of the field of risk management in financial institutions.

SECTION 1 Banking Risks

1 Banking Business Lines

The banking industry has a wide array of business lines. Risk management practices and techniques vary significantly between the main poles, such as retail banking, investment banking and trading, and within the main poles, between business lines. The differences across business lines appear so important, say between retail banking and trading for example, that considering using the same concepts and techniques for risk management purposes could appear hopeless. There is, indeed, a differentiation, but risk management tools, borrowing from the same core techniques, apply across the entire spectrum of banking activities generating financial risks. However, risks and risk management differ across business lines. This first chapter provides an overview of banking activities. It describes the main business poles, and within each pole the business lines. Regulations make a clear distinction between commercial banking and trading activities, with the common segmentation between ‘banking book’ and ‘trading book’. In fact, there are major distinctions within business lines of lending activities, which extend from retail banking to specialized finance. This chapter provides an overview of the banking business lines and of the essentials of financial statements.

BUSINESS POLES IN THE BANKING INDUSTRY The banking industry has a wide array of business lines. Figure 1.1 maps these activities, grouping them into main poles: traditional commercial banking; investment banking, with specialized transactions; trading. Poles subdivide into various business lines. Management practices are very different across and within the main poles. Retail banking tends to be mass oriented and ‘industrial’, because of the large number of transactions. Lending to individuals relies more on statistical techniques. Management reporting on such large numbers of transactions focuses on large subsets of transactions.

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Business Poles

Business Lines

Retail Financial Services Commercial Banking

Corporate−Middle Market Large Corporate

Lending and collecting deposits Individuals and small businesses Identified borrowers and relationship banking

Advisory Services Mergers & Acquisitions LBO ... Investment Banking

Banks & Financial Firms Assets Financing (Aircrafts ...)

'Structured Finance'

Commodities Securitization

Derivatives Trading Trading

Equity

Traded Instruments

Fixed Income

FIGURE 1.1

Private Private Banking Banking

Assets Management

Others ... Others..

Custody ...

Breaking down the bank portfolio along organizational dimensions

Criteria for grouping the transactions include date of origination, type of customer, product family (consumer loans, credit cards, leasing). For medium and large corporate borrowers, individual decisions require more judgment because mechanical rules are not sufficient to assess the actual credit standing of a corporation. For the middle market segment to large corporate businesses, ‘relationship banking’ prevails. The relation is stable, based on mutual confidence, and generates multiple services. Risk decisions necessitate individual evaluation of transactions. Obligors’ reviews are periodical. Investment banking is the domain of large transactions customized to the needs of big corporates or financial institutions. ‘Specialized finance’ extends from specific fields with standard practices, such as export and commodities financing, to ‘structured financing’, implying specific structuring and customization for making large and risky transactions feasible, such as project finance or corporate acquisitions. ‘Structuring’ designates the assembling of financial products and derivatives, plus contractual clauses for monitoring risk (‘covenants’). Without such risk mitigants, transactions would not be feasible. This domain overlaps with traditional ‘merchant’ banking and market business lines. Trading involves traditional proprietary trading and trading for third parties. In the second case,

BANKING BUSINESS LINES

5

Market Transactions

Specialized Finance

Off-balance Sheet

Markets

Corporate Lending

RFS

Product Groups

traders interact with customers and other banking business units to bundle customized products for large borrowers, including ‘professionals’ of the finance industry, banks, insurance, brokers and funds. Other activities do not generate directly traditional banking risks, such as private banking, or asset management and advisory services. However, they generate other risks, such as operational risk, similarly to other business lines, as defined in the next chapter. The view prevailing in this book is that all main business lines share the common goals of risk–expected return enhancement, which also drives the management of global bank portfolios. Therefore, it is preferable to differentiate business lines beyond the traditional distinctions between the banking portfolio and the trading portfolio. The matrix shown in Figure 1.2 is a convenient representation of all major lines across which practices differ. Subsequent chapters differentiate, whenever necessary, risk and profitability across the cells of the matrix, cross-tabulating main product lines and main market segments. Banking business lines differ depending on the specific organizations of banks and on their core businesses. Depending on the degree of specialization, along with geographical subdivisions, they may or may not combine one or several market segments and product families. Nevertheless, these two dimensions remain the basic foundations for differentiating risk management practices and designing ‘risk models’.

Consumers Corporate−Middle Market Large Corporate Firms Financial Institutions Specialized Finance

FIGURE 1.2 Main product–market segments RFS refers to ‘Retail Financial Services’. LBO refers to ‘Leveraged Buy-Out’, a transaction allowing a major fraction of the equity of a company to be acquired using a significant debt (leverage). Specialized finance refers to structured finance, project finance, LBO or assets financing.

Product lines vary within the above broad groups. For instance, standard lending transactions include overnight loans, short-term loans (less than 1 year), revolving facilities, term loans, committed lines of credit, or large corporate general loans. Retail financial services cover all lending activities, from credit card and consumer loans to mortgage loans. Off-balance sheet transactions are guarantees and backup lines of credit

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providing significant revenues to banks. Specialized finance includes project finance, commodities financing, asset financing (from real estate to aircraft) and trade financing. Market transactions cover all basic compartments, fixed income, equity and foreign exchange trading, including derivatives from standard swaps and options to exotic and customized products. Major market segments appear explicitly in the matrix. They also subdivide. Financial institutions include banks as well as insurance or brokers. Specialized finance includes various fields, including structured finance. The greyed cell represents a basic market–product couple. Risk management involves risk and expected return measuring, reporting and management for such transactions, for the bank portfolio as a whole and for such basic couples. The next section reverts to the basic distinction between banking book and trading book, which is essentially product-driven.

THE BANKING AND THE TRADING BOOKS The ‘banking book’ groups and records all commercial banking activities. It includes all lending and borrowing, usually both for traditional commercial activities, and overlaps with investment banking operations. The ‘trading book’ groups all market transactions tradable in the market. The major difference between these two segments is that the ‘buy and hold’ philosophy prevails for the banking book, contrasting with the trading philosophy of capital markets. Accounting rules differ for the banking portfolio and the trading portfolio. Accounting rules use accrual accounting of revenues and costs, and rely on book values for assets and liabilities. Trading relies on market values (mark-to-market) of transactions and Profit and Loss (P&L), which are variations of the mark-to-market value of transactions between two dates. The rationale for separating these ‘portfolios’ results from such major characteristics.

The Banking Book The banking portfolio follows traditional accounting rules of accrued interest income and costs. Customers are mainly non-financial corporations or individuals, although interbanking transactions occur between professional financial institutions. The banking portfolio generates liquidity and interest rate risks. All assets and liabilities generate accrued revenues and costs, of which a large fraction is interest rate-driven. Any maturity mismatch between assets and liabilities results in excesses or deficits of funds. Mismatch also exists between interest references, ‘fixed’ or ‘variable, and results from customers’ demand and the bank’s business policy. In general, both mismatches exist in the ‘banking book’ balance sheet. For instance, there are excess funds when collection of deposits and savings is important, or a deficit of funds whenever the lending activity uses up more resources than the deposits from customers. Financial transactions (on the capital markets) serve to manage such mismatches between commercial assets and liabilities through either investment of excess funds or long-term debt by banks. Asset–Liability Management (ALM) applies to the banking portfolio and focuses on interest rate and liquidity risks. The asset side of the banking portfolio also generates credit risk. The liability side contributes to interest rate risk, but does not generate credit risk, since the lenders or depositors are at risk with the bank. There is no market risk for the banking book.

BANKING BUSINESS LINES

7

The Trading Portfolio The market transactions are not subject to the same management rules. The turnover of tradable positions is faster than that of the banking portfolio. Earnings are P&L equal to changes of the mark-to-market values of traded instruments. Customers include corporations (corporate counterparties) or other financial players belonging to the banking industry (professional counterparties). The market portfolio generates market risk, defined broadly as the risk of adverse changes in market values over a liquidation period. It is also subject to market liquidity risk, the risk that the volume of transactions narrows so much that trades trigger price movements. The trading portfolio extends across geographical borders, just as capital markets do, whereas traditional commercial banking is more ‘local’. Many market transactions use non-tradable instruments, or derivatives such as swaps and options traded over-the-counter. Such transactions might have a very long maturity. They trigger credit risk, the risk of a loss if the counterparty fails.

Off-balance Sheet Transactions Off-balance sheet transactions are contingencies given and received. For banking transactions, contingencies include guarantees given to customers or to third parties, committed credit lines not yet drawn by customers, or backup lines of credit. Those are contractual commitments, which customers use at their initiative. A guarantee is the commitment of the bank to fulfil the obligations of the customer, contingent on some event such as failure to face payment obligations. For received contingencies, the beneficiary is the bank. ‘Given contingencies’ generate revenues, as either upfront and/or periodic fees, or interest spreads calculated as percentages of outstanding balances. They do not generate ‘immediate’ exposures since there is no outflow of funds at origination, but they do trigger credit risk because of the possible future usage of contingencies given. The outflows occur conditionally on what happens to the counterparty. If a borrower draws on a credit line previously unused, the resulting loan moves up on the balance sheet. ‘Off-balance sheet’ lines turn into ‘on-balance sheet’ exposures when exercised. Derivatives are ‘off-balance sheet’ market transactions. They include swaps, futures contracts, foreign exchange contracts and options. As other contingencies, they are obligations to make contractual payments conditional upon occurrence of a specified event. Received contingencies create symmetrical obligations for counterparties who sold them to the bank.

BANKS’ FINANCIAL STATEMENTS There are several ways of grouping transactions. The balance sheet provides a snapshot view of all assets and liabilities at a given date. The income statement summarizes all revenues and costs to determine the income of a period.

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RISK MANAGEMENT IN BANKING

Balance Sheet In a simplified view (Table 1.1), the balance sheet includes four basic levels, in addition to the off-balance sheet, which divide it horizontally: • • • •

Treasury and banking transactions. Intermediation (lending and collecting deposits). Financial assets (trading portfolio). Long-term assets and liabilities: fixed assets, investments in subsidiaries and equity plus long-term debt.

TABLE 1.1

Simplified balance sheet

Assets

Equity and liabilities

Cash Lending Financial assets Fixed assets

Short-term debt Deposits Financial assets Long-term debt Equity Off-balance sheet (contingencies given)

Off-balance sheet (contingencies received)

The relative weights of the major compartments vary from one institution to another, depending on their core businesses. Equity is typically low in all banks’ balance sheets. Lending and deposits are traditionally large in retail and commercial banking. Investment banking, including both specialized finance and trading, typically funds operations in the market. In European banks, ‘universal banking’ allows banking institutions to operate over the entire spectrum of business lines, contrasting with the separation between investment banking and commercial banking, which still prevails in the United States.

Income Statement and Valuation The current national accounting standards use accrual measures of revenues and costs to determine the net income of the banking book. Under such standards, net income ignores any change of ‘mark-to-market’ value, except for securities traded in the market or considered as short-term holdings. International accounting standards progress towards ‘fair value’ accounting for the banking book, and notably for transactions hedging risk. Fair value is similar to mark-to-market1 , except that it extends to non-tradable assets such as loans. There is a strong tendency towards generalizing the ‘fair value’ view of the balance sheet for all activities, which is subject to hot debates for the ‘banking book’. Accounting standards are progressively evolving in that direction. The implications are major in terms of profitability, since gains and losses of the balance sheet ‘value’ between two 1 See

Chapter 8 for details on calculating ‘mark-to-market’ values on non-tradable transactions.

BANKING BUSINESS LINES

9

dates would count as profit and losses. A major driving force for looking at ‘fair values’ is the need to use risk-adjusted values in modelling risks. In what follows, because of the existing variations around the general concept of ‘fair value’, we designate such values equivalently as ‘economic values’, ‘mark-to-market’ or ‘mark-to-model’ values depending on the type of valuation technique used. Fair values have become so important, if only because of risk modelling, that we discuss them in detail in Chapter 8. For the banking portfolio, the traditional accounting measures of earnings are contribution margins calculated at various levels of the income statement. They move from the ‘interest income’ of the bank, which is the difference between all interest revenues plus fees and all interest costs, down to net income. The total revenue cumulates the interest margin with all fees for the period. The interest income of commercial banking commonly serves as the main target for management policies of interest rate risk because it is entirely interest-driven. Another alternative target variable is the Net Present Value (NPV) of the balance sheet, measured as a Mark-to-Market (MTM) of assets minus that of liabilities. Commercial banks try to increase the fraction of revenues made up of fees for making the net income less interest rate-sensitive. Table 1.2 summarizes the main revenues and costs of the income statement. TABLE 1.2

Income statement and earnings

Interest margin plus fees Capital gains and losses −Operating costs =Operating income (EBTD)a −Depreciation −Provisions −Tax =Net income a EBTD

= Earnings Before Tax and Depreciation.

Provisions for loan losses deserve special attention. The provision policy should ideally be an indicator of the current credit risk of banking loans. However, provisions have to comply with accounting and fiscal rules and differ from economic provisions. Economic provisions are ‘ex ante’ provisions, rather than provisions resulting from the materialization of credit risk. They should anticipate the effective credit risk without the distortions due to legal and tax constraints. Economic provisioning is a debated topic, because unless new standards and rules emerge for implementation, it remains an internal risk management tool without impact on the income statement bottom line.

Performance Measures Performance measures derive directly from the income statement. The ratio of net income to equity is the accounting Return On Equity (ROE). It often serves as a target profitability measure at the overall bank level. The accounting ROE ratio is not the market return on equity, which is a price return, or the ratio of the price variation between two dates of

10

RISK MANAGEMENT IN BANKING

the bank’s stock (ignoring dividends). Under some specific conditions2 , it might serve as a profitability benchmark. Both the ROE and the market return on equity should be in line with shareholders’ expectations for a given level of risk of the bank’s stock. A current order of magnitude for the target ROE is 15% after tax, or about 25% before tax. When considering banking transactions, the Return On Assets (ROA) is another measure of profitability for banking transactions. The most common calculation of ROA is the ratio of the current periodical income, interest income and current fees, divided by asset balance. The current ROA applies both to single individual transactions and to the total balance sheet. The drawback of accounting ROE and ROA measures, and of the P&L of the trading portfolio, is that they do not include any risk adjustment. Hence, they are not comparable from one borrower to another, because their credit risk differs, from one trading transaction to another, and because the market risk varies across products. This drawback is the origin of the concept of risk-adjusted performance measures. This is an incentive for moving, at least in internal reports of risks and performances, to ‘economic values’, ‘mark-to-market’ or ‘mark-to-model’ values, because these are both risk- and revenue-adjusted3 .

2 It

can be shown that a target accounting ROE implies an identical value for the market return on the bank’s equity under the theoretical condition that the Price–Earnings Ratio (PER) remains constant. See Chapter 53. 3 See Chapter 8 to explain how mark-to-market values embed both risk and expected return.

2 Banking Risks

Risks are uncertainties resulting in adverse variations of profitability or in losses. In the banking universe, there are a large number of risks. Most are well known. However, there has been a significant extension of focus, from the traditional qualitative risk assessment towards the quantitative management of risks, due to both evolving risk practices and strong regulatory incentives. The different risks need careful definition to provide sound bases serving for quantitative measures of risk. As a result, risk definitions have gained precision over the years. The regulations, imposing capital charges against all risks, greatly helped the process. The underlying philosophy of capital requirement is to bring capital in line with risks. This philosophy implies modelling the value of risk. The foundation of such risk measures is in terms of potential losses. The capital charge is a quantitative value. Under regulatory treatment, it follows regulatory rules applying to all players. Under an economic view, it implies modelling potential losses from each source of risk, which turns out to be the ‘economic’ capital ‘adequate’ to risk. Most of the book explains how to assign economic values to risks. Therefore, the universal need to value risks, which are intangible and invisible, requires that risks be well-defined. Risk definitions serve as the starting point for both regulatory and economic treatments of risks. This book focuses on three main risks: interest rate risk for the banking book; market risk for the trading book; credit risk. However, this chapter does provide a comprehensive overview of banking risks.

BANKING RISKS Banking risks are defined as adverse impacts on profitability of several distinct sources of uncertainty (Figure 2.1). Risk measurement requires capturing the source of the uncertainty and the magnitude of its potential adverse effect on profitability. Profitability refers to both accounting and mark-to-market measures.

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RISK MANAGEMENT IN BANKING

Credit Interest rate Market Banking Risks Liquidity Operational Foreign exchange

Other risks: country risk, settlement risk, performance risk ...

FIGURE 2.1

Main bank risks

This book focuses on financial risks, or risks related to the market movements or the economic changes of the environment. Market risk is relatively easy to quantify thanks to the large volume of price observations. Credit risk ‘looked like’ a ‘commercial’ risk because it is business-driven. Innovation changed this view. Since credit risk is a major risk, the regulators insisted on continuously improving its measurement in order to quantify the amount of capital that banks should hold. Credit risk and the principle of diversification are as old as banks are. It sounds like a paradox that major recent innovations focus on this old and well-known risk. Operational risk also attracts attention1 . It covers all organizational malfunctioning, of which consequences can be highly important and, sometimes, fatal to an institution. Following the regulators’ focus on valuing risk as a capital charge, model designers developed risk models aimed at the quantification of potential losses arising from each source of risk. The central concept of such models is the well-known ‘Value at Risk’ (VaR). Briefly stated, a VaR is a potential loss due to a defined risk. The issue is how to define a potential loss, given that the loss can be as high as the current portfolio value. Of course, the probability of such an event is zero. In order to define the potential adverse deviation of value, or loss, a methodology is required to identify what could be a ‘maximum’ deviation. Under the VaR methodology, the worst-case loss is a ‘maximum’ bound not exceeded in more than a preset fraction (for instance 1%) of all possible states over a defined period. Models help to determine a market risk VaR and a credit risk VaR. The VaR concept also extends to other risks. Subsequent developments explain how to move from the following definitions of risk to VaR modelling and measuring. 1 The

New Basel Accord of January 2001 requires a capital charge against operational risk.

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13

CREDIT RISK Credit risk is the first of all risks in terms of importance. Default risk, a major source of loss, is the risk that customers default, meaning that they fail to comply with their obligations to service debt. Default triggers a total or partial loss of any amount lent to the counterparty. Credit risk is also the risk of a decline in the credit standing of an obligor of the issuer of a bond or stock. Such deterioration does not imply default, but it does imply that the probability of default increases. In the market universe, a deterioration of the credit standing of a borrower does materialize into a loss because it triggers an upward move of the required market yield to compensate the higher risk and triggers a value decline. ‘Issuer’ risk designates the obligors’ credit risk, to make it distinct from the specific risk of a particular issue, among several of the same issuer, depending on the nature of the instrument and its credit mitigants (seniority level and guarantees). The view of credit risk differs for the banking portfolio and the trading portfolio.

Banking Portfolio Credit risk is critical since the default of a small number of important customers can generate large losses, potentially leading to insolvency. There are various default events: delay in payment obligations; restructuring of debt obligations due to a major deterioration of the credit standing of the borrower; bankruptcies. Simple delinquencies, or payment delays, do not turn out as plain defaults, with a durable inability of lenders to face debt obligations. Many are resolved within a short period (say less than 3 months). Restructuring is very close to default because it results from the view that the borrower will not face payment obligations unless its funding structure changes. Plain defaults imply that the non-payment will be permanent. Bankruptcies, possibly liquidation of the firm or merging with an acquiring firm, are possible outcomes. They all trigger significant losses. Default means any situation other than a simple delinquency. Credit risk is difficult to quantify on an ‘ex ante’ basis, since we need an assessment of the likelihood of a default event and of the recoveries under default, which are contextdependent2 . In addition, banking portfolios benefit from diversification effects, which are much more difficult to capture because of the scarcity of data on interdependencies between default events of different borrowers are interdependent.

Trading Portfolio Capital markets value the credit risk of issuers and borrowers in prices. Unlike loans, the credit risk of traded debts is also indicated by the agencies’ ratings, assessing the quality of public debt issues, or through changes of the value of their stocks. Credit risk is also visible through credit spreads, the add-ons to the risk-free rate defining the required market risk yield on debts. The capability of trading market assets mitigates the credit risk since there is no need to hold these securities until the deterioration of credit risk materializes into effective losses. 2 Context

refers to all factors influencing loss under default, such as the outstanding balance of debt at default, the existence of guarantees, or the policy of all stakeholders with respect to existing debt.

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If the credit standing of the obligor declines, it is still possible to sell these instruments in the market at a lower value. The loss due to credit risk depends on the value of these instruments and their liquidity. If the default is unexpected, the loss is the difference between the pre- and post-default prices. The faculty of trading the assets limits the loss if sale occurs before default. The selling price depends on the market liquidity. Therefore, there is some interaction between credit risk and trading risk. For over-the-counter instruments, such as derivatives (swaps and options), whose development has been spectacular in the recent period, sale is not readily feasible. The bank faces the risk of losing the value of such instruments when it is positive. Since this value varies constantly with the market parameters, credit risk changes with market movements during the entire residual life of the instrument. Credit risk and market risk interact because these values depend on the market moves. Credit risk for traded instruments raises a number of conceptual and practical difficulties. What is the value subject to loss, or exposure, in future periods? Does the current price embed already the credit risk, since market prices normally anticipate future events, and to what extent? Will it be easy to sell these instruments when signs of deterioration get stronger, and at what discount from the current value since the market for such instruments might narrow when credit risk materializes? Will the bank hold these instruments longer than under normal conditions?

Measuring Credit Risk Even though procedures for dealing with credit risk have existed since banks started lending, credit risk measurement raises several issues. The major credit risk components are exposure, likelihood of default, or of a deterioration of the credit standing, and the recoveries under default. Scarcity of data makes the assessment of these components a challenge. Ratings are traditional measures of the credit quality of debts. Some major features of ratings systems are3 : • Ratings are ordinal or relative measures of risk rather than cardinal or absolute measures, such as default probability. • External ratings are those of rating agencies, Moody’s, Standard & Poor’s (S&P) and Fitch, to name the global ones. They apply to debt issues rather than issuers because various debt issues from the same issuer have different risks depending on seniority level and guarantees. Detailed rating scales of agencies have 20 levels, ignoring the near default rating levels. • By contrast, an issuer’s rating characterizes only the default probability of the issuer. • Banks use internal rating scales because most of their borrowers do not have publicly rated debt issues. Internal rating scales of banks are customized to banks’ requirements, and usually characterize both borrower’s risk and facility’s risk. • There are various types of ratings. Ratings characterize sovereign risk, the risk of country debt and the risk of the local currency. Ratings are also either short-term

3 Chapters

35 and 36 detail further the specifications of rating systems.

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15

or long-term. There are various types of country-related ratings: sovereign ratings of government sponsored borrowers; ratings of currencies; ratings of foreign currencies held locally; ratings of transfer risk, the risk of being unable to transfer cash out of the country. • Because ratings are ordinal measures of credit risk, they are not sufficient to value credit risk. Moreover, ratings apply only to individual debts of borrowers, and they do not address the bank’s portfolio risk, which benefits from diversification effects. Portfolio models show that portfolio risk varies across banks depending on the number of borrowers, the discrepancies in size between exposures and the extent of diversification among types of borrowers, industries and countries. The portfolio credit risk is critical in terms of potential losses and, therefore, for finding out how much capital is required to absorb such losses. Modelling default probability directly with credit risk models remained a major challenge, not addressed until recent years. A second challenge of credit risk measurement is capturing portfolio effects. Due to the scarcity of data in the case of credit risk, quantifying the diversification effect sounds like a formidable challenge. It requires assessing the joint likelihood of default for any pair of borrowers, which gets higher if their individual risks correlate. Given its importance for banks, it is not surprising that banks, regulators and model designers made a lot of effort to better identify the relevant inputs for valuing credit risk and model diversification effects with ‘portfolio models’. Accordingly, a large fraction of this book addresses credit risk modelling.

COUNTRY AND PERFORMANCE RISKS Credit risk is the risk of loss due to a deterioration of the credit standing of a borrower. Some risks are close to credit risk, but distinct, such as country risk and performance risk.

Country Risk Country risk is, loosely speaking, the risk of a ‘crisis’ in a country. There are many risks related to local crises, including: • Sovereign risk, which is the risk of default of sovereign issuers, such as central banks or government sponsored banks. The risk of default often refers to that of debt restructuring for countries. • A deterioration of the economic conditions. This might lead to a deterioration of the credit standing of local obligors, beyond what it should be under normal conditions. Indeed, firms’ default frequencies increase when economic conditions deteriorate. • A deterioration of the value of the local foreign currency in terms of the bank’s base currency.

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• The impossibility of transferring funds from the country, either because there are legal restrictions imposed locally or because the currency is not convertible any more. Convertibility or transfer risks are common and restrictive definitions of country risks. • A market crisis triggering large losses for those holding exposures in the local markets. A common practice stipulates that country risk is a floor for the risk of a local borrower, or equivalently, that the country rating caps local borrowers’ ratings. In general, country ratings serve as benchmarks for corporate and banking entities. The rationale is that, if transfers become impossible, the risk materializes for all corporates in the country. There are debates around such rules, since the intrinsic credit standing of a borrower is not necessarily lower than on that of the country.

Performance Risk Performance risk exists when the transaction risk depends more on how the borrower performs for specific projects or operations than on its overall credit standing. Performance risk appears notably when dealing with commodities. As long as delivery of commodities occurs, what the borrower does has little importance. Performance risk is ‘transactional’ because it relates to a specific transaction. Moreover, commodities shift from one owner to another during transportation. The lender is at risk with each one of them sequentially. Risk remains more transaction-related than related to the various owners because the commodity value backs the transaction. Sometimes, oil is a major export, which becomes even more strategic in the event of an economic crisis, making the financing of the commodity immune to country risk. In fact, a country risk increase has the paradoxical effect of decreasing the risk of the transaction because exports improve the country credit standing.

LIQUIDITY RISK Liquidity risk refers to multiple dimensions: inability to raise funds at normal cost; market liquidity risk; asset liquidity risk. Funding risk depends on how risky the market perceives the issuer and its funding policy to be. An institution coming to the market with unexpected and frequent needs for funds sends negative signals, which might restrict the willingness to lend to this institution. The cost of funds also depends on the bank’s credit standing. If the perception of the credit standing deteriorates, funding becomes more costly. The problem extends beyond pure liquidity issues. The cost of funding is a critical profitability driver. The credit standing of the bank influences this cost, making the rating a critical factor for a bank. In addition, the rating drives the ability to do business with other financial institutions and to attract investors because many follow some minimum rating guidelines to invest and lend. The liquidity of the market relates to liquidity crunches because of lack of volume. Prices become highly volatile, sometimes embedding high discounts from par, when counterparties are unwilling to trade. Funding risk materializes as a much higher cost

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17

of funds, although the cause lies more with the market than the specific bank. Market liquidity risk materializes as an impaired ability to raise money at a reasonable cost. Asset liquidity risk results from lack of liquidity related to the nature of assets rather than to the market liquidity. Holding a pool of liquid assets acts as a cushion against fluctuating market liquidity, because liquid assets allow meeting short-term obligations without recourse to external funding. This is the rationale for banks to hold a sufficient fraction of their balance sheet of liquid assets, which is a regulatory rule. The ‘liquidity ratio’ of banks makes it mandatory to hold more short-term assets than short-term liabilities, in order to meet short-run obligations. In order to fulfil this role, liquid assets should mature in the short-term because market prices of long-term assets are more volatile4 , possibly triggering substantial losses in the event of a sale. Moreover, some assets are less tradable than others, because their trading volume is narrow. Some stocks trade less than others do, and exotic products might not trade easily because of their high level of customization, possibly resulting in depressed prices. In such cases, any sale might trigger price declines, so that the proceeds from a one-shot or a progressive sale become uncertain and generate losses. To a certain extent, funding risk interacts with market liquidity and asset liquidity because the inability to face payment obligations triggers sales of assets, possibly at depressed prices. Liquidity risk might become a major risk for the banking portfolio. Extreme lack of liquidity results in bankruptcy, making liquidity risk a fatal risk. However, extreme conditions are often the outcome of other risks. Important unexpected losses raise doubts with respect to the future of the organization and liquidity issues. When a commercial bank gets into trouble, depositors ‘run’ to get their money back. Lenders refrain from further lending to the troubled institution. Massive withdrawals of funds or the closing of credit lines by other institutions are direct outcomes of such situations. A brutal liquidity crisis follows, which might end up in bankruptcy. In what follows, we adopt an Asset–Liability Management (ALM) view of the liquidity situation. This restricts liquidity risk to bank-specific factors other than the credit risk of the bank and the market liquidity. The time profiles of projected uses and sources of funds, and their ‘gaps’ or liquidity mismatches, capture the liquidity position of a bank. The purpose of debt management is to manage these future liquidity gaps within acceptable limits, given the market perception of the bank. This perspective does not fully address liquidity risk, and the market risk definitions below address this only partially. Liquidity risk, in terms of market liquidity or asset liquidity, remains a major issue that current techniques do not address fully. Practices rely on empirical and continuous observations of market liquidity, while liquidity risk models remain too theoretical to allow instrumental applications. This is presumably a field necessitating increased modelling research and improvement of practices.

INTEREST RATE RISK The interest rate risk is the risk of a decline in earnings due to the movements of interest rates. 4 Long-term

interest-bearing assets are more sensitive to interest rate movements. See the duration concept used for capturing the sensitivity of the mark-to-market value of the balance sheet in Chapter 22.

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Most of the items of banks’ balance sheets generate revenues and costs that are interest rate-driven. Since interest rates are unstable, so are earnings. Any one who lends or borrows is subject to interest rate risk. The lender earning a variable rate has the risk of seeing revenues reduced by a decline in interest rates. The borrower paying a variable rate bears higher costs when interest rates increase. Both positions are risky since they generate revenues or costs indexed to market rates. The other side of the coin is that interest rate exposure generates chances of gains as well. There are various and complex indexations on market rates. Variable rate loans have rates periodically reset using some market rate references. In addition, any transaction reaching maturity and renewed will stick to the future and uncertain market conditions. Hence, fixed rates become variable at maturity for loan renewals and variable rates remain fixed between two reset dates. In addition, the period between two rate resets is not necessarily constant. For instance, the prime rate of banks remains fixed between two resets, over periods of varying lengths, even though market rates constantly move. The same happens for the rates of special savings deposits, when they are subject to legal and fiscal rules. This variety makes the measure of interest rate sensitivity of assets and liabilities to market rates more complex. Implicit options in banking products are another source of interest rate risk. A wellknown case is that of the prepayment of loans that carry a fixed rate. A person who borrows can always repay the loan and borrow at a new rate, a right that he or she will exercise when interest rates decline substantially. Deposits carry options as well, since deposit holders transfer funds to term deposits earning interest revenues when interest rates increase. Optional risks are ‘indirect’ interest rate risks. They do not arise directly and only from a change in interest rate. They also result from the behaviour of customers, such as geographic mobility or the sale of their homes to get back cash. Economically, fixed rate borrowers compare the benefits and the costs of exercising options embedded in banking products, and make a choice depending on market conditions. Given the importance of those products in the balance sheets of banks, optional risk is far from negligible. Measuring the option risk is more difficult than measuring the usual risk which arises from simple indexation to market rates. Section 5 of this book details related techniques.

MARKET RISK Market risk is the risk of adverse deviations of the mark-to-market value of the trading portfolio, due to market movements, during the period required to liquidate the transactions. The period of liquidation is critical to assess such adverse deviations. If it gets longer, so do the deviations from the current market value. Earnings for the market portfolio are Profit and Loss (P&L) arising from transactions. The P&L between two dates is the variation of the market value. Any decline in value results in a market loss. The potential worst-case loss is higher when the holding period gets longer because market volatility tends to increase over longer horizons. However, it is possible to liquidate tradable instruments or to hedge their future changes of value at any time. This is the rationale for limiting market risk to the liquidation period. In general, the liquidation period varies with the type of instruments. It could be short (1 day) for foreign exchange and much longer for ‘exotic’ derivatives. The regulators

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provide rules to set the liquidation period. They use as reference a 10-day liquidation period and impose a multiple over banks’ internal measures of market value potential losses (see Chapter 3). Liquidation involves asset and market liquidity risks. Price volatility is not the same in high-liquidity and poor-liquidity situations. When liquidity is high, the adverse deviations of prices are much lower than in a poor-liquidity environment, within a given horizon. ‘Pure’ market risk, generated by changes of market parameters (interest rates, equity indexes, exchange rates), differs from market liquidity risk. This interaction raises important issues. What is the ‘normal’ volatility of market parameters under fair liquidity situations? What could it become under poorer liquidity situations? How sensitive are the prices to liquidity crises? The liquidity issue becomes critical in emerging markets. Prices in emerging markets often diverge considerably from a theoretical ‘fair value’. Market risk does not refer to market losses due to causes other than market movements, loosely defined as inclusive of liquidity risk. Any deficiency in the monitoring of the market portfolio might result in market values deviating by any magnitude until liquidation finally occurs. In the meantime, the potential deviations can exceed by far any deviation that could occur within a short liquidation period. This risk is an operational risk, not a market risk5 . In order to define the potential adverse deviation, a methodology is required to identify what could be a ‘maximum’ adverse deviation of the portfolio market value. This is the VaR methodology. The market risk VaR technique aims at capturing downside deviations of prices during a preset period for liquidating assets, considering the changes in the market parameters. Controlling market risk means keeping the variations of the value of a given portfolio within given boundary values through actions on limits, which are upper bounds imposed on risks, and hedging for isolating the portfolio from the uncontrollable market movements.

FOREIGN EXCHANGE RISK The currency risk is that of incurring losses due to changes in the exchange rates. Variations in earnings result from the indexation of revenues and charges to exchange rates, or of changes of the values of assets and liabilities denominated in foreign currencies. Foreign exchange risk is a classical field of international finance, so that we can rely on traditional techniques in this book, without expanding them. For the banking portfolio, foreign exchange risk relates to ALM. Multi-currency ALM uses similar techniques for each local currency. Classical hedging instruments accommodate both interest rate and exchange rate risk. For market transactions, foreign exchange rates are a subset of market parameters, so that techniques applying to other market parameters apply as well. The conversion risk resuls from the need to convert all foreign currency-denominated transactions into a base reference currency. This risk does exist, beyond accounting conversion in a single currency, if the capital base that protects the bank from losses 5 An

example is the failure of Baring Brothers, due to deficiencies in the control of the risk positions (see Leeson, 1996).

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is in local currency. A credit loss in a foreign country might result in magnified losses in local currency if the local currency depreciates relative to the currency of the foreign exposure.

SOLVENCY RISK Solvency risk is the risk of being unable to absorb losses, generated by all types of risks, with the available capital. It differs from bankruptcy risk resulting from defaulting on debt obligations and inability to raise funds for meeting such obligations. Solvency risk is equivalent to the default risk of the bank. Solvency is a joint outcome of available capital and of all risks. The basic principle of ‘capital adequacy’, promoted by regulators, is to define what level of capital allows a bank to sustain the potential losses arising from all current risks and complying with an acceptable solvency level. The capital adequacy principle follows the major orientations of risk management. The implementation of this principle requires: • Valuing all risks to make them comparable to the capital base of a bank. • Adjusting capital to a level matching the valuation of risks, which implies defining a ‘tolerance’ level for the risk that losses exceed this amount, a risk that should remain very low to be acceptable. Meeting these specifications drives the regulators’ philosophy and prudent rules. The VaR concept addresses these issues directly by providing potential loss values for various confidence levels (probability that actual losses exceed an upper bound).

OPERATIONAL RISK Operational risks are those of malfunctions of the information system, reporting systems, internal risk-monitoring rules and internal procedures designed to take timely corrective actions, or the compliance with internal risk policy rules. The New Basel Accord of January 2001 defines operational risk as ‘the risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events’. In the absence of efficient tracking and reporting of risks, some important risks remain ignored, do not trigger any corrective action and can result in disastrous consequences. In essence, operational risk is an ‘event risk’. There is a wide range of events potentially triggering losses6 . The very first step for addressing operational risk is to set up a common classification of events that should serve as a receptacle for data gathering processes on event frequencies and costs. Such taxonomy is still flexible and industry standards will emerge in the future. What follows is a tentative classification. Operational risks appear at different levels: 6 See

Marshall (2001) for a comprehensive view of operational risk and how to handle it for assessing potential losses.

BANKING RISKS

• • • •

21

People. Processes. Technical. Information technology.

‘People’ risk designates human errors, lack of expertise and fraud, including lack of compliance with existing procedures and policies. Process risk scope includes: • Inadequate procedures and controls for reporting, monitoring and decision-making. • Inadequate procedures on processing information, such as errors in booking transactions and failure to scrutinize legal documentation. • Organizational deficiencies. • Risk surveillance and excess limits: management deficiencies in risk monitoring, such as not providing the right incentives to report risks, or not abiding by the procedures and policies in force. • Errors in the recording process of transactions. • The technical deficiencies of the information system or the risk measures. Technical risks relate to model errors, implementation and the absence of adequate tools for measuring risks. Technology risks relate to deficiencies of the information system and system failure. For operational risks, there are sources of historical data on various incidents and their costs, which serve to measure the number of incidents and the direct losses attached to such incidents. Beyond external statistics, other proxy sources on operational events are expert judgments, questioning local managers on possible events and what would be their implications, pooling data from similar institutions and insurance costs that should relate to event frequencies and costs. The general principle for addressing operational risk measurement is to assess the likelihood and cost of adverse events. The practical difficulties lie in agreeing on a common classification of events and on the data gathering process, with several potential sources of event frequencies and costs. The data gathering phase is the first stage, followed by data analysis and statistical techniques. They help in finding correlations and drivers of risks. For example, business volume might make some events more frequent, while others depend on different factors. The process ends up with some estimate of worst-case losses due to event risks.

MODEL RISK Model risk is significant in the market universe, which traditionally makes relatively intensive usage of models for pricing purposes. Model risk is growing more important, with the extension of modelling techniques to other risks, notably credit risk, where scarcity of data remains a major obstacle for testing the reliability of inputs and models. This book details many modelling approaches, making model risk self-explanatory.

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Model risk materializes, for instance, as gaps between predicted values of variables, such as the VaR, and actual values observed from experience. Pricing models used for market instruments predict prices, which are readily comparable to observed prices. Often, modelled prices do not track actual prices accurately. When P&L calculations for the trading portfolio rely on ‘pricers’, there are ‘P&L leaks’ due to models. Such ‘P&L leaks’ represent the dollar value of model risk for ‘pricers’. Unfortunately, it is much easier to track gaps between predicted prices and actual prices than between modelled worst-case losses and actual large losses. The difficulty is even greater for credit risk, since major credit risk events remain too scarce. The sources of model risk, however, relate to common sense. Models are subject to misspecifications, because they ignore some parameters for practical reasons. Model implementation suffers from errors of statistical techniques, lack of observable data for obtaining reliable fits (credit risk) and judgmental choices on dealing with ‘outliers’, those observations that models fail to capture with reasonable accuracy. Credit risk models of interdependencies between individual defaults of borrowers rely on inferring such relations from ‘modelled’ events rather than observable events, because default data remains scarce. Consequently, measuring errors is not yet feasible until credit risk data becomes more reliable than it is currently. This is an objective of the New Basel Accord, from January 2001 (Chapter 3), making the data gathering process a key element in implementing the advanced techniques for measuring credit risk capital.

SECTION 2 Risk Regulations

3 Banking Regulations

Regulations have several goals: improving the safety of the banking industry, by imposing capital requirements in line with banks’ risks; levelling the competitive playing field of banks through setting common benchmarks for all players; promoting sound business and supervisory practices. Regulations have a decisive impact on risk management. The regulatory framework sets up the constraints and guidelines that inspire risk management practices, and stimulates the development and enhancement of the internal risk models and processes of banks. Regulations promote better definitions of risks, and create incentives for developing better methodologies for measuring risks. They impose recognition of the core concept of the capital adequacy principle and of ‘risk-based capital’, stating that banks’ capital should be in line with risks, and that defining capital requirements implies a quantitative assessment of risks. Regulations imposing capital charge against risks are a strong incentive to improve risk measures and controls. They set minimum standards for sound practices, while the banking industry works on improving the risk measures and internal models for meeting their own goals in terms of best practices of risk management. Starting from crude estimates of capital charges, with the initial so-called ‘Cooke ratio’, the regulators evolved towards increasingly ‘risk-sensitive’ capital requirements. The Existing Accord on credit risk dates from 1988. The Accord Amendment for market risk (1996) and the New Basel Accord (2001) provide very significant enhancements of risk measures. The existing accord imposed capital charge against credit risk. The amendment provided a standardized approach for dealing with market risk, and offered the opportunity to use internal models, subject to validation by the supervisory bodies, allowing banks to use their own models for assessing the market risk capital. The new accord imposes a higher differentiation of credit risk based on additional risk inputs characterizing banks’ facilities. By doing so, it paves the way towards internal credit risk modelling, already instrumental in major institutions.

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Regulations are inspired by economic principles, expanded on in subsequent chapters. However, regulations tend to strike a compromise between moving towards economic measures while being practical enough to be applicable by all players. This chapter describes the main features of regulations, and discusses the underlying economic rationale, as a transition towards subsequent developments. It does not provide a comprehensive and detailed presentation of regulations. Details are easily available from public documents on the Bank of International Settlements (BIS) site, and there is no need to replicate them. A number of dilemmas inspired the deregulation process in the eighties, followed by the reregulation wave. An ideal scheme would combine a minimal insurance mechanism without providing any incentive for taking more risk, and still foster competition among institutions. Old schemes failed to ensure system safety, as spectacular failures of the eighties demonstrate. New regulations make the ‘capital adequacy’ principle a central foundation, starting from the end of the eighties up to the current period. Regulatory requirements use minimum forfeit amounts of capital held against risks to ensure that capital is high enough to sustain future losses materializing from current risks. The main issue is to assess potential losses arising from current measures. Regulations provide sets of forfeits and rules defining the capital base as a function of banks’ exposures. The risk modelling approach addresses the same issue, but specifies all inputs and models necessary to obtain the best feasible economic estimates of the required capital. Regulations lag behind in sophistication because they need to set up rules that apply to all. As a result, regulatory capital does not coincide with modelled economic capital. The notable exception is market risk, since the 1996 Accord Amendment for market risk that allowed usage of internal models, subject to validation and supervision by the regulatory bodies. The New Accord of January 2001 progresses significantly towards more risksensitive measures of credit risk at the cost of an additional complexity. It proposes three approaches for defining capital requirements: the simplest is the ‘standardized’ approach, while the ‘foundation’ and the ‘advanced’ approach allow further refinements of capital requirements without, however, accepting internal models for credit risk. It has become obvious, when looking at regulatory accords, that the capital adequacy concept imposes continuous progress for quantifying potential losses, for measuring capital charges, making risk valuation a major goal and a major challenge for the industry. The regulatory system has evolved through the years. The 1996 Accord Amendment refined market risk measures, allowed internal models and is still in force. Capital requirements still do not apply to interest rate risk, although periodical enhanced reports are mandatory and the regulatory bodies can impose corrective actions. The last stage of the process is the New Basel Accord, published in early 2001, focusing on capital requirement definition (Pillar 1), with major enhancements to credit risk measures, a first coverage of operational risk, disclosure, supervisory review process (Pillar 2) and market discipline (Pillar 3). Simultaneously, it enhances some features of the 1988 Existing Accord, currently under implementation. The New Basel Accord target implementation date is early 2004. The first section discusses the basic issues and dilemmas that inspired the new regulation waves. The second section introduces the central concept of ‘capital adequacy’, which plays a pivotal role for regulations, quantitative risk analytics and risk management. The

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27

third section summarizes the rules of the Current Accord, which still applies, and the 1996 Amendment. Finally, the fourth section details the New Basel Accord in its January 2001 format, subject to revisions from banking industry feedback in mid-2001.

REGULATORY ISSUES The source of regulations lies in the differences between the objectives of banks and those of the regulatory authorities. Expected profitability is a major incentive for taking risks. Individual banks’ risks create ‘systemic risk’, the risk that the whole banking system fails. Systemic risk results from the high interrelations between banks through mutual lending and borrowing commitments. The failure of a single institution generates a risk of failure for all banks that have ongoing commitments with the defaulting bank. Systemic risk is a major challenge for the regulator. Individual institutions are more concerned with their own risk. The regulators tend to focus on major goals: • The risk of the whole system, or ‘systemic risk’, leading to pre-emptive actions against bank failures by the regulators, all inspired by the most important capital adequacy to risk principle. • Promoting a level playing field, without distortions in terms of unfair competitive advantages. • Promoting sound practices, contributing to the financial system safety. The regulators face several dilemmas when attempting to control risks. Regulation and competition conflict, since many regulations restrict the operations of banks. New rules may create unpredictable behaviour to turn around the associated constraints. For this reason, regulators avoid making brutal changes in the environment that would generate other uncertainties.

The Need for Regulation Risk taking is a normal behaviour of financial institutions, given that risk and expected return are so tightly interrelated. Because of the protection of bank depositors that exists in most countries, banks also benefit from ‘quasi-free’ insurance, since depositors cannot impose a real market discipline on banks. The banking system is subject to ‘moral hazard’: enjoying risk protection is an incentive for taking more risks because of the absence of penalties. Sometimes, adverse conditions are incentives to maximize risks. When banks face serious difficulties, the barriers that limit risks disappear. In such situations, and in the absence of aggressive behaviour, failure becomes almost unavoidable. By taking additional risks, banks maximize the chances of survival. The higher the risk, the wider the range of possible outcomes, including favourable ones. At the same time, the losses of shareholders and managers do not increase because of limited liability. In the absence of real downside risk, it becomes rational to increase risk. This is the classical attitude of a

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call option holder. He gets the upside of the bet with a limited downside. The potential for upside gain without the downside risk encourages risk taking because it maximizes the expected gains.

The Dilemmas of the Regulator A number of factors helped stabilize the banking environment in the seventies. Strong and constraining regulations weighed heavily on the banks’ management. Commercial banking meant essentially collecting resources and lending. Limited competition facilitated a fair and stable profitability. Concerns for the safety of the industry and the control of its money creation power were the main priorities for regulators. The rules limited the scope of the operations of the various credit institutions, and limited their risks as well. There were low incentives for change and competition. The regulators faced a safety–competition trade-off in addition to the negative side effects of risk insurance. In the period preceding deregulation, too many regulations segmented the banking industry, restraining competition. Deregulation increased competition between players unprepared by their past experiences, thereby resulting in increasing risks for the system. Reregulation aimed at setting up a regulatory framework reconciling risk control and fair competition. Regulation and Competition

Regulations limit the scope of operations of the various types of financial institutions, thereby interfering directly with free competition. Examples of inconsistency between competition and regulations have grown more and more numerous. A well-known example, in the United States in the early eighties, was the unfair competition between commercial banks, subject to regulation Q, imposing a ceiling on the interest paid on deposits, and investment bankers offering money market funds through interestearning products. Similar difficulties appear whenever rules enforce the segmentation of industry between those who operate in financial markets and others, savings and banking institutions, etc. The unfair competition stimulated the disappearance of such rules and the barriers to competition were progressively lifted. This old dilemma led to the deregulation of the banking industry. The seventies and the eighties were the periods of the first drastic waves of change in the industry. The disappearance of old rules created a vacancy. The regulators could not rely any longer on rules that segmented the industry as barriers to risk-taking behaviour. They started redefining new rules that could ensure the safety of the banking industry. The BIS (Bank for International Settlements) in Basel defined these new regulations, and national regulators relayed them for implementation within the national environments. Deregulation drastically widened the range of products and services offered by banks. Most credit institutions diversified their operations out of their original businesses. Moreover, the pace of creation of new products remained constantly high, especially for those acting in the financial markets, such as derivatives or futures. The active research for new

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market opportunities and products stimulated the growth of fields other than intermediation. Value-added services, such as advisory, structured transactions, asset acquisition, Leveraged Buy-Out (LBO), project finance, securitization of mortgages and credit card debts, derivatives and off-balance sheet operations, developed at a rapid pace. The banks entered new business fields and faced new risks. The market share of bank lending decreased with the development of capital markets and the competition within existing market shares rose abruptly. Those waves of change generated risks. Risks increased because of new competition, product innovations, a shift from commercial banking to capital markets, increased market volatility and the disappearance of old barriers. This was a radical change in the banking industry. Lifting existing constraints stimulates new competition and increases risks and failures for those players who are less ready to enter new businesses. Deregulation implies that players can freely enter new markets. The process necessitates an orderly and gradual progress to avoid bank management disruption. This is still an ongoing process, notably in the United States and Japan, where the Glass–Steagall Act and the Article 65 enforced a separation between commercial banking and the capital markets. For those countries where universal banking exists, the transition is less drastic, but leads to a significant restructuring of the industry. It is not surprising that risk management has emerged with such strong force at the time of these waves of transformations. Risk Control versus Risk Insurance

Risk control is pre-emptive, while insurance is passive, after the fact. The regulatory framework aims at making it an obligation to have a capital level matching risks. This is a pre-emptive policy. However, after the fact, there is still a need to limit the consequences of failures. Losses of depositors are the old consequences of bank failures. The consequences of bank failures are that they potentially trigger systemic risk, because of banks’ interdependencies, such as the recent LTCM collapse. Hence, there is a need for afterthe-fact regulatory action. If such actions rely on an insurance-like scheme, such as the insurance deposit scheme, it can foster risk-taking behaviour. Until reregulation through capital changes came into force, old regulations created insurance-based incentives for taking risks. Any rule that limits the adverse consequences of risk-taking behaviour is an incentive for risk taking rather than risk reduction. Regulators need to minimize adverse effects of failures for the whole system without at the same time encouraging the risk-taking behaviour. The ideal solution would be to control the risks without insuring them. In practice, there is no such ideal solution. The deposit insurance scheme protects depositors against bank failures. Nevertheless, it does generate the adverse side effect of encouraging risk taking. Theoretically, depositors should monitor the bank behaviour, just as any lender does. In practice, if depositors’ money were at risk, the functioning of the banking system would greatly deteriorate. Any sign of increased risks would trigger withdrawals of funds that would maximize the difficulties, possibly leading to failures. Such an adverse effect alone is a good reason to insure depositors, at least largely, against bank failure.

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The new regulation focuses on pre-emptive (ex ante) actions, while still enforcing after-the-fact (ex post) corrective actions.

CAPITAL ADEQUACY A number of rules, aimed at limiting risks in a simple manner, have been in force for a long time. For instance, several ratios are subject to minimum values. The liquidity ratio imposes that short-term assets be greater than short-term liabilities. Individual exposures to single borrowers are subject to caps. The main ‘pillar’ of the regulations is ‘capital adequacy’: by enforcing a capital level in line with risks, regulators focus on pre-emptive actions limiting the risk of failure. Guidelines are defined by a group of regulators meeting in Basel at the BIS, hence the name ‘Basel’ Accord. The process attempts to reach a consensus on the feasibility of implementing new complex guidelines by interacting with the industry. Basel guidelines are subject to some implementation variations from one country to another, according to the view of local supervisors. The first implemented accord focused on credit risk, with the famous ‘Cooke ratio’. The Cooke ratio sets up the minimum required capital as a fixed percentage of assets weighted according to their nature in 1988. The scope of regulations extended progressively later. The extension to market risk, with the 1996 Amendment, was a major step. The New Basel Accord of January 2001 considerably enhances the old credit risk regulations. The schedule of successive accords is as follows: • • • • • • •

1988 Current Accord published. 1996 Market Risk Amendment allowing usage of internal models. 1999 First Consultative Package on the New Accord. January 2001 Second Consultative Package. End-May 2001 Deadline for comments. End-2001 Publication of the New Accord. 2004 Implementation of the New Basel Capital Accord.

The next subsections refer to the 1988 ‘Current Accord’ plus the market risk capital regulations. The last section expands the ‘New Accord’ in its current ‘consultative’ form1 , which builds on the ‘Existing Accord’ on credit risk, while making capital requirements more risk-sensitive.

Risk-based Capital Regulations The capital base is not limited to equity plus retained earnings. It includes any debt subordinated to other commitments by the bank. Equity represents at least 50% of the total capital base for credit risk. Equity is also the ‘Tier 1’ of capital or the ‘core capital’. 1 The

consultation with industry expires by mid-2001, and a final version should follow.

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The Cooke ratio stipulates that the capital base should be at least 8% of weighted assets. The weights depend on the credit quality of the borrowers, as described below. For market risk, the rules are more complex because the regulations aim at a greater accuracy in capturing the economics of market risk, taking advantage of the widely available information on market parameters and prices. The 1996 Amendment for market risk opened the door to internal models that have the capability to value capital requirements with more accurate and sophisticated tools than simple forfeits. Regulations allow the usage of internal bank models for market risk only, subject to validation by supervisors. Market risk models provide ‘Value at Risk’ (VaR)-based measurements of economic capital, complying with specific guidelines for implementing models set by Basel, which substitute for regulatory forfeits. The New Accord of January 2001 paves the way for modelling credit risk, but recognizes that it is still of major importance to lay down more reliable foundations for inputs to credit risk models, rather than allowing banks to use model credit risk with unreliable inputs, mainly due to scarce information on credit risk.

The Implications of Capital Requirements Before detailing the regulations, the implications of risk-based capital deserve attention. Traditionally, capital represents a very small fraction of total assets of banks, especially when comparing the minimum requirements to similar ratios of non-financial institutions. A capital percentage of 8% of assets is equivalent to a leverage ratio (debt/equity ratio) of 92/8 = 11.5. This leverage ratio would be unsustainable with non-financial institutions, since borrowers would consider it as impairing too much the repayment ability and causing an increase in the bankruptcy risk beyond acceptable levels. The high leverage of banking institutions results from a number of factors. Economically, the discipline imposed by borrowers does not apply to depositors who benefit from deposit insurance programmes in all countries. Smooth operations of banks require easy and immediate access to financial markets, so that funding is not a problem, as long as the perceived risk by potential lenders remains acceptable. In addition, bank ratings by specialized agencies make risks visible and explicit.

The Function of Capital

The theoretical reason for holding capital is that it should provide protection against unexpected losses. Without capital, banks will fail at the first dollar of loss not covered by provisions. It is possible to compare regulatory forfeits to historical default rates for a first assessment of the magnitude of the safety cushion against losses embedded in regulatory capital. Default rates of corporate borrowers above 1% are speculative. Investment grade borrowers have yearly default rates between 0% and 0.1%. At the other extreme of the risk scale, corporate borrowers (small businesses) can have default rates ranging from 3% to 16% or more across the economic cycle. The 8% ratio of capital to weighted assets seems too high if compared to average observed default rates under good conditions, and perhaps too low with portfolios of high-risk borrowers when a downturn in economic conditions occurs. However, the 8%

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ratio does not aim at protecting banks against average losses. Provisions and reserves should, theoretically, take care of those. Rather, it aims at a protection against deviations from average. The ratio means that capital would cover deviations of losses, in excess of loan loss provisions, up to 8%, given the average portfolio structure of banks. However, without modelling potential portfolio losses, it is impossible to say whether this capital in excess of provisions is very or moderately conservative. Portfolio models provide ‘loss distributions’ for credit risk, the combinations of loss values plus the probability that losses hit each level, and allow us to see how likely various loss levels are, for each specific portfolio. In addition, loss provisioning might not be conservative enough, so that capital should also be sufficient to absorb a fraction of average losses in addition to loss in excess of average. Legal reserves restrain banks from economic provisioning, as well as window dressing or the need to maximize retained earnings that feed into capital. Economic provisioning designates a system by which provisions follow ex ante estimates of what the future average losses could be, while actual provisions are more ex post, after the fact. Still, regulatory capital is not a substitute for provisions. The Dilemma between Risk Controlling and Risk Taking

There is a trade-off between risk taking and risk controlling. Available capital puts a limit on risk taking which, in turn, limits the ability to develop business. This generates a conflict between business goals, in terms of volume and risks, and the limits resulting from available capital. The available capital might not suffice to sustain new business developments. Capital availability could result in tighter limits for account officers and traders, and for the volume of operations. On the other hand, flexible constraints allow banks to take more risks, which leads to an increase in risk for the whole system. Striking the right balance between too much risk control and not enough is not an easy task. Risk-based Capital and Growth

The capital ratio sets the minimum value of capital, given the volume of operations and their risks. The constraint might require raising new equity, liquidation of assets or risk reduction actions. Raising additional capital requires that the profitability of shareholders is in line with their expectations. When funding capital growth through retained earnings only, the profitability should be high enough for the capital to grow in line with the requirements. These are two distinct constraints. A common proxy for the minimum required return of shareholders is an accounting Return On Equity (ROE) of 15% after tax, or 25% before tax. When the target is funding capital, the minimum accounting ROE should be in line with this funding constraint. The higher the growth of the volume of operations, the higher the increase in required capital (if the ‘average’ risk level remains unchanged). Profitability limits the sustainable growth if outside sources of capital are not used. The ROE becomes equal to the growth rate of

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capital if no dividends are paid2 . Therefore, any growth above ROE is not sustainable when funding through retained earnings only, under constant average risk. The first implication of capital requirements is that the ROE caps the growth of banks when they rely on retained earnings only. The second implication is that capital should provide the required market return to stockholders given the risk of the bank’s stocks. Sometimes, the bank growth is too high to be sustainable given capital requirements. Sometimes, the required compensation to shareholders is the ‘effective’ constraint, while funding capital growth internally is not3 . The only options available to escape the capital constraints are to liquidate assets or to reduce risks. Direct sales of assets or securitizations serve this purpose, among others. Direct loan sales become feasible through the emerging loan trading market. Chapter 60 discusses structured transactions as a tool for capital management.

THE ‘CURRENT ACCORD’ CAPITAL REGULATIONS The regulation is quite detailed in order to cover all specific situations. Three sections covering credit risk, market risk and interest rate risk summarize the main orientations.

The Cooke Ratio and Credit Risk The 1988 Accord requires internationally active banks in the G10 countries to hold capital equal to at least 8% of weighted assets. Banks should hold at least half of its measured capital in Tier 1 form. The Cooke ratio addresses credit risk. Tier 1 capital is subject to a minimum constraint of 3% of total assets. The calculation of the ratio uses asset weights for differentiating the capital load according to quality in terms of credit standing. The weight scale starts from zero, for commitments with public counterparties, up to 100% for private businesses. Other weights are: 20% for banks and municipalities within OECD countries; 50% for residential mortgage backed loans. Off-balance sheet outstanding balances are weighted at 50%, in combination with the above weights scale. The 1988 Accord requires a two-step approach whereby banks convert their off-balance sheet positions into a credit equivalent amount through a scale of conversion factors, which are then weighted according to the counterparty’s risk weighting. The factor is 100% for direct credit substitutes such as guarantees and decreases for less stringent commitments. 2 If

the leverage is constant, capital being a fixed fraction of total liabilities, the growth rate of accounting capital is also the growth rate of the total liabilities. Leverage is the debt/equity ratio, or L = D/E where D is debt, E is equity plus retained earnings and L is leverage. The ROE is the ratio of net income (NI) to equity and retained earnings, or ROE = N I /E. ROE is identical to the sustainable growth rate of capital since it is the ratio of potential additional capital (net income NI) to existing capital. Since L is constant, total liabilities are always equal to (1 + L) × E. Hence, they grow at the same rate as E, which is identical to ROE under assumptions of no dividends and no outside capital. 3 In fact, both constraints are equivalent when the growth rate is exactly 15%, if this figure is representative of both accounting return on equity and the required market return to shareholders.

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The major strength of the Cooke ratio is its simplicity. Its major drawbacks are: • The absence of differentiation between the different risks of private corporations. An 8% ratio applying for both a large corporation rated ‘Aa’ (an investment grade rating) and a small business does not make much sense economically. This is obvious when looking at the default rates associated with high and low ratings. The accord is not risk-sensitive enough. • In addition, short facilities have zero weights, while long facilities have a full capital load, creating arbitrage opportunities to reduce the capital load. This unequal treatment leads to artificial arbitrage by banks, such as renewing short loans rather than lending long. • There is no allowance for recoveries if a default occurs, even in cases where recoveries are likely, such as for cash or liquid securities collateral. • Summing arithmetically capital charges of all transactions does not capture diversification effects. In fact, there is an embedded diversification effect in the 8% ratio since it recognizes that the likelihood of losses exceeding more than 8% of weighted assets is very low. However, the same ratio applies to all portfolios, whatever their degree of diversification. The regulators recognized these facts and reviewed these issues, leading to the New Basel Accord detailed in a subsequent dedicated section.

Market Risk A significant amendment was enacted in 1995–6, when the Committee introduced a measure whereby trading positions in bonds, equities, foreign exchange and commodities were removed from the credit risk framework and given explicit capital charges related to the bank’s open position in each instrument. Capital requirements extended explicitly to market risk. The amendment made explicit the notions of banking book and trading book, defined capital charges for market risk and allowed banks to use Tier 3 capital in addition to the previous two tiers. Scope

Market risk is the risk of losses in on- and off-balance sheet positions arising from movements in market prices. The risks subject to this requirement are: • The risks pertaining to interest rate-related instruments and equities in the trading book. • Foreign exchange risk and commodities risk throughout the bank. The 1995 proposal introduced capital charges to be applied to the current market value of open positions (including derivative positions) in interest rate-related instruments and equities in banks’ trading books, and to banks’ total currency and commodities positions.

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The extension to market risk provides two alternative techniques for assessing capital charges. The ‘standardized approach’ allows measurement of the four risks: interest rate, equity position, foreign exchange and commodity risks, using sets of forfeits. Under the standardized approach, there are specific forfeits and rules for defining to which base they apply, allowing some offsetting effects within portfolios of traded instruments. Offsetting effects reduce the base for calculating the capital charge by using a net exposure rather than gross exposures. Full netting effects apply only to positions subject to an identical underlying risk or, equivalently, a zero basis risk4 . For instance, it is possible to offset opposite positions in the same stocks or the same interest rates. The second method allows banks to use risk measures derived from their own internal risk management models, subject to a number of conditions, related to qualitative standards of models and processes. The April 1995 proposal allowed banks to use new ‘Tier 3’ capital, essentially made up of short-term subordinated debt to meet their market risks. Tier 3 capital is subject to a number of conditions, such as being limited to market risk capital and being subject to a ‘lock-in clause’, stipulating that no such capital can be repaid if that payment results in a bank’s overall capital being lower than a minimum capital requirement. The Basel extension to market risk relies heavily on the principles of sound economic measures of risks, although the standardized approach falls short of such a target by looking at a feasible compromise between simplicity and complexities of implementation. Underlying Economic Principles for Measuring Market Risks

The economics of market risk are relatively easy to grasp. Tradable assets have random variations of which distributions result from observable sensitivities of instruments and volatilities of market parameters. The standalone market risk of an instrument is a VaR (or maximum potential loss) resulting from Profit and Loss (P&L) distribution derived from the underlying market parameter variations. Things get more involved for portfolios because risks offset to some extent. The ‘portfolio effect’ reduces the portfolio risk, making it lower than the sum of all individual standalone risks. For example, long and short exposures to the same instrument are exactly offset. Long and short exposures to similar instruments are also partially offset. The price movements of a 4-year bond and a 5-year bond correlate because the 4-year and 5-year interest rates do. However, the correlation is not perfect since a 1% change in the former does not mechanically trigger a 1% change in the latter. There is a case for offsetting exposures of long and short positions in similar instruments, but only to a certain extent. A mechanical offset would leave out of the picture the residual risk resulting from the mismatch of the instrument characteristics. This residual risk is a ‘basis risk’ resulting from the mismatch between the reference rates applying to each instrument. Moving one step further, let us consider a common underlying flat rate across all maturities, with a maturity mismatch between opposing exposures in similar instruments. Any shift in the entire spectrum of interest rates by the same amount (a parallel shift of 1% for instance) does not result in the same price change for the two instruments because 4 ‘Basis

risk’ exists when the underlying sources of risk differ, even if only slightly.

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their sensitivities differ. The magnitudes of the two price changes vary across instruments as sensitivities to common parameters do. With different instruments, such as equity and bonds, a conservative view would consider that all prices move adversely simultaneously. This makes no sense because the market parameters—interest rates, foreign exchange rates and equity indexes—are interdependent. Some tend to move in the same direction, others in opposite directions. In addition, the strength of such ‘associations’ varies across pairs of market parameters. The statistical measure for such ‘associations’ is the correlation. Briefly stated, market parameters comply with a correlation structure, so that the assumption of simultaneous adverse co-movements is irrelevant. When trying to assess how prices change within a portfolio, the issue is: how to model co-movements of underlying parameters in conjunction with sensitivities differing across exposures. This leads to the basic concepts of ‘general’ versus ‘specific’ risk. General risk is the risk dependent on market parameters driving all prices. This dependence generates price co-movements, or price correlations, because all prices depend, to some extent, on a common set of market parameters. The price volatility unrelated to market parameters is the specific risk. By definition, it designates price variations unrelated to market parameter variations. Statistically, specific risk is easy to isolate over a given period. It is sufficient to relate prices to market parameters statistically to get a direct measure of general risk. The remaining fraction of the price volatility is the specific risk. The Basel Committee offer two approaches for market risk: • The standardized approach relies on relatively simple rules and forfeits for defining capital charges, which avoids getting into the technicalities of modelling price changes. • The proprietary model approach allows banks to use internal full-blown models for assessing market risk VaR, subject to qualifying conditions. The critical inputs include: • The current valuation of exposures. • Their sensitivities to the underlying market parameter(s)5 . • Rules governing offsetting effects between opposing exposures in ‘similar’ instruments and, eventually, across the entire range of instruments. Full-blown models should capture all valuation, sensitivity and correlation effects. The ‘standardized’ approach for market risk uses forfeits and allows partial diversification effects, striking a balance between complexities and measurement accuracy. The Standardized Approach

The standardized approach relies on forfeits for assessing capital charges as a percentage of current exposures, on grids for capturing the differences in sensitivities of various market instruments and on offsetting rules allowing netting of the risks within a portfolio 5 Sensitivities

are variations of values for a unit change in market parameter, as detailed in Chapter 6.

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whenever there is no residual ‘basis’ risk. Forfeits capture conservatively the potential adverse deviations of instruments. Forfeits vary across market compartments and products, such as bonds of different maturities, because their sensitivity changes. For stocks, forfeits are 8% of the net balances, after allowed offsetting of long and short comparable exposures. For bonds, forfeits are also 8%, but various weights apply to this common ratio, based upon maturity buckets. Adding the individual risks, without offsetting exposures, overestimates the portfolio risk because the underlying assumption is that all adverse deviations occur simultaneously. Within a given class of instruments, such as bonds, equity or foreign exchange, regulators allow offsetting risks to a certain extent. For instance, being long and short on the same stock results in zero risk, because the gain in the long leg offsets the loss in the short leg when the stock goes up, and vice versa. Offsetting is limited to exact matches of instrument characteristics. In other instances, regulators rely on the ‘specific’ versus ‘general’ risk distinction, following the principle of adding specific risk forfeits while allowing limiting offsetting effects for general risk. The rationale is that general risk refers to comovements of prices, while specific risk is unrelated to underlying market parameters. Such conservative measures provide a strong incentive to move to the modelling approach that captures portfolio effects. These general rules differ by product family. What follows provides only an overview of the main product families. All forfeit values and grids are available from the BIS documents. Interest Rate Instruments For interest rate instruments, the capital charge adds individual transaction forfeits, varying from 0% (government) to 8% for maturities over 24 months. Offsetting specific risks is not feasible except for identical debt issues. The capital requirement for general market risk captures the risk of loss arising from changes in market interest rates. There are two principal methods for measuring general risk: a ‘maturity’ method and a ‘duration’ method. The maturity method uses 13 time bands for assigning instruments. It aims at capturing the varying sensitivities across time bands of instruments. Offsetting long and short positions within time bands is possible, but there are additional residual capital charges applying because of intra-band maturity mismatches. Since interest rates across time bands correlate as well, partial offsets across time bands are possible. The procedure uses ‘zones’ of maturity buckets grouping the narrower time bands. Offsets decrease with the distance between the time bands. The accord sets percentages of exposures significantly lower than 1, to cap the exposures allowed to offset (100% implying a total offset of the entire exposure). The duration method allows direct measurement of the sensitivities, skipping the time band complexity by using the continuous spectrum of durations. It is necessary to assign sensitivities to a duration-based ‘ladder’, and within each slot, the capital charge is a forfeit. Sensitivities in values should refer to preset changes of interest rates, whose values are within the 0.6% to 1.00% range. Offsetting is subject to a floor for residual risk (basis risk, since there are duration mismatches within bands).

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Derivatives Derivatives are combinations of underlying exposures, and each component is subject to the same treatments as above. For example, a swap is a combination of two exposures, the receiving leg and the paying leg. Therefore, forfeit values add unless the absence of any residual risk allows full offsets. There is no specific risk for derivatives. For options, there are special provisions based on sensitivities to the various factors that influence their prices, and their changes with these factors. For options, the accord proposes the scenario matrix analysis. Each cell of the matrix shows the portfolio value given a combination of scenarios of underlying asset price and volatility (stressed according to regulators’ recommendations). The highest loss value in the matrix provides the capital charge for the entire portfolio of options. Equity The basic distinction between specific versus general risk applies to equity products. The capital charge for specific risk is 8%, unless the portfolio is both liquid and well diversified, in which case the charge is 4%. Offsetting long and short exposures is feasible for general risk, but not for specific risk. The general market risk charge is 8%. Equity derivatives should follow the same ‘decomposition’ rule as underlying exposures. Foreign Exchange Two processes serve for calculating the capital requirement for foreign exchange risk. The first is to measure the exposure in a single currency position. The second is to measure the risks inherent in a bank’s mix of long and short positions in different currencies. Structural positions are exempt of capital charges. They refer to exposures protecting the bank from movements of exchange rates, such as assets and liabilities in foreign currencies remaining matched. The ‘shorthand method’ treats all currencies alike, and applies the forfeited 8% capital charge over the greatest netted exposure (either long or short). Commodities Commodity risk is more complex than market instrument risk because it combines a pure commodity price risk with other risks, such as basis risk (mismatch of prices of similar commodities), interest rate risk (for carrying cost of exposures) and forward price risk. In addition, there is directional risk in commodities prices. The principle is to assign transactions to maturity buckets, to allow offsetting of matched exposures, and to assign a higher capital charge for risk. Proprietary Models of Market Risk VaR

The principles of the extension to proprietary market risk models are as follows:

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• Market risk is the risk of loss during the minimum period required to liquidate transactions in the market. • The potential loss is the 99th loss percentile6 (one-tailed) for market risk models. • This minimum period depends on the type of products, and on their sensitivities to a given variation of their underlying market parameters. However, a general 10-day period for liquidating position is the normal reference for measuring downside risk. • Potential losses depend on market movements during this period, and the sensitivities of different assets. Models should incorporate historical observation over at least 1 year. In addition, the capital charge is the higher of the previous day’s VaR or three times the average daily VaR of the preceding 60 days. A multiplication factor applies to this modelled VaR. It accounts for potential weaknesses in the modelling process or exceptional circumstances. In addition, the regulators emphasize: • Stress-testing, to see what happens under exceptional conditions. Stress-testing uses extreme scenarios maximizing losses to find out how large they can be. • Back-testing of models to ensure that models capture the actual deviations of the portfolios. Back testing implies using the model with past data to check whether the modelled deviations of values are in line or not with the historical deviations of values, once known. These additional requirements serve to modulate the capital charge, through the multiplier of VaR, according to the reliability of the model, in addition to fostering model improvements through time. Reliable models get a ‘premium’ in capital with a lower multiplier of VaR.

Derivatives and Credit Risk Derivatives are over-the-counter instruments (interest rate swaps, currency swaps, options) not liquid as market instruments. Theoretically, banks hold these assets until maturity, and bear credit risk since they exchange flows of funds with counterparties subject to default risk. For derivatives, credit risk interacts with market risk in that the mark-to-market (liquidation) value depends on market movements. It is the present value of all future flows at market rates. Under a ‘hold to maturity’ view, the potential future values over their life is the credit risk exposure because they are the value of all future flows that the defaulted counterparty will not pay. Future liquidation values are random because they are market-driven. To address random exposures over a long-term period, it is necessary to model the time profile of liquidation values. The principle is to determine the time profile of upper bounds that future values will not exceed with more than a preset low probability (confidence level) (see Chapter 39). From the regulatory standpoint, the issue is to define applicable rules as percentage forfeits of the notional of derivatives. The underlying principles for defining such forfeits are identical to those of models. The current credit risk exposure is the current liquidation value. There is the additional risk due to the potential upward deviations of liquidation 6A

loss percentile is the loss value not exceeded in more than the given preset percentage, such as 99%. See Chapter 7.

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value from the current value during the life of the instruments. Such drifts depend on the market parameter volatilities and on instrument sensitivities. It is feasible to calibrate forfeit values representative of such drifts on market observations. The regulators provided forfeits for capturing these potential changes of exposures. The forfeit add-ons are percentages of notional depending on the underlying parameter and the maturity buckets. Underlying parameters are interest rate, foreign exchange, equity and commodities. The maturity buckets are up to 1 year, between 1 and 5 years and beyond 5 years. The capital charge adds arithmetically individual forfeits. Forfeit add-ons are proxies and adding them does not allow offsetting effects, thereby providing a strong incentive for modelling derivative exposures and their offsetting effects. The G30 group, in 1993, defined the framework for monitoring derivative credit risk. The G30 report recommended techniques for assessing uncertain exposures, and for modelling the potential unexpected deviations of portfolios of derivatives over long periods. The methodology relies on internal models to capture worst-case deviations of values at preset confidence levels. In addition, it emphasized the need for enhanced reporting, management control and disclosure of risk between counterparties.

Interest Rate Risk (Banking Portfolio) Interest rate risk, for the banking portfolio, has generated debate and controversy. The current regulation does not require capital to match interest rate risk. Nevertheless, regulators require periodical internal reports to be made available. Supervising authorities take corrective actions if they feel them required. Measures of interest rate risk include the sensitivity of the interest income to shifts in interest rates and that of the net market value of assets and liabilities. The interest margin is adequate in the short-term. Market value sensitivities of both assets and liabilities capture the entire stream of future flows and provide a more long-term view. Regulators recommend using both. Subsequent chapters discuss the adequate models for such sensitivities, gaps and net present value of the balance sheet and its duration (see Chapters 21 and 22).

THE NEW BASEL ACCORD The New Basel Accord is the set of consultative documents that describes recommended rules for enhancing credit risk measures, extending the scope of capital requirements to operational risk, providing various enhancements to the ‘existing’ accord and detailing the ‘supervision’ and ‘market discipline’ pillars. The accord allows for a 3-year transition period before full enforcement, with all requirements met by banks at the end of 2004. Table 3.1 describes the rationale of the New Accord. The new package is very extensive. It provides a menu of options, extended coverage and more elaborate measures, in addition to descriptions of work in progress, with yet unsettled issues to be streamlined in the final package. The New Accord comprises three pillars: • Pillar 1 Minimum Capital Requirements.

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• Pillar 2 Supervisory Review Process. • Pillar 3 Market Discipline. TABLE 3.1 Rationale for a new accord: the need for more flexibility and risk sensitivity Existing Accord Focuses on a single risk measure One size fits all: only one option proposed to banks Broad brush structure (forfeits)

Proposed New Accord More emphasis on banks’ own internal methodologies, supervisory review and market discipline Flexibility, menu of approaches, incentives: banks have several options More credit risk sensitivity for better risk management

The Committee emphasizes the mutually reinforcing role of these three pillars—minimum capital requirements, supervisory review and market discipline. Taken together, the three pillars contribute to a higher level of safety and soundness in the financial system. Previous implementations of the regulations for credit and market risk, confirmed by VaR models for both risks, revealed that the banking book generates more risk than the trading book. Credit risk faces a double challenge. The measurement issues are more difficult to tackle for credit risk than for market risk and, in addition, the tolerance for errors should be lower because of the relative sizes of credit versus market risk.

Overview of the Economic Contributions of the New Accord The economic contributions of the New Accord extend to supervisory processes and market discipline. Economically, the New Accord appears to be a major step forward. On the quantitative side of risk measurements, the accord offers a choice between the ‘standardized’, the ‘foundation’ and the ‘advanced’ approaches, addresses and provides remedies for several critical issues that led banks to arbitrage the old capital requirements against more relevant economic measures. It remedies the major drawbacks of the former ‘Cooke ratio’, with a limited set of weights and a unique weight for all risky counterparties of 100%. Neither the definition of capital and Tier 1/Tier 2, nor the 8% coefficient change. The new set of ratios (the ‘McDonough ratios’) corrects this deficiency. Risk weights define the capital charge for credit risk. Weights are based on credit risk ‘components’, allowing a much improved differentiation of risks. They apply to some specific cases where the old weights appeared inadequate, such as for credit enhancement notes issued for securitization which, by definition, concentrate a major fraction of the securitized assets7 . Such distortions multiplied the discrepancies between economic risk-based policies and regulatory-based policies, leading banks to arbitrage the regulatory weights. In addition, banks relied on regulatory weights and forfeits not in line with risks, leading to distortion in risk assessment, prices, risk-adjusted measures of performance, etc. 7 The

mechanisms of securitizations are described in Chapter 34, and when detailing the economics of securitizations.

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The New Accord provides a more ‘risk-sensitive’ framework that should considerably reduce such distortions. In addition, it makes clear that data gathering on all critical credit risk drivers of transactions is a major priority, providing a major incentive to solve the ‘incompleteness’ of credit risk data. It also extends the scope of capital requirements to operational risk. Measures of operational risk remained in their ‘infancy’ for some time because of a lack of data, but the accord should stimulate rapid progress. From an economic standpoint, the accord suffers from the inaccuracy limitations of forfeit-based measures, a limitation mitigated by the need to strike a balance between accuracy and practicality. The accord does not go as far as authorizing the usage of credit risk VaR models, as it did for market risk. It mentions, as reasons for refraining from going that far, the difficulties of establishing the reliability of both inputs and outputs of such models. Finally, it still leaves aside interest rate risk for capital requirements and includes it in the supervision process (Pillar 2).

Pillar 1: Overall Minimum Capital Requirements The primary changes to the minimum capital requirements set out in the 1988 Accord are the approaches to credit risk and the inclusion of explicit capital requirements for operational risk. The accord provides a range of ‘risk-sensitive options’ for addressing both types of risk. For credit risk, this range includes the standardized approach, with the simplest requirements, and extends to the ‘foundation’ and ‘advanced’ Internal RatingsBased (IRB) approaches. Internal ratings are assessments of the relative credit risks of borrowers and/or facilities, assigned by banks8 . The Committee desires to produce neither a net increase nor a net decrease—on average—in minimum regulatory capital. With respect to the IRB approaches, the Committee’s ultimate goal is to improve regulatory capital adequacy for underlying credit risks and to provide capital incentives relative to the standardized approach through lower risk weights, on average, for the ‘foundation’ and the ‘advanced’ approaches. Under the New Accord, the denominator of the minimum total capital ratio will consist of three parts: the sum of all risk-weighted assets for credit risk, plus 12.5 times the sum of the capital charges for market risk and operational risk. Assuming that a bank has $875 of risk-weighted assets, a market risk capital charge of $10 and an operational risk capital charge of $20, the denominator of the total capital ratio would equal 875 + [(10 + 20) × 12.5] or $1250.

Risk Weights under Pillar 1 The New Accord strongly differentiates risk weights using a ‘menu’ of approaches designated as ‘standardized’, ‘foundation’ and ‘advanced’. 8 See

Chapters 35 and 36 on rating systems and credit data by rating class available from rating agencies.

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The Standardized Approach

In the standardized approach to credit risk, exposures to various types of counterparties—sovereigns, banks and corporates—have risk weights based on assessments (ratings) by External Credit Assessment Institutions (ECAIs or rating agencies) or Export Credit Agencies (ECAs) for sovereign risks. The risk-weighted assets in the standardized approach are the product of exposure amounts and supervisory determined risk weights. As in the current accord, the risk weights depend on the category of the borrower: sovereign, bank or corporate. Unlike in the current accord, there will be no distinction on the sovereign risk weighting depending on whether or not the sovereign is a member of the Organization for Economic Coordination and Development (OECD). Instead, the risk weights for exposures depend on external credit assessments. The treatment of off-balance sheet exposures remains largely unchanged, with a few exceptions. To improve risk sensitivity while keeping the standardized approach simple, the Committee proposes to base risk weights on external credit assessments. The usage of the supervisory weights is the major difference from the IRB approach, which relies on internal ratings. The approach is more risk-sensitive than the existing accord, through the inclusion of an additional risk bucket (50%) for corporate exposures, plus a 150% risk weight for low rating exposures. Unrated exposures have a 100% weight, lower than the 150% weight. The higher risk bucket (150%) also serves for certain categories of assets. The standardized approach does not allow weights to vary with maturity, except in the case of short-term facilities with banking counterparties in the mid-range of ratings, where weights decrease from 50% to 20% and from 100% to 50%, depending on the rating9 . The unrated class at 150% could trigger ‘adverse selection’ behaviour, by which lowrated entities give up their ratings to benefit from a risk weight of 100%, rather than 150%. On the other hand, the majority of corporates—and, in many countries, the majority of banks—do not need to acquire a rating in order to fund their activities. Therefore, the fact that a borrower has no rating does not generally signal low credit quality. The accord attempts to strike a compromise between these conflicting facts and stipulates that national supervisors have some flexibility in adjusting this weight. The 150% weight remains subject to consultation with the banking industry as of current date. For sovereign risks, the external ratings are those of an ECAI or ECA. For banks, the risk weights scale is the same, but the rating assignment might follow either one of two processes. Either the sovereign rating is one notch lower than the sovereign ratings, or it is the intrinsic bank rating. In Tables 3.2 and 3.3 we provide the grids of risk weights for sovereign ratings and corporates. The grid for banks under the first option is identical to that of sovereign ratings. TABLE 3.2

Risk weights

9 By

Risk weights of sovereigns AAA to AA−

A+ to A−

BBB+ to BBB−

BB+ to B−

Below B−

Unrated

0%

20%

50%

100%

150%

100%

contrast, the advanced IRB approach makes risk weights sensitive to maturity and ratings through the default probabilities.

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TABLE 3.3

Risk weights

Risk weights of corporates AAA to AA−

A+ to A−

BBB+ to BB−

Below BB−

Unrated

20%

50%

100%

150%

100%

Source: Basel Committee, January 2001.

Internal Ratings-based Framework

The IRB approach lays down the principles for evaluating economically credit risk. It proposes a treatment similar to the standardized approach for corporate, bank and sovereign exposures, plus separate schemes for retail banking, project finance and equity exposures. There are two versions of the IRB approach: the ‘foundation’ and the ‘advanced’ approaches. For each exposure class, the treatment uses three main elements called ‘risk components’. A bank can use either its own estimates for each of them or standardized supervisory estimates, depending on the approach. A ‘risk-weight function’ converts the risk components into risk weights for calculating risk-weighted assets. The risk components are the probability of default (PD), the loss given default (Lgd) and the ‘Exposure At Default’ (EAD). The PD estimate must represent a conservative view of a long-run average PD, ‘through the cycle’, rather than a short-term assessment of risk. Risk weights are a function of PD and Lgd. Maturity is a credit risk component that should influence risk weights, given the objective of increased risk sensitivity. However, the inclusion of maturity creates additional complexities and a disincentive for lending long-term. Hence, the accord provides alternative formulas for inclusion of maturity, which mitigate a simple mechanical effect on capital. The BIS proposes the Benchmark Risk Weight (BRW) for including the maturity effect on credit risk and capital weights. The function depends on the default probability DP. The benchmark example refers to the specific case of a 3-year asset, with various default probabilities and an Lgd of 50%. Three representative points show the sensitivity of risk weights to the annualized default probability (Table 3.4). TABLE 3.4 Sensitivity of risk weights with maturity: benchmark case (3-year asset, 50% Lgd) DP (%) BRW (%)

0.03 14

0.7 100

20 625

For DP = 0.7%, the BRW is 100% and the maximum risk weight, for DP = 20%, reaches 625%. This value is a cap for all maturities and all default probabilities. The weight profile with varying DP is more sensitive than the standardized approach weights, which vary in the range 20% to 150% for all maturities over 1 year. The weights increase less than proportionally with default probability until they reach the cap.

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The New Accord suggests using a forfeit maturity of 3 years for all assets for the ‘foundation’ approach, but leaves the door open to the usage of effective maturity. Risk weights adjusted for effective maturity apply to the ‘advanced’ approach. Retail Exposures

The Committee proposes a treatment of retail portfolios, based on the conceptual framework outlined above, modified to capture the specifics of retail exposures. Fixed rating scales and the assignment of borrower ratings are not standard practices for retail banking (see Chapters 37 and 38). Rather, banks commonly divide the portfolio into ‘segments’ made up of exposures with similar risk characteristics. The accord proposes to group retail exposures into segments. The assessment of risk components will be at the segment level rather than at the individual exposure level, as is the case for corporate exposures. In the case of retail exposures, the accord also proposes, as an alternative assessment of risk, to evaluate directly ‘expected loss’. Expected loss is the product of DP and Lgd. This approach bypasses the separate assessment, for each segment, of the PD and Lgd. The maturity (M) of the exposure is not a risk input for retail banking capital. Foundation Approach

The ‘foundation’ IRB approach allows banks meeting robust supervisory standards to input their own assessment of the PD associated with the obligor. Estimates of additional risk factors, such as loss incurred by the bank given a default (Lgd) and EAD, should follow standardized supervisory estimates. Exposures not secured by a recognized form of collateral will receive a fixed Lgd depending on whether the transaction is senior or subordinated. The minimum requirements for the foundation IRB approach relate to meaningful differentiation of credit risk with internal ratings, the comprehensiveness of the rating system, the criteria of the rating system and similar. There are a variety of methodologies and data sources that banks may use to associate an estimate of PD with each of its internal grades. The three broad approaches are: use of data based on a bank’s own default experience; mapping to external data such as those of ‘ECA’; use of statistical default models. Hence, a bank can use the foundation approach as long as it maps, in a sound manner, its own assessment of ratings with default probabilities, including usage of external data. Advanced Approach

A first difference with the ‘foundation’ approach is that the bank assesses the same risk components plus the Lgd parameter characterizing recoveries. The subsequent presentation of credit risk models demonstrates that capital is highly sensitive to this risk input. In general, banks have implemented ratings scales for some time, but they lack risk data on recoveries. The treatment of maturity also differs from the ‘foundation’ approach, which refers to a single benchmark for all assets. BRWs depend on maturity in the ‘advanced’ approach. The cap of 625% still applies as in the forfeit 3 years to maturity. The maturity effect depends on an annualized default probability as in the ‘foundation’ approach, plus a term b in a BRW function of DP and maturity depending on the effective maturity of assets. This

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is a more comprehensive response to the need to make capital sensitive to the maturity effect. It strikes a compromise between ‘risk sensitivity’ and the practical requirement of avoiding heavy capital charges for long-term commitments that would discourage banks from entering into such transactions. As a remark, credit risk models (see Chapter 42) capture maturity effects through revaluation of facilities according to their final risk class at a horizon often set at 1 year. The revaluation process at the horizon does depend on maturity since it discounts future cash flows from loans at risk-adjusted discount rates. Nevertheless, the relationship depends on the risk and excess return of the asset over market required yields, as explained in Chapter 8. Consequently, there is no simple relation between credit risk and maturity. Presumably, the BRW function combines contributions of empirical data of behaviour over time of historical default frequencies and relations between capital and maturity from models. However, the Committee is stopping short of permitting banks to calculate their capital requirements based on their own portfolio credit risk models. Reasons are the current lack of reliability of inputs required by such models, plus the difficulty of demonstrating the reliability of model capital estimates. However, by imposing the build-up of risk data for the next 3 years, the New Accord paves the way for a later implementation. Given the difficulty of assessing the implications of these approaches on capital requirements, the Committee imposes some conservative guidelines such as a floor on required capital. The Committee emphasizes the need for banks to anticipate regulatory requirements by performing stress testing and establishing additional capital cushions of their own (i.e. through Pillar 2) during periods of economic growth. In the longer run, the Committee encourages banks to consider the merits of building such stress considerations directly into their internal ratings framework. Guarantees

The New Accord grants greater recognition of credit risk mitigation techniques, including collateral, guarantees and credit derivatives, and netting. The new proposals provide capital reductions for various forms of transactions that reduce risk. They also impose minimum operational standards because a poor management of operational risks—including legal risks—would raise doubts with respect to the actual value of such mitigants. Further, banks are required to hold capital against residual risks resulting from any mismatch between credit risk hedges and the corresponding exposure. Mismatches refer to differences in amounts or maturities. In both cases, capital requirements will apply to the residual risks. Collateral The Committee has adopted for the standardized approach a definition of eligible collateral that is broader than that in the 1988 Accord. In general, banks can recognize as collateral: cash; a restricted range of debt securities issued by sovereigns, public sector entities, banks, securities firms and corporates; certain equity securities traded on recognized exchanges; certain mutual fund holdings; gold. For collateral, it is necessary to account for time changes of exposure and collateral values. ‘Haircuts’ define the required excess collateral over exposure to ensure effective

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credit risk protection, given time periods necessary for readjusting the collateral level (re-margining), recognizing the counterparty’s failure to pay or to deliver margin and the bank’s ability to liquidate collateral for cash. Two sets of haircuts have been developed for a comprehensive approach to collateral: those established by the Committee (i.e. standard supervisory haircuts); others based on banks’ ‘own estimates’ of collateral volatility subject to minimum requirements (see Chapter 41). There is a capital floor, denoted w, whose purpose is twofold: to encourage banks to focus on and monitor the credit quality of the borrower in collateralized transactions; to reflect the fact that, irrespective of the extent of over-collateralization, a collateralized transaction can never be totally without risk. A normal w value is 0.15. Guarantees and Credit Derivatives For a bank to obtain any capital relief from the receipt of credit derivatives or guarantees, the credit protection must be direct, explicit, irrevocable and unconditional. The Committee recognizes that banks only suffer losses in guaranteed transactions when both the obligor and the guarantor default. This ‘double default’ effect reduces the credit risk if there is a low correlation between the default probabilities of the obligor and the guarantor10 . The Committee considers that it is difficult to assess this situation and does not grant recognition to the ‘double default’ effect. The ‘substitution approach’ provided in the 1988 Accord applies for guarantees and credit derivatives, although an additional capital floor, w, applies. The substitution approach simply substitutes the risk of the guarantor for that of the borrower subject to full recognition of the enforceability of the guarantee. On-balance Sheet Netting On-balance sheet netting in the banking book is possible subject to certain operational standards. Its scope will be limited to the netting of loans and deposits with the same single counterparty. Portfolio Granularity

The Committee is proposing to make another extension of the 1988 Accord in that minimum capital requirements do not depend only on the characteristics of an individual exposure but also on the ‘concentration risk’ of the bank portfolio. Concentration designates the large sizes of exposures to single borrowers, or groups of closely related borrowers, potentially triggering large losses. The accord proposes a measure of granularity11 and incorporates this risk factor into the IRB approach by means of a standard supervisory capital adjustment applied to all exposures, except those in the retail portfolio. This treatment does not include industry, geographic or forms of credit risk concentration other than size concentration. The ‘granularity’ adjustment applies to the total risk-weighted assets at the consolidated bank level, based on the comparison of a reference portfolio with known granularity. 10 Chapter

41 provides details on the technique for assessing the default probability reduction resulting from ‘double’ or ‘joint’ default of the primary borrower and the guarantor. 11 An example is the ‘Herfindahl index’ calculation, measuring size concentration, given in Chapters 55–57.

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Other Specific Risks

The New Accord addresses various other risks: asset securitizations; project finance; equity exposures. The accord considers that asset securitizations deserve a more stringent treatment. It assigns risk weights more in line with the risks of structured notes issued by such structures and, notably, the credit enhancement note, subject to the first loss risk. These notes concentrate a large fraction of the risk of the pool of securitized assets12 . It also imposes the ‘clean break’ principle through which the non-recourse sale of assets should be unambiguous, limiting the temptation of banks to support sponsored structures for reputation motives (reputation risk13 ). Under the standardized approach, any invested amount in the credit enhancement note of securitization becomes deductible from capital. For banks investing in securitization notes, the Committee proposes to rely on ratings provided by an ECAI. Other issues with securitizations relate to operational risk. Revolving securitizations with early amortization features, or liquidity lines provided to structures (commitments to provide liquidity for funding the structure under certain conditions), generate some residual risks. There is a forfeited capital loading for such residual risk. The Committee considers that project finance requires a specific treatment. The accord also imposes risk-sensitive approaches for equity positions held in the banking book. The rationale is to remedy the possibility that banks could benefit from a lower capital charge when they hold the equity rather than the debt of an obligor. Interest Rate Risk

The accord considers it more appropriate to treat interest rate risk in the banking book under Pillar 2, rather than defining capital requirements. This implies no capital load, but an enhanced supervisory process. The guidance on interest rate risk considers banks’ internal systems as the main tool for the measurement of interest rate risk in the banking book and the supervisory response. To facilitate supervisors’ monitoring of interest rate risk across institutions, banks should provide the results of their internal measurement systems using standardized interest rate shocks. If supervisors determine that a bank is not holding capital commensurate with the level of interest rate risk, they can require that the bank reduces its risk, holds an additional amount of capital or combines the two.

Operational Risk The Committee adopted a standard industry definition of operational risk: ‘the risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events’. As a first approximation in developing minimum capital charges, the Committee estimates operational risk at 20% of minimum regulatory capital as measured under the 1988 12 See

details in subsequent descriptions of structures (Chapter 40). risk is the risk of adverse perception of the sponsoring bank if structure explicitly related to the bank suffers from credit risk deterioration or from a default event. 13 Reputation

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Accord. The Committee proposes a range of three increasingly sophisticated approaches to capital requirements for operational risk: basic indicator; standardized; internal measurement. The ‘basic indicator approach’ links the capital charge for operational risk to a single indicator that serves as a proxy for the bank’s overall risk exposure. For example, if gross income is the indicator, each bank should hold capital for operational risk equal to a fixed percentage (‘alpha factor’) of its gross income. The ‘standardized approach’ builds on the basic indicator approach by dividing a bank’s activities into a number of standardized business lines (e.g. corporate finance and retail banking). Within each business line, the capital charge is a selected indicator of operational risk times a fixed percentage (‘beta factor’). Both the indicator and the beta factors may differ across business lines. The ‘internal measurement approach’ allows individual banks to rely on internal data for regulatory capital purposes. The technique necessitates three inputs for a specified set of business lines and risk types: an operational risk exposure indicator; the probability that a loss event occurs; the losses given such events. Together, these components make up a loss distribution for operational risks. Nevertheless, the loss distribution might differ from the industry-wide loss distribution, thereby necessitating an adjustment, which is the ‘gamma factor’.

Pillar 2: Supervisory Review Process The second pillar of the new framework aims at ensuring that each bank has sound internal processes to assess the adequacy of its capital based on a thorough evaluation of its risks. Supervisors will be responsible for evaluating how well banks are assessing their capital needs relative to their risks. The Committee regards the market discipline through enhanced disclosure as a fundamental part of the New Accord. It considers that disclosure requirements and recommendations will allow market participants to assess key pieces of information for the application of the revised accord. The risk-sensitive approaches developed by the New Accord rely extensively on banks’ internal methodologies, giving banks more discretion in calculating their capital requirements. Hence, separate disclosure requirements become prerequisites for supervisory recognition of internal methodologies for credit risk, credit risk mitigation techniques and other areas of implementation. The Committee formulated four basic principles that should inspire supervisors’ policies: • Banks should have a process for assessing their overall capital in relation to their risk profile and a strategy for maintaining their capital levels. • Supervisors should review and evaluate banks’ internal capital adequacy assessments and strategies, as well as their ability to monitor and ensure their compliance with regulatory capital ratios. Supervisors should take appropriate supervisory actions if they are not satisfied with the results of this process. • Supervisors should expect banks to operate above the minimum regulatory capital ratios and should have the ability to require banks to hold capital in excess of this minimum.

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• Supervisors should intervene at an early stage to prevent capital from falling below the minimum levels required to support the risk of a particular bank, and should require corrective actions if capital is not maintained or restored.

Pillar 3: Market Discipline The third major element of the Committee’s approach to capital adequacy is market discipline. The accord emphasizes the potential for market discipline to reinforce capital regulations and other supervisory efforts in promoting safety and soundness in banks and financial systems. Given the influence of internal methodologies on the capital requirements established, it considers that comprehensive disclosure is important for market participants to understand the relationship between the risk profile and the capital of an institution. Accordingly, the usage of internal approaches is contingent upon a number of criteria, including appropriate disclosure. For these reasons, the accord is setting out a number of disclosure proposals as requirements, some of them being prerequisites to supervisory approval. Core disclosures convey vital information for all institutions and are important for market discipline. Disclosures are subject to ‘materiality’. Information is ‘material’ if its omission or misstatement could change or influence the assessment or decision of any user relying on that information. Supplementary disclosures may convey information of significance for market discipline actions with respect to a particular institution.

SECTION 3 Risk Management Processes

4 Risk Management Processes

The ultimate goal of risk management is to facilitate a consistent implementation of both risks and business policies. Classical risk practices consist of setting risk limits while ensuring that business remains profitable. Modern best practices consist of setting risk limits based on economic measures of risk while ensuring the best risk-adjusted performances. In both cases, the goal remains to enhance the risk–return profile of transactions and of the bank’s portfolios. Nevertheless, new best practices are more ‘risk-sensitive’ through quantification of risks. The key difference is the implementation of risk measures. Risks are invisible and intangible uncertainties, which might materialize into future losses, while earnings are a standard output of reporting systems complying with established accounting standards. Such differences create a bias towards an asymmetric view of risk and return, making it more difficult to strike the right balance between both. Characterizing the risk–return profile of transactions and of portfolios is key for implementing risk-driven processes. The innovation of new best practices consists of plugging new risk–return measures into risk management processes, enriching them and leveraging them with more balanced views of profitability and risks. The purpose of this chapter is to show how quantified risk measures feed the risk management processes. It does not address the risk measuring issue and does not describe the contribution of risk models, for which inputs are critical to enrich risk processes. Because quantifying intangible risks is a difficult challenge, concentrating on risk measures leaves in the shadow the wider view of risk processes implementing such risk measures. Since the view on risk processes is wider than the view on risk measuring techniques, we move first from a global view of risk processes before getting to the narrower and more technical view of risk measuring. The ‘risk–return profiles’ of transactions and portfolios are the centrepiece of the entire system and processes. For this reason, risk–return profiles are the interface between new

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risk models and risk processes. All risk models and measures converge to provide these profiles at the transaction, the business lines and the global portfolio levels. Risk models provide new risk measures as inputs for processes. Classical processes address risks without the full capability of providing adequate quantification of risks. Risk models provide new measures of return and risks, leveraging risk processes and extending them to areas that were previously beyond reach. New risk measures interact with risk processes. Vertical processes address the relationship between global goals and business decisions. The bottom-up and top-down processes of risk management allow the ‘top’ level to set up global guidelines conveyed to business lines. Simultaneously, periodical reporting from the business levels to the top allows deviations from guidelines to be detected, such as excess limits, and corrective actions to be taken, while comparing projected versus actual achievements. Transversal processes address risk and return management at ‘horizontal’ levels, such as the level of individual transactions, at the very bottom of the management ‘pyramid’, at the intermediate business line levels, as well as at the bank’s top level, for comparing risk and return measures to profitability target and risk limits. There are three basic horizontal processes: setting up risk–return guidelines and benchmarks; risk–return decision-making (‘ex ante perspective’); risk–return monitoring (‘ex post perspective’). The first section provides an overview of the vertical and horizontal processes. The subsequent sections detail the three basic ‘transversal’ processes (benchmarks, decisionmaking, monitoring). The last section summarizes some general features of ‘bank-wide risk management’ processes.

THE BASIC BUILDING BLOCKS OF RISK MANAGEMENT PROCESSES Processes cover all necessary management actions for taking decisions and monitoring operations that influence the risk and return profiles of transactions, subportfolios of business lines or the overall bank portfolio. They extend from the preparation of decisions, to decision-making and control. They include all procedures and policies required to organize these processes. Risk management combines top-down and bottom-up processes with ‘horizontal’ processes. The top-down and bottom-up views relate to the vertical dimension of management, from general management to individual transactions, and vice versa. The horizontal layers refer to individual transactions, business lines, product lines and market segments, in addition to the overall global level. They require to move back and forth from a risk–return view of the bank to a business view, whose main dimensions are the product families and the market segments.

Bottom-up and Top-down Processes The top-down process starts with global target earnings and risk limits converted into signals to business units. These signals include target revenues, risk limits and guidelines applicable to business unit policies. They make it necessary to allocate income and risks to business units and transactions. Otherwise, the global targets remain disconnected from

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55

operations. The monitoring and reporting of risks is bottom-up oriented, starting with transactions and ending with consolidated risks, income and volumes of transactions. Aggregation is required for supervision purposes and to compare, at all levels where decision-making occurs, goals with actual operations. In the end, the process involves the entire banking hierarchy from top to bottom, to turn global targets into signals to business units, and from bottom to top, to aggregate risks and profitability and monitor them. The pyramid image of Figure 4.1 illustrates the risk diversification effect obtained when moving up along the hierarchy. Each face of the pyramid represents a dimension of risk, such as credit risk or market risk. The overall risk is less than the simple arithmetic addition of all original risks generated by transactions (at the base of the pyramid) or subsets (subportfolios) of transactions. From bottom to top, risks diversify. This allows us to take more risks at the transaction level since risk aggregation diversifies away a large fraction of the sum of all individual transaction risks. Only post-diversification risk remains retained by the bank. Revenues and risk (capital) allocations

Global risk−return targets Poles, subsidiaries ... Business units Transactions

FIGURE 4.1

The pyramid of risk management

Risk models play a critical role in this ‘vertical’ process. Not only do they provide risk visibility at the transaction and business units level, but they also need to provide the right techniques for capturing risk diversification when moving up and down along the pyramid. Without quantification of the diversification effect, there are missing links between the sustainable risks, at the level of transactions, and the aggregated risk at the top of the pyramid that the bank capital should hedge. In other words, we do not know the overall risk when we have, say, two risks of 1 each, because the risk of the sum (1 + 1) is lower than 2, the sum of risks. There are missing links as well between the sustainable post-diversification risk, the bank’s capital and the risks tolerable at the bottom of the pyramid for individual transactions, business lines, market segments and product families. In other words, if a global post-diversification risk of 2 is sustainable at the top of the pyramid, it is compatible with a sum of individual risks larger than 2 at the bottom. How large can the sum of individual risks be (3, 4, 5 or more), compatible with an overall global risk limit of 2, remains unknown unless we have tools to allocate the global risk. The capital allocation system addresses these needs. This requires a unified risk management framework.

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Transversal Process Building Blocks Transversal processes apply to any horizontal level, such as business lines, product lines or market segments. The typical building transversal blocks of such processes (Figure 4.2) are: 1. Setting up risk limits, delegations and target returns. 2. Monitoring the compliance of risk–return profiles of transactions or subportfolios with guidelines, reporting and defining of corrective actions. 3. Risk and return decisions, both at the transaction and at the portfolio levels, such as lending, rebalancing the portfolio or hedging.

2. Decisionmaking Ex ante New transactions Hedging Portfolio R&R enhancement

1. Guidelines

Risk & Return (R&R)

3. Monitoring 3. Monitoring

FIGURE 4.2

Limits Delegations Target return

Ex post Follow-up Reporting R&R Corrective actions

The three-block transversal processes

These are integrated processes, since there are feedback loops between guidelines, decisions and monitoring. Risk management becomes effective and successful only if it develops up to the stage where it facilitates decision-making and monitoring.

Overview of Processes Putting together these two views could produce a chart as in Figure 4.3, which shows how vertical and transversal dimensions interact.

Risk Models and Risk Processes Risk models contribute to all processes because they provide them with better and richer measures of risk, making them comparable to income, and because they allow banks to enrich the processes using new tools such as risk-adjusted performance or valuing the risk reduction effects of altering the portfolio structure. Figure 4.4 illustrates how models provide the risk–return measures feeding transversal and vertical processes.

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57

Global Risk Management Guidelines

Decision-making Ex ante

Top Down

Limits Delegations Target returns ...

On- and offbalance sheet Excess limits Hedging Individual transactions Lending, trading ... Global portfolio Portfolio management

Monitoring Ex post Bottom - Up Business Unit Management

FIGURE 4.3

The three basic building blocks of risk management processes Risk Management 'Toolbox'

Risk Models

Risks Risks

Global

Return Return

Decisions

Guidelines

Business line Transaction Monitoring

FIGURE 4.4

Overall views of risk processes and risk–return

Risk Processes and Business Policy Business policy deals with dimensions other than risk and return. Attaching risk and returns to transactions and portfolios is not enough, if we cannot convert these views into the two basic dimensions of business policy: products and markets. This requires a third type of process reconciling the risk and return view with the product–market view. For

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Market Transactions

Specialized Finance

Off-balance Sheet

Corporate Lending

RFS

business purposes, it is natural to segment the risk management process across business lines, products and markets. The product–market matrices provide a synthethic business view of the bank. Broad business lines and market segments could look like the matrix in Figure 4.5. In each cell, it is necessary to capture the risk and return profiles to make economically meaningful decisions. The chart illustrates the combinations of axes for reporting purposes, and the need for Information Technology (IT) to keep in line with such multidimensional reports and analyses.

Consumers Corporate−Middle Market Large Corporate Firms Financial Institutions Specialized Finance

Risk & Return

FIGURE 4.5 Reporting credit risk characteristics within product–market couples RFS refers to ‘Retail Financial Services’. Specialized finance refers to structured finance, project finance, LBO or assets financing.

The next section discusses the three basic transversal processes: setting up guidelines; decision-making; monitoring.

PROCESS #1: SETTING UP RISK AND RETURN GUIDELINES Guidelines include risk limits and delegations, and benchmarks for return. The purpose of limits is to set up an upper bound of exposures to risks so that an unexpected event cannot impair significantly the earnings and credit standing of the bank. Setting up benchmarks of return refers to the target profitability goals of the bank, and how they translate into pricing. When setting up such guidelines, banks face trade-offs in terms of business volume versus risk and business volume versus profitability. Risk Benchmarks: Limits and Delegations

Traditional risk benchmarks are limits and delegations. For credit risk, limits set upper bounds to credit risk on obligors, markets and industries, or country risk. For market risk, limits set upper bounds to market risk sensitivity to the various market parameters.

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Delegations serve to decentralize the decision-making process. They mean business lines do not need to refer always to the central functions to make risk decisions as long as the risk is not too large. Classical procedures and policies have served to monitor credit risk for as long as banks have existed. Limit systems set caps on amounts at risk with any one customer, with a single industry or country. To set up limits, some basic rules are simple, such as: avoid any single loss endangering the bank; diversify the commitments across various dimensions such as customers, industries and regions; avoid lending to any borrower an amount that would increase debt beyond borrowing capacity. The equity of the borrower sets up some reasonable limit on debt given acceptable levels of debt/equity ratios and repayment ability. The capital of the bank sets up another limit to lending given the diversification requirements and/or the credit policy guidelines. Credit officers and credit committees reach a minimal agreement before making a credit decision by examining in detail credit applications. Delegations, at various levels of the bank, stipulate who has permission to take credit commitments depending on their size. Typical criteria for delegations are size of commitments or risk classes. Central reporting of the largest outstanding loans to customers serves to ensure that these amounts stay within reasonable limits, especially when several business units originate transactions with the same clients. This makes it necessary to have ‘global limit’ systems aggregating all risks on any single counterparty, no matter who initiates the risk. Finally, there are risk diversification rules across counterparties. The rationale for risk limits potentially conflicts with the development of business volume. Banking foundations are about trust and relationships with customers. A continuous relationship allows business to be carried on. The business rationale aims at developing such relationships through new transactions. The risk rationale lies in limiting business volume because the latter implies risks. ‘Name lending’ applies to big corporations, with an excellent credit standing. Given the high quality of risk and the importance of business opportunities with big customers, it gets harder to limit volume. This is the opposite of a limit rationale. In addition, allowing large credit limits is necessary in other instances. For instance, banks need to develop business relationships with a small number of big players, for their own needs, such as setting up hedges for interest rate risk. The same principles apply to controlling market risk. In these cases, to limit the potential loss of adverse markets moves, banks set upper bounds to sensitivities of instruments and portfolios. Sensitivities are changes in value due to forfeit shocks on market parameters such as interest rates, foreign exchange rates or equity indexes. Traders comply with such limits through trading and hedging their risks. Limits depend on expectations about market conditions and trading book exposures. To control the interest rate risk of the banking portfolio, limits apply to the sensitivities of the interest income or the Net Present Value (NPV, the mark-to-market valuation of assets minus liabilities) to shocks on interest rates. By bounding these values, banks limit the adverse movements of these target variables. Risk limits imply upper bounds on business volume, except when it is possible to hedge risks and avoid any excess exposure over limits. This is feasible for market transactions and for hedging interest rate risk, unless the cost of setting up the hedge is too high. For credit risk, there was no hedge until recently. Today, insurance, guarantees, credit derivatives and securitization offer a wide range of techniques for taking on more risk while still complying with risk limits.

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Return Benchmarks

The target profitability of the bank provides signals to business units for making business. Classical profitability measures include interest income and fees for the banking portfolio, Return On Equity (ROE) for the bank and for individual transactions, using the former Cooke ratio to determine the capital loading of a transaction or a portfolio, and Return On Assets (ROA) relating income to the size of the exposure. For the trading book, the profitability is in Profit and Loss (P&L), independent of whether sales of assets occur or not. Such measures have existed for a long time. Nevertheless, such measures fall short of addressing the issue of the trade-off between risk and return. However, only the risk and return profile of transactions or portfolios is relevant because it is easy to sacrifice or gain return by altering risk. Risk-based pricing refers to pricing differentiation based on risks. It implies that risks be defined at the transaction level, the subportfolio level and the entire bank portfolio level. Two systems are prerequisites for this process: the capital allocation system, which allocates risks; the funds transfer pricing system, which allocates income. The allocation of income and risks applies to transactions, business lines, market segments, customers or product lines. The specifications and roles of these two tools are expanded in Chapters 51 and 52 (capital allocation) and Chapters 26 and 27 (funds transfer pricing). Pricing benchmarks are based on economic transfer prices in line with the cost of funds. The Funds Transfer Pricing (FTP) system defines such economic prices and to which internal exchanges of funds they apply. Benchmark transfer prices generally refer to market rates. With internal prices, the income allocation consists simply of calculating the income as the difference between interest revenues and these prices. The target overall profitability results in target mark-ups over these economic prices. Risk allocation is similar to income allocation, except that it is less intuitive and more complex because risks do not add up arithmetically as income does. Capital allocation is a technique for allocating risks based on objective criteria. Risk allocations are capital allocations in monetary units. Capital refers to ‘economic capital’, or the amount of capital that matches the potential losses as measured by the ‘Value at Risk’ (VaR) methodology. Both transfer pricing and capital allocation systems are unique devices that allow interactions between risk management and business lines in a consistent bank-wide risk management framework. If they do not have strong economic foundations, they put the entire credibility of the risk management at stake and fail to provide the bottom-up and top-down links between ‘global’ targets and limits and ‘local’ business targets and limits.

PROCESS #2: DECISION-MAKING (EX ANTE PERSPECTIVE) The challenge for decision-making purposes is to capture risks upstream in the decision process, rather than downstream, once decisions are made. Helping the business decision process necessitates an ‘ex ante’ perspective, plus adequate tools for measuring and pricing risk consistently. Risk decisions refer to transactions or business line decisions, as well as portfolio decisions. New transactions, portfolio rebalancing and portfolio restructuring through securitizations (sales of bundled assets into the markets) are risk decisions. Hedging decisions effectively alter the risk–return profiles of transactions or of the entire portfolio. Decisions refer to both ‘on-balance sheet’, or business, decisions and

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‘off-balance sheet’, or hedging, decisions. Without a balanced risk–return view, the policy gets ‘myopic’ in terms of risks, or ignores the effect on income of hedging transactions.

On-balance Sheet Actions On-balance sheet actions relate to both new business and the existing portfolio of transactions. New business raises such basic questions as: Are the expected revenues in line with risks? What is the impact on the risk of the bank? Considering spreads and fees only is not enough. Lending to high-risk lenders makes it easy to generate margins, but at the expense of additional risks which might not be in line with income. Without some risk–return measures, it is impossible to solve the dilemma between revenues and volume, other than on a judgmental basis. This is what banks have done, and still do, for lack of better measures of risks. Banks have experience and perception of risks. Nevertheless, risk measures, combined with judgmental expertise, provide benchmarks that are more objective and help solve dilemmas such as volume versus profitability. Risk measures facilitate these decisions because they shed new light on risks. A common fear of focusing on risks is the possibility of discouraging risk taking by making risks explicit. In fact, the purpose is just the opposite. Risk monitoring can encourage risk taking by providing explicit information on risks. With unknown risks, prudence might prevail and prevent risk-taking decisions even though the profitability could well be in line with risks. When volume is the priority, controlling risks might become a second-level priority unless risks become more explicit. In both cases, risk models provide information for taking known and calculated risks. Similar questions arise ex post, once decisions are made. Since new business does not influence income and risks to an extent comparable to the existing portfolio, it is important to deal with the existing portfolio as well. Traditionally, periodical corrective actions, such as managing non-performing loans and providing incentives to existing customers to take advantage of new services, help to enhance the risk–return profile. Portfolio management extends this rationale further to new actions, such as securitizations that off-load risk in the market, syndications, loan trading or hedging credit risk, that were not feasible formerly. This is an emerging function for banks, which traditionally stick to the ‘originate and hold’ view of the existing portfolio, detailed further below.

Off-balance Sheet Actions Off-balance sheet recommendations refer mainly to hedging transactions. Asset–Liability Management (ALM) is in charge of controlling the liquidity and interest rate risk of the banking portfolio and is responsible for hedging programmes. Traders pursue the same goals when using off-balance instruments to offset exposures whenever they need to. Now, loan portfolio management and hedging policies also take shape, through credit derivatives and insurance. Hedging makes extensive use of derivative instruments. Derivatives include interest rate swaps, currency swaps and options on interest rates, if we consider interest rate only. Credit derivatives are new instruments. All derivatives shape both risk and return since they generate costs, the cost of hedging, and income as well because they capture the underlying market parameters or asset returns. For instance, a swap receiving the variable

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rate and paying a fixed rate might reduce the interest exposure and, simultaneously, generate variable rate revenues. A total return swap exchanges the return of a reference debt against a fixed rate, thereby protecting the buyer from capital losses of the reference asset in exchange for a return giving up the asset capital gains. Setting up such hedges requires a comprehensive view on how they affect both risk and return dimensions.

Loan Portfolio Management Even though banks have always followed well-known diversification principles, active management of the banking portfolio remained limited. Portfolio management is widely implemented with market transactions because diversification effects are obvious, hedging is feasible with financial instruments and market risk quantification is easy. This is not the case for the banking portfolio. The classical emphasis of credit analysis is at the transaction level, rather than the portfolio level, subject to limits defined by the credit department. Therefore, loan portfolio management is one of the newest fields of credit risk management. Incentives for the Development of Loan Portfolio Management

There are many incentives for developing portfolio management for banking transactions: • The willingness to make diversification (portfolio) effects more explicit and to quantify them. • The belief that there is a significant potential to improve the risk–return trade-off through management of the banking portfolio as a whole, rather than focusing only on individual banking transactions. • The growing usage of securitizations to off-load risk into the market, rather than the classical arbitrage between the on-balance sheet cost of funding versus the market cost of funds. • The emergence of new instruments to manage credit risk: credit derivatives. • The emergence of the loan trading market, where loans, usually illiquid, become tradable over an organized market. Such new opportunities generate new tools. Portfolio management deals with the optimization of the risk–return profile by altering the portfolio structure. Classical portfolio management relies more on commercial guidelines, on a minimum diversification and/or aims at limiting possible risk concentrations in some industries or with some big customers. New portfolio management techniques focus on the potential for enhancing actively the profile of the portfolio and on the means to achieve such goals. For example, reallocating exposures across customers or industries can reduce risk without sacrificing profitability, or increase profitability without increasing risks. This is the familiar technique of manipulating the relative importance of individual exposures to improve the portfolio profile, a technique well known and developed for market portfolios. The first major innovation in this area is the implementation of portfolio models providing measures of credit risk diversification. The second innovation is the new flexibility to shape the risk profile of portfolios through securitization, loan sales and usage of

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credit derivatives. Credit derivatives are instruments based on simple mechanisms. They include total return swaps, default options and credit spread swaps. Total return swaps exchange the return of any asset against some market index. Default options provide compensation in the event of default of a borrower, in exchange for the payment of a premium. Credit spread swaps exchange one market credit spread against another one. Such derivatives can be used, as other traditional derivatives, to hedge credit risk. The Loan Portfolio Management Function

The potential gains of more active portfolio management are still subject to debate. Manipulating the ‘weights’ of commitments for some customers might look like pure theory, given the practice of relationship banking. Banks maintain continuous relationships with customers that they know well, and are willing to keep doing business with. Volumes are not so flexible. Hence, a debate emerged on the relative merits of portfolio management and relationship banking. There might be an apparent conflict between stable relations and flexible portfolio management. What would be the benefit of trading loans if it adversely influenced ‘relationship banking’? In fact, the opposite holds. The ability to sell loans improves the possibility of generating new ones and develops, rather than restricts, relationship banking. Once we start to focus on portfolio management, the separation of origination from portfolio management appears a logical step, although this remains a major organizational and technical issue. The rationale of portfolio management is global risk–return enhancement. This implies moving away from the traditional ‘buy and hold’ policy and not sticking to the portfolio structure resulting from the origination business unit. Portfolio management requires degrees of freedom to be effective. They imply separation, to a certain extent, from origination. The related issues are numerous. What would be the actual role of a portfolio management unit? Should the transfer of transactions from origination to portfolio management be extensive or limited? What would be the internal transfer prices between both units? Can we actually trade intangible risk reductions, modelled rather than observed, against tangible revenues? Perhaps, once risk measures are explicit, the benefits of such trade-offs may become more visible and less debated. Because of these challenges, the development of the emerging portfolio management function is gradual, going through various stages: • • • • •

Portfolio risk reporting. Portfolio risk modelling. A more intensive usage of classical techniques such as syndication or securitizations. New credit derivatives instruments and loan trading. Separation of portfolio management and origination, since both functions differ in their perspective and both can be profit centres.

PROCESS #3: RISK–RETURN MONITORING (EX POST PERSPECTIVE) The monitoring and periodical reviews of risks are a standard piece of any controlling system. They result in corrective actions or confirmations of existing guidelines. For

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credit risk, the monitoring process has been in existence since banks started lending. Periodical reviews of risk serve to assess any significant event that might change the policy of the bank with respect to some counterparties, industries or countries. Monitoring systems extend to early warning systems triggering special reviews, including borrowers in a ‘watch list’, up to provisioning. Corrective actions avoid further deteriorations through restructuring of individual transactions. Research on industries and countries, plus periodical reviews result in confirmation of existing limits or adjustments. Reviews and corrective actions are also event-driven, for example by a sudden credit standing deterioration of a major counterparty. Analogous processes apply for market risk and ALM. A prerequisite for risk–return monitoring is to have measures of risk and return at all relevant levels, global, business lines and transactions. Qualitative assessment of risk is insufficient. The challenge is to implement risk-based performance tools. These compare ex post revenues with the current risks or define ex ante which pricing would be in line with the overall target profitability, given risks. The standard tools for risk-adjusted performance, as well as risk-based pricing, are the RaRoC (Risk-adjusted Return on Capital) and SVA (Shareholders Value Added) measures detailed in Chapters 53 and 54. Risk-based performance allows: • Monitoring risk–return profiles across business lines, market segments and customers, product families and individual transactions. • Making explicit the possible mispricing of subportfolios or transactions compared to what risk-based pricing would be. • Defining corrective or enhancement actions. Defining target risk-adjusted profitability benchmarks does not imply that such pricing is effective. Competition might not allow charging risk-based prices without losing business. This does not imply that target prices are irrelevant. Mispricing is the gap between target prices and effective prices. Such gaps appear as RaRoC ratios or SVA values not in line with objectives. Monitoring ex post mispricing serves the purpose of determining ex post what contributes to the bank profitability on a risk-adjusted basis. Without mispricing reports, there would be no basis for taking corrective actions and revising guidelines.

BANK-WIDE RISK MANAGEMENT The emergence of risk models allowed risk management to extend ‘bank-wide’. ‘Bankwide’ means across all business lines and risks. Bank-wide risk management implies using the entire set of techniques and models of the risk management ‘toolbox’. Risk management practices traditionally differ across risks and business lines, so that a bankwide scope requires a single unified and consistent framework.

Risk Management Differs across Risks Risk management appears more fragmented than unified. This contrasts with the philosophy of ‘bank-wide risk management’, which suggests some common grounds and common frameworks for different risks. Risk practice differs across business lines. The

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market risk culture and the credit culture seem not to have much in common yet. Market culture is quantitative in nature and model-driven. By contrast, the credit culture focuses on the fundamentals of firms and on the relationship with borrowers to expand the scope of services. Moreover, the credit culture tends to ignore modelling because of the challenge of quantifying credit risk. In the capital markets universe, trading risks is continuous since there is a market for doing so. At the opposite end of the spectrum, the ‘buy and hold’ philosophy prevails in the banking portfolio. ALM does not have much to do with credit and market risk management, in terms of goals as well as tools and processes. It needs dedicated tools and risk measures to define a proper funding and investment policy, and to control interest rate risk for the whole bank. Indeed, ALM remains very different from market and credit risk and expands over a number of sections in this book. Indeed, risk models cannot yet pretend to address the issues of all business lines. Risk-adjusted performance is easier to implement in the middle market than with project finance or for Leveraged Buy-Outs (LBOs), which is unfortunate because risk-adjusted performances would certainly facilitate decision-making in these fields. On the other hand, the foundations of risk measures seem robust enough to address most risks, with techniques based on the simple ‘potential loss’, or VaR, concept. Making the concept instrumental is quite a challenge, as illustrated by the current difficulties of building up data for credit risk and operational risk. But the concept is simple enough to provide visibility on which roads to follow to get better measures and understand why crude measures fail to provide sound bases for decision-making and influencing, in general, the risk management processes. Because of such limitations, new best practices will not apply to all activities in the near future, and will presumably extend gradually.

Different Risks Fit into a Single Framework The view progressively expanded in this book is that risk management remains differentiated, but that all risks tend to fit in a common basic framework. The underlying intuition is that many borders between market and credit risk tend to progressively disappear, while common concepts, such as VaR and portfolio models, apply gradually to all risks. The changing views in credit risk illustrate the transformation:

• Credit risk hedges now exist with credit derivatives. • The ‘buy and hold’ culture tends to recede and the credit risk management gets closer to the ‘trading philosophy’. • The portfolio view of credit risk gains ground because it brings some new ways of enhancing the risk–return profile of the banking portfolio. • Accordingly, the ‘model culture’ is now entering the credit risk sphere and increasingly interacts with the credit culture. • The building up of data has been productive for those banks that made it a priority. • Regulations using quantitative measures of risk perceived as intangibles gained acceptance in the industry and stimulate progress.

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Credit Risk

Interest Rate Risk

Global Integration

Liquidity Risk

FIGURE 4.6

Market Risk

+ + Other Risks

Global integration of differentiated risks and risk management techniques

A preliminary conclusion is that there are now sufficient common grounds across risks to capture them within a common framework. This facilitates the presentation of concepts, methodologies, techniques and implementations, making it easier to wrap around a framework showing how tools and techniques capture differentiated risks and risk management processes in ways that facilitate their global and comprehensive integration (Figure 4.6).

5 Risk Management Organization

The development of bank-wide risk management organization is an ongoing process. The original traditional commercial bank organization tends to be dual, with the financial sphere versus the business sphere. The business lines tend to develop volume, sometimes at the expense of risks and profitability, while the financial sphere tends to focus on profitability, with dedicated credit and market risk monitoring units. This dual view is fading away with the emergence of new dedicated functions implemented bank-wide. Bank-wide risk management has promoted the centralization of risk management and a clean break between risk-taking business lines and risk-supervising units. Technical and organizational necessities foster the process. Risk supervision requires separating the supervising units from the business units, since these need to take more risk to achieve their profitability targets. Risk centralization is also a byproduct of risk diversification. Only post-diversification risks are relevant for the entire bank’s portfolio, and required capital, once the aggregation of individual risks diversified away a large fraction of them. The risk department emerged from the need for global oversight on credit risk, market risk and interest rate risk, now extending to operational risk. Separating risk control from business, plus the need for global oversight, first gave birth to the dedicated Asset–Liability Management (ALM) function, for interest rate risk, and later on stimulated the emergence of risk departments, grouping former credit and market risk units. While risk measuring and monitoring developed, other central functions with different focuses, such as management control, accounting, compliance with regulations, reporting and auditing, differentiated. This chapter focuses on the risk management functions, and illustrates how they emerged and how they relate to other functions. The modern risk management organization separates risk management units from business units. The banking portfolio generates interest rate risk, transferred from business lines, which have no control over interest rate movements, to the ALM unit which is in charge

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of managing it. The banking portfolio and ALM, when setting up hedges with derivatives, generate credit risk, supervised by the credit risk unit. The market risk unit supervises the trading portfolio. The portfolio management unit deals with the portfolio of loans as a whole, for risk control purposes and for a more active restructuring policy through actions such as direct sales or securitization. The emerging ‘portfolio management’ unit might necessitate a frontier with origination through loans management post-origination. It might be related to the risk department, or be a profit-making entity as well. The first section maps risk origination with risk management units supervising. The second section provides an overview of central functions, and of the risk department. The third section details the ALM role. The fourth section focuses on risk oversight and supervision by the risk department entity. The fifth section details the emerging role of the ‘loan portfolio management’ unit. The last section emphasizes the critical role of Information Technology (IT) for dealing with a much wider range of models, risk data warehouses and risk measures than before.

MAPPING ORGANIZATION WITH RISK MANAGEMENT NECESSITIES Figure 5.1 shows who originates what risks and which central functions supervise them. Various banks have various organizations. There is no unique way of organizing the RISKS

Credit

Market

Liquidity

Interest Rate

Risk Management General Management

Guidelines and goals Guidelines and goals

Portfolio Management

Portfolio risk−return profile Portfolio risk- return profile

Risk Risks Department Department ALM ALM

Risk Origination Commercial Bank Investment Bank Market

FIGURE 5.1

Mapping risk management with business lines and central functions

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risk management processes. Available risk models and management tools and techniques influence organization schemes, as various stages of differentiation illustrate.

THE DIFFERENTIATION OF CENTRAL FUNCTIONS The functions of the different central units tend to differentiate from each other when going further into the details of risk and income actions. Figure 5.2 illustrates the differentiation process of central functions according to their perimeter of responsibilities. ALM Control

- Accounting - Regulatory compliance - Cost accounting - Budgeting and planning - Monitoring and control - Monitoring performances - Reporting to general management

- Liquidity and interest rate risk management - Measure and control of risks - Compliance - Hedging - Recommendations: balance sheet actions - Transfer pricing systems - ALCO - Reporting to general management

Credit Risk

- Credit policy - Setting credit risk limits and delegations - Assigning internal ratings - Credit administration (credit applications and documentation) - Credit decisions (credit committees) - Watch lists - Early warning systems - Reporting to general management

Risk Department

Risks Department

- Monitoring and control of all risks - Credit & market, in addition to ALM - Decision-making - Credit policy - Setting limits - Development of internal tools and risk data warehouses - Risk-adjusted performances - Portfolio actions - Reporting to general management

FIGURE 5.2

Market Risk

Portfolio Management

- Trading credit risk - Portfolio reporting - Portfolio restructuring - Securitizations - Portfolio actions - Reporting to general management

- Limits - Measure and control of risks - Compliance - Monitoring - Hedging - Business actions - Reporting to general management

Functions of central units and of the risk department

THE ALM FUNCTION ALM is the unit in charge of managing the interest rate risk and liquidity of the bank. It focuses essentially on the commercial banking pole, although the market portfolio also generates liquidity requirements.

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The ALM Committee (ALCO) is in charge of implementing ALM decisions, while the technical unit prepares all analyses necessary for taking decisions and runs the ALM models. The ALCO agenda includes ‘global balance sheet management’ and the guidelines for making the business lines policy consistent with the global policy. ALM addresses the issue of defining adequate structures of the balance sheet and the hedging programmes for liquidity and interest rate risks. The very first mission of ALM is to provide relevant risk measures of these risks and to keep them under control given expectations of future interest rates. Liquidity and interest rate policies are interdependent since any projected liquidity gap requires future funding or investing at an unknown rate as of today, unless setting up hedges today. ALM scope (Figure 5.3) varies across institutions, from a limited scope dedicated to balance sheet interest and liquidity risk management, to being a profit centre, in addition to its mission of hedging the interest rate and liquidity risks of the bank. In between, ALM extends beyond pure global balance sheet management functions towards a better integration and interaction with business policies.

FIGURE 5.3

Traditional

Tools, procedures and processes for managing globally the interest rate risk and the liquidity risk of commercial banking.

Wider scope

Tools, models and processes facilitating the implementation of both business and financial policy and an aid to decision-making at the global and business unit levels.

ALM policies vary across institutions

- Sometimes subordinated to business policy. - Sometimes active in business policy making. - Sometimes defined as a P&L unit with its own decisions, in addition to the hedging (risk reward) policy for the commercial bank.

ALM scope

The ALCO is the implementation arm of ALM. It groups heads or business lines together with the general bank management, sets up the guidelines and policies with respect to interest rate risk and liquidity risk for the banking portfolio. The ALCO discusses financial options for hedging risks and makes recommendations with respect to business policies. Both financial and business policies should ensure a good balance between ‘on-balance sheet’ actions (business policy and development) and ‘off-balance sheet’ actions (hedging policies). Financial actions include funding, investing and hedging. Since these are market transactions, they always have an influence on the interest income, creating a trade-off between risk and interest income. Technical analyses underlie the recommendations, usually conducted by a technical unit which prepares ALCO meetings and recommendations.

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Other units interact with ALM, such as a dedicated unit for funding, in charge of raising funds, or the treasury, which manages the day-to-day cash flows and derivatives, without setting the basic guidelines of interest rate risk management. Management control and accounting also interact with ALM to set the economic transfer prices, or because of the Profit and Loss (P&L) incidence of accounting standards for hedging operations. The transfer pricing system is in charge of allocating income between banking business units and transactions, and is traditionally under ALM control.

THE RISK DEPARTMENT AND OVERSIGHT ON RISKS Dedicated units normally address credit risk and market risk, while ALM addresses the commercial banking side of interest rate risk and funding. Risk supervisors should be independent of business lines to ensure that risk control is not under the influence of business and profit-making policies. This separation principle repeatedly appears in guidelines. Separation should go as far as setting up limits that preclude business lines from developing new transactions because they would impair the bank’s risk too much. The functions of the risk department include active participation in risk decision-making, with a veto power on transactions, plus the ability to impose restructuring of transactions to mitigate their risk. This empowers the risk-supervising unit to full control of risks, possibly at the cost of restricting risky business. Getting an overview of risks is feasible with separate entities dedicated to each main risk. However, when reaching this stage, it sounds natural to integrate the risk supervision function into a risk department, while still preserving the differentiation of risk measures and management. There are several reasons for integration in a single department, some of them organizational and others technical. From an organizational standpoint, integration facilitates the overview of all risks across all business lines. In theory, separate risk functions can do the same job. In practice, because of process harmonization, lack of interaction between information systems and lack of uniform reporting systems, it is worth placing the differentiated entities under the same control. Moreover, separate risk control entities, such as market and credit risk units, could possibly deal separately with business lines, to the detriment of the global policy. When transactions get sophisticated, separating functions could result in disruptions. A single transaction can trigger multiple risks, some of them not obvious at first sight. Market transactions and ALM hedges create credit risk. Investment banking activities or structured finance generate credit risk, interest rate risk and operational risk. Multiple views on risks might hurt the supervisory process, while a single picture of all risks is a simple practical way to have a broader view on all risks triggered by the same complex transactions. For such reasons the risk department emerged in major institutions, preserving the differentiation of different risks, but guaranteeing the integration of risk monitoring, risk analyses and risk reporting. Under this scheme, the risk department manages and controls risks, while business lines generate risks (Figure 5.4). Each major business pole can have its own risk unit, interacting with the risk department. Few banks set up a portfolio management unit, but many progress along such lines. The portfolio management unit is in charge of restructuring the portfolio, after origination, to enhance its risk–return profile. The first stages of portfolio management extend from

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Risk Department Credit risk

Market risk

Markets

ALM

Business Lines

Other risks Products

FIGURE 5.4

Risk department, risk units and business lines

reporting on risks and portfolio structure, to interaction with the limit setting process and the definitions of credit policy guidelines. At further stages, the portfolio management unit trades credit risk after origination through direct sales of loans, securitizations or credit risk hedging. Active portfolio management is a reality, with or without a dedicated unit, since all techniques and instruments allowing us to perform such tasks have been developed to a sufficient extent. The full recognition of the portfolio management unit would imply acting as a separate entity, with effective control over assets post-origination, and making it a profit centre. The goal is to enhance the risk–return profile of the entire portfolio. Transfers of assets from origination should not result in lost revenues by origination without compensation. The ultimate stage would imply full separation and transfer prices between origination and portfolio management, to effectively allocate income to each of these two poles. The risk department currently plays a major role in the gradual development of portfolio management. It has an overview of the bank’s portfolio and facilitates the innovation process of hedging credit risk. It has a unique neutral position for supervising rebalancing of the portfolio and altering of its risk–return profile. Nevertheless, the risk department cannot go as far as making profit, as a portfolio management unit should ultimately do, since this would negate its neutral posture with respect to risk. It acts as a facilitator during the transition period when risk management develops along these new dimensions.

INFORMATION TECHNOLOGY Information technology plays a key role in banks in general, and particularly in risk management. There are several reasons for this: • Risk data is continuously getting richer. • New models, running at the bank-wide scale, produce new measures of risk. • Bringing these measures to life necessitates dedicated front-ends for user decisionmaking. Risk data extends from observable inputs, such as market prices, to new risk measures such as Value at Risk (VaR). New risk data warehouses are required to put together data and to organize the data gathering process, building up historical data on new inputs, such as those required by the 2001 Basel Accord. VaR necessitates models that did not exist a few years ago, running at the entire bank scale to produce market risk and credit risk measures. IT scope extends to the implementation of these models, and to assembling their outputs in a usable form for

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end-users. Due to the scale of operations of bank systems, creating the required risk data warehouse with the inputs and outputs of these models, and providing links to front-ends and reporting modules for end-users, are major projects. Bringing the information to life is an IT challenge because it requires new generation tools capable of on-line queries and analyses embedded in front-ends and reporting. Without such aids, decision-makers might simply ignore the information because of lack of time. Since new risk measures are not intuitive, managers not yet familiar with them need tools to facilitate their usage. ‘On-Line Analysis and Processing’ (OLAP) systems are critical to forward relevant information to end-users whenever they need it. Multiple risk measures generate several new metrics for risks, which supplement the simple and traditional book exposures for credit risk, for example. The risk views now extend to expected and unexpected losses, capital and risk allocations, in addition to mark-to-market measures of loan exposures (the major building blocks of risk models are described in Chapter 9). Profitability measures also extend to new dimensions, from traditional earnings to risk-adjusted measures, ex ante and ex post, at all levels, transactions, subportfolios and the bank’s portfolio. Simultaneously, business lines look at other dimensions, transactions, product families, market segments or business unit subportfolios. Business Reporting

Risk−Return Reporting

Business units

Exposures

Markets

Expected loss Ratings

Product families Transactions

Capital RaRoC ...

.....

..... Reports + 'OLAP' Slicing & Dicing

Drill-Down What-If

FIGURE 5.5

IT and portfolio reporting

Combining several risk dimensions with profitability dimensions and business dimensions has become a conceptual and practical challenge. Multidimensional views of the bank’s portfolio have become more complex to handle. New generation IT systems can handle the task. However, IT still needs to design risk, profitability and business reports so that they integrate smoothly within the bank’s processes. Multidimensional reporting requires a more extensive usage of new tools:

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• Slicing and dicing the portfolio across any one of these dimensions, or combinations of them, such as reporting the risk-adjusted profitability by market segment, business unit or both. • A drilling-down function to find out which transactions are the source of risk for subsets of transactions. The simplest example would be to find which transactions and obligors contribute most to the risk of a business unit, or simply understanding what transactions make a risk metric (exposure, expected loss, risk allocation) higher than expected. • ‘What if’ simulation capabilities to find out the outcomes of various scenarios such as adding or withdrawing a transaction or a business line, conducting sensitivity analyses to find which risk drivers influence risk more, or when considering rebalancing the bank’s subportfolios. New risk softwares increasingly embed such functions for structuring and customizing reports. Figure 5.5 illustrates the multiple dimensions and related reporting challenges. To avoid falling into the trap of managing reports rather than business, on-line customization is necessary to produce the relevant information on time. Front-end tools with ‘what if’ and simulation functions, producing risk–return reports both for the existing portfolio and new transactions, become important for both the credit universe and the market universe.

SECTION 4 Risk Models

6 Risk Measures

Risk management relies on quantitative measures of risks. There are various risk measures. All aim at capturing the variation of a given target variable, such as earnings, market value or losses due to default, generated by uncertainty. Quantitative indicators of risks fall into three types: • Sensitivity, which captures the deviation of a target variable due to a unit movement of a single market parameter (for instance, an interest rate shift of 1%). Sensitivities are often market risk-related because they relate value changes to market parameters, which are value drivers. The interest rate gap is the sensitivity of the interest margin of the banking portfolio to a forfeit move of the yield curve. Sensitivities are variations due to forfeit moves of underlying parameters driving the value of target variables. • Volatility, which captures the variations around the average of any random parameter or target variable, both upside and downside. Unlike forfeit movements, volatility characterizes the varying instability of any uncertain parameters, which forfeit changes ignore. Volatility measures the dispersion around its mean of any random parameter or of target variables, such as losses for credit risk. • Downside measures of risk, which focus on adverse deviations only. They characterize the ‘worst-case’ deviations of a target variable, such as earnings, market values or credit losses, with probabilities for all potential values. Downside risk measures require modelling to have probability distributions of target variables. The ‘Value at Risk’ (VaR) is a downside risk measure. It is the adverse deviation of a target variable, such as the value of a transaction, not exceeded in more than a preset fraction of all possible future outcomes. Downside risk is the most ‘comprehensive’ measure of risk. It integrates sensitivity and volatility with the adverse effect of uncertainty (Figure 6.1). This chapter details related techniques and provides examples of well-known risk measures. In spite of the wide usage of risk measures, risks remain intangible, making

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Quantitative indicators of risk

FIGURE 6.1

1

Sensitivity

2

Volatility

3

Downside risk or VaR

Risk measures

the distinction between risks and risk measures important. The first section draws a line between intangible risks and quantitative measures. The next sections provide examples of sensitivity and volatility measures. The final section discusses VaR and downside risk.

MEASURING UNCERTAINTY Not all random factors that alter the environment and the financial markets—interest rates, exchange rates, stock indexes—are measurable. There are unexpected and exceptional events that radically and abruptly alter the general environment. Such unpredictable events might generate fatal risks and drive businesses to bankruptcy. The direct way to deal with these types of risks is stress scenarios, or ‘worst-case’ scenarios, where all relevant parameters take extreme values. These values are unlikely, but they serve the purpose of illustrating the consequences of such extreme situations. Stress testing is a common practice to address highly unlikely events. For instance, rating agencies use stress scenarios to assess the risk of an industry or a transaction. Most bank capital markets units use stress scenarios to see how the market portfolio would behave. Extreme Value Theory (see Embrechts et al., 1997 plus the discussion on stress testing market VaR models) helps in modelling extreme events but remains subject to judgmental assessment. Quantitative risk measures do not capture all uncertainties. They depend on assumptions, which can underestimate some risks. Risks depend on qualitative factors and on intangible events, which quantification does not fully capture. Any due diligence of risks combines judgments and quantitative risk assessment. Quantitative techniques address only measurable risks, without being substitutes for judgment. Still, the current trend is to focus on quantitative measures, enhancing them and extending their range throughout all types of risks. First, when data becomes available, risks are easier to measure and some otherwise intangible risks might become more prone to measurement. Second, when it is difficult to quantify a risk, it might be feasible to qualify it and rank comparable risks, as ratings agencies do. Finally, risk measures became more critical when regulators made it clear that they should provide the basis for capital requirements protecting the bank against unfavourable conditions. Quantitative measures gain feasibility and credibility for all these reasons. This chapter looks at the basic quantified risk measures, before tackling VaR in the next chapter.

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SENSITIVITY Sensitivities are ratios of the variation of a target variable, such as interest margin or change in mark-to-market values of instruments, to a forfeit shock of the underlying random parameter driving this change. This property makes them very convenient for measuring risks, because they link any target variable of interest to the underlying sources of uncertainty that influence these variables. Examples of underlying parameters are interest rates, exchange rates and stock prices. Market risk models use sensitivities widely, known as the ‘Greek letters’ relating various market instrument values to the underlying market parameters that influence them. Asset–Liability Management (ALM) uses gaps, which are sensitivities of the interest income of the banking portfolio to shifts of interest rates. Sensitivities have well-known drawbacks. First, they refer to a given forfeit change of risk drivers (such as a 1% shift of interest rates), without considering that some parameters are quite unstable while others are not. Second, they depend on the prevailing conditions, the value of the market parameters and of assets, making them proxies of actual changes. This section provides definitions and examples.

Sensitivity Definitions and Implications Percentage sensitivities are ratios of relative variations of values to the same forfeit shock on the underlying parameter. For instance, the sensitivity of a bond price with respect to a unit interest rate variation is equal to 5. This sensitivity means that a 1% interest rate variation generates a relative price variation of the bond of 5 × 1% = 5%. A ‘value’ sensitivity is the absolute value of the change in value of an instrument for a given change in the underlying parameters. If the bond price is 1000, its variation is 5% × 1000 = 50. Let V be the market value of an instrument. This value depends on one or several market parameters, m, that can be prices (such as indexes) or percentages (such as interest rates). By definition: s(% change of value) = (V /V ) × m S(value) = (V /V ) × V × m Another formula is s(% change of value) = (V /V ) × (m/m), if the sensitivity measures the ‘return sensitivity’, such as for stock return sensitivity to the index return. For example, if a stock return varies twice as much as the equity index, this last ratio equals 2. A high sensitivity implies a higher risk than a low sensitivity. Moreover, the sensitivity quantifies the change. The sensitivity is only an approximation because it provides the change in value for a small variation of the underlying parameter. It is a ‘local’ measure because it depends on current values of both the asset and the market parameter. If they change, both S and s do1 . 1 Formally,

the sensitivity is the first derivative of the value with respect to m. The next order derivatives take care of the change in the first derivative.

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Sensitivities and Risk Controlling Most of the random factors that influence the earnings of a bank are not controllable by the bank. They are random market or environment changes, such as those that increase the default probabilities of borrowers. A bank does not have any influence on market or economic conditions. Hence, the sources of uncertainty are beyond control. By contrast, it is possible to control the ‘exposure’ or the ‘sensitivities’ to these outside sources of uncertainties. There are two ways to control risk: through risk exposures and through sensitivities. Controlling exposures consists of limiting the size of the amount ‘at risk’. For credit risk, banks cap individual exposures to any obligor, industry or country. In doing so, they bound losses per obligor, industry or country. However, the obvious drawback lies in limiting business volume. Therefore, it should be feasible to increase business volume without necessarily increasing credit risk. An alternative technique for controlling risk is by limiting the sensitivity of the bank earnings to external random factors that are beyond its control. Controlling risk used to be easier for ALM and market risk, using derivatives. Derivatives allow banks to alter the sensitivities to interest rates and other market risks and keep them within limits. For market risk, hedging exposures helps to keep the various sensitivities (the ‘Greeks’) within stated limits (Chapter 30 reviews the main sensitivities). Short-selling bonds or stocks, for example, offsets the risk of long positions in stocks or bonds. For ALM, banks control the magnitude of ‘gaps’, which are the sensitivities of the interest margin to changes in interest rates. Sensitivities have straightforward practical usages. For example, a positive variable interest rate gap of 1000 implies that the interest margin changes by 1000 × 1% = 10 if there is a parallel upward shift of the yield curve. The same techniques now apply with credit derivatives, which provide protection against both the deterioration of credit standing and the default of a borrower. Credit derivatives are insurances sold to lenders by sellers of these protections. The usage of credit derivatives extends at a fast pace because there was no way, until recently, to limit credit risk without limiting size, with a mechanical adverse effect on business volume. Other techniques, notably securitizations, expand at a very fast pace when players realize they could off-load risk and free credit lines for new business.

VOLATILITY In order to avoid using a unique forfeit change in underlying parameters, independently of the stability or instability of such parameters, it is possible to combine sensitivities with measures of parameter instability. The volatility characterizes the stability or instability of any random variables. It is a very common statistical measure of the dispersion around the average of any random variable such as market parameters, earnings or mark-to-market values. Volatility is the standard deviation of the values of these variables. Standard deviation is the square root of the variance of a random variable (see Appendix).

Expectations, Variance and Volatility The mean, or the expectation, is the average of the values of a variable weighted by the probabilities of such values. The variance is the sum of the squared deviations around

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the mean weighted by the probabilities of such deviations. The volatility is the square root of the variance2 . When using time series of historical observations, the practice is to assign to each observation the same weight. The arithmetic mean is an estimator of the expectation. The arithmetic average of the squared deviations from this mean is the historical variance. Standard formulas apply to obtain these statistics. The appendix to this chapter illustrates basic calculations. Here, we concentrate on basic definitions and their relevance to volatility for risk measures.

Probability and Frequency Distributions The curve plotting the frequencies of occurrences for each of the possible values of the uncertain variable is a frequency distribution. It approximates the actual probability distribution of the random variable. The x-axis gives the possible values of the parameter. The y-axis shows either the number of occurrences of this given value, or the percentage over the total number of observations. Such frequency distributions are historical or modelled distributions. The second class of distributions uses well-known curves that have attractive properties, simplifying considerably the calculation of statistics characterizing these distributions. Theoretical distributions are continuous, rather than discrete3 . Continuous distributions often serve as good approximations of observed distributions of market values. A probability distribution of a random variable is either a density or a cumulative distribution. The density is the probability of having a value within any very small band of values. The cumulative function is the probability of occurrence of values between the lowest and a preset upper bound. It cumulates all densities of values lower than this upper bound. ‘pdf’ and ‘cdf’ designate respectively the probability density function and the cumulative density function. A very commonly used theoretical distribution is the normal curve, with its familiar bell shape (Figure 6.2). This theoretical distribution actually looks like many observed Probability The dispersion of values around the mean increases with the 'width' of the curve

Mean

FIGURE 6.2 2 Expectation

Values of the random variable

Mean and dispersion for a distribution curve

and variance are also called the first two ‘moments’ of a probability distribution. example of a simple discrete distribution is the distribution of loss under default risk. There are only two events: default or no default, with assigned probabilities such as 1% and 99%. There is a 1% chance that the lender suffers from the loss under default, while the loss remains zero under no default.

3 An

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distributions of a large number of random phenomena. The two basic statistics, mean and standard deviation, are sufficient to fully determine the entire normal distribution. The normal distribution N(µ, σ ) has mean µ and standard deviation σ . The pdf of the normal distribution is: √ Pr(X = X) = (1/σ 2π) exp[−(X − µ)2 /2σ 2 ] Bold letters (X) designate random variables and italic letters (X) a particular value of the random variable. The standardized normal distribution N(0, 1) has a mean of 0 and a variance of 1. The variable Z = (X − µ)/σ follows a standardized normal distribution N(0, 1), with probability density: √ Pr(Z = Z) = (1/ 2π) exp[−Z 2 /2] The cumulative normal distribution is available from tables or easily calculated with proxy formulas. The cumulative standardized normal distribution is (0, 1). The normal distribution is parametric, meaning that it is entirely defined by its expectation and variance. The normal distribution applies notably to relative stock price return, or ratio of price variations to the initial price over small periods. However, the normal distribution implies potentially extreme negative values, which are inconsistent with observed market prices. It can be shown that the lognormal distribution, which does not allow negative values, is more consistent with the price behaviour. The lognormal distribution is such that the logarithm of the random variable follows a normal distribution. Analytically, ln(X) follows N(µ, σ ), where ln is the Napierian logarithm and N(µ, σ ) the pdf of the normal distribution. The lognormal distribution is asymmetric, unlike the normal curve, because it does not allow negatives at the lower end of values, and permits unlimited positive values4 . Other asymmetric distributions have become popular when modelling credit risk. They are asymmetric because the frequency of small losses is much higher than the frequency of large losses within a portfolio of loans. The moments of a distribution characterize its shape. The moments are the weighted averages of the deviations from the mean, elevated to power 2, 3, 4, etc., using the discrete probabilities of discrete values, or the probability densities as weights. The first moment is the expectation, or mean, of the function. The second moment is the variance. It characterizes dispersion around the mean. The square root of the variance is the standard deviation. It is identical to the ‘volatility’. The third moment is skewness, which characterizes departure from symmetry. The fourth moment is kurtosis, which characterizes the flatness of the distribution.

Volatility The volatility measures the dispersion of any random variable around its mean. It is feasible to calculate historical volatility using any set of historical data, whether or not they follow a normal distribution. It characterizes the dispersion of market parameters, such as that of interest rates, exchange rate and equity index, because the day-to-day observations are readily available. Volatilities need constant updating when new observations are available. An alternative measure of volatility is the implicit volatility embedded in options prices. It derives 4 When

X takes extreme negative values, the logarithm tends towards zero but remains positive.

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the value of volatility from the theoretical relationship (the Black–Scholes formula) of observed option prices with all underlying parameters, one of them being the volatility of the underlying. A benefit of implicit volatilities is that they are forward looking measures, as prices are. A drawback of this approach is that implicit volatilities are fairly volatile, more than historical volatility. For other than market data, such as accounting earnings, the frequency of observations is more limited. It is always possible to calculate a standard deviation with any number of observations. Nevertheless, a limited number of observations might result in a distorted image of the dispersion. The calculation uses available observations, which are simply a sample from the entire distribution of values. If the sample size is large, statistics calculated over the sample are good proxies of the characteristics of the underlying distributions. When the sample size gets smaller, there is a ‘sampling error’. Obviously, the sampling error might get very large when we have very few observations. The calculation remains feasible but becomes irrelevant. The ‘Earnings at Risk’ (EaR) approach for estimating economic capital (Chapter 7) uses the observed volatility of earning values as the basis for calculating potential losses, hence the capital value capable of absorbing them. Hence, even though accounting data is scarcer than market data, the volatility of earnings might serve some useful purpose in measuring risk.

Historical Volatilities The calculation of historical mean and volatility requires time series. Defining a time series requires defining the period of observation and the frequency of observations. For example, we can observe daily stock returns for an entire year, roughly 250 working days. The choice of frequency determines the nature of volatility. A daily volatility results from daily observations, a weekly volatility from weekly observations, and so on. We are simply sampling from an underlying distribution a variable number of daily observations. When a distribution does not change over time, it is ‘stationary’. Daily volatilities are easier to calculate since we have more information. Because the size of the sample varies greatly according to the length of the period, the sampling error—the difference between the unobservable ‘real’ value of volatility and that calculated—is presumably greater with short observation periods such as 1 or 3 months. Convenient rules allow an easy conversion of daily volatilities into monthly or yearly volatilities. These simple rules rely on assumptions detailed in Chapter 30. In essence, they assume that the random process is stable through consecutive periods. For instance, for a random stock return, this would mean that the distribution of the stock return is exactly the same from one period to another. Monthly volatilities are larger than daily volatilities because the changes over 1 month are likely to be larger than the daily √ changes. A practical rule states that the volatility over horizon T , σT , is equal to σ1day T , when T is in days. For example, the daily volatility of a stock return is 1%, measured from 252√daily observations over a 1-year period. The monthly volatility would be σ1month = σ1day 30 = 1% × 5.477 = 5.477%. The volatility increases with time, but less than proportionally5 . According to this rule, the multiples used √ 5 This formula σ = σ t 1 t applies when the possible values of a variable at t do not depend on the value at t − 1.

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TABLE 6.1 Period Multiple

Time coefficients applicable to volatility 1

2

3

4

5

6

7

8

9

10

1.000

1.414

1.732

2.000

2.236

2.449

2.646

2.828

3.000

3.162

to convert a one-period (1 day, for instance) volatility into a multiple-period volatility (1 year, rounded to 250 days, for example) are as given in Table 6.1. The formula is known as the ‘square root of time rule’. It requires specifying the base period, whether 1 day or 1 year. For instance, the 1-year volatility when the yearly volatility is, say, 10% is 14.14%. The ‘square root’ rule is convenient but applies only under restrictive assumptions. Techniques for modelling volatilities as a function of time and across different periods have developed considerably. The basic findings are summarized in the market risk chapter (Chapter 30). Because the rule implies an ever-increasing volatility over time, it does not apply beyond the medium term to random parameters that tend to revert to a long-term value, such as interest rates.

VOLATILITY AND DOWNSIDE RISK Risk materializes only when earnings deviate adversely, whereas volatility captures both upside and downside deviations. The purpose of downside risk measures is to capture loss, ignoring the gains. Volatility and downside risk relate to each other, but are not equivalent. Volatility of earnings increases the chances of losses, and this is precisely why it is a risk measure. However, if downside changes are not possible, there is volatility but no downside risk at all. A case in point is options. The buyer of an option has an uncertain gain, but no loss risk, when looking forward, once he has paid the price for acquiring the option (the premium). A call option on stock provides the right to purchase the stock at 100. If the stock price goes up to 120, exercise generates a profit of 120 − 100 = 20. The potential profit is random just as the stock price is. However, the downside risk is zero since the option holder does not exercise his option when the price falls below 100, and does not incur any loss (besides the upfront premium paid). However, the seller of the call option has a downside risk, since he needs to sell the stock to the buyer at 100, even when he has to pay a higher price to purchase the stock, unless he holds it already. The downside risk actually has two components: potential losses and the probability of occurrence. The difficulty is to assess these probabilities. Worst-case scenarios serve to quantify extreme losses. However, the chances of observing the scenarios are subjective. If someone else has a different perception of the environment uncertainty, he or she might consider that another scenario is more relevant or more likely. The measure of risk changes with the perception of uncertainty. For these reasons, downside risk measures necessitate the prior modelling of the probability distributions of potential losses.

APPENDIX: STATISTICS There are well-known standard formulas for calculating the mean, variance and standard deviation. Some definitions may appear complex at first sight. Actually, these statistics

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are easy to calculate, as shown in the examples below6 . Let X be the random variable, with particular values X. The formulas for calculating the mean and standard deviation of discrete observed values of a random variable, the usual case when using time series of observations, are given below. The random variable is X, the mean is E(X). With historical time series, probabilities are the frequencies of the observed values, eventually grouped in narrow bands. The assigned probability to a single value among n is 1/n. The expectation becomes: E(X) = Xi /n i

The volatility, or standard deviation, is: σ (X) = (1/n) [Xi − E(X)]2 i

In general, probabilities have to be assigned to values, for instance by assuming that the distribution curve is given. The corresponding formulas are given in any statistics textbook. The mean and the standard deviation depend on the probabilities pi assigned to each value Xi of the random variable X. The total of all probabilities is 100% since the distribution covers all feasible values. The mean is: E(X) = pi Xi /n i

The variance is the weighted average by the probabilities of the squared deviations from the mean. The volatility is the square root of this value. The volatility is equal to: σ (X) = pi [Xi − E(X)]2 i

The variance V (X) is identical to σ 2 . With time series, all pi are equal to 1/n, which results in the simplified formulas above. The example in Table 6.2 shows how to calculate a yearly volatility of earnings over a 12-year time series of earnings observations. The expectation is the mean of all observed values. The variance is the sum of squared deviations from the mean, and the standard deviation is the square root. The table gives a sample of calculations using these definitions. Monthly observations of accounting earnings are available for 1 year, or 12 observed values. Volatilities are in the same unit as the random variable. If, for instance, the exchange rate of the euro against the dollar is 0.9 USD/EUR, the standard deviation of the exchange rate is also expressed in USD/EUR, for instance 0.09 USD/EUR. The percentage volatility is the ratio of the standard deviation to the current value of the variable. For instance, the above 0.09 USD/EUR volatility is also equal to 10% of the current exchange rate since 6 Since

the average algebraic deviation from the mean is zero by definition. Squared deviations do not cancel out. Dispersion should preferably be in the same unit as the random variable. This is the case with the standard deviation, making it directly comparable to the observed values of a random parameter.

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TABLE 6.2 Calculation of mean and volatility with a time series of observed data: example Dates

Earnings (dollars)

Deviations from mean

Squared deviations

1 2 3 4 5 6 7 8 9 10 11 12 Sum

15.00 12.00 8.00 7.00 2.00 −3.00 −7.00 −10.00 −5.00 0.00 5.00 11.00 35.00

12.08 9.08 5.08 4.08 −0.92 −5.92 −9.92 −12.92 −7.92 −2.92 2.08 8.08

146.01 82.51 25.84 16.67 0.84 35.01 98.34 166.84 62.67 8.51 4.34 65.34 712.92

Mean

2.92

Sum Statistics:a Variance Volatility

59.41 7.71

a The mean is the sum of observed values divided by the number of observations (12). The variance is the sum of squared deviations divided by 12. The volatility is the square root of variance.

0.09/0.9 = 10%. For accounting earnings, the volatility could either be in dollars or as a percentage of current earnings. For interest rates, the volatility is a percentage, as the interest rate, or a percentage of the current level of the interest rate. Continuous distributions are the extreme case when there is a probability that the variable takes a value within any band of values, no matter how small. Formally, when X is continuous, for each interval [X, X + dX], there is a probability of observing values, which depends on X. The probability density function provides this probability. There are many continuous distributions, which serve as proxies for representing actual phenomena, the most well known being the normal distribution. The pdf is such that all probabilities sum to 1, as with the frequency distribution above. A useful property facilitating the calculation of the variance is that: σ 2 (X) = E(X2 ) − [E(X)]2 Another property of the variance is: σ 2 (aX) = a 2 × σ 2 (X) This also implies that σ (aX) = a × σ (X).

7 VaR and Capital

The ‘Value at Risk’ (VaR) is a potential loss. ‘Potential losses’ can theoretically extend to the value of the entire portfolio, although everyone would agree that this is an exceptional event, with near-zero probability. To resolve this issue, the VaR is the ‘maximum loss’ at a preset confidence level. The confidence level is the probability that the loss exceeds this upper bound. Determining the VaR requires modelling the distribution of values at some future time point, in order to define various ‘loss percentiles’, each one corresponding to a confidence level. VaR applies to all risks. Market risk is an adverse deviation of value during a certain liquidation period. Credit risk materializes through defaults of migrations across risk classes. Defaults trigger losses. Migrations trigger risk-adjusted value changes. VaR for credit risk is an adverse deviation of value, due to credit risk losses or migrations, at a preset confidence level. VaR applies as long as we can build up a distribution of future values of transactions or of losses. The VaR methodology serves to define risk-based capital, or economic capital. Economic capital, or ‘risk-based capital’, is the capital required to absorb potential unexpected losses at the preset confidence level. The confidence level reflects the risk appetite of the bank. By definition, it is also the probability that the loss exceeds the capital, triggering bank insolvency. Hence, the confidence level is equivalent to the default probability of the bank. The VaR concept shines for three major reasons: it provides a complete view of portfolio risk; it measures economic capital; it assigns fungible values to risks. Unlike intuition would suggest, the average loss is not sufficient to define portfolio risk because portfolio losses vary randomly around this average. Because VaR captures the downside risk, it is the basis for measuring economic capital, the ultimate safety cushion for absorbing losses. Finally, instead of capturing risks through multiple qualitative indicators (sensitivities, ratings, watch lists, excess limits, etc.), VaR assigns a dollar value to risk. Valuation makes

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all risks fungible, whatever the sources of uncertainty. By contrast, classical indicators do not add up as dollar values do. ‘Earnings at Risk’ (EaR) is a simple and practical version of VaR. EaR measures, at preset confidence levels, the potential adverse deviations of earnings. EaR is not VaR but shares the same underlying concept, and has the benefit of being relatively easy to measure. Although similar to VaR, EaR does not relate the adverse deviations of earnings to the underlying risks because EaR aggregates the effects of all risks. By contrast, VaR requires linking losses to each risk. Relating risk measures to the sources of risk is a prerequisite for risk management, because the latter aims at controlling risk ex ante, rather than after its materialization into losses. VaR models the value of risk and relates it to the instrumental variables, allowing ex ante control of risk using such parameters as sensitivities to market risk, exposure limits, concentration for credit risk, and so on. The first section describes the potential uses of VaR, and shows how VaR ‘synthesizes’ other traditional measures of risk. The second section details the different levels of potential losses of interest. They include: expected loss; unexpected and exceptional losses from loss distributions. The next sections detail the loss distribution and relate VaR to loss percentiles (or ‘loss at a preset confidence level’), using the normal distribution as an example to introduce further developments. The last section discusses benefits and drawbacks of EaR.

VAR AND RISK MANAGEMENT VaR is a powerful concept for risk management because of the range and importance of its applications. It is also the foundation of economic capital measures, which underlie all related tools, from risk-based performance to portfolio management.

The Contributions of VaR-based Measures VaR provides the measure of economic capital defined as an upper bound of future potential losses. Once defined at the bank-wide level, the capital allocation system assigns capital, or a risk measure after diversification effect, to any subset of the bank’s portfolio, which allows risk-adjusted performances to be defined, using both capital allocation and transfer pricing systems. Economic capital is a major advance because it addresses such issues as: • Is capital adequate, given risks? • Are the risks acceptable, given available capital? • With given risks, any level of capital determines the confidence level, or the bank’s default probability. Both risks and capital should adjust to meet a target confidence level which, in the end, determines the bank’s risk and solvency.

VaR and Common Indicators of Risk VaR has many benefits when compared to traditional measures of risk. It assigns a value to risk, it is synthetic and it is fungible. In addition, the VaR methodology serves to define risk-based capital. The progress is significant over other measures.

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89

Figure 7.1 illustrates the qualitative gap between traditional risk measures and VaR. It describes the various indicators of risk serving various purposes for measuring or monitoring risks. Such indicators or quantified measures are not fungible, and it is not possible to convert them, except for market instrument sensitivities, into potential losses. By contrast, VaR synthesizes all of them and represents a loss, or a risk value. Because VaR is synthetic, it is not a replacement for such specific measures, but it summarizes them. Credit risk Market risk - Risk measures Volatilities Sensitivities 'Greek letters' - Market values

- Ratings / Maturities / Industries - Watch lists - Concentration - Portfolio monitoring

VaR

Interest rate risk - Gaps Liquidity Interest rate Duration gaps

FIGURE 7.1

Other risks

From traditional measures of risk to VaR

POTENTIAL LOSS This section further details the VaR concept. There are several types of potential losses: Expected Loss (EL); Unexpected Loss (UL); exceptional losses. The unexpected loss is the upper bound of loss not exceeded in more than a limited given fraction of all outcomes. Such potential loss is also a loss percentile defined with the preset confidence level. Since the confidence level might take various values, it is necessary to be able to define all of them. Hence, modelling the unexpected loss requires modelling the loss distribution of the bank portfolio, which provides the frequencies of all various possible values of losses. Obtaining such loss distributions is the major challenge of risk models. VaR is the unexpected loss set by the confidence level. The exceptional loss, or extreme loss, is the loss in excess of unexpected loss. It ranges from the unexpected loss, as a lower bound, up to the entire portfolio value, but values within this upper range have extremely low probability of occurrence.

Expected Loss The expected loss serves for credit risk. Market risk considers only deviations of values as losses, and ignores expected Profit and Loss (P&L) gains for being conservative. Expected loss represents a statistical loss over a portfolio of a large number of loans. The law of

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large numbers says that losses will sometimes be high or low. Intuition suggests that they revert to some long-term average. This is the foundation for economic provisioning and ‘expected loss risk management’. Intuition suggests that provisioning the expected loss should be enough to absorb losses. This might be true in the long-term. By definition, statistical losses average losses over a number of periods and yearly losses presumably tend to revert to some long-term mean. The intuition is misleading, however, because it ignores the transitory periods when losses exceed a long-term average. Lower than long-term average loss in good years could compensate, in theory, higher losses in bad years. There is no guarantee that losses will revert quickly to some long-run average. Deviations might last longer than expected. Therefore, economic provisioning will result in transitory excess losses over the long-term average. Unless there is capital to absorb such excesses, it cannot ensure bank solvency. The first loss above average would trigger default. However, the choice of reference period for calculating the average loss counts. Starting in good years, we might have an optimistic reference value for expected loss and vice versa. Regulators insist on measuring ‘through the cycle’ to average these effects. This is a sound recommendation, so that economic provisions, if implemented, do not underestimate average losses in bad years because they refer to loss observed during the expansion phase of the economic cycle. Statistical losses are more a portfolio concept rather than an individual transaction concept. For one single transaction, the customer may default or not. However, for a single exposure, the real loss is never equal to the average. On the other hand, for a portfolio, the expected loss is the mean of the distribution of losses. It makes sense to charge to each transaction this average, because each one should contribute to the overall required provision. The more diversified a portfolio is, the lower is the loss volatility and the closer losses tend to be to the average value. However, this does not allow us to ignore the unexpected loss. One purpose of VaR models is to specify precisely both dimensions of risk, average level and chances/magnitudes of deviations from this average. Focusing on only one does not provide the risk profile of a portfolio. In fact, characterizing this profile requires the entire loss distribution to see how likely are large losses of various magnitudes. The EL, as a long-term average, is a loss value that we will face it sooner or later. Therefore, it makes sense to deduct the EL from revenues, since it represents an overall averaged charge. If there were no random deviations around this average, there would be no need to add capital to economic provisions. This rationale implies that capital should be in excess of expected loss under economic provisioning.

Unexpected Loss and VaR Unexpected losses are potential losses in excess of the expected value. The VaR approach defines potential losses as loss percentiles at given confidence levels. The loss percentile is the upper bound of loss not exceeded in more than a given fraction of all possible cases, this fraction being the confidence level. It is L(α), where α is the one-tailed1 probability 1 Only

adverse deviations count as losses. Opposite deviations are gains and do not value risk.

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of exceeding L(α). For example, L(1%) = 100 means that the loss exceeds the value of 100 in no more than 1% of cases (one out of 100 possible scenarios, or two to three days within a year)2 . The purpose of VaR models is to provide the loss distribution, or the probability of each loss value, to derive all loss percentiles for various confidence levels. The unexpected loss is the excess of the loss percentiles over the expected loss, L(α) − EL. Economic capital is equal to unexpected loss measured as a loss percentile in excess of expected loss (under economic provisioning).

Exceptional Losses Unexpected loss does not include exceptional losses beyond the loss percentile defined by a confidence level. Exceptional losses are in excess of the sum of the expected loss plus the unexpected loss, equal to the loss percentile L(α). Only stress scenarios, or extreme loss modelling when feasible, help in finding the order of magnitude of such losses. Nevertheless, the probability of such scenarios is likely to remain judgmental rather than subject to statistical benchmarks because of the difficulty of inferring extreme losses which, by definition, are almost unobservable.

MEASURING EXPECTED AND UNEXPECTED LOSSES The two major ingredients for defining expected and unexpected losses are the loss distribution and the confidence level. The confidence level results from a management choice reflecting the risk appetite, or the tolerance for risk, of the management and the bank’s policy with respect to its credit standing. Modelling loss distributions raises major technical challenges because the focus is on extreme deviations rather than on the central tendency. Since downside risk characterizes VaR and economic capital, loss volatility and the underlying loss distribution are critical.

Loss Distributions In theory, historical loss distributions are observable historically. For market risk, loss distributions are simply the distributions of adverse price deviations of the instruments. Since there are approximately as many chances that values increase or decrease, such deviations tend to be bell-shaped, with some central tendency. Of course, the loss distribution is a distribution of negative earnings truncated at the zero level. The bell-shaped distribution facilitates modelling, especially when using the normal or the lognormal distributions as approximations. Unfortunately, historical data is scarce for credit risk and does not necessarily reflect the current risk of banks. Therefore, it is necessary to model loss distributions. For credit risk, losses are not negative earnings. They result from defaults, or loss of asset value because of credit standing deterioration. Such distributions are highly skewed to the left, because the most frequent losses are very small. Both types of distributions are shown in Figure 7.2. 2 An

alternative notation is L(99%), where 99% represents the probability that the loss value does not exceed the same upper bound.

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Market risk Probability

Probability

Credit risk

Gains

Losses

Losses - Large losses - Low probability

FIGURE 7.2

Fat tails and extreme losses

With distributions, the visual representation of losses is simple. In Figure 7.3, losses appear at the right-hand side of the zero level along the x-axis. The VaR at a given confidence level is such that the probability of exceeding the unexpected loss is equal to this confidence level. The area under the curve at the right of VaR represents this probability. The maximum total loss at the same confidence level is the sum of the expected loss plus unexpected loss (or VaR). Losses at the extreme right-hand side and beyond unexpected losses are ‘exceptional’. The VaR represents the capital in excess of expected loss necessary for absorbing deviations from average losses. Mode (most frequent)

Expected loss

Expected + Unexpected loss

Probability

Probability of loss < UL

Probability of loss > UL: Confidence level

Losses = 0

Expected loss

FIGURE 7.3

Losses

Unexpected loss = VaR

Exceptional loss

Unexpected loss and VaR

A well-known characteristic of loss distributions is that they have ‘fat tails’. Fat tails are the extreme sections of the distribution, and indicate that large losses, although unlikely because their probabilities remain low, still have some likelihood to occur that is not negligible. The ‘fatness’ of the tail refers to the non-zero probabilities over the long end of the distributions.

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The VaR is easy to determine under simplifying assumptions on the distribution curve of losses. With normal curves, the VaR is a multiple of loss volatility that depends on the confidence level. For example, the 2.5% one-tailed confidence level corresponds to a multiple of loss volatility of 1.96. Therefore, if the loss volatility is 100, the unexpected loss will not exceed the upper bound of 196 in more than two or three cases out of 100 scenarios. Unfortunately, such multiples do not apply when the distribution has a different shape, for instance for credit risk. When implementing techniques based on confidence levels and loss percentiles, there is a need for common benchmarks, such as confidence levels, for all players. With a very tight confidence level, the VaR could be so high that business transactions would soon become limited by authorizations, or not feasible at all. If competitors use different VaR models or confidence levels, banks will not operate on equal grounds. Tighter confidence levels than competitors’ levels would reduce the volume of business of the most prudent banks and allow competitors having more risk appetite to take advantage of an overly prudent policy.

LOSS PERCENTILES OF THE NORMAL DISTRIBUTION The normal distribution is a proxy for market random P&L over a short period, but it cannot apply to credit risk, for which loss distributions are highly asymmetrical. In this section, we use the normal distribution to illustrate the VaR concept and the confidence levels. The VaR at a confidence level α is the ‘loss percentile’ L(α). In Figure 7.4, the area under the curve, beyond the boundary value on the left-hand side, represents the probability that losses exceed this boundary value. Visually, a higher volatility means that the curve dispersion around its mean is wider. Hence, the chances that losses exceed a given boundary value grow larger. The confidence intervals are probabilities that losses exceed an upper bound (negative earnings, beyond the zero level). They are ‘one-tailed’ because only one-sided negative deviations materialize downside risk. Probability of earnings Probability 5%

Losses

FIGURE 7.4

Maximum loss at the 5% level

Gains

Volatility and downside risk

When both upside and downside deviations of the mean are considered, the confidence interval is ‘two-tailed’. With a symmetric distribution, the two-tailed probability is twice the one-tailed probability. When the probability of losses exceeding a maximum value is

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lev

els

2.5%, the probability that losses exceed either the lower or the upper bounds is 5%. Unless otherwise stated, we will stick to the ‘one-tailed’ rule for specifying confidence intervals. Confidence intervals are boundary values corresponding to a specified confidence level. In the case of the normal curve, confidence intervals are simply multiples of the volatility. Figure 7.5 shows their values. With the normal curve, the upper bounds of (negative) deviations corresponding to the confidence levels of 16%, 10%, 5%, 2.5% and all other values are in the normal distribution table. They correspond respectively to deviations from the mean of 1, 1.28, 1.65 and 1.96 times the standard deviation σ of the curve. Any other confidence interval corresponds to deviations expressed as multiples of volatilities for this distribution. 16% Normal distribution

id

en

ce

10%

Co

nf

5% 2.5%

1.0% 0

Mean 1.00 σ

Losses

Earnings

1.28 σ 1.65 σ 1.96 σ 2.33 σ

FIGURE 7.5

Confidence levels with the normal distribution

ISSUES AND ADVANCES IN MODELLING VAR AND PORTFOLIO RISKS When we characterize an individual asset, independent of a portfolio context, we adopt a ‘standalone’ view and calculate a ‘standalone’ VaR. This serves only as an intermediate step for moving from ‘standalone’ loss distributions of individual assets to the portfolio loss distribution, which combines losses from all individual assets held in the portfolio. Standalone loss distributions list all possible values of losses for an asset with their probabilities. For instance, a loan whose value is outstanding balance under no default and zero under default has a loss distribution characterized by a 0% loss with probability of, say, 98% and a 100% loss with probability of 2%. The obvious difficulty in VaR measures is the modelling of the loss distribution of a portfolio. The focus on high losses implies modelling the ‘fat tail’ of the distributions rather than looking at the central tendency. Even with market-driven P&L, the normal distribution does a poor job of modelling distribution tails. For credit risk, the issue is worse since the loss distributions are highly skewed. Loss distributions depend on portfolio structure, size discrepancies (concentration risk on big exposures) and the interdependencies between individual losses (the fact that a loss occurrence increases or

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decreases the likelihood of occurrence of other losses). In subsequent chapters, ‘portfolio models’ designate models providing the loss distribution of portfolios. Diversification and granularity effects within a portfolio were recognized long ago but banks could not quantify them. What is new with portfolio models is that they provide the ability to quantify concentration and diversification effects on the portfolio risk. The added complexity is the price to pay for this single most important value added of the new portfolio models. ‘Fat tails’ of actual distributions make the quantification of extreme losses and their probability of occurrence hazardous. The main VaR modelling drawback is that they are highly demanding in terms of data. Because of the technicalities of modelling loss distribution for market and credit risk, several dedicated chapters address the various building blocks of such models: for market risk, see Chapter 32; for credit risk, see Chapters 44–50.

RISK-BASED CAPITAL The VaR methodology applies to measure risk-based capital. The latter differs from regulatory capital or from available capital in that it measures actual risks. Regulatory capital uses forfeits falling short of measuring actual risks. Economic capital necessitates the VaR methodology, with the modelling of loss distribution, with all related complexities. The EaR concept is an alternative and simpler route to capital than VaR. For this reason, it is useful to detail the technique and to contrast the relative merits of EaR versus VaR.

The Limitations of Simple Approaches to Capital The simplest way to define the required capital is to use regulatory capital. This is a common practice in the absence of any simple and convincing measure of capital. In addition, the regulatory capital is a requirement. At first sight, there seems to be no need to be more accurate than the regulators. However, regulatory capital has many limitations, even after the enhancements proposed by the New Accord. Using regulatory capital as a surrogate for economic capital generates important distortions because of the divergence between the real risks and the forfeited risks of regulatory capital. For regulation purposes, credit risk is dependent on outstanding balances (book exposures) and on risk weights. Such forfeits are less risk-sensitive than economic measures. In addition, standardized regulatory approaches measure risk over a portfolio by a simple addition of individual risks for credit risk. This ignores the diversification effect and results in the same measure of risk for widely diversified portfolios and for highly concentrated portfolios. The shortcomings of forfeit measures have implications for the entire risk management system. What follows applies to credit risk, since market models are allowed. Visibility on actual risks remains limited. Credit risk limits remain based on book exposures since regulatory capital depends on these. The target performance also uses forfeit measures of capital. The allocation of this capital across business units does not depend on their ‘true’ risks. Any risk-based policy for measuring risk-adjusted performances, or for risk-based pricing, suffers from such distortions. The most important benefit of economic capital is to correct such distortions.

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Earnings at Risk EaR is an important methodology for measuring capital. A simple way of approaching risk is to use the historical distributions of earnings. The wider the dispersion of time series of earnings, the higher the risk of the bank. Principle

Several measures of earnings can be used to capture their instability over time: accounting earnings; interest margins; cash flows; market values, notably for the trading portfolio. The volatility is the adequate measure of such dispersion. It is the standard deviation of a series of observations. Such calculation always applies. For instance, even with few observations, it remains possible to calculate volatility. Of course, the larger the data set, the more relevant the measure will be. The concept applies to any subportfolio as well as to the entire bank portfolio. When adding the earning volatility across subportfolios, the total should exceed the loss volatility of the entire portfolio because of diversification. Once earnings distributions are obtained, it is easy to use loss volatility as the unit for measuring capital. Deriving capital follows the same principle as VaR. It implies looking for some aggregated level of losses that is not likely to be exceeded in more than a given fraction of all outcomes. Benefits

The major benefit of EaR is in providing a very easy overview of risks. This is the quicker approach to risk measurement. It is not time intensive, nor does it require much data. It relies on existing data since incomes are always available. EaR requires few Information Technology (IT) resources and does not imply major investments. There is no need to construct a risk data warehouse, since existing databases provide most of the required information and relate it easily to transactions. It is easy to track major variations of earnings to some specific events and to interpret them. EaR provides a number of outputs: the earnings volatility, the changes of earnings volatility when the perimeter of aggregation increases, measuring the diversification effect. The level of capital is an amount that losses are not likely to exceed. This is a simple method of producing a number of outputs without too much effort. This is so true that EaR has attracted attention everywhere, since it is a simple matter to implement it. Drawbacks

There are a number of drawbacks to this methodology. Some of them are purely technical. A volatility calculation raises technical issues, for instance when trends of time series increase volatility. In such cases, the volatility comes from the trend rather than instability. There is no need to detail further technical difficulties, since they are easy to identify. The technique requires assumptions, but the number of options remains tractable and easily managed. In general, the drawbacks of simplicity are that EaR provides only crude measures. However, the major drawbacks relate to risk management. It is not possible to define the sources of the risk making the earnings volatile. Presumably, various types of risks

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materialize simultaneously and create adverse deviations of earnings. The contributions of these risks to the final volatility remain unknown. Unlike VaR models, EaR captures risks as an outcome of all risks, not at their source. Without links to the sources of risk, market, credit or interest rates, EaR serves to define aggregated capital, but it does not allow us to trace back risks to their sources. A comprehensive and integrated risk management system links measures with specific sources of risk. The VaR for market risk and credit risk, and the ALM measures of interest rate, are specific to each risk. They fit better bank-wide risk management systems because they allow controlling each risk upstream, rather than after the fact. The EaR methodology does not meet such specifications. On the other hand, it is relatively easy to implement compared to full-blown systems. EaR appears to be an additional tool for risk management, but not a substitute.

8 Valuation

Under the traditional accounting framework applying to the banking portfolio, loans are valued at book value and earnings are interest income accrued over a period. Although traditional accounting has undisputed merits, it has economic drawbacks. Periodical measures of income ignore what happens in subsequent periods. The book value of a loan does not change when the revenues or the risks are higher or lower than average. Hence, book values are not revenue- or risk-adjusted, while ‘economic values’ are. In essence, an ‘economic value’ is a form of mark-to-market measure. It is a discounted value of future contractual cash flows generated by assets, using appropriate discount rates. The discounting process embeds the revenues from the assets in the cash flows, and embeds the risk also in the discount rates. This explains the current case for ‘fair value’. Marking-to-market does not imply that assets are actually tradable. The mark-to-market values of the trading portfolio are market prices. However, a loan has a mark-to-market value even though it is not tradable. Since the risk–return trade-off is universal in the banking universe, a major drawback of book values is that they are not faithful images of such risks and returns. Another drawback is that book values are not sensitive to the actual ‘richness’ or ‘poorness’ of transactions, and of their risks, which conflicts with the philosophy of a ‘faithful’ image. Risk models also rely on mark-to-market valuations. ‘Mark-to-model’ valuations are similar to mark-to-market, but exclude some value drivers. For example, isolating the effect of credit risk on value does not require using continuously adjusted interest rates. ‘Mark-to-future’ is a ‘mark-to-model’ valuation at future time points, differentiated across scenarios characterizing the random outcomes from current date up to a future time point. Value at Risk (VaR) risk models use revaluations of assets at future dates for all sets of random outcomes, to provide the value distribution from which VaR derives. This makes the ‘revaluation block’ of models critical for understanding VaR.

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However, moving to ‘fair value’ is a quantum leap because of the implication in terms of profitability. The relevant measure of profitability becomes Profit and Loss (P&L) rather than accrual income. Traditional measures of performance for the banking portfolio are interest income plus fees. For the market portfolio they are P&L or the change of markto-market values of assets traded between any two dates. Moving to fair values would imply P&L measures, and P&L volatility blurring the profitability view of the bank. The debate on accounting rules follows. Whatever the outcome on accounting standards, economic values will be necessary for two main reasons: they value both risks and returns; valuation is a major building block of credit models because all VaR models need to define the distribution of future values over the entire range of their future risk states. Because of the current growing emphasis on ‘fair value’, we review here valuation issues and introduce ‘mark-to-model’ techniques. The first section is a reminder of accounting standards. The next section details mark-to-market calculations and market yields. The third section summarizes why economic valuation is a building block of risk models. The fourth discusses the relative merits of book versus economic valuations. The fifth section introduces risk-adjusted performance measures, which allow separation of the risk and return components of economic value, rather than bundling them in a single fair value number.

ACCOUNTING STANDARDS Accounting standards have evolved progressively following the guidelines of several boards. The Financial Accounting Standard Board (FASB) promotes the Generally Accepted Accounting Practices (GAAP) that apply to all financial institutions. The International Accounting Standard (IAS) Committee promotes guidelines to which international institutions, such as the multinational banks, abide. All committees and boards promote the ‘fair value’ concept, which is a mark-to-market value. ‘Fair value’ is identical to market prices for traded instruments. Otherwise, it is an economic value. It is risk-adjusted because it uses risky market yields for valuation. It is revenue-adjusted because it discounts all future cash flows. Economic values might be either above or below face value depending on the gap between asset returns and the market required yield applicable to assets of the same risk class. The main implication of fair value accounting is that earnings would result from the fluctuations of values, generating volatile P&L. The sensitivity of value to market movements is much higher for interest-earning assets with fixed rates than with floating rates, as subsequent sections illustrate. Fair value is implemented partially. The IAS rules recommended that all derivatives should be valued at mark-to-market prices. The rule applies to derivatives serving as hedges to bank exposures. The implication is that a pair made up of a banking exposure valued at book value matched with a derivative could generate a value fluctuating because of the valuation of the derivative while the book value remains insensitive to market movements. The pair would generate a fluctuating value and, accordingly, a volatile P&L due to the valuation of the derivative. This is inconsistent with the purpose of the hedge, which is to stabilize the earnings for the couple ‘exposure plus hedge’. To correct the effect of the valuation mismatch, the rule allows valuation of both exposure and hedge at the fair value.

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It is too early to assess all implications of fair values. Moreover, many risk models use mark-to-model valuation for better measuring risk, notably credit risk models that apply to the banking portfolio. Nevertheless, it is necessary to review fair value calculations in order to understand them. The subsequent sections develop gradually all concepts for determining mark-to-market ‘fair values’.

MARK-TO-MARKET VALUATION This section develops the basic mark-to-market model step-by-step. First, it explains the essentials of principles for discounting, the interpretation of the discounting process and the implication for selecting a relevant discount rate. Second, it introduces the identity between market interest rates, or yields, and required rates of return on market investments in assets providing interest revenues. Third, it explains the relationship between market prices and yield to maturity, and explains how this applies to non-marketable assets as well, such as loans. For the sake of clarity, we use a one-period example. Then, using the yield to maturity (Ytm) concept, we extend the same conclusions to a multiple-period framework with simple examples.

The Simple ‘Discounted Cash Flow’ Valuation and the ‘Time Value of Money’ The familiar Discounted Cash Flow (DCF) model explains how to convert future values into present values and vice versa, using market transactions. Therefore, it explains which are the discount rates relevant for such actual time translations. It is also the foundation of mark-to-market approaches, although it is not a full mark-to-market because the plain DCF model does not embed factors embedded in actual mark-to-market values, such as the credit spreads specific to each asset. The present value of an asset is the discounted value of the stream of future flows that it generates. When using market rates as discount rates, the present value is a mark-to-market value. The present value of a stream of future flows Ft is: Ft /[1 + y(t)]t V = t

The market rates are rates applying to a single flow at date t, or the zero-coupon rates y(t). The formula applies to any asset that generates a stream of contractual and certain flows. Cash flows include, in addition to interest and principal repayments, fees such as upfront flat fees, recurring fees and non-recurring fees (Figure 8.1). With floaters, interest payments are indexed to market rates, and the present value is equal to the face value. For instance, an asset with a face value of 100 generates a flow over the next period that includes both the interest and the principal repayment. This final flow is equal to 100(1 + r), r also being the market rate that applies to the period. When this flow is discounted with these market rates, the present value is constant and equal to 100(1 + r)/(1 + r) = 100. The same result applies when the horizon extends beyond one period with the same assumptions. Nevertheless, when the asset pays more than the discount rate, say y + m, with m being a constant percentage, while the discount rate is y, the one-period flow becomes 100(1 + y + m) with discounted value at y: 100(1 + y +

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Discounting at y (t)

Ft

0

FIGURE 8.1

t

Time

The present value of assets with the DCF model

m)/(1 + y) = 100[1 + m/(1 + y)]. The value becomes higher than face value, by a term proportional to m/(1 + y), and sensitive to the yield y. The relevant discount rates are those that allow flows to be transferred across time through borrowing and lending. This is why market rates are relevant. For instance, transferring a future flow of 1000 to today requires borrowing an amount equal to 1000/(1 + yb ), where yb is the borrowing rate. Transferring a present flow of 1000 to a future date requires lending at the market rate yl . The final flow is 1000(1 + yl ). Hence, market rates are relevant for the DCF model (Figure 8.2).

0

1

Borrow X/(1+yb)

FIGURE 8.2

Repay X

0

1

Lend X

Receive X(1+yl)

Discounting and borrowing or lending at market rates

The simple DCF model refers to borrowing and lending rates of an entity. The actual market values of traded assets use different discount rates, which are market rates given the risk of the asset. Full mark-to-market valuation implies using the DCF model with these rates.

Continuous and Discrete Compounding or Discounting Pricing models use continuous compounding instead of discrete compounding, even though actual product valuation uses discrete compounding. Because some future examples use continuous calculations, we summarize the essentials here. The basic formulas for future and present values are very simple, with a discount rate y (continuous) and a horizon n: FVc (y, n) = exp(yn) = eyn

PVc (y, n) = exp(−yn) = e−yn

The index c stands for continuous. Using discounted cash flow formulas to value the present value of a stream of cash flows, simply substitutes e−rt in the discount factor

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1/(1 + y)t . The formulas do not give the same numerical results, although there is a continuous rate that does. If the discrete rate is y, the discrete time and the continuous time formulas are respectively: PVc (y) = PVc (y) =

n t=1

n

CFt /(1 + y)t CFt exp(−yt)

t=1

We provide details and, notably, the rule for finding continuous rates equivalent to discrete rates in the appendix to this chapter.

Market Required Rates and Asset Yields or Returns The yield of a tradable asset is similar to a market interest rate. However, banking assets and liabilities generally do not provide or pay the market rates because of positive spreads over market rates for assets and costs of deposits lower than market rates on the liability side. Valuation depends on the spread between asset and market yields. The asset fixed return is r = 6% and y is the required market yield for this asset. The market required yield depends on maturity and the risk of the asset. We assume both are given, and the corresponding y is 7%. We receive a contractual cash flow, 1 year from now, equal to 1 + r in 1 year, r being the interest payment. For instance, an asset of face value 1000 provides contractually 6% for 1 year, with a bullet repayment in 1 year. The contractual risk-free flow in 1 year is 1060. The 1-year return on the asset is 6%, if we pay the asset at face value 1000 since r = (1060 − 1000)/1000 = 6%. If the market rate y is 7%, the investors want 7%. If they still pay 1000 for this asset, they get only 6%, which is lower. Therefore, they need to pay less, say a price P unknown. The price P is such that the return should be 7%: (1060 − P )/P = 7%. This equation is identical to: P = 1060/(1 + 7%) = 990.65 If y = 7% is the 1-year market required rate, the present value, or mark-to-market value, of the future flow is 990.65 = 1060(1 + y) = 1000(1 + r)/(1 + y). If we pay the face value, the 1-year return becomes equal to the market yield because [(1 + y) − 1]/1 = y. Hence, any market interest rate has the dimension of a return, or a yield. Getting 1 + y, one period from now, from an investment of 1 means that its yield is y. Market rates are required returns from an investor perspective. Since the market provides y, a rate resulting from the forces of supply and demand for funds, any investor requires this return (Figure 8.3). The implication is that the investor pays the price that provides the required market return whatever the actual contractual flows are. Paying 990.65 generates the required market return of 7%. This price is lower than the face value. If the market return is identical to the contractual asset rate of 6%, the investor actually gets exactly 6% by paying the face value 1000 since (1060 − P )/P = 6%. If the market return is lower than the contractual asset rate of 6%, say 5%, the investor actually gets more than 5%

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Asset

Cash flow = 1060

Face value = 1000 0

1

Value of asset cash flow given market required yields

V(0) = 990.65 0

FIGURE 8.3

Time

Return r = 6%

Time

Market required yield y = 7%

1

Asset return versus market required yield

by paying 1000 since (1060 − P )/P = 6%. For the investor to get exactly 5%, he pays exactly P = 1060/(1 + 5%) = 1009.52. The price becomes higher than the face value. The general conclusions are that the value is identical to the face value only when the asset return is in line with the required market return. Table 8.1 summarizes these basic conclusions. TABLE 8.1

Asset return versus market required yield

Asset return r > market required return y Asset return r < market required return y Asset return r = market required return y

value > book value value > book value value = book value

Asset Return and Market Rate Since the asset value is the discounted value of all future cash flows at market rates, there is an inverse relationship between interest rate and asset value and the market rate. This inverse relationship appears above when discounting at 6% and 7%. The asset value drops from 1000 to 990.65. The value–interest rate profile is a downward sloping curve. This relationship applies to fixed rate assets. It serves constantly when looking at the balance sheet economic value (Net Present Value, NPV) in Asset–Liability Management (ALM) (Figure 8.4). The value of a floater providing interest revenues calculated at a rate equal to the discounting rate does not depend on prevailing rates. This is obvious in the case of a single period, since the asset provides 1 + r at the end of the period and this flow discounted at i has a present value of (1 + r)/(1 + r) = 1. Although this is not intuitive, the property extends to any type of floater, for instance, an amortizing asset generating revenue calculated with the same rate as the discount rate. This can be seen through examples. An amortizing loan of 1000, earning 10%, repaid in equal amounts of 200 for 5 years, provides the cash flow stream, splitting flows into capital and interest: 200 + 100, 200 + 80, 200 + 60, 200 + 40, 200 + 20. The discounted value of these flows at 10% is equal to 1000. If the rate floats, and becomes 12% for instance, the cash flow stream then

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Asset Value Fixed rate asset

Variable rate asset

Interest Rate

FIGURE 8.4

Fixed rate asset value and interest rate

becomes: 200 + 120, 200 + 96, 200 + 72, 200 + 48, 200 + 241 . Discounting these flows at 12% results in a present value of 1000 again. When the asset yield is above the discount rates, there is an ‘excess spread’ in the cash flows. For example, if the asset provides 10.5% when rates are 10%, the spread is 0.5%. When rates are 12%, the spread also remains at 0.5% because the asset provides 12.5%. A floater is the sum of a zero-spread floater (at 12%) whose value remains equal to the face value whatever the rates, plus a smaller fixed rate asset providing a constant 0.5% of the outstanding balance, which is sensitive to rate changes. Since the amount is much smaller than the face value, the value change of such a variable rate asset is much smaller than that of a similar fixed rate asset.

Interest Rates and Yields to Maturity These results extend to any number of periods through the yield to maturity concept. When considering various maturities, we have different market yields for each of them. The entire spectrum of yields across maturities is the ‘yield curve’, or the term structure of interest rates2 . In practice, two different types of yields are used: zero-coupon yields and yields to maturity. The zero-coupon rates apply to each individual future flow. The yields to maturity are ‘compounded averages’ across all periods. We index current yields by the maturity t, the notation being y(t). The date t is the end of period t. For instance, y(1) is the yield at the end of period 1. If the period is 1 year, and if we are at date 0 today, y(1) is the spot rate for 1 year, y(2) the spot rate for 2 years, etc. Using zero-coupon rates serves to price an asset starting from the contractual stream of cash flows and the market zero-coupon rates. Market prices of bonds are the discounted cash flows of various maturities using these yields. Instead of using these various rates, it is convenient to use a single discount rate, called the yield to maturity (or ‘Ytm’). Using the Ytm addresses the issue of finding the return for an investor between now and 1 The

interest flows are 12% times the outstanding balance, starting at 1000 and amortized by 200 at each period. 2 See Chapter 12 for details on the term structure of rates and zero-coupon rates.

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maturity, given the contractual stream of cash flows and the price of the asset3 . The Ytm is the unique discount rate making the present value of all future flows identical to the observed price. Yield to Maturity and Asset Return

For a loan, there is also a yield to maturity. It is the discount rate making the borrowed amount, net of any fees, identical to the discounted contractual cash flows of the loan. The yield to maturity derives from the stream of contractual cash flows and the current price of a bond or of the net amount borrowed. The following example illustrates the concept with a bullet bond (principal repayment is at maturity) generating coupons equal to 6% of face value, with a face value equal to 1000, and maturing in 3 years. The stream of cash flows is therefore 60, 60 and 1060, the last one including the principal repayment. In this example, we assume that cash flows are certain (risk-free) and use market yields to maturity applicable to the 3year maturity. The asset rate is 6%. Discounting all flows at 6% provides exactly 1000, implying that the yield to maturity of the bond valued at 1000 is 6%. It is equal to its book return of 6%. This is a general property. When discounting at a rate equal to the asset contractual return r = 6%, we always find the face value 1000 as long as the bond repays without any premium or discount to the principal borrowed. If the market yield to maturity for a 3-year asset rises to 7%, the value of the asset cash flow declines to 973.77. This is identical to what happens in our former example. If the required yield is above the book return of 6%, the value falls below par. Back to the 7% market yield to maturity, the asset mark-to-market value providing the required yield to maturity to investors has to be 973.77. An investor paying this value would have a 7% yield to maturity equal to the market required yield. The reverse would happen with a yield to maturity lower than 6%. The value would be above the book value of 1000. These conclusions are identical to those of the above example of a 1-year asset (Table 8.2). TABLE 8.2

Value and yield to maturity

Date Rate Cash flows Discounted cash flows Current value Rate Discounted cash flows Current value

3 The

0

1

2

3

6% 60 56.60

6% 60 53.40

6% 1060 890.00

7% 56.07

7% 52.41

7% 865.28

1000

973.77

second application relies on assumptions limiting the usage of the Ytm, which we ignore at this stage. It is easy to show that the Ytm is the effective yield obtained by the investor if he holds the asset to maturity and if all intermediate flows received, whatever their nature, interest or principal repayments, are reinvested in the market at the original yield to maturity. The second assumption is unrealistic since there is no way to guarantee that the future market rate prevailing will be in line with the original Ytm calculation, except by an extraordinary coincidence.

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Value and Term Structure of Yields

When using different yields to maturity, the discounting calculations serve to determine the mark-to-market value of any asset, including loans, even though these are not traded in the market. In this latter case, the value is a theoretical mark-to-market, rather than an actual price in line with market rates. We can compare the theoretical price with actual prices for traded assets, which we cannot do for non-traded assets. Using the market zero-coupon yields of Table 8.3, we find that the discounted value of the bond flows is: V = 60/(1 + 5.00%) + 60/(1 + 6.00%)2 + 1060/(1 + 7.00%)3 V = 57.14 + 53.40 + 865.28 = 975.82 TABLE 8.3 rates

Term structure of ‘zero-coupon’ market

End of period Market rate, date t

1

2

3

5.00%

6.00%

7.00%

The value has no reason to be identical to 1000. It represents a mark-to-market valuation of a contractual and certain stream of cash flows. There is a yield to maturity making the value identical to the discounted value of cash flows, using it as a unique discount rate across all periods. It is such that: 975.82 = 60/(1 + y) + 60/(1 + y)2 + 1060/(1 + y)3 The value is y = 6.920%4 . It is higher than the 6% on the face value, because we acquire this asset at a value below par (975.82). The 6.92% yield is the ‘averaged’ return of an investor buying this asset at its market price, with these cash flows and holding it to maturity.

Risk-free versus Risky Yields Some yields are risk-free because the investors are certain of getting the contractual flows of assets, whereas others are risky because there is a chance that the borrower will default on his obligations to pay debt. Risk-free debt is government debt. The risky yield and the risk-free yield at date t are y and yf . The risky required yield has to be higher than the risk-free rate for the same maturity to compensate the investors for the additional risk borne by acquiring risky debt. For each maturity t, there is a risk-free yield yf (t) and a risky yield y(t) for each risk class of debt. The market provides both yield curves, derived from observed bond prices. The difference is the credit spread. Risky yields are the sum of the risk-free yield plus a ‘credit spread’ corresponding to the risk class of the asset and the maturity. The credit spread is cs(t) = y(t) − yf (t). Both zero-coupon yields and 4 It

is easy to check that discounting all contractual flows at this rate, we actually find the 949.23 value. The rate y is the internal rate of return of the stream of flows when using 949.23 as initial value.

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yields to maturity are such that y(t) = yf (t) + cs(t). When using full mark-to-market, the discount rates depend on both the maturity of flows and their credit risk5 (Table 8.4). Credit spreads vary with maturity and other factors. The spreads theoretically reflect the credit risk of assets under normal conditions. Since the prices of risky bonds might also depend on other factors such as product and market liquidity, spreads might not depend only on credit risk. Credit spreads are observable from yields and increase with credit risk, a relationship observed across rating classes. TABLE 8.4 risky yields

Term structure of risk-free and

End of period Risky yield Risk-free yield Credit spreads

1

2

3

5.00% 4.50% 0.50%

6.00% 5.00% 1.00%

7.00% 5.90% 1.10%

There are two basic ways to discount future flows: using zero-coupon yields for each date, or using a unique yield to maturity. Zero-coupon yields embed credit spreads for each maturity. The yield to maturity embeds an average credit spread for risky debts. V (0) is the current value of an asset at date 0. By definition of the risky yield to maturity, the mark-to-market value of the entire stream of contractual cash flows is: V (0) =

T

Ft /(1 + y)t

V (0) =

T

Ft /[1 + yf (t) + cs(t)]t

t=1

t=1

Risk and Relative Richness of Facilities Marking-to-model differentiates facilities according to relative richness and credit risk irrespective of whether assets are traded or illiquid banking facilities. Rich facilities providing revenues higher than market have a value higher than book, and conversely with low revenue facilities. Richness results from both revenues and risk. The ‘excess spread’ concept characterizes richness. It is the spread between the asset return over the sum of the risk-free rate plus the credit spread corresponding to the risk class of the asset. To make this explicit, let us consider a zero-coupon maturing at T , with a unit face value, providing the fixed yield r to the lender, and a risk such that the risky discount rate is y, including the credit spread: V = (1 + r)T /(1 + y)T It is obvious that V > 1 and V ≤ 1 depending on whether r > y or r ≤ y. From the above, valuation is at par value (book value) when the excess spread is zero. Deviations 5 There

is an alternative valuation technique for risky debt using so-called risk-neutral probabilities of default. In addition, credit spreads provide information on these probabilities. The details are in Chapter 42 dedicated to the valuation of credit risk.

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from par result from both revenues and risks. Mark-to-model valuation reflects risk and excess revenues compared to the market benchmark. Figure 8.5 shows two cases, with fair value above and below book value, from these two return and risk adjustments. The risk adjustment results from discounting at a rate y > yf , with a risk premium over the risk-free rate. The return adjustment corresponds to the excess return r − yf of the asset above the risk-free rate. The overall adjustment nets the two effects and results in the excess return of the asset over the risky discount rate, or r − y. If both adjustments compensate, implying y = r, the value remains at par. Excess return adjustment

Book value

Fair value

Book value

Fair value

Risk adjustment

FIGURE 8.5

From book value to fair value: excess return and risk adjustments

The excess spread is r − y. The gap between r and y is the ‘relative richness’ of the asset, relative to the required market return. A rich asset provides a higher return than the market requires given risk. Its mark-to-market value is above book value. A poor asset pays less than the required market return y. Its value is below book value. These properties apply for all risky assets, except that, for each risk class, we need to refer to market rates corresponding to this specific risk class. As an example, we assume that a risky asset provides a 6% yield, with a principal of 100 and a 1-year maturity. The market required risky yield is 5.5% and the risk-free yf required yield is 5.5%. The excess spread is 6% −5.5% = 0.5%. The value of the asset at an excess spread of zero implies a zero excess spread, or an asset return of 5.5%. With this return, the market value of the asset is obviously the face value, or 1000. The value of risk is the difference between the value at the required risky rate and the value at the risk-free yield, or 1060/(1+5.5%) − 1060/(1+5%) = 1004.74 − 1009.52 = −4.78. The value of the excess spread is the difference between the value of an asset providing exactly the required risky yield, or 100, and the value of the actual asset with a positive excess spread, or 1004.74, which is +4.74.

MARK-TO-MODEL VERSUS FULL MARK-TO-MARKET VALUATION In many risk models, the valuation building block plays a critical role: • For ALM models, the NPV of the balance sheet is the present value of assets minus the value of liabilities. It serves as a target variable of ALM models because it captures

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the entire stream of cash flows generated by assets and liabilities (see Chapter 22). Intuitively, if assets provide excess return over market yields, and liabilities cost less than market yields, assets are above par and liabilities below par value. The difference should capture the expected profitability of the balance sheet in terms of present value. • For market risk VaR models, the loss percentile derives from the distribution of future values of all assets at a future time point. The future value is random because of market movements and the distribution implies revaluation of assets at the future time point as a function of random market moves. • For credit risk models, the same principle applies, except that we focus on random credit risk events to find the distribution of uncertain values at a future horizon. The credit events include defaults and changes of credit standing, implying a value adjustment with market risky yields. The revaluation building block provides the spectrum of future values corresponding to all future credit states. The valuation calculation depends on various options. Full mark-to-market valuation uses all parameters influencing prices to find out the distribution of random values at future time points. This is not so for ALM and credit risk. The NPV of the balance sheet uses a single set of market rates differentiated by maturity, but not by risk class. The calculation differs from mark-to-market and fair value, in that it does not differentiate the discount rates according to the risk of each individual asset. The interpretation of the NPV is straightforward if we consider the bank as ‘borrowing’ from the market at its average cost of funding, which depends on its risk class. The bank could effectively bring back to present all future asset cash flows to present by borrowing against them. Borrowing ‘against’ them means borrowing the exact amount that would require future repayments, inclusive of interest, identical to the cash flows of all assets at this future date. This is a mark-to-model rather than a full mark-to-market. Mark-to-model calculations eliminate some of the drawbacks of the full MTM. For instance, when isolating credit risk, day-to-day variations due to interest rate fluctuations are not necessarily relevant because the aim is to differentiate future values according to their future random credit states. One option is to price the credit risk using the current ‘crystallized’ rates allowing credit spreads only to vary according to the future credit states at the horizon. The process avoids generating unstable values due to interest rate risk, which NPV scenarios capture for ALM. Mark-to-future refers to the forward valuation of the assets. It is different from markto-market because it addresses the issue of unknown future values using models to determine the possible future risk states of assets. The valuation block of VaR models uses both current valuation and mark-to-future to generate the value distribution at a future time point.

BENEFITS AND DRAWBACKS OF ECONOMIC VALUES The theoretical answer to the problem of considering both revenues and risk in exposures is to mark-to-market transactions. Pure mark-to-market applies to market transactions only. Economic values, or ‘fair values’, are similar and apply to all assets, including banking portfolio loans. The economic values discount all future flows with rates that reflect the risk of each transaction.

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The arguments for accounting standards for the banking book relate to stability of earnings. A major drawback of mark-to-market is that the P&L would change every day, as the market does, thereby generating earnings instability. This is more the case, however, with fixed rate assets than with floating rate assets. The latter have values closer to par when interest rates vary. Moreover, it is not possible to trade the assets and liabilities of the banking portfolio, so that marking them to market does not make much sense. Lack of liquidity could result in large discounts from the theoretical mark-to-market values. In spite of their traditional strengths, accounting values suffer from a number of deficiencies. Accounting flows capture earnings over a given period. They do not give information about the long-term profitability of existing facilities. They ignore the market conditions, which serve as an economic benchmark for actual returns. For example, a fixed rate loan earning an historical interest rate has less value today if interest rates increased. It earns less than a similar loan based on current market conditions. Therefore, accounting measures of performance do not provide an image of profitability relative to current interest rates. In addition, different spreads apply to different credit risks. Hence, without risk adjustment, the interest incomes are not comparable from one transaction to another. Contractual interest income is not risk-adjusted. The same applies to book value. Whether a loan is profitable or not, it has the same accounting value. Whether it is risky or not, it also has the same value. On the other hand, ‘fair value’ is economic and richer in terms of information. A faithful image should provide the ‘true’ value of a loan, considering its richness as well as its risk. Book values do not. Economic or fair values do. They have informational and reporting value added for both the bank and outside observers.

RISK-ADJUSTED PERFORMANCES AND ECONOMIC VALUES Risk-adjusted performances are a compromise between accounting measures of performance and the necessity of adjusting income for risk. Although not going the full way to economic valuation, since they are accrual income-based measures, they do provide a risk adjustment. The popular risk-adjusted measures of accounting profitability are the Risk-adjusted Return on Capital (RaRoC) and the Shareholders Value Added (SVA)6 . Both combine profitability and risk. RaRoC nets expected losses from accrued revenues and divides them by the capital allocated to a transaction, using a capital (risk) allocation system. The RaRoC ratio should be above an appropriate hurdle rate representative of a minimum profitability. SVA nets both the expected loss from revenues and the cost of the required capital, valued with an appropriate price in line with shareholder’s required return. A positive SVA implies creation of value, while a negative SVA implies destruction of value from the bank’s shareholders’ standpoint. The same principle applies for market transactions, except for expected loss. Capital results from either market risk or credit risk. It is either regulatory capital or economic capital, the latter being conceptually a better choice. Both calculations use the accrued income flow of a period, instead of discounting to present all future flows. Economic valuation, or ‘mark-to-market’, is not a substitute for risk-adjusted performances because it combines both the revenue and the risk adjustments 6 Both

are detailed in Chapters 53 and 54. Here, we focus on the difference between such measures and valuation.

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in a single figure. The excess spread, or spread between contractual return and the market required yield, also combines both adjustments. Instead of bundling these two adjustments in a single value, RaRoC or SVA make both revenues and risk adjustments explicit.

APPENDIX: DISCRETE TIME AND CONTINUOUS TIME COMPOUNDING This appendix details the calculations of continuous time compounding and discounting and how to derive the continuous rate equivalent to a discrete time rate. Continuous time compounding serves for the pricing model literature.

Future Values The future value FV(y, n, 1) of an investment of 1 at y for n years, compounding only once per year, is (1 + y)n . Dividing the year into k subperiods, we have FV(y, n, k) = (1 + y/k)kn . For instance, at y = 10% per year, the future value is: FV(10%, 1, 1) = (1 + 10%) = 1.1000 for one subperiod

FV(10%, 1, 2) = (1 + 10%/2)2 = 1.1025 FV(10%, 1, 4) = (1 + 10%/4)4 = 1.1038

for two subperiods for four subperiods

The limit when k tends towards infinity provides the ‘continuous compounding’ formula. When k increases, kn grows and y/k decreases. Mathematically, the limit becomes: FVc (y, n) = exp(yn) = eyn The exponential function is exp(x) = ex . For n = 1 and y = 10%, the limit is: FVc (10%, 1) = exp(10% × 1) = 1.1052

Present Values The present value PVc (y, n, k) of a future flow in n years results from FVc (y, n, k). If FVc (n) = eyn , the present value of 1 dated n at the current date is: PVc (y, n) = exp(−yn) = e−yn For one period: PV(10%, 1) = (1 + 10%)−1 = 0.90909 PVc (10%, 1) = exp(−10% × 1) = 0.90484 When the period n comprises k distinct subperiods, the present value formula uses the actual number of subperiods for n. If n = 1 year, the present value for one subperiod is PVc (10%, 1) and becomes PVc (10%, 0.5) if k = 2.

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Equivalence between Continuous and Discrete Rates There is a discrete rate y such that the future values are identical when compounding with discrete subperiods and continuously. It is such that the future value at the continuous rate yc is identical to the future value at the discrete rate yd . The equivalence results from: (1 + y/k)kn = exp(yc n) This rate y does not depend on n: (1 + y/k)k = exp(yc ). Hence, taking the Napierian logarithm (ln) of both sides: yc = k ln(1 + y/k) y = k[exp(yc /k) − 1] As an example, y = 10% is equivalent for 1 year and k = 1 to a continuous rate such that: 10% = exp(yc ) − 1 or

exp(yc ) = 1 − 10% = 0.9

It is equivalent to state that: yc = k ln(1 + y/k) = ln(1 + 10%) = 9.531% This allows us to move back and forth from continuous to discrete rates.

9 Risk Model Building Blocks

The development of risk models for banks has been extremely fast and rich in the recent period, starting with Asset–Liability Management (ALM) models, followed by market Value at Risk (VaR) models for risk and the continuous development of credit risk models, to name the main trends only. The risk models to which we refer in this book are ‘instrumental’ models, which help in implementing better risk management measures and practices in banks. Looking at the universe of models—the variety of models, the risks they address and their techniques—raises real issues over how to structure a presentation of what they are, what they achieve and how they do it. Looking closely at risk models reveals a common structure along major ‘blocks’, each of them addressing essential issues such as: What are the risk drivers of a transaction? How do we deal with risk diversification within portfolios? How do we obtain loss distributions necessary for calculating VaR? The purpose of this chapter is to define the basic ‘building blocks’ of risk models. The definition follows from a few basic principles: • The primary goal of risk management is to enhance the risk–return profiles of transactions and portfolios. The risk–return profiles of transactions or portfolios are a centrepiece of risk management processes. Traditional accounting practices provide returns. By contrast, risks raise measurement challenges because they are intangible and invisible until they materialize. • Risk–return measures appear as the ultimate goal of risk models. Their main innovation is in risk measurement. Models achieve this goal by assembling various techniques and tools that articulate to each other. Together, they make up the risk management ‘toolbox’. • Risk models differ along two main dimensions: the nature of the risk (interest rate risk, market, credit, etc.) that they address; whether they apply to isolated transactions or portfolios, because portfolios raise the issue of measuring diversification in addition

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to measuring the standalone risk of transactions. Accordingly, techniques to address various risks for standalone transactions or portfolios differ. Such principles suggest a common architecture for adopting a ‘building block’ structure of risk models. Summarizing, we find that risk models combine four main building blocks: I. Risk drivers and standalone risk of transactions. Risk drivers are all factors influencing risks, which are necessary inputs for measuring the risk of individual transactions. When considered in isolation, the intrinsic of a transaction is ‘standalone’. II. Portfolio risk. Portfolio models aim to capture the diversification effect that makes the risk of a portfolio of transactions smaller than the sum of the risks of the individual transactions. They serve to measure the economic capital under the VaR methodology. III. Top-down and bottom-up links. These links relate global risk to individual transaction risks, or subportfolio risks. They convert global risk and return targets into risk limits and target profitability for business lines (top-down) and, conversely, for facilitating decision-making at the transaction level and for aggregating business line risks and returns for global reporting purposes (bottom-up). IV. Risk-adjusted performance measuring and reporting, for transactions and portfolios. Both risk–return profiles feed the basic risk processes: profitability and limit setting, providing guidelines for risk decisions and risk monitoring. The detailed presentation uses these four main blocks as foundations for structuring the entire book. Each main block subdivides into smaller modules, dedicated to intermediate tasks, leading to nine basic blocks. The book presents blocks I and II for the three main risks: interest rate risk, market risk and credit risk. On the other hand, tools and techniques of blocks III and IV appear transversal to all risks and do not require a presentation differentiated by risk. After dealing with blocks I and II separately for each main risk, we revert to a transversal presentation of blocks III and IV. The first section focuses on the double role of risk–return profiles: as target variables of risk policies for decision-making; as a key interface between risk models and risk processes. The second section reviews the progressive development stages of risk models. The third section provides an overview of the main building blocks structure. The next sections detail each of the blocks. The last section provides a view of how sub-blocks assemble and how they map to different risks.

RISK MODELS AND RISK PROCESSES Risk–return is the centrepiece of risk management processes. All bank systems provide common measures of income, such as accrual revenues, fees and interest income for the banking portfolio, and Profit and Loss (P&L) for the market portfolio. Measuring risk is a more difficult challenge. Without a balanced view on both risk and return, the management of banks is severely impaired since these two dimensions are intertwining in every process, setting guidelines, making risk decisions or monitoring performance. Therefore, the major contributions of models are to capture risks in all instances where it is necessary to set targets, make risk decisions, both at the transaction level and at the portfolio level.

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There are multiple risk models. ALM models aims to model the interest rate risk of banks at the global balance sheet level. Market risk and credit risk models serve both to measure the risk of individual transactions (standalone risk) and to measure the portfolio risk. To reach this second goal, they need to address the intrinsic risk of individual transactions, independent of the portfolio context. This requires identifying all risk drivers and modelling how they alter the risk of transactions. When reaching the portfolio risk stage, complexities appear because of the diversification effect, making the risk of the portfolio—the risk of the sum of individual transactions—much less than the sum of all individual risks. The issue is to quantify such diversification effects that portfolio models address. The applications of portfolio models are critical and numerous. They serve to determine the likelihood of various levels of potential losses of a portfolio. The global portfolio loss becomes the foundation for allocating risk to individual transactions, after diversification effects. Once this stage is reached, it becomes relatively simple to characterize the risk–return profile of the portfolio as well as those of transactions, since we have the global risk and the risk allocations, plus the income available from the accounting systems. Moving to portfolio models is a quantum leap because they provide access to the risk post-diversification effect for any portfolio, the global portfolio or any subset of transactions. This is a major innovation, notably for credit risk. Measuring risks allows the risk–return view to be extended bank-wide, across business lines and from transactions to the entire bank portfolio. This faculty considerably leverages the efficiency of the bank processes that previously dealt with income without tangible measures of portfolio risks. The building blocks of bank processes respond to common sense needs: setting guidelines, making decisions, monitoring risk and return, plus the ‘vertical’ processes of sending signals to business lines, complying with overall targets and obtaining a consolidated view of risk and return. Risk models provide risk, the otherwise ‘missing’ input to processes. Figure 9.1 summarizes the overview of interaction between risk models and risk processes.

Risk Models

Risk drivers

Risk & Return View

Risk & Return Processes

Risks Return Transaction risk models

Portfolio risk models

From Risk Models to Risk Processes through Risk & Return Measures From Risk Models to Risk Processes through Risk & Return Measures

FIGURE 9.1

From risk models to risk processes through risk and return measures

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GENERATIONS OF BANK RISK MODELS There are a range of banking risk models differing in their purposes and techniques, and the risks that they address. The ALM models were the first to appear. They aimed to capture the interest rate risk of the banking book of commercial banks. Later on, they developed into several building blocks, some of them addressing the interest rate risk of the interest income, others the interest rate risk of the mark-to-market value of the balance sheet, and others aiming to value embedded prepayment options in banking products. Regulations focused on capital requirements and banks focused on economic capital. The capital adequacy principle led to the VaR concept and to VaR models for market risk and credit risk. Market risk models aim to capture the market portfolio risk and find the capital requirement adequate for making this risk sustainable. They address the behaviour of risk drivers, interest rates, foreign exchange rates and equity indexes and, from this first block, derive market risk for the portfolio as VaR. There are several generations of risk models. The academic community looked at modelling the borrowers’ risk, captured through ratings or observed default events, from observable characteristics such as financial variables a long time ago. There is renewed interest in such models because of the requirements of the New Basel Accord to provide better measures of credit risk drivers at the borrower and the transaction levels. The newest generation of models addresses both transaction risk and portfolio risk with new approaches. Risk drivers are exposures, default probabilities and recoveries. Modelling default probability became a priority. New models appeared relating observable attributes of firms to default and ratings, using new techniques, such as the neural network approach. The implementation of the conceptual view of default as an option held by stockholders1 is a major advance that fostered much progress. Various techniques address the portfolio diversification issue, trying to assess the likelihood of simultaneous adverse default or risk deterioration events depending on the portfolio structure. Modelling credit risk and finding the risk of portfolios remained a major challenge because of the scarcity of data, in contrast to market models, which use the considerable volume of market price information. Recent models share common modules: identifying and modelling risk drivers in order to model the standalone risk of individual transactions; modelling diversification effects within portfolios to derive the portfolio risk. Other tools serve for linking risk measures with the business processes, by allocating income and risks to transactions and portfolios. Allocating income is not a pure accounting problem since it is necessary to define the financial cost matching the transaction revenues. Risks raise a more difficult challenge. The Funds Transfer Pricing (FTP) system and the capital—or risk—allocation system perform these two tasks. Once attached to risk models, they allow us to reach the ultimate stage of assigning a risk–return profile to any transaction or portfolio. Then, the bank processes can take over and use these risks–returns as inputs for management and business purposes. 1 This

is the Merton (1974) model, later implemented by the KMV Corporation to measure the ‘expected default frequency’ (Edf ). The default option is the option held by stockholders to give up the assets of the firm if their value falls below that of debt. See Chapter 38.

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RISK MODEL BUILDING BLOCKS The risk management toolbox comprises four main building blocks that Figure 9.2 maps with nine basic risk modelling building blocks, each related to a specific issue. The four main blocks are: I. Risk measurement of transactions on a standalone basis, or ‘intrinsic risk’, without considering the portfolio context that diversifies away a large fraction of the sum of individual risks. II. Risk modelling of portfolios, addressing the issue of portfolio risk diversification and its measurement through basic characteristics such as expected (statistical) losses and potential losses. III. Risk allocation and income allocation within portfolios. The risk allocation mechanisms should capture the diversification effect within the portfolio. The diversification effect results in a lower risk of the portfolio than the sum of the intrinsic risks of transactions. Since risks do not add arithmetically as income does, the mechanism is more complex than earnings allocations to subsets of the portfolio. IV. Determination of the risk–return profiles of transactions, subsets of the bank portfolio, such as business lines, and the overall bank portfolio. The risk–return profile is the ultimate aim of modelling, which risk management attempts to enhance for individual transactions and portfolios.

FIGURE 9.2

I

Risk drivers and transaction risk (standalone)

II

Portfolio risk

III

Top-down & bottom-up tools

IV

Risk & return measures

The main modelling building blocks

This basic structure applies to all risks. The contents of the first two blocks differ with risks, although all modelling techniques need similar inputs and produce similar outputs. The third and fourth blocks apply to all risk, and there is less need to differentiate them for credit and market risks, for instance. When considering all risk models, it appears they share the same basic structure. The four main building blocks each differentiate into sub-blocks dedicated to specific tasks. Various specific ‘modules’ provide the necessary information to progress step-by-step towards the ultimate goal of assigning risk–return profiles to transactions and portfolios. Figure 9.3 breaks down these building blocks into modules addressing intermediate issues. The top section describes the main blocks. The bottom section illustrates the linkages between risk models and risk processes. All modules need to fit with each other, using

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MODELS Basic Modelling Building Blocks Basic Modeling Building Blocks

Main Building Blocks Main Building Blocks

Standalone Risk

I

I.

II

II

III III

IV IV

Portfolio Portfolio Risk Risk Models Models

1

Risk Drivers

2

Risk Exposures and Values

3

Risk Valuation (Standalone)

4

Correlations between Risk Events

5

Portfolio Risk Profile (Loss Distribution)

6

Capital (Economic)

7

Funds Transfer Pricing

8

Capital Allocation

II.

Top-down & Top-Down & Bottom-up Bottom Tools - up Tools

III.

Risk−Return Risk- Return

IV.

9

Risk-adjusted Performance

PROCESSES Risk−Return Measuring and Monitoring

Risks Return

FIGURE 9.3

Risk−Return Decisions, Hedging & Enhancement

Main risk modelling building blocks

the inputs of the previous ones, and providing outputs that serve as inputs to the next ones. This structure conveniently organizes the presentation of various modelling contributions. Vendors’ models bundle several modules together. Other modules serve to fill out gaps in inputs that other building blocks need to proceed further, such as the modelling of the market risk or the credit drivers of transactions. This ‘four/nine’ block structures the entire presentation expanded in this book. The book expands this structure for each of the three risks: interest rate risk for ALM, market risk and credit risk. What follows describes in detail the basic modelling blocks.

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BLOCK I: STANDALONE RISK Block I requires risk and transaction data to capture individual transaction risks.

Risk Drivers Block I starts by identifying ‘risk drivers’, those parameters whose uncertainty alters values and revenues. These random risk drivers are identical to value drivers because measures of risk are variations of value. For ALM, the main risk drivers are the interest rates, which drive the interest income. For market risk, risk drivers are all the market parameters that trigger changes in value of the transactions. They differ across the market compartments: fixed income, foreign exchange or equity. The credit risk drivers are all parameters that influence the credit standing of counterparties and the loss for the bank if it deteriorates. They include exposure, recoveries under default, plus the likelihood of default, or default probability (DP) and its changes (migration risk) when time passes. Later on, we make a distinction between risk drivers and risk factors. Risk factors alter risk drivers, which then directly influence risk events, such as default or the change in value of transactions. The distinction is purely technical. In risk models, risk drivers relate directly to risk events, while risk factors serve for modelling these direct drivers. The distinction is expanded in Chapters 31 and 44 addressing correlation modelling with ‘factor’ models, hence the above distinction. The basic distinction for controlling risk is between risk drivers, which are the sources of uncertainty, and the exposure and sensitivities of transaction values to such sources. Risk drivers remain beyond the control of the bank, while exposures and sensitivities to risk drivers result from the bank’s decisions.

Exposures Block I’s second sub-block relates to exposure size and economic valuation. Book value, or notional amount (for derivatives), characterizes size. Economic valuation is the ‘fair value’, risk and revenue-adjusted. Economic valuation requires mapping exposures to the relevant risk drivers and deriving values from them. Because it is risk-adjusted, any change in the risk drivers materializes in a value change, favourable or unfavourable. Exposure serves as a basis for calculating value changes. For market risk, exposures are mark-to-market and change continuously with market movements. Exposures for credit risk are at book values for commercial banking. Since book values do not change when the default probabilities change, economic values serve for full valuation of credit risk. Exposures data include, specifically, the expected size under a default event (Exposure At Default, EAD), plus the recoveries, which result in loss given default (Lgd) lower than exposure. For ALM, exposures are the ‘surfaces’ serving to calculate interest revenues and costs. ALM uses both book values and economic values. Note that ALM’s first stage is to consolidate all banking portfolio exposures by currency and type of interest rates, thereby capturing the entire balance sheet view from the very beginning. By contrast, credit and market risks modelling starts at the transaction level, to characterize the individual risk, and the global portfolio risk is dealt with in other blocks.

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Standalone Risk and Mark-to-Future Sub-block 3 is forward looking. It looks at future events materializing risks today for each individual transaction, or on a ‘standalone’ basis. Such events include deviations of interest rates for the banking portfolio, all market parameters for market risk and all factors influencing the credit standing of borrowers for credit risk. Sub-block 3 assesses risk as any downside deviation of value at a future time point. Since future values are uncertain, they follow a distribution at this horizon. Adverse deviations are the losses materializing risk. ‘Marking-to-future’ designates the technical processes of calculating potential values at a future date, and deriving losses from this distribution. The principle is the same for transactions and portfolios. The difference is the effect of diversification within a portfolio, which standalone risk ignores.

Obtaining the Risk Driver Inputs Risk drivers and exposure are ingredients of risk models. Sometimes, it is possible to observe them, such as interest rates whose instability over time is readily observable. Sometimes, they are assigned a value because they result from a judgment on risk quality, such as ratings from rating agencies or from internal rating systems. A rating is simply a grade assigned to the creditworthiness combining a number of factors that influence the credit standing of borrowers. Several models help in modelling the uncertainty of the risk drivers. For instance, interest rate models allow us to derive multiple scenarios of interest rates that are consistent with their observed behaviour. Rating models attempt to mimic ratings from agencies through some observable characteristics of firms, such as financial data. Default probability models, such as KMV’s Credit Monitor or Moody’s RiskCalc, provide modelled estimates of default probabilities.

BLOCK II: PORTFOLIO RISK Block I focuses on a standalone view of risk, ignoring the portfolio context and the diversification effects. Block II’s goal is to capture the risk profile of the portfolio rather than the risk of single transactions considered in isolation.

Diversification Effects Diversification requires new inputs characterizing the portfolio risk, mainly the sizes of individual exposures and the correlations between risk events of individual transactions. The idea is intuitive. Should all transactions suffer losses simultaneously, the portfolio loss would be the sum of all simultaneous losses. Fortunately, not all losses materialize at the same time, and the portfolio loss varies according to the number and magnitude of individual losses occurring within a given period. Intuitively, a small number of losses are much more frequent than a simultaneous failure of many of them. This is the essence of diversification: not all risks materialize together. Which losses are going to materialize and

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how many will occur within the period are uncertain. This is the challenge of quantifying diversification effects, which requires ‘portfolio models’. The technical measure of association between transaction risks is the ‘correlation’. For instance, the adverse deviations of interest income, a usual target variable for ALM policies, result from interest rate variations. The correlation between interest rates tends to reduce the diversification effect. All interest revenues and costs tend to move simultaneously in the same direction. Asset and liability exposures to interest rates are offset to the extent that they relate to the same interest rate references. The adverse deviations of the market value of the trading portfolio correlate because they depend on a common set of market parameters, equity indexes, interest rates and foreign exchange rates. However, diversification effects are important because some of these risk drivers are highly correlated, but others are not, and sometimes they vary in opposite directions. Moreover, positions in opposite directions offset. Credit defaults and migrations tend to correlate because they depend on the same general economic conditions, although with different sensitivities. Models demonstrated that portfolio risk is highly sensitive to such correlations. In spite of such positive correlations, diversification effects are very significant for credit risk. The opposite of ‘diversification’ is ‘concentration’. If all firms in the same region tend to default together, there is a credit risk concentration in the portfolio because of the very high default correlation. A risk concentration also designates large sizes of exposures. A single exposure of 1000 might be riskier than 10 exposures of 100, even if borrowers’ defaults correlate, because there is no diversification for this 1000, whereas there is some across 10 different firms. ‘Granularity’ designates risk concentration due to size effect. It is higher with a single exposure of 1000 than with 10 exposures of 100.

Correlation Modelling Obtaining correlations requires modelling. Correlations between risk events of each individual transaction result from the correlation between their risk and value drivers. Correlations for market risk drivers, and for interest rates for ALM, are directly observable. Correlations between credit risk events, such as defaults, suffer from such a scarcity of data that it is difficult to infer correlations from available data on yearly defaults. In fact, credit risk remains largely ‘invisible’. Credit risk models turn around the difficulty by inferring correlations between credit risk events from the correlations of observable risk factors that influence them. For all main risks, correlation modelling is critical because of the high sensitivity of portfolio risk to correlations and size discrepancies between exposures.

Portfolio Risk and VaR Once correlations are obtained, the next step is to characterize the entire spectrum of outcomes for the portfolio, and its downside risk. Since a portfolio is subject to various levels of losses, the most comprehensive way of characterizing the risk profile is with the distribution of losses, assigning a probability to each level of portfolio loss. Portfolio losses are declines in value triggered by various correlated risk events. From a risk perspective, the main risk statistics extracted from loss distributions are the expected loss, the loss volatility characterizing the dispersion around the mean and

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the loss percentiles at various confidence levels for measuring downside risk and VaR. The VaR is also the ‘economic capital’ at the same preset confidence level.

BLOCK III: REVENUE AND RISK ALLOCATION Blocks I and II concentrate on individual and global risk measures. For the bank as a whole, only portfolio risk and overall return count. However, focusing only on these aggregated targets would not suffice to develop risk management practices within business lines and for decision-making purposes. Moving downwards to risk processes requires other tools. They are the top-down links, sending signals to business lines, and the bottomup links, consolidating and reporting, making up block III. Two basic devices serve for linking global targets and business policy (Figure 9.4): • The FTP system. The system allocates income to individual transactions and to business lines, or any other portfolio segment, such as product families or market segments. • The capital allocation system, which allocates risks to any portfolio subset and down to individual transactions. Global Risk Management

Transfer prices

Risk allocation

Business Policy

FIGURE 9.4

From global risk management to business lines

Funds Transfer Pricing Transfer pricing systems exist in all institutions. Transfer prices, are internal references used to price financial resources across business units. Transfer pricing applies essentially to the banking book. The interest income allocated to any transaction is the difference between the customer rate and the internal reference rate, or ‘transfer price’, that applies for this transaction. Transfer rates also serve as a reference for setting customer rates. They should represent the economic cost of making the funds available. Without such references, there is no way to measure the income generated by transactions or business units (for commercial banking) and no basis for setting customer rates.

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Transfer prices serve other interrelated purposes in a commercial bank. Chapters 26 and 27 detail the specifications making the income allocation process consistent with all risk and return processes.

Capital Allocation Unlike revenues and costs, risks do not add arithmetically to portfolio risk because of diversification. Two risks of 1 each do not add up to 2, but add up in general to less than 2 because of the diversification effect, a basic pillar for the banking industry. Portfolio risk models characterize the portfolio risk as a distribution of loss values with a probability attached to each value. From this distribution, it is possible to derive all loss statistics of interest after diversification effects. This represents a major progress but, as such, it does not solve the issue of moving back to the risk retained by each individual transaction, after diversification effects. Since we cannot rely on arithmetic summation for risks, there is no obvious way to allocate the overall risk to subportfolios or transactions. To perform this task, we need a capital (risk) allocation model. The capital allocation model assigns to any subset of a portfolio a fraction of the overall risk, called the ‘risk contribution’. Risk contributions are much smaller than ‘standalone’ risk measures, or measures of the intrinsic risk of transactions considered in isolation, pre-diversification effects, because they measure only the fraction of risk retained by a transaction post-diversification. The capital allocation system performs this risk allocation and provides risk contributions for all individual transactions as well as for any portfolio subsets. The underlying principle is to turn the non-arithmetic properties of risks into additive risk contributions that sum up arithmetically to the overall bank’s portfolio risk. The capital allocation system allocates capital in proportion to the risk contributions provided by portfolio models. For example, using some adequate technique, we can define the standalone risk of three transactions, for example 20, 30 and 30. The sum is 80. However, the portfolio model shows that VaR is only 40. The capital allocation module tells us how much capital to assign to each of the three transactions, for example 10, 15 and 25. They sum up to the overall risk of 40. These figures are ‘capital allocations’, or ‘risk contributions’ to the portfolio risk. The FTP and capital allocation systems provide the necessary links between the global portfolio management of the bank and the business decisions. Without these two pieces, there would be missing links between the global policy of the bank and the business lines. The two pieces complement each other for obtaining risk-adjusted performance measures. Block III leads directly to the ultimate stage of defining risk and return profiles and risk-based performance and pricing.

Aggregating VaR for Different Risks Note that the overall portfolio management for the bank includes integrating all risks in a common unified framework. The VaR concept solves the issue by valuing risks in dollar value. Adding the VaR of different risks remains a problem, however, since the different

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main risks may correlate more or less, or be independent. A sample global report of a VaR implementation for a bank illustrates the overview (Table 9.1). TABLE 9.1

Example of a VaR report

Risks Credit Market Interest rate (ALM) Operational risk Total

VaR capital 260 50 50 80 400

65.0% 12.5% 12.5% 20.0%

In this example, total potential losses are valued at 40. The reports aggregate the VaR generated by different risks, which is a prudent rule. However, it is unlikely that unexpected losses will hit their upper bounds at the same time for credit risk, market risk and other risks. Therefore, the arithmetic total is an overestimate of aggregated VaR across risks.

BLOCK IV: RISK AND RETURN MANAGEMENT Block IV achieves the ultimate result of providing the risk–return profiles of the overall bank portfolio, of transactions or subportfolios. It also allows breaking down aggregated risk and return into those of transactions or subportfolios. Reporting modules take over at this final stage for slicing risks and return along all relevant dimensions for management. The consistency of blocks I to IV ensures that any breakdown of risks and returns reconciles with the overall risk and return of the bank. At this stage, it becomes possible to plug the right measures into the basic risk management processes: guidelines-setting, decision-making and monitoring. In order to reach this final frontier with processes, it is necessary to add a risk-based performance ‘module’. The generic name of risk-adjusted profitability is RaRoC (Risk-adjusted Return on Capital). Risk-Adjusted Performance Measurement (RAPM) designates the risk adjustment of individual transactions that allows comparison of their profitability on the same basis. It is more an ex post measure of performance, once the decision of lending is made, that applies to existing transactions. Risk-Based Pricing (RBP) relates to ex ante decision-making, when we need to define the target price ensuring that the transaction profitability is in line with both risk and the overall target profitability of the bank. For reasons detailed later in Chapters 51 and 52, we differentiate the two concepts, and adopt this terminology. RBP serves to define ex ante the risk-based price. RAPM serves to compare the performance of existing transactions. RAPM simply combines the outputs of the previous building blocks: the income allocation system, which is the FTP system, and the capital allocation system. Once both exist, the implementation is very simple. We illustrate the application for ex post RAPM. Transaction A has a capital allocation of 5 and its revenue, given the transfer price, is 0.5. Transaction B has a capital allocation of 3 and a revenue of 0.4. The RaRoC of transaction A is 0.5/5 = 10% and that of transaction B is 0.4/3 = 13.33%. We see that the less risky

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transaction B has a better risk-adjusted profitability than the riskier transaction A, even though its revenue is lower.

THE RISK MANAGEMENT TOOLBOX: AN OVERVIEW This section summarizes: the nature of risk and return decisions that models aim to facilitate; how the several building blocks assemble to provide such risk–return profiles; how they differentiate across the main risks, interest rate risk, credit risk and market risk.

Risk Modelling and Risk Decisions All model blocks lead ultimately to risk and return profiles for transactions, subportfolios of business units, product families or market segments, plus the overall bank portfolio. The purpose of characterizing this risk–return profile is to provide essential inputs for decision-making. The spectrum of decisions includes: • • • •

New transactions. Hedging, both individual transactions (micro-hedges) and subportfolios (macro-hedges). Risk–return enhancement of transactions, through restructuring for example. Portfolio management, which essentially consists of enhancing the portfolio risk–return profile.

Note that this spectrum includes both on-balance sheet and off-balance sheet decisions. On-balance sheet actions refer to any decisions altering the volume, pricing and risk of transactions, while off-balance sheet decisions refer more to hedging risk.

How Building Blocks Assemble In this section, we review the building block structure of risk models and tools leading to the ultimate stage of modelling the risk–return profiles of transactions and portfolios. It is relatively easy to group together various sub-blocks to demonstrate how they integrate and fit comprehensively to converge towards risk and return profiles. An alternative view is by nature of risk, interest rate risk for commercial banking (ALM), market risk or credit risk, plus other risks. The structure by building blocks proves robust across all risks because it distinguishes the basic modules that apply to all. The differences are that modules differ across risks. For instance, market risk uses extensively data-intensive techniques. Credit risk relies more on technical detours to remedy the scarcity of data. ALM focuses on the global balance sheet from the very start simply because the interest rate risk exposure does not raise a conceptual challenge, but rather an information technology challenge. We provide below both views of the risk models: • The building blocks view, across risks. • The view by risk, with the three main risks ‘view’ developed in this text: interest rate risk in commercial banking (ALM), market risk and credit risk.

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Figure 9.5 summarizes how basic blocks link together to move towards the final risk and return block. Making available this information allows the main risk management processes to operate. The figure uses symbols representative of the purpose of risk modelling blocks and of risk process blocks.

II. Portfolio Risk

III. TopIII. Topdown & Down & Bottom-up Bottom Tools - up tools

IV. Risk− IV. RiskReturn Return Decisions Decisions

ALM

Risk Drivers

Market Risk

Exposures

Credit Risk

Standalone Risk

Correlations

Frequency

I. Risk Data and Risk Measures

Portfolio Risk Loss

Capital

Global Business Line Transaction

Capital Allocation Funds Transfer Pricing

R&R Measures R&R Decisions R&R Monitoring

Risks Risks

Return Return

Risk Processes

FIGURE 9.5

From modelling building blocks to risk and return profiles

How Building Blocks Map to the Main Risks All building blocks map to different risks: interest rate risk, market risk and credit risk. The book uses this mapping as the foundation for structuring the outline of the subsequent detailed presentations. In practice, the organization by risk type applies essentially for building blocks I and II. The FTP system and the capital allocation system are ‘standalone modules’. The ALM is in charge of the FTP module. The capital (or risk) allocation module applies to market and credit risk with the same basic rules. Finally, the ‘risk–return’ module uses

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all contributions of upper blocks. The book uses the matrix in Table 9.2, developing the modelling for each main risk. The book structure is shown by the cells of the table. TABLE 9.2

Cross-tabulating model building blocks with main risks ALM

I. II. III. IV.

Market risk

Credit risk

Standalone risk of transactions Portfolio risk Top-down and bottom-up tools Risk and return

Figure 9.6 combines the risk view with the building block view, and shows how all blocks articulate to each other. Credit Risk

Market Risk

Interest Rate Risk (Banking book - ALM)

I

Credit risk

Market risk

IR risk

II

Portfolio risk models

ALM models

III

Capital allocation

FTP

Risk Contributions

IV

Market P&L

Banking Spreads

Risk & Return

Risk Processes

FIGURE 9.6

Building blocks view and risks view of the risk management toolbox

SECTION 5 Asset–Liability Management

10 ALM Overview

Asset–Liability Management (ALM) is the unit in charge of managing the Interest Rate Risk (IRR) and the liquidity of the bank. Therefore, it focuses essentially on the commercial banking pole. ALM addresses two types of interest rate risks. The first risk is that of interest income shifts due to interest rate movements, given that all balance sheet assets and liabilities generate revenues and costs directly related to interest rate references. A second risk results from options embedded in banking products, such as the prepayment option of individual borrowers who can renegotiate the rate of their loans when the interest rates decline. The optional risk of ALM raises specific issues discussed in a second section. These two risks, although related, need different techniques. ALM policies use two target variables: the interest income and the Net Present Value (NPV) of assets minus liabilities. Intuitively, the stream of future interest incomes varies in line with the NPV. The main difference is that NPV captures the entire stream of future cash flows generated by the portfolio, while the interest income relates to one or several periods. The NPV view necessitates specific techniques, discussed in different chapters from those relating to the interest income sensitivity to interest rates. This chapter presents a general overview of the main building blocks of ALM. For ALM, the portfolio, or balance sheet, view dominates. It is easier to aggregate the exposures and the risk of individual transactions in the case of ALM than for other risks, because all exposures share interest rates as common risk drivers.

ALM BUILDING BLOCKS In the case of ALM, the global view dominates because it is easier to aggregate the exposures and the risk of individual transactions than for other risks. Figure 10.1

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summarizes the main characteristics of ALM building blocks using the common structure proposed for all risks. The next sections detail blocks I: standalone risk and II: banking book—portfolio—risk. Building Blocks: ALM Building Blocks: ALM I.

I. 1

2

3

II. II.

FIGURE 10.1

4

Risk Drivers

Interest rates by currency, entire term structure. Mapped to all assets and liabilities.

Risk Exposures

Outstanding balances of assets and liabilities, at book or mark-to-model values. Banking products also embed implicit options.

Standalone Risk

Valuation of adverse deviations of interest margin and NPV, inclusive of option risk.

Correlations

Gaps fully offset exposures to the same interest rates.

5.1

Portfolio Risk

5.2

Capital (Economic)

Using projected portfolios, valuation of margins and NPV under both interest rate and business scenarios. Forward NPV and interest income distributions from scenarios. Not required. EaR results from adverse deviations of interest margins. An NPV VaR captures the interest rate risk, given business risk.

Major building blocks of ALM

BLOCK I: STANDALONE RISK The risk drivers are all interest rates by currency. Exposures in the banking portfolio are the outstanding balances of all assets and liabilities, except equity, at book value. To capture the entire profile of cash flows generated by all assets and liabilities, the NPV of assets and liabilities is used at market rates. Risk materializes by an adverse change of either one of these two targets. The standalone risk of any individual exposure results from the sensitivity to interest rate changes of the interest income or of its value. The downside risk is that of adverse variations of these two target variables due to interest rate moves. Hence, the valuation of losses results from revaluation at future dates of interest income or of the NPV for each scenario of interest rates. There are two sources of interest rate exposure: liquidity flows and interest rate resets. Each liquidity flow creates an interest rate exposure. For a future inflow, for instance from a future new debt or a repayment of loans, the interest rate risk results from the unknown rate of the utilization of this inflow for lending or investing. Interest rate risk arises also from interest rate resets in the absence of liquidity flows. A floating rate long-term debt

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has several reset dates. For example, a 10-year bullet loan with a floating LIBOR 1-year rate has rate resets every year, while the main cash flow occurs at maturity. Interest rate risk depends on the nature, fixed or variable, of interest rates attached to transactions. When a transaction earns a variable rate, rate resets influence interest revenues both positively and negatively. When it earns a fixed rate, the asset has less value if rates rise above its fixed rate, which makes sense since it generates revenues lower than the market does, and vice versa. When a debt pays a fixed rate above market rate, there is an economic cost resulting from borrowing at a rate higher than the rate. This is an ‘opportunity cost’, the cost of giving up more profitable opportunities. Interest rate variations trigger changes of either accounting revenues or costs, or opportunity (economic) costs. The exposures are sensitive to the entire set of common interest rates. Each interest rate is a common index for several assets and liabilities. Since they are sensitive to the same rate, ‘gaps’ offset these exposures to obtain the net exposure for the balance sheet for each relevant rate. ALM also deals with the ‘indirect’ interest rate risk resulting from options embedded in retail banking products. The options are mainly prepayment options that individuals hold, allowing them to renegotiate fixed rates when rates move down. These options have a significant value for all retail banking lines dealing with personal mortgage loans. Should he exercise his option, a borrower would borrow at a lower rate, thereby reducing interest revenues. Depositors also have options to shift their demand deposits to higher interest-earning liabilities, which they exercise when rates increase. The value of options is particularly relevant for loans since banks can price them rather than give them away to customers. The expected value of options is their price. The potential risk of options for the lender is the potential upside of values during their life, depending on how rates behave. Dealing with optional risk addresses two questions: What is the value of the option for pricing it to customers? What is the downside risk of options given all possible time paths of rates until expiration? The techniques for addressing these questions differ from those of classical ALM models. They are identical to the valuation techniques for market options. The main difference with market risk is that the horizon is much longer. Technical details are given in Chapters 20 and 21.

BLOCK II: PORTFOLIO RISK ALM exposures are global in nature. They result from mismatches of the volumes of assets and liabilities or from interest rate mismatches. The mismatch, or ‘gap’, is a central and simple concept for ALM. A liquidity gap is a difference between assets and liabilities, resulting in a deficit or an excess of funds. An interest rate gap is a mismatch between the size of assets and liabilities indexed to the same reference interest rate. The NPV is the mismatch between liability values and asset values. The sensitivity of the NPV to interest rate changes results from a mismatch of the variations of values of assets and liabilities. For these reasons, ALM addresses from the very beginning the aggregated and netted exposure of the entire balance sheet. Implicit options embedded in banking products add an additional complexity to ALM risk. The interest rate risk of the balance sheet depends on correlations between interest rates. Often, these correlations are high, so that there is no need to differentiate the movements of similar interest rates, although this is only an approximation. In addition, only netted

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exposures are relevant for assets and liabilities indexed to the same, or to similar, interest rates. Treasurers manage netted cash flows, and ALM controls net exposures to interest rate risk. Measuring risk consists of characterizing the downside variations of target variables under various scenarios. ALM uses the simulation methodology to obtain the risk–return profile of the balance sheet using several interest rate scenarios. The expected return is the probability weighted average of profitability across scenarios, measured by interest income or NPV. The downside risk is the volatility of the target variables, interest income or NPV, across scenarios, or their worst-case values. ALM models serve to revalue at future time points the target variables as a function of interest rate movements. To achieve this purpose, ALM models recalculate all interest revenues and costs, or all cash flows for NPV, for all scenarios. The process differs from the market risk and the credit risk techniques in several ways: • The revaluation process focuses on interest income and NPV, instead of market values for market risk and credit risk-adjusted values at the horizon for credit risk. • Since the ALM horizon is medium to long-term, ALM does not rely only on the existing portfolio as a ‘crystallized’ portfolio over a long time. Simulations also extend to projections of the portfolio, adding new assets and liabilities resulting from business scenarios. By contrast, market risk and credit models focus on the crystallized portfolio as of today. • Because of business projections, ALM scenarios deal with business risk as well as interest rate risk, while the market risk and credit risk models ignore business risk. • Because ALM discusses periodically the bank position with respect to interest and liquidity risk in conjunction with business activities, it often relies on a discrete number of scenarios, rather than attempting to cover the entire spectrum of outcomes as market and credit models do. In addition, it is much easier to define worst-case scenarios for ALM risk than it is for market risk and credit risk, because only interest rates influence the target variables in ways relatively easy to identify. • Since there is no regulatory capital for ALM risk, there is no need to go as far as fullblown simulations to obtain a full distribution of the target variables at future horizons. However, simulations do allow us to derive an NPV ‘Value at Risk’ (VaR) for interest rate risk. An ALM VaR is the NPV loss percentile at the preset confidence level. In the case of interest rate risk, large deviations of interest rates trigger optional risk, from the implicit options (prepayment options) embedded in the banking products. This makes it more relevant to extend simulations to a wide array of possible interest rate scenarios. Distribution of forward values of interest income or NPV

Expected value Current value

FIGURE 10.2 NPV)

Loss

Forward valuation of target variables (interest income or balance sheet

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135

To value options for the banking portfolio, the same technique as for single options applies. The NPV is more convenient in order to have a broader view of both loans and options values as of today. Visualizing the portfolio risk–return profile is easy because there is a one-to-one correspondence between interest rate scenarios and the interest income or the NPV, if we ignore business risk (Figure 10.2). Moreover, assigning probabilities to interest rate scenarios allows us to define the Earnings at Risk (EaR) for interest income, as the lower bound of interest income at a preset confidence level, or the NPV VaR as the lower bound of the NPV at the same preset confidence level.

11 Liquidity Gaps

Liquidity risk results from size and maturity mismatches of assets and liabilities. Liquidity deficits make banks vulnerable to market liquidity risk. Liquid assets protect banks from market tensions because they are an alternative source of funds for facing the near-term obligations. Whenever assets are greater than resources, deficits appear, necessitating funding in the market. When the opposite occurs, the bank has available excess resources for lending or investing. Both deficits and excesses are liquidity gaps. A time profile of gaps results from projected assets and liabilities at different future time points. A bank with long-term commitments and short-term resources generates immediate and future deficits. The liquidity risk results from insufficient resources, or outside funds, to fund the loans. It is the risk of additional funding costs because of unexpected funding needs. The standard technique for characterizing liquidity exposure relies on gap time profiles, excesses or deficits of funds, starting from the maturity schedules of existing assets and liabilities. This time profile results from operating assets and liabilities, which are loans and deposits. Asset–Liability Management (ALM) structures the time schedule of debt issues or investments in order to close the deficits or excess liquidity gaps. Cash matching designates the reference situation such that the time profile of liabilities mirrors that of assets. It is a key benchmark because it implies, if there are no new loans and deposits, a balance sheet without any deficit or excess of funds. Future deficits require raising funds at rates unknown as of today. Future excesses of funds require investing in the future. Hence, liquidity risk triggers interest rate risk because future funding and investment contracts have unknown rates, unless ALM sets up hedges. The next chapters discuss the interest rate risk. This chapter focuses on the liquidity position of the banking portfolio. The first section introduces the time profile of liquidity gaps, which are the projected differences between asset and liability time profiles. The second section discusses ‘cash matching’ between assets and liabilities. The third

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section discusses the limitations of liquidity gaps, which result mainly from lines without contractual time profiles.

THE LIQUIDITY GAP Liquidity gaps are the differences, at all future dates, between assets and liabilities of the banking portfolio. Gaps generate liquidity risk, the risk of not being able to raise funds without excess costs. Controlling liquidity risk implies spreading over time amounts of funding, avoiding unexpected important market funding and maintaining a ‘cushion’ of liquid short-term assets, so that selling them provides liquidity without incurring capital gains and losses. Liquidity risk exists when there are deficits of funds, since excess of funds results in interest rate risk, the risk of not knowing in advance the rate of lending or investing these funds. There are two types of liquidity gaps. Static liquidity gaps result from existing assets and liabilities only. Dynamic liquidity gaps add the projected new credits and new deposits to the amortization profiles of existing assets.

The Definition of Liquidity Gaps Liquidity gaps are differences between the outstanding balances of assets and liabilities, or between their changes over time. The convention used below is to calculate gaps as algebraic differences between assets and liabilities. Therefore, at any date, a positive gap between assets and liabilities is equivalent to a deficit, and vice versa.

L A

Gap

Time profile of gaps

Time

A = Assets L = Liabilities

FIGURE 11.1

Liquidity gaps and time profile of gaps

Marginal, or incremental, gaps are the differences between the changes in assets and liabilities during a given period (Figure 11.1). A positive marginal gap means that the algebraic variation of assets exceeds the algebraic variation of liabilities. When assets and liabilities amortize over time, such variations are negative, and a positive gap is equivalent to an outflow1 . The fixed assets and the equity also affect liquidity gaps. Considering equity and fixed assets given at any date, the gap breaks down as in Figure 11.2, isolating the gap between 1 For

instance, if the amortization of assets is 3, and that of liabilities is 5, the algebraic marginal gap, as defined above, is −3 − (−5) = +2, and it is an outflow. Sometimes the gaps are calculated as the opposite difference. It is important to keep a clear convention, so that the figures can easily be interpreted.

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Liabilities (50)

Fixed Assets − Equity = −15

Assets (95) Equity (30)

Fixed Assets (15) Total = 110

FIGURE 11.2

Liquidity Gap Liquidity Gap 110 − 80 = 30 110 - 80 = 30 Total = 110

The structure of the balance sheet and liquidity gap

fixed assets and equity and the liquidity gap generated by commercial transactions. In what follows, we ignore the bottom section of the balance sheet, but liquidity gaps should include the fixed assets equity gap.

Static and Dynamic Gaps Liquidity gaps exist for all future dates. The gaps based on existing assets and liabilities are ‘static gaps’. They provide the image of the liquidity posture of the bank. ‘Static gaps’ are time profiles of future gaps under cessation of all new business, implying a progressive melt-down of the balance sheet. When new assets and liabilities, derived from commercial projections, cumulate with existing assets and liabilities, the gap profile changes completely. Both assets and liabilities, existing plus new ones, tend to increase, rather than amortize. This new gap is the ‘dynamic gap’. Gaps for both existing and new assets and liabilities are required to project the total excesses or deficits of funds. However, it is a common practice to focus first on the existing assets and liabilities to calculate the gap profile. One reason is that there is no need to obtain funds in advance for new transactions, or to invest resources that are not yet collected. Instead, funding the deficits or investing excesses from new business occurs when they appear in the balance sheet. Another reason is that the dynamic gaps depend on commercial uncertainties.

Example As mentioned before, both simple and marginal gaps are calculated. The marginal gaps result from variations of outstanding balances. Therefore, the cumulated value over time of the marginal gaps is equal to the gap based on the outstanding balances of assets and liabilities. Table 11.1 is an example of a gap time profile.

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TABLE 11.1 Time profiles of outstanding assets and liabilities and of liquidity gaps Dates Assets Liabilities Gapa Asset amortization Liability amortization Marginal gapb Cumulated marginal gapc

1

2

3

4

5

6

100 100 0

900 800 100 −10 −20 100 100

700 500 200 −20 −30 100 200

650 400 250 −50 −10 50 250

500 350 150 −15 −50 −10 150

300 100 200 −20 −25 50 200

a Calculated

as the difference between assets and liabilities. A positive gap is a deficit that requires funding. A negative gap is an excess of resources to be invested. b Calculated as the algebraic variation of assets minus the algebraic variation of liabilities between t and t − 1. With this convention, a positive gap is an outflow and a negative gap is an inflow. c The cumulated marginal gaps are identical to the gaps calculated with the outstanding balances of assets and liabilities.

In this example, assets amortize quicker than liabilities. Therefore, the inflows from repayments of assets are less than the outflows used to repay the debts calculated with the outstanding balances of assets and liabilities. Hence, a deficit cumulates from one period to the next, except in period 5. Figures 11.3 and 11.4 show the gaps time profile. Assets

1000 800

Liabilities

Gaps

600 400 200 0 1

2

3

4 Dates

FIGURE 11.3

5

6

Gaps

Time profile of liquidity gaps

Liquidity Gaps and Liquidity Risk When liabilities exceed assets, there is an excess of funds. Such excesses generate interest rate risk since the revenues from the investments of these excess assets are uncertain. When assets exceed liabilities, there is a deficit. This means that the bank has long-run commitments, which existing resources do not fund entirely. There is a liquidity risk generated by raising funds in the future to match the size of assets. The bank faces the

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Assets 300 200

Gaps

100 0 −100

1

2

3

4

5

6

−200 Marginal gaps

−300 −400 Liabilities amortization

FIGURE 11.4

Dates

Time profile of marginal liquidity gaps

risk of not being able to obtain the liquidity on the markets, and the risk of paying higher than normal costs to meet this requirement. In addition, it has exposure to interest rate risk. Liquidity gaps generate funding requirements, or investments of excess funds, in the future. Such financial transactions occur in the future, at interest rates not yet known, unless hedging them today. Liquidity gaps generate interest rate risk because of the uncertainty in interest revenues or costs generated by these transactions. In a universe of fixed rates, any liquidity gap generates an interest rate risk. A projected deficit necessitates raising funds at rates unknown as of today. A projected excess of resources will result in investments at unknown rates as well. With deficits, the margin decreases if interest rates increase. With excess resources, the margin decreases when interest rates decrease. Hence, the liquidity exposure and the interest rate exposure are interdependent in a fixed rate universe. In a floating rate universe, there is no need to match liquidity to cancel out any interest rate risk. First, if a short debt funds a long asset, the margin is not subject to interest rate risk since the rollover of debt continues at the prevailing interest rates. If the rate of assets increases, the new debt will also cost more. This is true only if the same reference rates apply on both sides of the balance sheet. However, any mismatch between reset dates and/or a mismatch of interest references, puts the interest margin at risk. For instance, if the asset return indexed is the 3-month LIBOR, and the debt rate refers to 1-month LIBOR, the margin is sensitive to changes in interest rate. It is possible to match liquidity without matching interest rates. For instance, a long fixed rate loan funded through a series of 3-month loans carrying the 3-month LIBOR. In this case, there is no liquidity risk but there is an interest rate risk. Sometimes, future excesses or deficits of liquidity motivate early decisions, before they actually appear. It might be advisable to lock in an interest rate for these excesses and deficits without waiting. Such decisions depend on interest rate expectations, plus the bank policy with respect to interest rate risk.

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Gap Profiles and Funding In the example of deficits, gaps represent the cumulated needs for funds required at all dates. Cumulated funding is not identical to the new funding required at each period because the debt funding gaps of previous periods do not necessarily amortize over subsequent periods. For instance, a new funding of 100, say between the dates 2 and 3, is not the cumulated deficit between dates 1 and 3, for example 200, because the debt contracted at date 2, 100, does not necessarily amortize at date 3. If the debt is still outstanding at date 3, the deficit at date 3 is only 100. Conversely, if the previous debt amortizes after one period, the total amount to fund would be 200, that is the new funding required during the last period, plus any repayment of the previous debt. The marginal gaps represent the new funds required, or the new excess funds available for investing. A positive cumulated gap is the cumulated deficit from the starting date, without taking into consideration the debt contracted between this date and today. This is the rationale for using marginal gaps. These are the amounts to raise or invest during the period, and for which new decisions are necessary. The cumulated gaps are easier to interpret, however.

Liquidity Gaps and Maturity Mismatch An alternative view of the liquidity gap is the gap between the average maturity dates of assets and liabilities. If all assets and liabilities have matching maturities, the difference in average maturity dates would be zero. If there is a time mismatch, the average maturity dates differ. For instance, if assets amortize slower than liabilities, their average maturity is higher than that of liabilities, and vice versa. The average maturity date calculation weights maturity with the book value of outstanding balances of assets and liabilities (Figure 11.5).

Time profile of gaps: Assets > Liabilities

FIGURE 11.5 liabilities

Time

Asset Average Maturity

Liability Average Maturity Gap

Liquidity gaps and gaps between average maturity dates of assets and

In a universe of fixed rates, the gap between average maturities provides information on the average location of assets and liabilities along the yield curve, which provides the market rates for all maturities. Note that averages can be arithmetic or, better, weighted by the outstanding volumes of assets and liabilities.

CASH MATCHING Cash matching is a basic concept for the management of liquidity and interest rate risks. It implies that the time profiles of amortization of assets and liabilities are identical. The

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nature of interest applicable to assets and maturities might also match: fixed rates with fixed rates, floating rates revised periodically with floating rates revised at the same dates using the same reference rate. In such a case, the replication of assets’ maturities extends to interest characteristics. We focus below on cash matching only, when the repayment schedule of debt replicates the repayment schedule of assets. Any deviation from the cash matching benchmark generates interest rate risk, unless setting up hedges.

The Benefits of Cash Matching With cash matching, liquidity gaps are equal to zero. When the balance sheet amortizes over time, it does not generate any deficit or excess of funds. If, in addition, the interest rate resets are similar (both dates and nature of interest rate match), or if they are fixed interest rates, on both sides, the interest margin cannot change over time. Full matching of both cash and interest rates locks in the interest income. Cash matching is only a reference. In general, deposits do not match loans. Both result from the customers’ behaviour. However, it is feasible to structure the financial debt in order to replicate the assets’ time profile, given the amortization schedule of the portfolio of commercial assets and liabilities. Cash matching also applies to individual transactions. For instance, a bullet loan funded by a bullet debt having the same amount and maturity results in cash matching between the loan and the debt. However, there is no need to match cash flows of individual transactions. A new deposit with the same amount and maturity can match a new loan. Implementing cash matching makes sense after netting assets and liabilities. Otherwise, any new loan will necessitate a new debt, which is neither economical nor necessary.

The Global Liquidity Posture of the Balance Sheet The entire gap profile characterizes the liquidity posture of a bank. The zero liquidity gaps are the benchmark. In a fixed rate universe, there would not be any interest rate risk. In a floating rate universe, there is no interest risk either if reset dates and the interest index are identical for assets and liabilities. Figures 11.6 to 11.8 summarize various typical situations. The assumption is that any excess funds, or any deficits, at the starting date are fully funded or invested. Therefore, Outstanding balances

Assets

Liabilities

Time

FIGURE 11.6

Gaps profile close to zero

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Outstanding balances Assets

Liabilities Time

FIGURE 11.7

Deficits Outstanding balances

Liabilities Assets

Time

FIGURE 11.8

Excess funds

the current gap is zero, and non-zero gaps appear only in the future. In the figures, the liability time profile hits the level of capital once all debts amortize completely. The figures ignore the gap between fixed assets and equity, which varies across time because both change.

New Business Flows With new business transactions, the shapes of the time profiles of total assets and liabilities change. The new gap profile is dynamic! Total assets can increase over time, as well as depositors’ balances. Figure 11.9 shows the new transactions plus the existing assets and liabilities. The amortization schedules of existing assets and liabilities are close to each other. Therefore, the static gaps, between existing assets and liabilities, are small. The funding requirements are equal to netted total assets and liabilities. Since total assets increase more than total resources from customers, the overall liquidity gap increases with the horizon. The difference between total assets and existing assets, at any date, represents the new business. To be accurate, when projecting new business, volumes should be net from the amortization of the new loans and the new deposits, since the amortization starts from the origination date of these transactions. When time passes, the gap profile changes: existing assets and liabilities amortize and new business comes in. The gap, which starts from 0 at t, widens up to some positive

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Total assets Outstanding balances

New assets Total liabilities New liabilities Existing assets

Existing liabilities Time

FIGURE 11.9

Gap profile with existing and new business

value at t + 1. The positive gap at t + 1 is a pre-funding gap, or ex ante. In reality, the funding closes the gap at the same time that new business and amortization open it. The gap at t + 1 is positive before funding, and it is identical to the former gap projected, at this date, in the prior periods (Figure 11.10). Gap Assets

Liabilities

t

FIGURE 11.10

t+1

t+1

Time

The gap profile when time passes

LIQUIDITY MANAGEMENT Liquidity management is the continuous process of raising new funds, in case of a deficit, or investing excess resources when there are excesses of funds. The end of this section briefly discusses the case of investments. The benchmark remains the cash matching case, where both assets and liabilities amortize in parallel.

Funding a Deficit In the case of a deficit, global liquidity management aims at a target time profile of gaps after raising new resources. In Figure 11.10, the gap at date t is funded so that, at the current date, assets and liabilities are equal. However, at date t + 1, a new deficit appears, which requires new funding. The liquidity decision covers both the total amount required to bridge the gap at t + 1 and its structuring across maturities. The funding decision reshapes the entire amortization profile of liabilities. The total amount of the new debt is the periodic gap. Nevertheless, this overall amount piles up ‘layers’ of bullet debts with varying maturities. Whenever a layer amortizes, there is a downside step in the amortization profile of the debt.

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The amount of funds that the bank can raise during a given period is limited. There is some upper bound, depending on the bank’s size, credit standing and market liquidity. In addition, the cost of liquidity is important, so that the bank does not want to raise big amounts abruptly for fear of paying the additional costs if market liquidity is tight. If 1000 is a reasonable monthly amount, the maximum one-period liquidity gap is 1000. When extending the horizon, sticking to this upper bound implies that the cumulated amount raised is proportional to the monthly amount. This maximum cumulated amount is 2000 for 2 months, 3000 for 3 months, and so on. The broken line shows what happens when we cumulate a constant periodical amount, such as 1000, to the existing resources profile. The length of vertical arrows increases over time. Starting from existing resources, if we always use the ‘maximum’ amount, we obtain an upper bound for the resources profiles. This is the upper broken line. The upper bound of the resources time profile should always be above the assets time profile, otherwise we hit the upper bound and raising more funds might imply a higher cost. In Figure 11.11, the upper bound for resources exceeds the assets profile. Hence there is no need to raise the full amount authorized.

Outstanding balances

Maximum resource profile

Total assets

Total resources New New funding funding Time

FIGURE 11.11

The gap and resources time profiles

Cash Matching in a Fixed Rate Universe This subsection refers to a situation where resources collected from customers are lower than loans, generating a liquidity deficit. The target funding profile depends on whether ALM wishes to close all liquidity gaps, or maintain a mismatch. Any mismatch implies consistency with expectations on interest rates. For instance, keeping a balance sheet underfunded makes sense only when betting on declining interest rates, so that deferring funding costs less than now. Making the balance sheet overfunded implies expectations of raising interest rates, because the investment will occur in the future. Sometimes the balance sheet remains overfunded because there is an excess of deposits over loans. In such a case, the investment policy should follow guidelines with respect to target interest revenue and interest rate risk on future investments. Structuring the Funding

Below we give two examples of profiles of existing assets and liabilities. In both cases, the resources amortize quicker than assets. The amortization of the balance sheet generates

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new deficits at each period. In a universe of fixed rates, such a situation could make sense if the ALM unit decided to try to bet on decreasing interest rates. We assume that ALM wishes to raise the resources profile up to that of assets because expectations have changed, and because the new goal is to fully hedge the interest margin. The issue is to define the structuring of the new debts consistent with such a new goal. The target resources profile becomes that of assets. The process needs to define a horizon. Then, the treasurer piles up ‘layers’ of debts, starting from the longest horizon. Figure 11.12 illustrates two different cases. On the left-hand side, the gap decreases continuously until the horizon. Then, layers of debts, raised today, pile up, moving backwards from the horizon, so that the resources’ amortization profile replicates the asset profile. Of course, there are small steps, so this hedge is not perfect. In the second case (right-hand side), it is not feasible to hit the target assets’ profile with debt contracted as of today (spot debt). This is because the gap increases with time, peaks and then narrows up to the management horizon. Starting from the end works only partially, since filling completely the gap at the horizon would imply raising excess funds today. This would be inconsistent with cash matching. On the other hand, limiting funds to what is necessary today does not fill the gap in intermediate periods between now and the horizon. Horizon

Amount

Horizon

Amount Assets

Assets

Resources

Resources Time

FIGURE 11.12

Time

Neutralizing the liquidity gap (deficit)

In a fixed rate universe, this intermediate mismatch generates interest rate risk. Hence, ‘immunization’ of the interest margin requires locking in as of today the rates on the future resources required. This is feasible with forward hedges (see Chapter 14). The hedge has to start at different dates and correspond to the amounts of debts represented in grey. Numerical Example

Table 11.2 shows a numerical example of the first case above with three periods. The existing gap is 200, and the cash matching funding necessitates debts of various maturities. The ‘layer 1’ is a bullet debt extending from now to the end of period 3. Its amount is equal to the gap at period 3, or 50. A second bullet debt from now to period 2, of amount 50, bridges the gap at period 2. Once these debts are in place, there is still a 100 gap left for period 1. In the end, the treasurer contracts three bullet debts: 50 for three periods, 50 for two periods, and 100 for one period. Once this is done, the time profile of resources becomes identical to that of assets over the total horizon. In the second case of the preceding subsection, the same process does not apply because the gap increases and, after a while, decreases until the horizon. It is not possible to reach

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TABLE 11.2

Bridging the liquidity gap: example

Period Assets Resources Gap New funding: Debt 1 Debt 2 Debt 3 Total funding Gap after funding

1

2

3

1000 800 200

750 650 100

500 450 50

50 50 100 200 0

50 50

50

100 0

50 0

cash matching with resources raised today. One bullet debt, of 100, bridges partially the gaps from now up to the final horizon. A second 100 bullet debt starts today until the end of period 2. Then, the treasurer needs a third forward starting debt of 150. In a fixed rate universe, a forward contract should lock in its rate as of today. However, effective raising of liquidity occurs only at the beginning of period date 2 up to the end of period 2 (Table 11.3). TABLE 11.3

Bridging the liquidity gap: example

Period Assets Resources Gap New funding: Debt 1 Debt 2 Debt 3 Total funding Gap after funding

1

2

3

1000 800 200

750 400 350

500 400 100

100 100 0 200 0

100 100 150 350 0

100

100 0

Structural Excesses of Liquidity Similar problems arise when there are excesses of funds. Either some funds are re-routed to investments in the market, or there are simply more resources than uses of funds. ALM should structure the timing of investments and their maturities according to guidelines. This problem also arises for equity funds because banks do not want to expose capital to credit risk by lending these funds. It becomes necessary to structure a dedicated portfolio of investments matching equity. The expectations about interest rates are important since the interest rates vary across maturities. Some common practices are to spread securing investments over successive periods. A fraction of the available funds is invested up to 1 year, another fraction for 2 years, and so on until a management horizon. Spreading each periodical investment over all maturities avoids locking in the rate of a single maturity and the potential drawback of renewing entirely the investment at the selected maturity at uncertain rates. Doing so for periodical investments avoids crystallizing the current yield curve in the investment

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portfolio. This policy smooths the changes of the yield curve shape over time up to the selected longest maturity. There are variations around this policy, such as concentrating the investments at both short and long ends of the maturity structure of interest rates. Policies that are more speculative imply views on interest rates and betting on expectations. We address the issue again when dealing with interest rate risk.

THE COST OF THE LIQUIDITY RATIO The cost of liquidity for banks often refers to another concept: the cost of maintaining the liquidity ratio at a minimum level. The liquidity ratio is the ratio of short-term assets to short-term liabilities, and it should be above one. When a bank initiates a combination of transactions such as borrowing short and lending long, the liquidity ratio deteriorates because the short liabilities increase without a matching increase in short-term assets. This is a typical transaction in commercial banking since the average maturity of loans is often longer than the average maturity of resources. In order to maintain the level of the liquidity ratio, it is necessary to have additional short-term assets, for instance 1000 if the loan is 1000, without changing the short-term debt, in order to restore the ratio to the value before lending (1000). Borrowing long and investing short is one way to improve the liquidity ratio. The consequence is a mismatch between the long rate, of the borrowing side, and the short rate, of the lending side. The cost of restoring the liquidity ratio to its previous level is the cost of neutralizing the mismatch. Bid–ask spreads might increase that cost, since the bank borrows and lends at the same time. Finally, borrowing implies paying a credit spread, which might not be present in the interest of short-term assets, such as Treasury bills. Therefore, the cost of borrowing includes the bank’s credit spread. The first cost component is the spread between long-term rates and short-term rates. In addition, borrowing long at a fixed rate to invest short also generates interest rate risk since assets roll over faster than debt. Neutralizing interest rate risk implies receiving a fixed rate from assets rather that a rate reset periodically. A swap receiving the fixed rate and paying the variable rate makes this feasible (see Chapter 14). The swap eliminates the interest rate risk and exchanges the short-term rate received by assets for a long-term rate, thereby eliminating also any spread between long-term rates and short-term rates. The cost of the swap is the margin over market rates charged by the issuer of the swap. Finally, the cost of the liquidity ratio is the cost of the swap plus the credit spread of borrowing (if there is none on investments) plus the bid–ask spread. The cost of maintaining the liquidity ratio is a component of the ‘all-in’ cost of funds, that serves for pricing loans. Any long-term loan implies swapping short-term assets for a loan, thereby deteriorating the liquidity ratio.

ISSUES FOR DETERMINING THE LIQUIDITY GAP TIME PROFILE The outstanding balances of all existing assets and liabilities and their maturity schedules are the basic inputs to build the gap profile. Existing balances are known, but not necessarily their maturity. Many assets and liabilities have a contractual repayments schedule.

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However, many others have no explicit maturity. Such assets without maturity are overdrafts, credit card consumer loans, renewed lines of credit, committed lines of credit or other loans without specific dates. Demand deposits are liabilities without maturity. Deriving liquidity gaps necessitates assumptions, conventions or projections for the items with no explicit maturity. These are: • Demand deposits. These have no contractual maturity and can be withdrawn instantly from the bank. They can also increase immediately. However, a large fraction of current deposits is stable over time, and represents the ‘core deposit base’. • Contingencies such as committed lines of credit. The usage of these lines depends on customer initiative, subject to the limit set by the lender. • Prepayment options embedded in loans. Even when the maturity schedule is contractual, it is subject to uncertainty due to prepayment options. The effective maturity is uncertain.

Demand Deposits There are several solutions to deal with deposits without maturity. The simplest technique is to group all outstanding balances into one maturity bucket at a future date, which can be the horizon of the bank or beyond. This excludes the influence of demand deposit fluctuations from the gap profile, which is neither realistic nor acceptable. Another solution is to make conventions with respect to their amortization, for example, using a yearly amortization rate of 5% or 10%. This convention generates an additional liquidity gap equal to this amortization forfeit every year which, in general, is not in line with reality. A better approach is to divide deposits into stable and unstable balances. The core deposits represent the stable balance that remains constantly as a permanent resource. The volatile fraction is treated as short-term debt. Separating core deposits from others is closer to reality than the above assumptions, even though the rule for splitting deposits into core deposits and the remaining balance might be crude. The last approach is to make projections modelled with some observable variables correlated with the outstanding balances of deposits. Such variables include the trend of economic conditions and some proxy for their short-term variations. Such analyses use multiple regression techniques, or time series analyses. There are some obvious limitations to this approach. All parameters that have an impact on the market share deposits are not explicitly considered. New fiscal regulations on specific tax-free interest earnings of deposits, alter the allocation of customers’ resources between types of deposits. However, this approach is closer to reality than any other.

Contingencies Given (Off-balance Sheet) Contingencies generate outflows of funds that are uncertain by definition, since they are contingent upon some event, such as the willingness of the borrower to use committed lines of credit. There are many such lines of credit, including rollover short-term debts, or any undrawn fraction of a committed credit line. Only the authorization and its expiry date are fixed. Statistics, experience, knowledge of customers’ accounts (individuals or corporate borrowers) and of their needs help to make projections on the usage of such

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lines. Otherwise, assumptions are required. However, most of these facilities are variable rate, and drawings are necessarily funded at unknown rates. Matching both variable rates on uncertain drawings of lines eliminates interest rate risk.

Amortizing Loans When maturities are contractual, they can vary considerably from one loan to another. There are bullet loans, and others that amortize progressively. Prepayment risk results in an effective maturity different from the contractual maturity at origination. The effective maturity schedule is more realistic. Historical data help to define such effective maturities. Prepayment models also help. Some are simple, such as the usage of a constant prepayment ratio applicable to the overall outstanding balances. Others are more sophisticated because they make prepayments dependent on several variables, such as the interest rate differential between the loan and the market, or the time elapsed since origination, and so on. The interest income of the bank is at risk whenever a fixed rate loan is renegotiated at a lower rate than the original fixed rate. Section 7 of this book discusses the pricing of this risk.

Multiple Scenarios Whenever the future outstanding balances are uncertain, it is tempting to use conventions and make assumptions. The problem with such conventions and assumptions is that they hide the risks. Making these risks explicit with several scenarios is a better solution. For instance, if the deposit balances are quite uncertain, the uncertainty can be captured by a set of scenarios, such as a base case plus other cases where the deposit balances are higher or lower. If prepayments are uncertain, multiple scenarios could cover high, average and low prepayment rate assumptions. The use of multiple scenarios makes more explicit the risk with respect to the future volumes of assets and liabilities. The price to pay is an additional complexity. The scenarios are judgmental, making the approach less objective. The choices should combine multiple sources of uncertainty, such as volume uncertainty, prepayment uncertainty, plus the uncertainty of all commercial projections for new business. In addition, it is not easy to deal with interest rate uncertainty with more than one scenario. There are relatively simple techniques for dealing with multiple scenarios, embedded in ALM models detailed in Chapter 17.

12 The Term Structure of Interest Rates

The yield curve is the entire range of market interest rates across all maturities. Understanding the term structure of interest rates, or yield curve, is essential for appraising the interest rate risk of banks because: • Banks’ interest income is at risk essentially because of the continuous movements of interest rates. • Future interest rates of borrowing or lending–investing are unknown, if no hedge is contracted before. • Commercial banks tend to lend long and borrow short. When long-term interest rates are above short-term interest rates, this ‘natural exposure’ of commercial banks looks beneficial. Often, banks effectively lend at higher rates than the cost of their debts because of a positive spread between long-term rates and short-term rates. Unfortunately, the bank’s interest income is at risk with the changes of shape and slope of the yield curve. • All funding or investing decisions resulting from liquidity gaps have an impact on interest rate risk. Banks are net lenders, when they have excess funds, or net borrowers, when they have future deficits. As any lender or borrower, they cannot eliminate interest rate risk. A variable rate borrower faces the risk of interest rate rises. A fixed rate borrower faces the risk of paying a fixed rate above declining rates. The exposure of the lender is symmetrical. The consequence is that there is no way to neutralize interest rate risk. The only available options are to choose the best exposure according to management criteria. The purpose of controlling interest rate risk is to adopt this ‘best’ exposure. The prerequisite is measuring the interest rate exposure. Defining this ‘best’ exposure requires

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understanding the mechanisms driving the yield curve and making choices on views of what future rates could be. This chapter addresses both issues. Subsequent chapters describe the management implications and the techniques for controlling interest rate risk. The expectation theory of interest rates states that there are no risk-free arbitrage opportunities for investors and lenders ‘riding’ the spot yield curve, which is the term structure of interest rates. The theory has four main applications: • Making lending–borrowing decisions based on ‘views’ of interest rates compared to ‘forward rates’ defined from the yield curve. • Locking in a rate at a future date, as of the current date. • Extending lending–borrowing opportunities to the forward transactions, future transactions of which terms are defined today, based on these forward rates. The first section discusses the nature of interest rate references, such as variable rates, fixed rates, market rates and customer rates. The second section discusses interest rate risk and shows that any lender–borrower has exposure to this risk. The third section discusses the term structure of market interest rates, or the ‘spot’ yield curve. It focuses on the implications of the spreads between short-term rates and long-term rates for lenders and borrowers. The fourth section defines forward rates, rates set as of today for future transactions, and how these rates derive from the spot yield curve. The fifth section presents the main applications for taking interest rate exposures. The last section summarizes some practical observations on economic drivers of interest rates, providing a minimum background underlying ‘view’ on future interest rates, which guide interest rate policies.

INTEREST RATES To define which assets and liabilities are interest-sensitive, interest rates qualify as ‘variable’ or ‘fixed’. This basic distinction needs some refining. Variable rates change periodically, for instance every month if they are indexed to a 1-month market rate. They remain fixed between any two reset dates. The distinction, ‘fixed’ or ‘variable’, is therefore meaningless unless a horizon is specified. Any variable rate is fixed between now and the next reset date. A variable rate usually refers to a periodically reset market rate, such as the 1-month LIBOR, 1-year LIBOR or, for the long-term, bond yields. The frequency of resets can be as small as 1 day. The longer the period between two reset dates, the longer the period within which the value of the ‘variable’ rate remains ‘fixed’. In some cases, the period between resets varies. The prime rate, the rate charged to the best borrowers, or the rates of regulated saving accounts (usually with tax-free interest earnings) can change at any time. The changes depend on the decision of banks or regulating bodies. Such rates are not really fixed. In some cases, they change rather frequently, such as the prime rate of banks. In others, the changes are much less frequent and do not happen on a regular basis. In the recent period, it has become permitted to make regulated rates variable according to market benchmarks. The motivation is that rates in Euroland have decreased to unprecedented levels, comparable to regulated rates. This phenomenon impairs the profitability of banks collecting this kind of resource.

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Finally, among rates periodically reset to market, some are determined at the beginning of the period and others at the end. Most rates are set at the beginning of the period. However, in some countries, some market indexes are an average of the rates observed during the elapsed period. For instance, a 1-month rate can be the average of the last month’s daily rates. In such cases, the uncertainty is resolved only at the end of the current period, and these rates remain uncertain until the end of the period. On the other hand, predetermined rates are set at the beginning of the current period and until the next reset date. From an interest rate risk perspective, the relevant distinction is between certain and uncertain rates over a given horizon. A predetermined variable rate turns fixed after the reset date and until the next reset date. The ‘fixed rate’ of a future fixed rate loan is actually interest rate-sensitive since the rate is not set as of today, but some time in the future. All rates of new assets and liabilities originated in the future are uncertain as of today, even though some will be those of fixed rate transactions. The only way to fix the rates of future transactions as of today is through hedging. The nature of rates, fixed or variable, is therefore not always relevant from an interest risk standpoint. The important distinction is between unknown rates and interest rates whose value is certain over a given horizon. As shown in Figure 12.1, there are a limited number of cases where future rates are certain and a large number of other rates indexed to markets, or subject to uncertain changes. In the following, both distinctions ‘fixed versus variable’ and ‘certain versus uncertain’ are used. Without specifying, we use the convention that fixed rates are certain and variable rates are uncertain as of today.

FIGURE 12.1

Today

Horizon until next reset

Fixed

Fixed

Variable

Variable

To maturity Certain

Uncertain

Interest rates, revision periods and maturity

INTEREST RATE RISK Intuition suggests that fixed rate borrowers and lenders have no risk exposure because they have certain contractual rates. In fact, fixed rate lenders could benefit from higher rates. If rates rise, lenders suffer from the ‘opportunity cost’ of not lending at these higher rates. Fixed rate borrowers have certain interest costs. If rates decline, they suffer from not taking advantage of lower rates. They also face the ‘opportunity cost’ because they do not take advantage of this opportunity. Variable rate borrowers and lenders have interest costs or revenues indexed to market rates. Borrowers suffer from rising rates and lenders from declining rates. Future (forward) borrowers benefit from declining rates, just as variable rate borrowers do. Future (forward) lenders suffer from declining rates, just as variable rate lenders do. The matrix in Table 12.1 summarizes these exposures, with ‘+’

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TABLE 12.1 Rate

Exposure to Interest Rate Risk (IRR) Change of rates

Lender

Borrower

Existing exposure Floating rate Fixed ratea

Rate ↑ Rate ↓ Rate ↑ Rate ↓

+ − − +

− + + −

Projected exposure (same as current + floating) — —

Rate ↑ Rate ↓

+ −

− +

a Uncertainty

with respect to opportunity cost of debt only. + gain, − loss.

representing gains and ‘−’ losses, both gains and losses being direct or opportunity gains and costs. A striking feature of this matrix is that there is no ‘zero’ gain or loss. The implication is that there is no way to escape interest rate risk. The only options for lenders and borrowers are to choose their exposure. The next issue is to make a decision with respect to their exposures.

THE TERM STRUCTURE OF INTEREST RATES Liquidity management decisions raise fundamental issues with respect to interest rates. First, funding or investing decisions require a choice of maturity plus the choice of locking in a rate, or of using floating rates. The choice of a fixed or a floating rate requires a comparison between current and expected future rates. The interest rate structure is the basic input to make such decisions. It provides the set of rates available across the maturities range. In addition, it embeds information on expected future rates that is necessary to make a decision on interest rate exposure. After discussing the term structure of interest rates, we move on to the arbitrage issue between fixed and variable rates.

The Spot Yield Curve There are several types of rates, or yields. The yield to maturity of a bond is the discount rate that makes the present value of future cash flows identical to its price. The zerocoupon rates are those discount rates that apply to a unique future flow to derive its present value. The yield to maturity embeds assumptions with respect to reinvestment revenues of all intermediate flows up to maturity1 . The yield to maturity also changes 1 The yield to maturity is the value of y that makes the present value of future flows equal to the price of a bond: price = nt=1 CFt /(1 + y)t where CFt are the cash flows of date t and y is the yield to maturity. This equation can also be written as: price = nt=1 CFt (1 + y)n−t /(1 + y)n . The equation shows that the price is also the present value of all flows reinvested until maturity at the yield y. Since the future reinvestment of

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Rates

with the credit standing of the issue. The spread between riskless yields, such as those applicable to government bonds, and the yields of private issues is the credit spread. Yields typically derive from bond prices, given their future cash flows. The zero-coupon yields can be derived from yields to maturity observed for a given risk class using an iterative process2 . The set of yields from now to any future maturity is the yield curve. The yield curve can have various shapes, as shown in Figure 12.2.

Maturity

FIGURE 12.2

The term structure of interest rates

Interest Rate Arbitrage for Lenders and Borrowers The difference in rates of different maturities raises the issue of the best choice of maturity for lending or borrowing over a given horizon. An upward sloping yield curve means that long rates are above short rates. In such a situation, the long-term lender could think that the best solution is to lend immediately over the long-term. The alternative choice would be to lend short-term and renew the lending up to the horizon of the lender. The loan is rolled over at unknown rates, making the end result unknown. This basic problem requires more attention than the simple comparison between short-term rates and long-term rates. We make the issue more explicit with the basic example of a lender whose horizon is 2 years. Either he lends straight over 2 years, or he lends for 1 year and rolls the loan over after 1 year. The lender for 2 years has a fixed interest rate over the horizon, which is above the current 1-year rate. If he lends for 1 year, the first year’s revenue will be lower since the current spot rate for 1 year is lower than the 2-year rate. At the beginning of the second year, he reinvests at the prevailing rate. If the 1-year rate, 1 year from now, increases sufficiently, it is conceivable that the second choice beats the first. There is a minimum value of the yearly rate, 1 year from now, that will provide more revenue at the end of the 2-year horizon than the straight 2-year loan. intermediate flows will not occur at this yield y, but at the prevailing rates at the date when reinvestments occur, this assumption is unrealistic. 2 Zero-coupon rates apply to a single flow. Such rates derive from the observed bond yields. The one-period yield to maturity is a zero-coupon rate since the first flows occur at the end of the first period. In the following, this period is 1 year. Starting with this first zero-coupon rate, we can derive the others. A 2-year instrument generates two flows, one at date 1, CF1 , and one at date 2, CF2 . The present value of the flow CF1 uses the 1-year rate z1 . This value is CF1 /(1 + z1 ). By borrowing this amount at date 0, we cancel the flow CF1 of the bond. Then, the present value of CF2 discounts this flow at the unknown zero-coupon rate for date 2. This present value should be equal to the price (for example 1) minus the present value of the flow CF1 . Using the equality: 1 − CF1 /(1 + z1 ) = CF2 /(1 + z2 )2 , we obtain the value of the zero-coupon rate for date 2, z2 .

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The same reasoning applies to a borrower. A borrower can borrow short-term, even though he has a long-term liquidity deficit because the short-term rates are below the long-term rates. Again, he will have to roll over the debt for another year after 1 year. The outcome depends on what happens to interest rates at the end of the first year. If the rates increase significantly, the short-term borrower who rolls over the debt can have borrowing costs above those of a straight 2-year debt. In the above example, we compare a long transaction with a series of short-term transactions. The two options differ with respect to interest rate risk and liquidity risk. Liquidity risk should be zero to have a pure interest rate risk. If the borrower needs the funds for 2 years, he should borrow for 2 years. There is still a choice to make since he can borrow with a floating or a fixed rate. The floating rate choice generates interest rate risk, but no liquidity risk. From a liquidity standpoint, short-term transactions rolled over and long-term floating rate transactions are equivalent (with the same frequency of rollover and interest rate reset). The subsequent examples compare long to short-term transactions. The issue is to determine whether taking interest risk leads to an expected gain over the alternative solution of a straight 2-year transaction. The choice depends first on expectations. The short-term lender hopes that interest rates will increase. The short-term borrower hopes that they will decrease. Nevertheless, it is not sufficient that interest rates change in the expected direction to obtain a gain. It is also necessary that the rate variation be large enough, either upward or downward. There exists a break-even value of the future rate such that the succession of short-term transactions, or a floating rate transaction—with the same frequency of resets as the rollover transaction—is equivalent to a long fixed rate transaction. The choice depends on whether the expected rates are above or below such break-even rates. The break-even rate is the ‘forward rate’. It derives from the term structure of interest rates.

INTEREST RATE EXPECTATIONS AND FORWARD RATES The term structure of interest rates used hereafter is the set of zero-coupon rates derived from government, or risk-free, assets. The zero-coupon rates apply to single future flows and are pure fixed rate transactions. The expectation theory considers that rates of various maturities are such that there is no possible risk-free arbitrage across rates. Forward rates result directly from such an absence of risk-free arbitrage opportunities. A risk-free arbitrage generates a profit without risk, based upon the gap between observed rates and their equilibrium value. Today, we observe only current spot rates and future spot rates are uncertain. Even though they are uncertain, they have some expected value, resulting from the expectations of all market players. This principle allows us to derive the relation between today’s spot rates and future expected rates. The latter are the implicit, or forward, rates. The standard example of a simple transaction covering two periods, such as investing for 2 years, illustrates the principle. For simplicity, we assume this is a risk-neutral world.

Risk-neutrality Risk-neutrality is a critical concept for arbitrage issues. When playing heads and tails, we toss a coin. Let’s say we gain if we have heads and lose in the case of tails. There is a 50% chance of winning or losing. The gain is 100 if we win and −50 if we lose. The expected gain is 0.5 × 100 + 0.5 × (−50) = 25. A risk-neutral player is, by definition,

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indifferent, between a certain outcome of 25 and this uncertain outcome. A risk-adverse investor is not. He might prefer to hold 20 for sure rather than betting. The difference between 25 and 20 is the value of risk aversion. Since investors are risk averse, we need to adjust expectations for risk aversion. For the sake of simplicity, we explain arbitrage in a risk-neutral world.

Arbitrage across Maturities Two alternative choices are investigated: investing straight for 2 years or twice for 1 year. These two choices should provide the same future value to eliminate arbitrage opportunities. The mechanism extends over any succession of periods for any horizon. Forward rates depend on the starting date and the horizon after the starting date, and require two indexes to define them. We use the following notations: • The future uncertain spot rate from date t up to date t + n is t it+n 3 . The current spot rates for 1 and 2 years are known and designated by 0 i1 and 0 i2 respectively. Future spot rates are evidently random. • The forward rate from date t up to date t + n is t Ft+n 4 . This rate results from current spot rates, as shown below, and has a certain value as of today. The forward rate for 1 year and starting 1 year from now is 1 F2 , and so on. For instance, the spot rates for 1 and 2 years could be 10% and 11%. The future value, 2 years from now, should be the same using the two paths, straight investment over 2 years or a 1-year investment rolled over an additional year. The condition is: (1 + 11%)2 = (1 + 10%)[1 + E(1 i2 )]. Since 1 i2 remains uncertain as of today, we need the expectation of this random variable. The rate 1 year from now for 1 year 1 i2 is uncertain, and its expected value is E(1 i2 ). According to this equation, this expected value is around 12% (the accurate value is 12.009%). If not, there is an arbitrage opportunity. The reasoning would be the same for borrowing. If the 1-year spot rate is 10% and the future rate is 12% (approximately), it is equivalent to borrow for 2 years directly or to borrow for 1 year and roll over the debt for another year. This applies under risk-neutrality, since we use expectations as if they were certain values, which they are not. The expected value that makes the arbitrage unprofitable is the expected rate by the market, or by all market players. Some expect that the future rate 1 i2 will be higher, others that it will be lower, but the net effect of expectations is the 12% value. It is also the value of 1 i2 making arbitrage unprofitable between a 1-year transaction and a 2-year transaction in a risk-neutral world. The determination of forward rates extends to any maturity through an iterative process. For instance, if the spot rate 0 i1 is 10% and the spot rate 0 i2 is 11%, we can derive the forward rate 2 F3 from the spot rate for date 3, or 0 i3 . The return over the last year would be the forward rate 2 F3 = E(2 i3 ). Therefore, the condition of non-profitable arbitrage becomes: (1 + 0 i3 )3 = (1 + 0 i2 )2 [1 + E(2 i3 )] = (1 + 0 i2 )2 (1 + 2 F3 ) It gives 2 F3 from the two spot rates. 3 Bold

letter i indicates a random variable, italic letter i indicates any particular value it might have. alternative notation would be n it , n being the period and t the starting date. We prefer to use date notations, t and t + n, to avoid any ambiguity between dates and periods. 4 An

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Expected Rates

Rates

Rates

The implications of the expectation theory of the term structure are important to choose the maturity of transactions and/or the nature, fixed or variable, of interest rates. The long-run rates are geometric averages of short-run rates5 following the rule of no riskfree arbitrage across spot rates. If the yield curve is upward sloping, the forward rates are above the spot rates and expectations are that interest rates will rise. If the yield curve is downward sloping, forward rates are below the spot rates and expectations are that interest rates will fall (Figure 12.3).

Time Increasing rates

FIGURE 12.3

Time Decreasing rates

Term structure of interest rates and future expected rates

If the market expectations are true, the outcome is identical for long transactions and a rollover of short transactions. This results from the rule of no profitable risk-free arbitrage. The future rates consistent with this situation are the forward rates, which are also the expected values of uncertain future rates. The limit of the expectation theory is that forward rates are not necessarily identical to future spot rates, which creates the possibility of profitable opportunities. The expectation theory does not work too well when expectations are heterogeneous. The European yield curve was upward sloping in recent years, and the level of rates declined for all maturities over several years, until 1999. This demonstrates the limitations of the predictive power of the expectation theory. It works necessarily if expectations are homogeneous. If everyone believes that rates will increase, they will, because everyone tries to borrow before the increase. This is the rationale of the ‘self-fulfilling prophecy’. On the other hand, if some believe that rates might increase because they seem to have reached a bottom line and others predict further declines, the picture becomes fuzzy. In the late nineties, while Europe converged towards the euro, expectations were unclear.

FORWARD RATES APPLICATIONS The forward rates, embedded in the spot yield curves, serve a number of purposes: • They are the break-even rates for comparing expectations with market rates and arbitraging the market rates. • It is possible to effectively lock in forward rates for a future period as of today. 5 This

is because the reasoning can extend from two periods to any number of periods. The average is a geometric average.

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• They provide lending–borrowing opportunities rather than simply lending–borrowing cash at the current spot rates.

Forward Rates as Break-even Rates for Interest Rate Arbitrage The rule is to compare, under risk-neutrality, the investor’s expectation of the future spot rate with the forward rate, which serves as a break-even rate. In the example of two periods: • If investor’s expectation is above the forward rate 1 F1 , lending short and rolling over the loan beats the straight 2-year investment. • If investor’s expectation is below the forward rate 1 F1 , lending long for 2 years beats the short-term loan renewed for 1 year. • If investor’s expectation matches the break-even rate, investors are indifferent between going short and going long. If a lender believes that, 1 year from now, the rate will be above the forward rate for the same future period, he expects to gain by rolling over short transactions. If the borrower expects that, 1 year from now, the rate will be below the forward rate, he expects to gain by rolling over short-term transactions. Whether the borrower or the lender will actually gain or not depends on the spot rates that actually prevail in the future. The price to pay for expected gains is uncertainty. For those willing to take interest rate risk, the benchmarks for their decisions are the forward rates. The lender makes short transactions if expected future rates are above forward rates. The lender prefers a long transaction at the spot rate if expected future rates are below the forward rates. Similar rules apply to borrowers. Hence, expectations of interest rises or declines are not sufficient for making a decision. Expectations should state whether the future rates would be above or below the forward rates. Otherwise, the decision to take interest rate risk or not remains not fully documented (Figure 12.4). These are simplified examples, with two periods only. Real decisions cover multiple periods of varying lengths6 . Spot yield curve

Forward rates

Rates

Long or Short

Expectations Time

FIGURE 12.4 6 For

Expectations and decisions

instance, over a horizon of 10 years, the choice could be to use a 1-year floating rate, currently at 4%, or to borrow at a fixed rate for 10 years at 6%. Again, the choice depends on expectations with respect to the 9-year rate 1 year from now. The break-even rate, or forward rate, for 9 years 1 year from now is such that: (1 + 4%)(1 + 1 F9 %)9 = (1 + 6%)10 . The yearly forward rate 1 F9 is 6.225%. If the borrower believes

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Locking in a Forward Rate as of Today It is possible to lock in a rate for a future date. This rate is the forward rate derived from the spot yield curve. We assume that an investor has 1000 to invest 1 year from now and for 1 year. He will receive this flow in 1 year. The investor fears that interest rates will decline. Therefore, he prefers to lock in a rate as of today. The trick is to make two transactions that cancel out as of today and result, by netting, in a forward investment 1 year from now and for 1 year. The spot rates are 4% for 1 year and 6% for 2 years. In addition, borrowing and lending are at the same rate. Since the investor wants to lend, he lends today for 2 years. In order to do so, he needs to borrow the money invested for 1 year, since he gets an inflow 1 year from now. The amount borrowed today against the future inflow of 1000 is 1000/(1 + 4%) = 961.54. This results in a repayment of 1000 in 1 year. Simultaneously, he lends for 2 years resulting in an inflow of 961.54(1 + 6%)2 = 1080.385 2 years from now. The result is lending up to date 2 a flow of 1000 at date 1 to get 1080.385. This is a yearly rate of 8.0385%. This rate is also the forward rate resulting from the equation: (1 + 4%)(1 + 1 F2 ) = (1 + 6%)2 Since (1 + 1 F2 ) = (1 + 6%)2 /(1 + 4%) = 1.1236/1.04 = 1.80385. Therefore forward rates ‘exist’ and are more than the result of a calculation (Figure 12.5).

Dates

961.54

1000

961.54

1000

0

1

Forward Rate = 8.0385%

FIGURE 12.5

2

1080.385

Locking in a forward rate 1 year from now for 1 year

that 1 year from now the yearly rate for a maturity of 9 years will be above 6.225%, he or she should borrow fixed. Otherwise, he or she should use a floating rate debt. With a spread of 2% between short-run and long-run rates, intuition suggests that borrowing fixed today is probably very costly. A fixed debt has a yearly additional cost over short-run debt, which is 6% − 4% = 2%. However, the difference in yearly rate does not provide a true picture because we are comparing periods of very different lengths. The 2% additional cost is a yearly cost. If the long rate (9 years) actually increases to 6.225%, the additional cost of borrowing fixed 1 year later will be 0.225% per year. This additional cost cumulates over the 9 years. In other words, the loss of 2% over 1 year should be compared to a potential loss of 0.225% over 9 years.

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The Spot and Forward Yield Curves There is a mechanical relationship between spot rates and forward rates of future dates. The Relative Positions of the Spot and Forward Yield Curves

Spot yield curve

Spot yield curve Forward yield curve

Maturity

FIGURE 12.6

Interest Rates

Forward yield curve

Interest Rates

Interest Rates

When spot yield curves are upward sloping, the forward rates are above the spot yield curve. Conversely, with an inverted spot yield curve, the forward rates are below the spot yield curve. When the spot yield curve is flat, both spot and forward yields become identical (Figure 12.6).

Maturity

Forward yield curve Spot yield curve Maturity

The relative positions of the spot and forward yield curves

When investing and borrowing, it is important to consider whether it is more beneficial to ride the spot or the forward yield curves, depending on expectations about future spot rates. Since banks often have non-zero liquidity gaps, they are net forward borrowers when they have positive liquidity gaps (assets higher than liabilities) and net forward lenders when they have negative liquidity gaps (assets lower than liabilities). The spot yield curve plus the forward yield curves starting at various future dates chart all funding–investment opportunities. Example of Calculating the Forward Yield Curves

The spot yield curve provides all yields by maturity. Forward yield curves derive from the set of spot yields for various combinations of dates. The calculations in Table 12.2 provide an example of a forward yield curve 1 year from now and all subsequent maturities, as well as 2 years from now and all subsequent maturities. In the first case, we plot the forward rates starting 1 year from now and for 1 year (date 2), 2 years (date 3), 3 years (date 4), and so on. In the second case, the forward rates are 2 years from now and for maturities of 1 year (date 3), 2 years (date 4), and so on. The general formulas use the TABLE 12.2 Date t Spot yield Forward t F1 Forward t+2 F1

The spot and forward yield curves 0

1

2

3

4

5

3.50%

4.35% 5.21%

5.25% 6.14% 7.98%

6.15% 7.05% 8.28%

6.80% 7.64% 8.57%

7.30% 8.08% 8.82%

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term structure relationships. For instance, the forward rates between dates 1 and 3 (1 year from now for 2 years) and between dates 1 and 4 (1 year from now for 3 years) are such that: (1 + 1 F3 )2 = (1 + 0 i3 )3 /(1 + 0 i1 )

(1 + 1 F4 )3 = (1 + 0 i4 )4 /(1 + 0 i1 )

If we deal with yields 2 years from now, we divide all terms such as (1 + 0 i1 ) by (1 + 0 it )2 to obtain 1 + 2 Ft , when t ≥ 2. All forward rates are above the spot rates since the spot yield curve is upward sloping (Figure 12.7). In addition, the further we look forward, the lower is the slope of the forward curve. 10% 9% 8% 7% 6% 5% 4% 3% 2% 1% 0%

Forward t + 2 Forward t + 1

Spot t

0

FIGURE 12.7

1

2

3

4

5

6

Spot and forward yield curves

Forward yield curves start later than the spot yield curves because they show forward investment opportunities. The graph provides a broad picture of spot and forward investment opportunities.

VIEWS ON INTEREST RATES A major Asset–Liability Management (ALM) choice is in forming views about future rates. Interest rate models do not predict what future rates will be. However, most ALM managers will face the issue of taking views of future rates. Not having views on future rates implies a ‘neutral’ ALM policy considering all outcomes equally likely and smoothing variations in the long-term. Any deviations from that policy imply an opinion on future interest rates. Various theories on interest rates might help. However, forming ‘views’ on future rates is more a common sense exercise. The old segmentation theory of interest rates is a good introduction. The segmentation approach considers that rates of different maturities vary independently of each other because there are market compartments, and because the drivers of rates in each different compartment are different. For instance, short-term rates depend on the current economic conditions and monetary policy, while long-term rates are representative of the fundamentals of the economy and/or depend on budget deficits. This view conflicts with the

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arbitrage theory and with practice, which makes it obvious that rates of different maturities are interdependent. Drifts of interest rates follow the law of supply and demand. Large borrowings shift yields upwards, and conversely. Hence, large budget deficits tend to increase rates. Still, arbitraging long and short interest rates happens continuously, interlinking all yields across maturities. The shape of the yield curve changes. Monetary policy mechanisms tend to make short interest rates more volatile. They work through the ‘open market’ policy. When the central bank wants to raise interest rates, it simply increases its borrowing, drying up the market. This explains the high levels of interest rates in some special situations. Traditional policies fight inflation through higher interest rates in the United States. In Europe, some countries used short-term interest rates to manage exchange rates before ‘Euroland’ formed as a monetary entity. If a local currency depreciates against the USD, for example, a rise in interest rates makes it more attractive to investors to buy the local currency for investing. This increases the demand for the local currency, facilitating its appreciation, or reducing its depreciation. The capability of the central banks to influence strongly interest rates results from their very large borrowing capacity. Long-term interest rates are more stable in general. They might relate more to the fundamentals of the economy than short-term rates. Such observations help us to understand why the yield curve is upward sloping or downward sloping. Whenever short rates jump to high levels because of monetary policy, the yield curve tends to be ‘inverted’, with a downward slope. Otherwise, it has a flat shape or a positive slope. However, there is no mechanism making it always upward sloping. Unfortunately, this is not enough to predict what is going to happen to the yield curve. The driving forces might be identifiable, but not their magnitude, or the timing of central interventions. All the above observations are consistent with arbitraging rates of different maturities. They also help in deciding whether there is potential for future spot rates to be above or below forward rates. Expectations and comparisons with objective benchmarks remain critical for making sound decisions.

13 Interest Rate Gaps

The interest ‘variable rate gap’ of a period is the difference between all assets and liabilities whose interest rate reset dates are within the period. There are as many interest rate gaps as there are interest rate references. The ‘fixed rate gap’ is the difference between all assets and liabilities whose interest rates remain fixed within a period. There is only one fixed rate gap, which is the mirror image of all variable rate gaps. Gaps provide a global view of the balance sheet exposure to interest rates. The underlying rationale of interest rate gaps is to measure the difference between the balances of assets and liabilities that are either ‘interest rate-sensitive’ or ‘insensitive’. If assets and liabilities have interest revenues and costs indexed to the same rate, this interest rate drives the interest income mechanically. If the balances of these assets and liabilities match, the interest income is insensitive to rates because interest revenues and costs vary in line. If there is a mismatch, the interest income sensitivity is the gap. The rationale for interest rate gaps is that they link the variations of the interest margin to the variations of interest rates. This single property makes interest rate gaps very attractive. The ‘gap’ concept has a central place in Asset–Liability Management (ALM) for two main reasons: • It is the simplest measure of exposure to interest rate risk. • It is the simplest model relating interest rate changes to interest income. Interest rate derivatives directly alter the interest rate gaps, providing easy and straightforward techniques for altering the interest rate exposure of the banking portfolio. However, gaps have several drawbacks due to: • Volume and maturity uncertainties as for liquidity gaps: the solution is multiple scenarios, as suggested for liquidity gaps.

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• The existence of options: implicit options on-balance sheet and optional derivatives off-balance sheet. A fixed rate loan is subject to potential renegotiation of rates, it remains a fixed rate asset whether or not the likelihood of renegotiation is high or low. Obviously, when current rates are well below the fixed rate of the loan, the asset is closer to a variable rate asset than a fixed rate one. Such options create ‘convexity risk’, developed later. • Mapping assets and liabilities to selected interest rates rather than using the actual rates of individual assets and liabilities. • Intermediate flows within time bands selected for determining gaps. The first section introduces the gap model. The second section explains how gaps link interest income to the level of interest rates. The third section explains the relationship between liquidity and interest rate gaps. The fourth section details the direct calculation of interest income and of their variations with interest rates. The fifth section discusses hedging. The last section discusses the drawbacks and limitations of interest rate gaps.

THE DEFINITION OF INTEREST RATE GAPS The interest rate gap is a standard measure of the exposure to interest rate risk. There are two types of gaps: • The fixed interest rate gap for a given period: the difference between fixed rate assets and fixed rate liabilities. • The variable interest rate gap: the difference between interest-sensitive assets and interest-sensitive liabilities. There are as many variable interest rate gaps as there are variable rates (1-month LIBOR, 1-year LIBOR, etc.). Both differences are identical in absolute value when total assets are equal to total liabilities. However, they differ when there is a liquidity gap, and the difference is the amount of the liquidity gap. The convention in this text is to calculate interest rate gaps as a difference between assets and liabilities. Therefore, when there is no liquidity gap, the variable rate gap is the opposite of the fixed rate gap because they sum up to zero. Specifying horizons is necessary for calculating the interest rate gaps. Otherwise, it is not possible to determine which rate is variable and which rate remains fixed between today and the horizon. The longer the horizon, the larger the volumes of interest-sensitive assets and liabilities, because longer periods allow more interest rate resets than shorter periods (Figure 13.1). An alternative view of the interest rate gap is the gap between the average reset dates of assets and liabilities. With fixed rate assets and liabilities, the difference between the reset dates is the difference between the average maturity dates of assets and liabilities. If the variable fraction of assets grows to some positive value, whether all liabilities remain fixed rate or not, the average reset maturity of assets shortens because rate resets occur before asset maturity. With variable rate assets and liabilities, the variable interest rate gap, for a given variable rate reference, roughly increases with the gap between reset dates of assets and liabilities, if average reset dates are weighted by sizes of assets and

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Fixed rate assets (FRA) 400

Variable rate assets (VRA) 600

Fixed rate liabilities (FRL) 200

Variable rate liabilities (VRL) 800

Fixed IR gap = FRA − FRL = 400 − 200 = + 200 Variable IR gap = VRA − VRL = 600 − 800 = − 200

Fixed IR Gap = − Variable IR Gap

FIGURE 13.1

Interest rate gap with total assets equal to total liabilities

liabilities. The gap between average reset dates of assets and maturities is a crude measure of interest rate risk. Should they coincide, there would be no interest rate risk on average since rate resets would occur approximately simultaneously. In Figure 13.2 the positive variable rate gaps imply that variable rate assets are larger than variable rate liabilities. This is equivalent to a faster reset frequency of assets. The average reset dates show approximately where asset and liability rates are positioned along the yield curve.

IR gaps: Assets reset dates > Liabilities reset dates

IR Gap Time

Asset average reset date

Liability average reset date

FIGURE 13.2 Interest rate gaps and gaps between average reset dates Note: For a single variable rate reference for assets and liabilities.

INTEREST RATE GAP AND VARIATIONS OF THE INTEREST MARGIN The interest rate gap is the sensitivity of the interest income when interest rates change. When the variable rate gap (interest rate-sensitive assets minus interest rate-sensitive liabilities) is positive, the base of assets that are rate-sensitive is larger than the base of liabilities that are rate-sensitive. If the index is common to both assets and liabilities, the interest income increases mechanically with interest rates. The opposite happens when the variable rate gap is negative. When the interest rate gap is zero, the interest income is insensitive to changes in interest rates. In this specific case, the interest margin is ‘immune’ to variations of rates. In the above example, the gap between interest-sensitive assets and liabilities is +200. The interest income is the difference between interest revenues and charges over the period. The calculation of the variation of the margin is simple under the set of assumptions below:

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• The gap is constant over the period. This implies that the reset dates are identical for assets and liabilities. • The rates of interest-sensitive assets and liabilities are sensitive to a common rate, i. We use the notation: IM, Interest Margin; VRA, Variable Rate Assets; VRL, Variable Rate Liabilities; i, interest rate. The change in IM due to the change in interest rate, i, is: IM = (VRA − VRL)i = +200i VRA and VRL are the outstanding balances of variable rate assets and liabilities. The variation of the margin in the above example is 2 when the rate changes by 1%. The basic formula relating the variable rate gap to interest margin is: IM = (VRA − VRL)i = (interest rate gap)i The above formula is only an approximation because there is no such thing as a single interest rate. It applies however when there is a parallel shift of all rates. The formula applies whenever the variable interest rate gap relates to a specific market rate. A major implication for hedging interest rate risk is that making the interest margin ‘immune’ to interest rate changes simply implies neutralizing the gap. This makes the sensitivity of the margin to interest rate variations equal to zero.

INTEREST GAP CALCULATIONS Interest rate and liquidity gaps are interrelated since any future debt or investment will carry an unknown rate as of today. The calculation of gaps requires assumptions and conventions for implementation purposes. Some items of the balance sheet require a treatment consistent with the calculation rules. Interest rate gaps, like liquidity gaps, are differences between outstanding balances of assets and liabilities, incremental or marginal gaps derived from variations of volumes between two dates, or gaps calculated from flows.

Fixed versus Variable Interest Rate Gaps The ‘fixed rate’ gap is the opposite of the ‘variable rate’ gap when assets and liabilities are equal. A future deficit implies funding the deficit with liabilities whose rate is still unknown. This funding is a variable rate liability, unless locking in the rate before funding. Accordingly, the variable rate gap decreases post-funding while the fixed rate gap does not change. The gap implications are symmetrical when there is an excess of liquidity: the interest rate of the future investment is unknown. The variable rate gap increases while the fixed rate gap remains unchanged. The fixed rate gap is consistent with an ex ante view, when no hedge is yet in place. Hedging modifies the interest rate structure of the balance sheet ex post. In addition, the fixed rate calculation is convenient because there is no need to deal with the variety of interest rates that are ‘variable’. It excludes all interest rate-sensitive assets and liabilities within the period whatever the underlying variable rates. However, this is a drawback of the fixed rate calculation since interest rate-sensitive assets and liabilities are

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sensitive to different indexes. Therefore, deriving the interest margin sensitivity from the fixed rate gap is only a crude proxy because it synthesizes the variations of all variable rates into a single index. In reality, accuracy would require calculating all variable interest rate gaps corresponding to each one of these market rates. The interest rate gap is similar to the liquidity gap, except that it isolates fixed rate from variable rate assets and liabilities. Conversely, the liquidity gap combines all of them, whatever the nature of rates. Another difference is that any interest rate gap requires us to define a period because of the fixed rate–variable rate distinction. Liquidity gaps consider amortization dates only. Interest rate gaps require all amortization dates and all reset dates. Both gap calculations require the prior definition of time bands.

Time Bands for Gap Calculation Liquidity gaps consider that all amortization dates occur at some conventional dates within time bands. Interest rate gaps assume that resets also occur somewhere within a time band. In fact, there are reset dates in between the start and end dates. The calculation requires the prior definition of time bands plus mapping reset and amortization dates to such time bands. Operational models calculate gaps at all dates and aggregate them over narrow time bands for improving accuracy. In addition, calculations of interest revenues and interest costs require exact dates, which is feasible with the detailed data on individual transactions independent of the necessity of providing usable reports grouping dates in time bands.

Interdependency between Liquidity Gaps and Interest Rate Gaps For future dates, any liquidity gap generates an interest rate gap. A projected deficit of funds is equivalent to an interest-sensitive liability. An excess of funds is equivalent to an interest-sensitive asset. However, in both cases, the fixed rate interest gap is the same. This is not the case with variable rate gaps if they isolate various interest rates. Liquidity gaps have no specific variable rate ex ante, since the funding or the investment is still undefined. In the example below, there is a projected deficit. Variable interest rate gaps pre-funding differ from post-funding gaps. The interest rate gap derived from interest-sensitive assets and liabilities is the gap before funding minus the liquidity gap, as Figure 13.3 shows. Specific items deserve some attention, the main ones being equity and fixed assets. These are not interest-earning assets or liabilities and should be included from interest rate gaps, but not from liquidity gaps. They influence the interest rate gaps through the liquidity gaps. Some consider equity as a fixed rate liability, perhaps because equity requires a fixed target return. In fact, this compensation does not have much to do with a fixed rate. Since it depends on the interest margin, the effective return is not the target return and it is interest rate-sensitive. We stick to the simple rule that interest rate gaps use only interest rate assets and liabilities and exclude non-interest-bearing items. We illustrate below the detailed calculations of both liquidity and interest rate gaps:

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Fixed Assets (10)

Fixed Assets − Equity = −10

Fixed Rate Assets (75)

Equity (20)

Fixed Rate Liabilities (30) Variable Rate Liabilities (40)

Variable Rate Gap (VRA − VRL) = −5 Pre-funding

Variable Rate Assets (35)

Liquidity Gap (Variable Rate) (30)

Total = 120

Total = 120

Variable Rate Gap Post-funding = −35

FIGURE 13.3

Variable Rate Gap Pre-funding = −5

=

−

Fixed Rate Gap (FRA − FRL) = 45

Liquidity Gap 120 − 90 = + 30

Liquidity Gap = + 30

The balance sheet structure and gaps

Liquidity gaps = total assets − total liabilities = 10 + 75 + 35 − (20 + 30 + 40) = 120 − 90 = +30 (a deficit of liquidity) a = (interest rate assets − interest rate liabilities) + (fixed assets − equity) = (75 + 35 − 30 − 40) + (10 − 20) = +30 Fixed interest rate gap = fixed rate assets − fixed rate liabilities (excluding non-interest-bearing items) = 75 − 30 = 45 Variable interest rate gap = variable rate assets − variable rate liabilities = 35 − 40 = −5 without the liquidity gap = −5 − 30 = −35 with the liquidity gap Therefore, the general formula applies: Variable interest rate gap post-funding = fixed interest rate gap − liquidity gap In the example, −35 = −5 − 30.

Cumulative and Marginal Gaps Gaps are differences between outstanding balances at one given date, or differences of variations of those balances over a period. Gaps calculated from variations between two

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dates are ‘marginal gaps’, or differences of flows rather than volumes. For a unique given period, the variation of the interest margin is equal to the gap times the interest rate variation. For a succession of periods, it becomes necessary to break down the same types of calculations by period. The gaps change with time. When the gap is assumed constant over one period, it means that the volume of interest-sensitive assets and liabilities does not change. If not, splitting the period into smaller subperiods makes the constant gap assumption realistic. The example below demonstrates the impact of varying gaps over time (Table 13.1). There are three balance sheets available at three dates. The variations of assets and liabilities are differences between two consecutive dates. The data are as follows: TABLE 13.1

Interest rate gaps, cumulative and marginal

End of period

Month 1

Month 2

Month 3

0 200

250 200

300 200

−200 −200

+50 +250

+100 +50

Variable rate assets Variable rate liabilities Cumulative gaps Marginal gaps

• The total period is 3 months long. • The total assets and liabilities are 1000 and do not change. All items mature beyond the 3-month horizon and there is no new business. The liquidity gap is zero for all 3 months. • The interest rate increases during the first day of the first month and remains stable after. • The interest-sensitive liabilities are constant and equal to 200. • The interest-sensitive assets change over time. The fluctuation of interest-sensitive assets is the only source of variation of the gap.

MARGIN VALUES AND GAPS All the above calculations aim to determine variations of margins generated by a variation of interest rates. They do not give the values of the margins. The example below shows the calculation of the margins and reconciles the gap analysis with the direct calculations of margins. The margins, expressed in percentages of volumes of assets and liabilities, are the starting point. They are the differences between the customer rates and the market rates. The market rate is equal to 10% and changes by 1%. If the commercial margins are 2% for loans and −4% for deposits, the customer rates are, on average, 10 + 2 = 12% for loans and 10 − 4 = 6% for liabilities. The negative commercial margins for liabilities mean that rates paid to customers are below market rates. The values of margins are given in Table 13.2. The initial margin in value is 1000(10% + 2%) − 1000(10% − 4%) = 60. If the interest rate changes by 1%, only the rates of interest rate-sensitive assets and liabilities change, while others keep their original rates. New customer rates result from the variation of the reference rate plus constant percentage margins. The change in interest margin is +1. This is consistent with an interest rate gap of 100, combined with a 1% variation of interest rate since 100 × 1% = 1 (Table 13.3).

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TABLE 13.2

TABLE 13.3

Balance sheets and margins: example

Assets

Outstanding balances

Margins

Fixed rate Variable rate Total assets Liabilities

700 300 1000 Outstanding balances

+2% +2% Margins

Fixed rate Variable rate Total liabilities

800 200 1000

−4% −4%

Variable rate gap

+100

Calculation of interest margin before and after an interest rate rise

Rate

10%

Fixed rate Variable rate

700 300

Total assets Fixed rate Variable rate Total liabilities Margin

800 200

11%

Assets: 700 × 12% 84 300 × 12% 36 120 Liabilities: 800 × 6% 48 200 × 6% 12 60 60

700 × 12% 300 × 13%

800 × 6% 200 × 7%

Variation 84 39

0 +3

123

+3

48 14

0 +2

62 61

+2 +1

The Interest Margin Variations and the Gap Time Profile Over a given period, with a constant gap and a variation of interest rate i, the revenues change by VRA × i and the costs by VRL × i. Hence the variation in IM is the difference: IM = (VRA − VRL)i = gap × i Over a multi-periodic horizon, subperiod calculations are necessary. When interest rates rise, the cost increase depends on the date of the rate reset. At the beginning of month 1, an amount of liabilities equal to 200 is interest rate-sensitive. It generates an increase in charges equal to 200 × 1%1 (yearly value) over the first month only. At the beginning of month 2, the gap becomes 50, generating an additional revenue of 50 × 1% = 0.5 for the current month. Finally, the last gap generates a change in margin equal to 100 × 1% for the last month. The variation of the cumulative margin over the 3-month period is the total of all monthly variations of the margin. Monthly variations are 1/12 times the yearly variations. This calculation uses cumulative gaps (Table 13.4). In practice, the gaps should be available on a monthly or a quarterly basis, to capture the exposure time profile with accuracy. Table 13.4 also shows that the calculation of the cumulative variation of the margin over the total 3-month period can follow either one of two ways. The first calculation is 1 Calculations

are in yearly values. The actual revenues and charges are calculated over a quarter only.

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TABLE 13.4 Cumulative gaps and monthly changes in interest margins Month 1 2 3

Cumulative gaps −200 +50 +100

Monthly variation of the margin −200 × 1% +50 × 1% +100 × 1%

−2.0 +0.5 +1.0

Total

−0.5

as above. Using cumulative gaps over time, the total variation is the sum of the monthly variations of the margins. Another equivalent calculation uses marginal gaps to obtain the margin change over the residual period to maturity. It calculates the variation of the margin over the entire 3-month period due to each monthly gap. The original gap of −200 has an impact on the margin over 3 periods. The second marginal gap of +250 alters the margin during 2 periods. The third marginal gap influences only the last period’s margin. The cumulative variation of the 3-period margin is the product of the periodical marginal gaps times the change in rate, and times the number of periods until maturity2 . Monthly calculations derive from the above annual values.

INTEREST RATE GAPS AND HEDGING One appealing feature of gaps is the simplicity of using them for hedging purposes. Hedging often aims at reducing the interest income volatility. Since the latter depends linearly on the interest rate volatility through gaps, controlling the gaps suffices for hedging or changing the exposure using derivatives. Simple examples and definitions suffice to illustrate the simplicity of gap usage.

Hedging over a Single Period It is possible to neutralize the interest rate gap through funding, without using any interest rate derivative. We assume that the liquidity gap is +30 and that the variable rate gap is 2 The

calculation with marginal gaps would be as follows. The marginal gaps apply to residual maturity: [(−200 × 3) + (250 × 2) + (50 × 1)] × 1% = −0.5

With cumulated gaps, the calculation is: [(−200) + (+250 − 200) + (+50 + 250 − 200)] × 1% = −0.5 These are two equivalent calculations, resulting from the equality of the sum of marginal gaps, weighted by residual maturities, with the sum of cumulated gaps of each period. The periodic gaps are g1 , g2 and g3 . The cumulated gaps are G1 = g1 , G2 = g1 + g2 , G3 = g1 + g2 + g3 . The total of cumulated gaps is: G1 + G2 + G3 = g1 + (g1 + g2 ) + (g1 + g2 + g3 ) which is identical to: 3 × g1 + 2 × g2 + 1 × g3

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+20 before funding and −10 after funding with a variable rate debt. The bank wishes to hedge its sensitivity to interest rates. If the bank raises floating rate funds, it has too many floating rate liabilities, exactly 10. If the bank raises floating rate debt for 20 and locks in the rate for the remaining 10, as of today, the interest rate gap post-funding and hedging becomes 0. The interest rate margin is ‘immune’ to interest rate changes. In order to lock in a fixed rate for 10 at the future date of the gap calculation, it is necessary to use a Forward Rate Agreement (FRA). If we change the data, this type of hedge does not work any more. We now assume that we still have a funding gap of +30, but that variable interest rate gap is +45 pre-funding and +15 post-funding with floating rate debt. The bank has too many floating rate assets, and even though it does not do anything to lock in a rate for the debt, it is still interest rate-sensitive. The excess of floating rate assets post-funding is +15. The bank receives too much floating rate revenue. To neutralize the gap, it is necessary to convert the 15 variable rate assets into fixed rate assets. Of course, the assets do not change. The only way to do this conversion is through a derivative. The derivative should receive the fixed rate and pay the floating rate. The asset plus the derivative generates a neutralized net floating rate flow and a net fixed rate flow. This derivative is an Interest Rate Swap (IRS) receiving the fixed rate and paying the floating rate (Figure 13.4). Receive floating Floating rate Asset

Receive fixed

Fixed rate Asset

Receive fixed IRS Pay floating

FIGURE 13.4

Hedging interest rate risk with an interest rate swap

Hedging over Multiple Periods Figure 13.5 shows the time profiles of both marginal and cumulated gaps over multiple periods for the previous example. The cumulated gap is negative at the end of the first month, and becomes positive later. Marginal gap

300

Cumulated gap

200

Gap

100 0 −100

Month 1

Month 2

−200 −300

FIGURE 13.5

Interest rate gap profiles

Dates

Month 3

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Immunization of the interest margin requires setting the gap to zero at all periods. For instance, at period 1, a swap with a notional of 200, paying the fixed rate and receiving the floating rate, would set the gap to zero at period 1. The next period, a new swap paying the floating rate and receiving the fixed rate, with a notional of 250, neutralizes the interest rate gap. Actually, the second swap should be contracted from date 0 and take effect at the end of month 1. This is a forward swap3 starting 1 month after date 0. A third swap starting at the end of month 2, and maturing at the end of month 3, neutralizes the gap of month 3. Since the forward swap starting at the beginning of month 2 can extend to hedge the risk of month 3, the third new swap has a notional of only 50. The marginal gaps represent the notionals of the different swaps required at the beginning of each period, assuming that the previous hedges stay in place. The usage of marginal interest rate gaps is similar to that of marginal liquidity gaps, these representing the new debts or investments assuming that the previous ones do not expire. The gap model provides the time profile of exposures to interest rate risk. Very simple rules suffice for hedging these exposures. The start dates and the termination dates of such hedges result from the gap profile. The hedge can be total or partial. The residual exposure, after contracting a hedge for a given amount, is readily available from the gap profile combined with the characteristics of the hedge.

LIMITATIONS OF INTEREST RATE GAPS Gaps are a simple and intuitive tool. Unfortunately, gaps have several limitations. Most of them result from assumptions and conventions required to build the gap profile. The main difficulties are: • Volume and maturity uncertainties as for liquidity gaps: the solution lies in assumptions and multiple scenarios, as suggested for liquidity gaps. • Dealing with options: implicit options on-balance sheet and optional derivatives offbalance sheet. These options create ‘convexity risk’, developed later. • Mapping assets and liabilities to selected interest rates as opposed to using the actual rates of individual assets and liabilities. • Dealing with intermediate flows within time bands selected for determining gaps. We deal with the latter three issues, the first one having been discussed previously with liquidity gaps.

Derivatives and Options Some derivative instruments do not raise any specific difficulties. A swap substitutes a variable rate for a fixed rate. The impact on the interest rate gap follows immediately. Other optional instruments raise issues. Caps and floors set maximum and minimum values of interest rates. They change the interest sensitivity depending on whether the level 3A

forward swap is similar to a spot swap, but starts at a forward date.

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of the interest rate makes them in-the-money or not. When they are out-of-the-money, options do not change the prevailing rates. When they are in-the-money, caps make assets interest rate-insensitive. In other words, options behave sometimes as variable rate items and sometimes as fixed rate items, when the interest rates hit the guaranteed rates. The interest rate gap changes with the level of interest rates. This is a serious limitation to the ‘gap picture’, because the gap value varies with interest rates! There are many of embedded or implicit options in the balance sheet. Prepayments of fixed rate mortgage loans when interest rates decrease are options held by the customers. Some regulated term deposits offer the option of entering into a mortgage with a subsidized rate. Customers can shift funds from non-interest-bearing deposits to interest-earning deposits when rates increase. When sticking to simple gap models, in the presence of options, assumptions with respect to their exercise are necessary. The simplest way to deal with options is to assume that they are exercised whenever they are in-the-money. However, this ignores the value of waiting for the option to get more valuable as time passes. Historical data on the effective behaviour might document these assumptions and bring them close to reality. The prepayment issue is a well-known one, and there are techniques for pricing implicit options held by customers. Another implication is that options modify the interest rate risk, since they create ‘convexity risk’. Measuring, modelling and controlling this optional risk are dealt with in related chapters (see Chapters 20 and 21 for pricing risk, Chapter 24 for measuring optional risk at the balance sheet level).

Mapping Assets and Liabilities to Interest Rates Interest rate gaps assume that variable rate assets and liabilities carry rates following selected indexes. The process requires mapping the actual rates to selected rates of the yield curve. It creates basis risk if both rates differ. Sensitivities, relating actual rates to selected rates, correct basis risk. The technique serves for calculating ‘standardized gaps’4 . The alternative solution is to use directly the contractual rate references of contracts at the level of individual transactions, raising Information Technology (IT) issues. Sensitivities

The average rate of return of a subportfolio, for a product family for instance, is the ratio of interest revenues (or costs) to the total outstanding balance. It is feasible to construct time series of such average rates over as many periods as necessary. A statistical fit to observed data provides the relationship between the average rate of the portfolio and the selected market indexes. A linear relation is such that: Rate = a0 + a1 × index1 + ‘random residual’ The coefficient a1 is the sensitivity of the loan portfolio rate with respect to index1 . The residual is the random deviation between actual data and the fitted model. A variation of 4 Sensitivities

measure correlations between actual and selected rates. A basis risk remains because correlations are never perfect (equal to 1). See Chapter 28.

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the market index of 1% generates a variation of the rate of the loan portfolio of a1 %. The same methodology extends notably to transaction rates, which are not market rates, such as the prime rate. The prime rate variations depend on the variations of two market rates. The loan balances indexed with the prime rate behave as a composite portfolio of loans indexed with the market index1 and a portfolio of loans indexed with the market index2 . The statistical relationship becomes: Prime rate = a0 + a1 × index1 + a2 × index2 + ‘random residual’ Such statistical relations can vary over time, and need periodical adjustments. The sensitivities are the coefficients of the statistical model. Standardized Gaps

With sensitivities, the true relationships between the interest margin and the reference interest rates are the standardized gaps, rather than the simple differences between assets and liabilities. Standardized gaps are gaps calculated with assets and liabilities weighted by sensitivities. For instance, if the sensitivity of a loan portfolio is 0.5, and if the outstanding balance is 100, the loan portfolio weighted with its sensitivity becomes 50. With interest rate-sensitive assets of 100 with a sensitivity of 0.5, and interest rate-sensitive liabilities of 80 with a sensitivity of 0.8, the simple gap is 100 − 80 = 20 and the standardized gap is: 0.5 × 100 − 0.8 × 80 = 50 − 64 = −14 This is because a variation of the interest rate i generates a variation of the interest on assets which is 0.5i and a variation of the rate on liabilities which is 0.8i. The resulting variation of the margin is: IM = 100 × 0.5 + 80 × 0.8i = (0.5 × 100 − 0.8 × 80)i IM = (standardized gap)i In this example, the simple gap and the standardized gap have opposite signs. This illustrates the importance of sensitivities when dealing with aggregated portfolios of transactions rather than using the actual rate references.

Intermediate Flows and Margin Calculations The gap model does not date accurately the flows within a period. It does not capture the effect of the reinvestment or the funding of flows across periods. In some cases, both approximations can have a significant impact over margins. This section expands possible refinements. Intra-periodic Flows

Gaps group flows within time bands as if they were simultaneous. In reality, there are different reset dates for liquidity flows and interest rates. They generate interest revenues

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Reinvestment at the rate at t t′ t Borrowing at the rate at t′

Reference period for the gap calculation

FIGURE 13.6

Interest rate risk depends on the positioning of flows across time

or costs, which are not valued in the gap–margin relationship. Figure 13.6 shows that revenues and costs assigned to intermediate reinvestments or borrowings depend on the length of the period elapsed between the date of a flow and the final horizon. A flow with reset date at the end of the period has a negligible influence on the current period margin. Conversely, when the reset occurs at the beginning of the period, it has a significant impact on the margin. The simple gap model considers that these flows have the same influence. For instance, a flow of 1000 might appear at the beginning of the period, and another flow of 1000, with an opposite sign, at the end of the period. The periodic gap of the entire period will be zero. Nevertheless, the interest margin of the period will be interest-sensitive since the first flow generates interest revenues over the whole period, and such revenues do not match the negligible interest cost of the second flow (Figure 13.7). 1000 Reinvestment at the rate at 0

End of period (1) Beginning of period (0) 1000

Borrowing at the rate at 1

Reference period for the gap calculation

FIGURE 13.7

Zero gap combined with an interest-sensitive margin

The example below illustrates the error when the goal is to immunize the interest margin. In the preceding example, the gap was zero, but the margin was interest-sensitive. In the example below, we have the opposite situation. The gap differs from zero, but the interest margin is immune to interest rate changes. The gap is negative, which suggests that the margin should increase when the interest rate decreases, for example from 10% to 8%. However, this ignores the reinvestment of the positive intermediate flow of date 90 at

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a lower rate for 270 days. On the other hand, the negative flow generates a debt that costs less over the remaining 180 days. The margin is interest rate-sensitive if we calculate interest revenues and costs using the accurate dates of flows (Figure 13.8). 1000 270 days

180 days 0

1

90 days 180 days −1536

FIGURE 13.8

Negative cumulative gap and rate-insensitive margin

The changes of interest revenues and costs, due to a shift of rates from 8% to 10%, are: Inflow at day 90 : Outflow at day 180 :

1000(1.10270/360 − 1.08270/360 ) = −14.70

−1536(1.10180/360 − 1.08180/360 ) = +14.70

In this example, the decline of funding costs matches the decline of interest revenues. This outcome is intuitive. The interest revenues and costs are proportional to the size of flows and to the residual period over which reinvestment or funding occurs. The first flow is less important than the second flow. However, it generates interest revenues over a longer period. The values are such that the residual maturity differential compensates exactly the size differential. The outflow is ‘equivalent’ to a smaller outflow occurring before, or the inflow is ‘equivalent’ to a smaller inflow occurring after. The example shows how ‘gap plugging’, or direct gap management, generates errors. In the first example, the margin is interest-sensitive, which is inconsistent with a zero gap. In the second example, the gap model suggests plugging another flow of 536 to hedge the margin. Actually, doing so would put the margin at risk! The exact condition under which the margin is insensitive to interest rates is relatively easy to derive for any set of flows. It differs from the zero gap rule (see appendix to Chapter 23). The condition uses the duration concept, discussed in Chapter 22. Interest Cash Flows and Interest Rate Gaps

The flows used to calculate interest rate gaps are variations of assets and liabilities only. However, all assets and liabilities generate interest revenues (or costs), received (or paid), at some dates. Only the cash payments of interest count, not the accrued accounting interest flows. Gap profiles based on capital flows only ignore these flows for the sake of simplicity. Gaps should include such interest payments, as the mark-to-market value of the balance sheet does.

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Floating rate assets or liabilities generate interest flows as well. A common practice is to project the stream of interest flows using the current value of the floating rate. It ‘crystallizes’ the current rates up to the horizon. Forward rates would serve the same purpose and are conceptually more appropriate since they are the image of the market expectations. Nevertheless, when the interest rate changes, floating rate flows change as well, and change both the liquidity and interest rate gap profiles.

14 Hedging and Derivatives

Derivatives alter interest rate exposures and allow us to hedge exposures and make interest income independent of rates. This does not eliminate risk, however, since hedges might have opportunity costs because they set the interest income at lower levels than rates could allow. There are two main types of derivatives: forward instruments are forward rate agreements or swaps; optional instruments allow capping the interest rate (caps) or setting a minimum guaranteed rate (floors). Forward instruments allow us to change and reverse exposures, such as turning a floating rate exposure into a fixed rate one, so that the bank can turn an adverse scenario into a favourable one. However, hedging with forward instruments implies a bet on the future and still leaves the bank unprotected against adverse interest rate movements if the bet goes wrong. Optional instruments allow us to capture the best of both worlds: getting the upside of favourable interest rate movements and having protection against the downside of adverse deviations, with a cost commensurable with such benefits. Derivatives serve to alter the exposure, to resize or reverse it, and to modify the risk–return profile of the balance sheet as a whole. Futures are instruments traded on organized markets. They perform similar functions. Since there is no way to eliminate interest rate risk, the only option is to modify the exposure according to the management views. To choose an adequate hedging policy, Asset–Liability Management (ALM) needs to investigate the payoff profiles of alternative policies under various interest rate scenarios. Examples demonstrate that each policy implies a bet that rates will not hit some break-even value depending on the particular hedge examined and its cost. This break-even value is such that the payoff for the bank is identical with and without the hedge. Decision-making necessitates comparing the break-even value of rates with the management views on interest rates.

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Foreign exchange derivatives perform similar functions, although the mechanisms of forward contracts differ from those of interest rate contracts. There is a mechanical connection between interest rates in two currencies, spot and forward exchange rates. This allows shifting exposures from one currency to the other, and facilitates multicurrency ALM. This chapter provides the essentials for understanding the implementation of these hedging instruments. The first section discusses alternative ‘hedging’ policies. The next two sections explain the mechanisms that allow derivatives to alter the exposure profile, starting with forward instruments and moving on to options. Both sections detail the payoffs of hedges depending on interest rate scenarios. The fourth section introduces definitions about future markets, explaining how they serve for hedging. The last section deals with currency risk, using the analogy with interest rate hedging instruments. It details the mechanisms of foreign exchange forward contracts and of options on currency exchange rates.

INTEREST RATE RISK MANAGEMENT This section discusses interest rate exposure and hedging policies. There are two alternative objectives of hedging policies: • To take advantage of opportunities and take risks. • A prudent management aimed at hedging risks, totally or partially (to avoid adverse market movements and benefit others). In the second case, policies aim to: • Lock in interest rates over a given horizon using derivatives or forward contracts. • Hedge adverse movements only, while having the option to benefit from other favourable market movements. There are two types of derivative instruments: • Forward hedges lock in future rates, but the drawback lies in giving up the possibility of benefiting from the upside of favourable movements in rates (opportunity cost or risk). • Optional hedges provide protection against adverse moves and allow us to take advantage of favourable market movements (no opportunity cost), but their cost (the premium to pay when buying an option) is much higher. In addition, futures also allow us to implement hedging policies. The futures market trades standard contracts, while derivatives might be ‘plain vanilla’ (standard) or customized1 . 1 See

Figlewski (1986) for both theoretical and practical aspects of hedging with futures for financial institutions.

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FORWARD INTEREST RATE INSTRUMENTS Forward instruments aim at locking the rate of exposure. Whether the hedge is profitable or not depends on the subsequent variations of interest rates. Forward hedges lock in rates, but they do not allow borrowers and lenders to benefit from the upside, as options do. Therefore, there are benchmarks to consider when setting up such hedges. The universal principle that applies is to make sure, as far as possible, that expected interest rates would be above, or below, some break-even values. These break-even values are the interest rates making the hedge unfavourable, since contracting a hedge implies contracting an interest rate. We present the basic mechanisms with examples only. The most common instruments are forward rate agreements and swaps2 .

Interest Rate Swaps The objective is to exchange a floating rate for a fixed rate, and vice versa, or one floating rate for another (a 1-month rate against a 1-year rate). The example is that of a firm that borrows fixed rate and thinks rates will decline. The firm swaps its fixed rate for a floating rate in order to take advantage of the expected decline. The fixed rate debt pays 11%, originally contracted for 5 years. The firm would prefer to borrow at the 1-year LIBOR. The residual life of the debt is 3 years. The current 3-year spot rate is 10%. The 1-year LIBOR is 9%. The swap exchanges the current 3-year fixed rate for the 1-year LIBOR. The treasurer still pays the old fixed rate of 11%, but receives from the counterparty (a bank) the current 3-year fixed rate of 10% and pays LIBOR to the bank. Since the original debt is still there, the borrower has to pay the agreed rate. The cost of the ‘original debt plus swap’ is: 1-year LIBOR (paid to bank) + 11% (original debt) − 10% (received from bank) The borrower now pays LIBOR (9%) plus the rate differential between the original fixed rate (5 years, 11%) and the current spot rate (3 years, 10%), or 1%. The total current cost is LIBOR + 1% = 10%. The treasurer has an interest rate risk since he pays a rate reset every 3 months. As long as LIBOR is 9%, he now saves 1% over the original cost of the debt. However, if LIBOR increases over 10%, the new variable cost increases beyond the original 11%. The 10% value is the break-even rate to consider before entering into the swap. The treasurer bets that LIBOR will remain below 10%. If it moves above, the cost of the joint combination ‘original debt plus swap’ increases above the original cost. If this happens, and lasts for a long period, the only choice left is to enter into a reverse swap. The reverse swap exchanges the floating rate with the current fixed rate. Forward hedges are reversible, but there is a cost in doing so. The cost of the swap is the spread earned by the counterparty, a spread deducted from the rate received by the swap. The above swap receives the fixed rate and pays the floating rate. Other swaps do the reverse. In addition, swaps might exchange one floating rate for another. There are 2 See

Hull (2000) for an overview of derivative definitions and pricing.

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also differed swaps, or forward starting swaps. These swaps use the forward market rates, minus any retained spread by the counterparty. Swaptions are options on swaps, or options of entering into a swap contract at a deferred date. For a bank, the application is straightforward. Banks have both assets and liabilities, which result in interest rate gaps. The interest rate gap is like a debt or an investment. When the variable rate gap is positive, the bank is adversely exposed to interest rate declines and conversely. If the variable rate gap is negative, the bank fears an increase in interest rates. This happens when the variable debt is larger than the variable rate assets. The exposure is similar to that of the above borrower. Therefore, banks can swap fixed rates of assets or liabilities to adjust the gap. They can swap different variable rates and transform a floating rate gap, say on 1-month LIBOR, into a floating rate gap, for instance 3-month LIBOR.

Forward Contracts Such contracts allow us to lock in a future rate, for example locking in the rate of lending for 6 months in 3 months from now. These contracts are Forward Rate Agreements (FRAs). The technology of such contracts is similar to locking in a forward rate through cash transactions, except that cash transactions are not necessary. To offer a guaranteed rate, the basic mechanism would be that the bank borrows for 9 months, lends to a third party for 3 months, and lends to the borrower when the third party reimburses after 3 months. The cost of forward contracts is the spread between the 9-month and 3-month interest rates. Since this would imply cash transactions increasing the size of the bank balance sheet, they are lending costs for the bank because of regulations. Hence, the technology for FRAs is to separate liquidity from interest rate exposure, and the bank saves the cost of carrying a loan. A future borrower might buy an FRA if he fears that rates will increase. A future lender might enter into an FRA if he fears that rates will decline. As usual, the FRA has a direct cost and an opportunity cost. When the guaranteed rate is 9%, the bank pays the difference between the current rate, for instance 9.25%, and 9% to the borrower. Should the rate be 8%, the borrower will have to pay to the counterparty the 1% differential. In such a case, the borrower loses with the FRA. The break-even rate is that of the FRA.

OPTIONAL INTEREST RATE INSTRUMENTS An option is the right, but not the obligation, to buy (call) or sell (put) at a given price a given asset at (European option), or up to (American option), a given date. Options exist for stocks, bonds, interest rates, exchange rates and futures.

Terminology The following are the basic definitions for understanding options. A call option is the right, but not the obligation, to buy the underlying at a fixed exercise price. The exercise price is the price to pay the underlying if the holder of the option exercises it.

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The payoff, or liquidation value, of the option is the gain between the strike price and the underlying price, netted from the premium paid. The option value is higher by its ‘time value’, or the value of waiting for better payoffs. A put option is the right, but not the obligation, to sell the underlying. Options are in-the-money when they provide a gain, which is when the liquidation value is higher than the exercise price. Options are out-of-the-money when the underlying value is below the exercise price. Such options have only time value, since their payoff would be negative. Options have various underlyings: stock, interest rate or currency. Hence, optional strategies serve for hedging interest rate risk and foreign exchange risk, market risk and, more recently, credit risk3 . Figure 14.1 illustrates the case of a call option on stock that is in-the-money. Option Value Time value

Liquidation value Underlying value

Premium Strike Strike Price Price

FIGURE 14.1

Option value, liquidation value and payoff

Interest Rate Options Common interest rate options are caps and floors. A cap sets up a guaranteed maximum rate. It protects the borrower from an increase of rates, while allowing him to take advantage of declines of rates. The borrower benefits from the protection against increasing rates without giving up the upside gain of declining rates. This benefit over forward hedges has a higher cost, which is the premium to pay to buy the cap. A floor sets up a minimum guaranteed rate. It protects the lender from a decline of rates and allows him to take advantage of increasing rates. As for caps, this double benefit has a higher cost than forwards do. Figure 14.2 visualizes the gains and the protection. Since buying caps and floors is expensive, it is common to minimize the cost of an optional hedge by selling one and buying the other. For instance, a floating rate borrower buys a cap for protection and sells a put to minimize the cost of the hedge. Of course, the rate guaranteed by the cap has to be higher than the rate that the borrower guarantees to the buyer of the floor. Hence, there is no benefit from a decline in rates below the guaranteed rate of the put sold. Figure 14.2 visualizes the collar together with the cap and floor guaranteed rates. 3 See

Hull (2000), Cox and Rubinstein (1985) and Chance (1990) for introductions to options.

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Interest Rate

Cap Collar Floor

Time

FIGURE 14.2

Cap, floor and collar

Optional Strategies Optional strategies might be directional, on volatility, or aim to minimize the cost of hedges. Directional policies are adequate when a trend appears likely. Buying a cap is consistent with a bet on interest rate increases. Betting on volatility makes sense when interest rates are unstable and when the current information does not permit identification of a direction. Both an increase and a decrease of rates might occur beyond some upper and lower bounds. For instance, buying a cap at 10% combined with the purchase of a put at 8% provides a gain if volatility drives the interest rate beyond these boundary values. The higher the volatility, the higher are the chances that the underlying moves beyond these upper and lower bounds. Minimizing costs requires collars. The benefit is a cost saving since X buys one option (cap) and sells the other (floor). The net cost is: premium of cap–premium of floor. The drawback is that the interest rate might decline beyond the strike of the floor. Then, the seller of the floor pays the difference between the interest rate and the strike of the floor.

Example: Hedging an Increase of Rates with a Cap The borrower X has 200 million USD floating rate debt LIBOR 1-month + 1.50% maturing in 1 year. X fears a jump in interest rates. He buys a cap with a guaranteed maximum rate of 8% yearly, costing a premium of 1.54%. After 1 year, the average value of LIBOR becomes 10%. Then X pays 11.50% to the original lender and 1.54% to the seller of the cap. He receives from the seller of the cap: 10% − 8% = 2%, the interest rate differential between the guaranteed rate of 8% and the current rate of 10%. The effective rate averaged is 10% + 1.5% + 1.54% − 2% = 11.04% instead of 11.50% without the cap. This is a winning bet. Payments are the notional 200 million USD times the percentages above. Now assume that after 1 year the average value of LIBOR becomes 7%. X pays 8.50% to the lender and 1.54% to the seller of the cap. X does not receive anything from the seller of the cap. The effective rate averaged over a year is: 7% + 1.50% + 1.54% = 10.04% instead of 8.50% without the cap! The break-even value of the yearly average of the LIBOR is 9.54% to have a winning optional hedge, or ‘guaranteed rate

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8% + premium 1.54%’. Before contracting the cap, the borrower should bet that rates would be above 9.54%.

Example: Hedging an Increase of Rates with a Collar The collar minimizes the direct cost of the optional hedge. The drawback is that the rates should not drop down to the minimum guaranteed rate to the buyer of the put, since the seller has to pay this minimum rate. Hence, there is no gain below that rate. For example, the debt rate is the LIBOR 1-year, with a residual maturity of 5 years, and an amount of 200 million USD. X buys a 5-year cap, with a strike at 10%, costing 2% and sells a floor with a strike at 7%, receiving the premium of 1%. The net cost is 1%. If the interest rate declines below 7%, the seller of the floor pays the buyer 7%. The maximum (but unlikely) loss is 7% if rates fall close to zero.

Exotic Options There are plenty of variations around ‘plain vanilla’ derivatives, called ‘exotic’ derivatives. There is no point in listing them all. Each one should correspond to a particular view of the derivative user. For instance, ‘diff swaps’, or ‘quanto swaps’, swap the interest percentage in one currency for an interest rate percentage in another currency, but all payments are in the local currency. Such options look very exotic at first, but it makes a lot of sense in some cases. Consider the passage to the euro ex ante. Sometimes before, this event was uncertain. We consider the case of a local borrower in France. Possible events were easy to identify. If the euro did not exist, the chances are that the local French interest rate would increase a lot, because the FRF tended to depreciate. In the event of depreciation, the French rate would jump to very high levels to create incentives for not selling FRF. With the euro, on the other hand, the rates would remain stable and perhaps converge to the relatively low German rates, since the DEM was a strong currency. Without the euro, a floating rate borrower faced the risk of paying a very high interest rate. Swapping to a fixed rate did not allow any advantage to be taken of the decline in rates that prevailed during the years preceding the euro. Swapping a French interest rate for a German interest rate did not offer real protection, because, even though the German rate was expected to remain lower than the French rate, the interest payment in depreciated FRF would not protect the borrower against the euro risk. The ‘diff swap’ solves this issue. It swaps the French rate into the German rate, but payments are in FRF. In other words, the French borrower pays the German percentage interest rate in FRF, using as notional the FRF debt. The diff swap split the currency risk from the interest rate risk. It allowed to pay in FRF the low German rate without suffering from the potential FRF depreciation. Hence, the view on risks makes the choice of this exotic product perfectly clear. Of course, should the euro come into force, the borrower does not have to use the diff swap. The general message on derivatives, whether plain vanilla or exotic, is that such hedges require taking ‘views’ on risks. Plain vanilla derivatives have implied break-even values of rates turning potential gains into losses if expectations do not materialize. This applies also to exotic products.

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FUTURES MARKETS Unlike over-the-counter derivatives, futures are standard contracts sold in organized markets. The major difference between over-the-counter derivatives and futures is the customization of derivatives versus standardized future contracts.

Future Contracts A future contract is a double transaction consisting of: • A commitment over a price and a delivery at a given maturity for a given asset. • Settling the transaction at the committed date and under the agreed terms. For instance, it is possible to buy today a government debt delivered in 1 year at a price of 90 with a future. If, between now and 1 year, the interest rate declines, the transaction generates a profit by buying at 90 a debt valued higher in the market. The reverse contract is to sell today a government debt at a price of 90 decided today. If, between now and 1 year, the interest rate increases, the transaction generates a profit by selling at 90 a debt valued lower in the market. The main features of futures markets4 are: • • • •

Standard contracts. Supply and demand centralized. All counterparties face the same clearinghouse. The clearinghouse allocates the gains and losses among counterparties and hedges them with the collaterals (margin calls) provided by counterparties.

A typical contract necessitates the definition of the ‘notional’ underlying the contract. The notional is a fictitious bond, or Treasury bill, continuously revalued. Maturities extend over variable periods, with intermediate dates (quarterly for instance). Note that forward prices and futures should relate to each other since they provide alternative ways to provide the same outcome, allowing arbitrage to bring them in line for the same reference rate and the same delivery date5 .

Hedging with Futures The facility to buy or sell futures allows us to make a profit or loss depending on the interest rate movements. Adjusting this profit to compensate exactly for the initial exposure makes the contract a hedge. The adjustment involves a number of technicalities, such as defining the number of contracts that matches as closely as possible the amount to be hedged and limiting basis risk, the risk that the underlying of the contract does not track perfectly the underlying risk of the transactions to be hedged (interest rates are different for example). 4 See 5 See,

Hull (2000) for a review of the futures market. for example, French (1983) for a comparison of forward and futures prices.

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Since a contract consists of buying or selling a notional debt at a future date, the value of the debt changes with interest rates. These changes of values should match the changes of the transaction to hedge. For instance, for a borrower fearing interest rate increases, buying a contract at a preset price is a hedge because the value of the notional declines if interest rates increase. For a lender fearing that interest rates will decline, selling a contract is a hedge because the value of the notional increases if they do, providing a gain (Table 14.1). TABLE 14.1

Hedging interest rate transactions on the futures market Hedge againsta :

Increase of rates Lender buys a contract • Buy forward at 100 what should be valued more (110) if interest rate decreases • Hence, there is a gain if interest rate decreases a Lender

Decrease of rates Borrower sells a contract • Sell forward at 100 what should be valued less (90) if interest rate increases • Hence, there is a gain if interest rate increases

and borrower have variable rate debts.

CURRENCY RISK As with interest rates, there are forward and optional hedges. Most of the notions apply, so that we simply highlight the analogies, without getting into any further detail.

The Mechanisms of Forward Exchange Rates The forward exchange rate is the exchange rate, for a couple of currencies, with market quote today for a given horizon. Forward rates serve to lock in an exchange rate for a future commercial transaction. Forward rates can be above or below spot exchange rates. There is a mechanical relationship between spot and forward exchange rates. The mechanisms of the ‘forex’ (foreign exchange) and the interest rates markets of two countries create these relationships between the spot and forward exchange rates, according to the interest rate differential between currencies. The only major difference with interest rate contracts is that there is a market of forward exchange rates, resulting from the interaction of interest rate differentials differing from the mechanisms of forward interest rates (Figure 14.3).

Hedging Instruments Forward rates lock in the exchange rate of a future flow. Swaps exchange one currency for another. This is equivalent to changing the currency of debt or investment into another one. Options allow us to take advantage of the upside together with protection against downside risk. In the case of currencies, the receiver of a foreign currency flow will buy a put option to sell this foreign currency against the local currency at a given strike price. Being long

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Lend at 4%

1 USD

−1.04 USD

+1.04 USD

Convert USD in FRF spot

5 FRF

FIGURE 14.3

+

Convert 5.25 FRF in USD forward (0.198 FRF per USD)

Lend at 5%

5.25 FRF

Spot, forward exchange rates and interest rates

in a currency means receiving the foreign currency, while being short means having to pay in a foreign currency. Hedging with derivatives is straightforward. For instance, X is long 200 million USD. A decline of USD is possible. The spot rate is 0.95 EUR/USD. The forward rate is 0.88 EUR/USD. The alternative choices are: • Sell forward at 0.88 EUR/USD. This generates an opportunity cost if the USD moves up rather than down because the sale remains at 0.88. • Buy a put (sale option) with a premium of 2% at strike 6.27 EUR/USD. The put allows the buyer to benefit from favourable upward movements of USD without suffering from adverse downside moves. On the other hand, it costs more than a forward hedge (not considering the opportunity cost of the latter). Embedded in the put are break-even values of the exchange rate. In order to make a gain, the buyer of the put needs to make a gain higher or equal to the premium paid. The premium is 2%, so the USD has to move by 2% or more to compensate the cost. Otherwise, X is better off without the put than with it. As usual, X needs a view on risk before entering into a hedge. There are plenty of variations around these plain vanilla transactions. We leave it to the reader to examine them.

SECTION 6 Asset–Liability Management Models

15 Overview of ALM Models

Asset–Liability Management (ALM) decisions extend to hedging, or off-balance sheet policies, and to business, or on-balance sheet policies. ALM simulations model the behaviour of the balance sheet under various interest rate scenarios to obtain the risk and the expected values of the target variables, interest income or the mark-to-market value of the balance sheet at market rates. Without simulations, the ALM Committee (ALCO) would not have a full visibility on future profitability and risk, making it difficult to set up guidelines for business policy and hedging programmes. ALM simulation scope extends to both interest rate risk and business risk. ALM policies are medium-term, which implies considering the banking portfolio changes due to new business over at least 2 to 3 years. Simulations have various outputs: • They provide projected values of target variables for all scenarios. • They measure the exposure of the bank to both interest rate and business risk. • They serve to ‘optimize’ the risk and return trade-off, measured by the expected values and the distributions of the target variables across scenarios. Simulation techniques allow us to explore all relevant combinations of interest rate scenarios, business scenarios and alternative hedging policies, to find those which enhance the risk–return profile of the banking portfolio. Three chapters detail the simulation methodology applied to ALM. The first chapter (Chapter 17) explains the specifics of typical exposures to interest rate risk of commercial banks. The second chapter (Chapter 18) explains how to derive risk and expected profitability profiles from simulations, ignoring business risk. A simplified example illustrates the entire process. The third chapter (Chapter 19) extends the approach to business risk,

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building on the same example. The examples use the interest income as target variable, but they apply to the balance sheet mark-to-market value (‘NPV’) as well. These chapters ignore optional risk, dealt with subsequently. This chapter provides an overview of the process. The first section describes the various steps of simulations. The second section discusses the definition of interest rate scenarios, which are critical inputs to the simulation process. The third section discusses the joint view on interest rate risk and business risk. The fourth section explains how ALM simulations help enhance the risk–return profile of the bank’s portfolio, by combining interest rate scenarios, business scenarios and alternative hedging policies. Finally, the last section briefly describes the inputs for conducting ALM simulations.

OVERVIEW OF ALM SIMULATIONS Simulations serve to construct the risk–return profile of the banking portfolio. Therefore, full-blown ALM simulations proceed through several steps: • Select the target variables, interest income and the balance sheet NPV. • Define interest rate scenarios. • Build business projections of the future balance sheets. The process either uses one single base business scenario, or extends to several business scenarios. For each business scenario, the goal is to project the balance sheet structure at different time points. • Project margins and net income, or the balance sheet NPV, given interest rates and balance sheet scenarios. The process necessitates calculations of interest revenues and costs, with all detailed information available on transactions. • When considering optional risk, include valuing options using more comprehensive interest rate scenarios than for ‘direct’ interest rate risk. Simulations consider optional risks by making interest income, or balance sheet NPV, calculations dependent on the time path of interest rates. • Combine all steps with hedging scenarios to explore the entire set of feasible risk and return combinations. • Jointly select the best business and hedging scenarios according to the risk and return goals of the ALCO. Once the risk–return combinations are generated by multiple simulations, it becomes feasible to examine which are the best hedging policies, according to whether or not they enhance the risk–return profile, and whether they fit the management goals. The last step requires an adequate technique for handling the large number of simulations resulting from this process. A simple and efficient technique consists of building up through simulations the expected profitability and its volatility, for each hedging policy, under business risk, in order to identify the best solutions. The same technique applies to both interest income and NPV as target variables. Figure 15.1 summarizes the process.

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Business scenarios

Interest rate scenarios

Target Target variables: variables: Interest Interest income & income & NPV NPV

Simulations Simulations

Target variable (Income & NPV) distributions

Current value

Loss

Funding and hedging scenarios Feedback loop for optimizing hedging

FIGURE 15.1

Risk Return

Overview of the ALM simulation process

Multiple simulations accommodate a large number of scenarios. Manipulating the large number of scenarios requires a methodology. This chapter and the next develop such methodology gradually using examples. The examples are simplified. Nevertheless, they are representative of real world implementations. The latter involve a much larger amount of information, but the methodology is similar. The technical tools for exploring potential outcomes range from simple gaps to fullblown simulations. Gaps are attractive but have drawbacks depending on uncertainties with respect to the volumes of future assets and liabilities, which relate to business risks. There are as many gaps as there are multiple balance sheet scenarios. Gaps do not capture the risk from options embedded in banking products. Optional risks make asset liabilities sensitive to interest rates in some ranges of values and insensitive in others. Simulations correct the limitations of the simple gap model in various ways. First, they extend to various business scenarios if needed, allowing gaps to depend on the volume uncertainties of future assets and liabilities. Second, they calculate directly the target variable values, without necessarily relying on gaps only to derive new values of margins when interest rates change. This is feasible by embedding in the calculation any specific condition that influences the interest revenues and costs. Such conditions apply to options, when it is necessary to make rates dependent on the possibility of exercising prepayment options. ALM simulations focus on selected target variables. They include the interest margins over several periods, the net income, the mark-to-market value of the balance sheet (NPV). For a long-term view, the NPV is the best target variable. Medium to long-term horizons make the numerous periodical interest margins more difficult to handle. Because the NPV summarizes the entire stream of these interest incomes over future years (see Chapter 21), it provides an elegant and simple way to handle all future margins. Since both short and long-term views are valuable, it makes sense to use interest income as a target variable for

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the near future, and the NPV for the long-term view. Technically, this implies conducting both calculations, which ALM models excel in doing.

INTEREST RATE SCENARIOS AND ALM POLICIES Interest rate scenarios are a critical input for ALM simulations since they aim to model the interest income or the balance sheet NPV. The discussion on interest rate scenarios is expanded further in subsequent chapters, through examples. The current section is an overview only. ALM simulations might use only a small number of yield curve scenarios, based on judgmental views of future rates, or consider a large number of values to have a comprehensive view of all possible outcomes. The definitions of interest rate scenarios depend on the purpose. Typical ALM simulations aim to model the values of the target variables and discuss the results, the underlying assumptions and the views behind interest rate scenarios. This does not necessitate full-blown simulation of rates. On the other hand, valuing interest rate options or finding full distributions of target variable values requires generating a large number of scenarios. Discrete scenarios imply picking various yield curve scenarios that are judgment based. The major benefit of this method is that there are few scenarios to explore, making it easier to understand what happens when one or another materializes. The drawback is that the method does not provide a comprehensive coverage of future outcomes. Increasing the number of scenarios is the natural remedy to this incomplete coverage. Multiple simulations address valuation of options, which requires modelling the entire time path of interest rates. Options embed a hidden risk that does not materialize until there are large variations of interest rates. Valuing implicit options necessitates exploring a large spectrum of interest rate variations. Models help to generate a comprehensive array of scenarios. Chapter 20 illustrates the valuation process of implicit options using a simple ‘binomial’ model of interest rates. Finding an ALM ‘Value at Risk’ (VaR), defined as a downside variation of the balance sheet NPV at a preset confidence level, also requires a large number of interest rate values for all maturities, in order to define the downside variation percentiles. When considering changes of interest rates, it is also important to consider shifts over all maturities and changes of slopes. ‘Parallel shifts’ assume that the yield curve retains the same shape but its level varies. Parallel shifts are a common way to measure sensitivity to interest rate variations. Changing slopes of the yield curve drive the spread between long and short rates, a spread that is critical in commercial banking, since banks borrow short to lend long. During the recent history of rates in Euroland, up to the date when the euro substituted all currencies in the financial industry, the yield curve shifted downwards and its slope flattened. These moves adversely affected the interest income of positive gap banks, because it embeds the spread between the long and short rates. Interest rate modelling raises a number of issues, ranging from model specifications to practical implementations. First-generation ALM models used a small number of interest rate scenarios to project the ALM target variables. New models integrate interest rate models, allowing us to perform a full valuation of all possible outcomes, while complying with the consistency of rates across the term structure and the observed rate volatilities. Interest rate models serve to provide common interest rate scenarios for ALM simulations,

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for the valuation of implicit options and consistent with market risk models used for calculating market risk VaR1 . Models help because they make the entire structure of interest rates a function of one or several factors, plus a random error term measuring the discrepancy between modelled values and actual values. The high sensitivity to both the level and slope of the yield curve of the interest margin rules out some models that tend to model rates based on the current curve2 . In many instances, as indicated above, the shape of the yield curve changes significantly. Some models help by generating a large number of yield curve scenarios without preserving the current yield curve structure3 . On the other hand, they do not allow us to use yield curve scenarios that match the management views on interest rate, which is a drawback for ALCO discussions of future outcomes. In addition, business risk makes the volumes of assets and liabilities uncertain, which makes it less important to focus excessively on interest rate uncertainty. For such reasons, traditional ALM models use judgmental yield curve scenarios as inputs rather than full-blown simulations of interest rates. Models excel for valuing implicit options, and for exploring a wide array of interest outcomes, for instance when calculating an ALM VaR. Historical simulations use interest data from observed time series to draw scenarios from these observations. Models allow the generation of random values of interest rates for all maturities, mimicking the actual behaviour of interest rates. Correlation matrices allow the associations between rates of different maturities to be captured. Principal component analysis allows modelling of the major changes of the yield curves: parallel shifts, slopes and bumps. Chapter 30 provides further details on Monte Carlo simulations of rates. Chapters 20 and 21 show how models help in valuing implicit options. In what follows, we stick to discrete interest rate scenarios, which is a common practice.

JOINT VIEWS ON INTEREST RATES AND BUSINESS UNCERTAINTIES Restricting the scope to pure ‘financial’ policies leads to modelling the risk–return profile of the banking portfolio under a given business policy, with one base scenario for the banking portfolio. Simulations allow us to explore the entire set of risk–return combinations under this single business scenario. They show all feasible combinations making up the risk–return profile of the banking portfolio without hedging, or without modifying the existing hedging programme. Obtaining this risk–return profile is a prerequisite for defining hedging policies. When considering a single business scenario, and a single period, the interest margin is a linear function of the corresponding gap if there are no options. For a single gap value, the interest income varies linearly with the interest rates. The distribution of net income values provides a view on its risk, while its expected value across all interest rate scenarios is the expected profitability. Modifying the gap through derivatives changes the interest 1 Market

risk VaR models need to simulate interest rates, but it is not practical to use full-blown models with all interest rates. The principal component technique, using a factor model of the yield curve, is easier to handle (Chapter 30). The same technique would apply to ALM models. 2 An example is the Ho and Lee (1986) model. 3 See Hull (2000) for a review of all interest rate models.

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income distribution, as well as its expected profitability. Each gap value corresponds to a ‘risk plus expected return’ combination. It becomes possible to characterize the risk with simple measures, such as the volatility of the target variable, or its downside deviation at a preset confidence level. This converts the distribution of target variable values resulting from interest rate uncertainty into a simple risk–return pair of values. Varying the gap allows us to see how the risk–return combination moves within this familiar space. The last step consists in selecting the subset of ‘best’ combinations and retaining the corresponding hedging solution (Figure 15.2). A full example of this technique is developed in Chapter 17, dealing with business risk. Distribution of forward values of margins or NPV

Expected value Current value

FIGURE 15.2

Loss

Risk−return space Expected value

Future time point

Risk

Forward valuation of ALM target variables

When the gap becomes a poor measure of exposure because there is optional risk, simulations require calculating the interest income for various interest rate scenarios, without using gaps. The calculation requires assumptions with respect to the behaviour of borrowers in exercising their right to reset their borrowing rate. The ALCO might find none of these risk–return profiles acceptable, because of the risk–return trade-off. Hedging risk might require sacrificing too much expected profitability, or the target profitability might require taking on too much risk, or simply remain beyond reach when looking at a single business scenario and altering risk and return through hedging policies only. The ALCO might wish to revise the business policy in such cases. Extending the simulation process to business scenarios has a twofold motivation: • Addressing ALCO demand for exploring various business scenarios, and allowing recommendations to be made for on-balance sheet policies as well as for off-balance sheet policies. Under this view, ALCO requires additional ‘what if’ simulations for making joint on and off-balance sheet decisions. • Addressing the interaction between business risk and interest rate risk by exploring business uncertainties with several business scenarios. Under this second view, the goal is to find the optimal hedging solutions given two sources of uncertainties, interest rate and business.

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ALM simulations address business risk through projections of the banking portfolio. This is a specific feature of ALM, compared to similar techniques applied to market risk and credit risk. When dealing with market or credit risk, simulations apply to a ‘crystallized’ portfolio, as of current date, ignoring medium-term business scenarios. Because of the ALCO willingness to explore various projections, and because of business risk in the medium and long-term, it is necessary to extend simulations to explicit projections of the banking portfolio over 2 to 3 years. Scenarios differ because they consider various new business assumptions and various product–market mixes. Since there are many business scenarios, there are many distributions of the target variables. Exposure uncertainty is another motive for multiple balance sheet scenarios. Existing assets and liabilities have no contractual maturities making gaps undefined. New business originates new assets and liabilities, whose volume is uncertain. Dealing with a single projection of the balance sheet for future periods ‘hides’ such uncertainties. A simple way to make them explicit consists of defining various scenarios. Some of them differ because they make different assumptions, for example to deal with lines without maturity.

ENHANCING AND OPTIMIZING RISK–RETURN PROFILES When the ALCO considers alternative business policies, ALM simulations address both on and off-balance sheet management. The goal becomes the joint optimization of hedging policies and on-balance sheet actions. This requires combining interest rate scenarios with several balance sheet scenarios and hedging solutions. The methodology for tackling the resulting increase in the number of simulations as illustrated in Chapter 18. For each pair of business and interest rate scenarios, ALM models determine the value of the target profitability variables. When the interest rate only varies, there is one distribution of values, with an expected value and the dispersion around this average. When there are several balance sheet projections, there are several distributions. This addresses the first goal of conducting ‘what if’ simulations to see what happens and make sure that the profitability and the risk remain in line with goals. However, it is not sufficient to derive a hedging policy. This issue requires testing various hedging policies, for instance various values of gaps at different time points, to see how they alter the distribution of the target variable. The process generates new distributions of values of the target variable, each one representing the risk–return profile of the balance sheet given the hedging solution. Each hedging solution value corresponds to a ‘risk plus expected return’ combination. The last step consists in selecting the subset of ‘best’ combinations and retaining the corresponding hedging solutions. The technique accommodates various measures of risks. Since we deal with distributions of values of the target variables at various time points, the volatility or the percentiles of downside deviations of target variables quantify the risk. This allows us to determine an ALM VaR as a maximum downside deviation of the target variable at a preset confidence level. Changing the hedging policy allows us to move in the ‘risk–return space’, risk being volatility or downside deviations and return being the expected value of interest income or the NPV.

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SIMULATIONS AND INFORMATION Actual projections of the balance sheet require a large set of information. In order to project margins, assumptions are also required on the interest rate sensitivity of transactions and the reference rates. The availability of such information is a critical factor to build an ALM information system. Table 15.1 summarizes the main information types, those relating to the volume of transactions and rates, plus the product–market information necessary to move back and forth from the purely financial view to the business view of future outcomes. TABLE 15.1

Information by transaction

Transaction

Type of transaction and product

Volume

Initial outstanding balance Amortization schedule Renewal and expected usage Optionsa Currency

Rates

Reference rate Margin, guaranteed historical rate, historical target rate Nature of rate (calculation mode) Interest rate profileb Sensitivity to reference rates Options characteristics

Customers

Customers, product–market segments

a Prepayment, b Frequency

etc. and reset dates.

The nature of the transactions is necessary to project exposure and margins. Amortizing loans and bullet loans do not generate the same gap profiles. Projected margins also depend on differentiated spreads across bank facilities. In addition, reporting should be capable of highlighting the contributions of various products or market segments to both gaps and margins. The information on rates is obviously critical for interest rate risk. Margin calculations require using all formulas to calculate effective rates (pre-determined, postdetermined, number of days, reset dates, options altering the nature of the rate, complex indexation formulas over several rates, and so on). The rate level is important for margin calculations and to assess the prepayment, or renegotiation, likelihood, since the gap with current rates commands the benefit of renegotiating the loan for mortgages. Sensitivities are required when balance sheet items map to a restricted set of selected reference rates. Options characteristics refer to strike price and the nature of the options for off-balance sheet hedging and embedded options in banking products, such as contractual caps on variable rates paid by borrowers.

16 Hedging Issues

Typically, banks behave as net lenders when they have positive variable rate gaps and as net borrowers when they have negative variable rate gaps. A net lender, like a lender, prefers interest rate increases and needs to hedge against declines. A net borrower, like a borrower, prefers interest rate declines and needs protection against increases. However, the sensitivity to variations of rates is not sufficient to characterize the exposure of a ‘typical’ commercial bank and to decide which hedging policy is the best. A ‘typical’ commercial bank pays short-term rates, or zero rates, on deposits and receive longer-term rates from loans. An upward sloping curve allows these banks to earn more from assets than they pay for liabilities, even if commercial margins are zero. Commercial banks capture a positive maturity spread. Simultaneously, the interest income is sensitive to both shifts and steepness of the yield curve. This variable gap depends on the nature of the rate paid on the core deposit base. If the rates are zero or fixed (regulated rates), the variable rate gap tends to be positive. If the rate is a short-term rate, the variable rate gap tends to become negative. In both cases, banks still capture the positive maturity spread, but the sensitivity to parallel shifts of the curve changes. The joint view of variable rate gap plus the gap between liability average rates and asset average rates characterizes the sensitivity of interest income of banks to shifts and slope changes of the yield curve. Deposits blur the image of liquidity and interest rate gaps because they have no maturity. Core deposits remain for a long period, earn either a short-term or a zero rate and tend to narrow the liquidity gap. Zero-rate deposits widen the variable rate gap, and interest-earning deposits narrow it. Such an economic view contrasts with the legal (zero maturity) or conventional (amortizing deposits) views. Net lender banks, with a large deposit base, are adversely exposed to declining rates, and vice versa. High long-term rates increase income. They are also adversely exposed to a decline in steepness of the yield curve. When rates decline, net lenders face the dilemma

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of waiting versus hedging. Early hedges protect them while late hedges are inefficient because they lock the low rates into interest income. Moreover, banks with excess funds face arbitrage opportunities between riding the spot or the forward yield curves, dependent on both the level and slope of yield curves. For such reasons, hedging policies are highly dependent on interest rate ‘views’ and goals. This chapter addresses these common dilemmas of traditional commercial banks. The first section details the joint view on interest rate gaps and yield curve, with a joint sensitivity to shifts of yield curve level and steepness. The second section shows how conventions on deposits alter the economic view of the bank’s exposure. The third section details hedging issues.

INTEREST RATE EXPOSURE The sensitivity to parallel shifts of the yield curve depends on the sign of the variable interest rate gap, or the gap between the average reset period of assets and that of liabilities. With a positive variable rate gap, banks behave as net lenders. With a negative variable rate gap, banks behave as net borrowers. Variable rate gaps increase with the gap between the asset reset period and the liability reset period. With more variable rate assets than variable rate liabilities, the reset period of assets becomes shorter than the average reset period of liabilities. With more variable rate liabilities than variable rate assets, liability resets are faster than asset resets. The relation between interest rate gaps and average reset dates is not mechanical. We assume, in what follows, that a positive variable rate gap implies a negative gap between the asset average reset date and the liability average reset date, and conversely. The difference between the two views is that the gap between average reset dates positions the average rates of assets and liabilities along the yield curve. The average dates indicate whether the bank captures a positive or negative yield curve spread, depending on the slope of the yield curve and how sensitive it is to this slope. Hence, the two views of gaps, variable rate gap and average reset dates gap, complement each other. The first refers to parallel shifts of the curve, while the second relates to its steepness. Figure 16.1 shows the two cases of positive and negative variable rate gaps. If variable rate gaps and reset date gaps have the same sign, the posture of the bank is as follows: • With a positive variable rate gap, the assets average reset period is shorter than the liabilities average reset period. The bank borrows long and lends short. With an upward sloping yield curve, the interest income of the bank suffers from the negative market spread between short and long rates. • With a negative variable rate gap, the liabilities average reset period is shorter than the assets average reset period. The bank borrows short and lends long. With an upward sloping yield curve, the interest income of the bank benefits from the positive market spread between short and long rates. The sensitivity to a change in slope of the yield curve depends on the gap between average reset periods of assets and liabilities. The hedging policy results from such views of the bank’s position. Banks with positive variable rate gaps have increasing interest income with increasing rates and behave as net

HEDGING ISSUES

FRA

203

Positive Positive variable variable rate gap rate gap

FRL

FRA

Negative Negative variable rate gap variable rate gap

FRL VRL

VRA VRA VRL

VRA VRL

VRA VRL

VR Gap > 0 i A

VR Gap < 0

IM

i L

L

IM

A

Average reset period of assets Average reset period assets Average reset period of of liabilities Average reset period of liabilities

FIGURE 16.1 liabilities

Interest rate gaps and gaps between average reset periods of assets and

Forward Yield Curve

Spot Yield Curve

A

FIGURE 16.2

L

Maturity

Interest Rate

Interest Rate

lenders. If they expect rates to rise, they benefit from maintaining their gaps open. If not, they should close their gaps. Banks with negative variable rates have increasing interest income with decreasing rates and behave as net borrowers. The average reset period of assets is longer than the average reset period of liabilities. If they expect rates to rise, they benefit from closing their gaps. If not, they should open their gaps. Rules that are more specific follow when considering the forward rates. A bank behaving as a net lender benefits from higher rates. Therefore, if it does not hedge fully its exposures, views on rates drive the hedging policies (Figure 16.2):

Spot Yield Curve Forward Yield Curve A

L

Maturity

Bank as net lender: asset reset period is shorter than liability reset period

• With an upward sloping yield curve, forward rates are above spot rates. The bank needs to lock in the current forward rates for future investments if it believes that future spot rates will be below the current forward rates, since these are the break-even rates.

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• With an inverted yield curve, forward rates are below spot rates. The bank profits from forward rates for future investments only if it believes that the future spot rates will decrease to a level lower than the current forward rates. Obviously, if banks expect short rates to be equal to future spot rates are equal to forward rates, they become indifferent since locking in a forward rate or not doing anything are, by definition, equivalent. When dealing with banks having large deposit bases, the situation differs. Essentially, this is because core deposits are long-term sources of funds earning a short-term rate or a zero rate. The discrepancy between the legal maturity of deposits and their effective maturity blurs the image of the bank’s exposure to interest rates.

THE ‘NATURAL EXPOSURE’ OF A COMMERCIAL BANK The ‘natural liquidity exposure’ of a bank results from the maturity gap between assets and liabilities because banks transform maturities. They collect short and lend longer. However, the core demand deposit base is stable. Hence, overfunded banks tend to have a large deposit base. Underfunded banks tend to have a small deposit base. Since deposits have no maturity, conventions alter the liquidity image. Deposits are stable and earn either a small fixed rate or a short-term rate. Therefore, their rate does not match their effective long maturity. This section discusses the ‘natural exposure’ of commercial banks having a large deposit base and draws implications for their hedging policies.

Liquidity

Volume

Core deposits

Volume

Volume

The views on the liquidity posture depend on the conventions used to determine liquidity gaps. Figure 16.3 shows three different views of the liquidity profile of a bank that has a large deposit base. Core deposits have different economic, conventional and legal maturities. Conventional rules allow for progressive amortization, even though deposits do not amortize. The legal maturity is short-term, as if deposits amortized immediately. This is also unrealistic. The economic maturity of core deposits is long by definition. This is the only valid economic view and we stick to it. Note that this economic view is the

Assets

Assets Deposits

Assets

Deposits Time Economic View

FIGURE 16.3

Time Conventional View

Liquidity views of a commercial bank

Time Legal View

HEDGING ISSUES

205

Liabilities + core deposits

Assets Time Overfunded

FIGURE 16.4

Assets

Time Underfunded

Volume

Liabilities + core deposits

Volume

Volume

reverse of the view that banks transform maturities by lending long and collecting short resources. Figure 16.4 shows two possible economic views: an overfunded bank that collects excess deposits and liabilities; an underfunded bank collecting fewer deposits and liabilities than it lends. The situation is either stable through time or reverses after a while. We assume that the total liabilities decrease progressively and slowly because there are some amortizing debts in addition to the large stable deposit base.

Assets

Liabilities + core deposits Time Mixed: underfunded then overfunded

Underfunded versus overfunded commercial bank

Banks with a large core deposit base have excess funds immediately or become progressively overfunded when assets amortize faster than the core deposit base. Banks with a small deposit base are underfunded, at least in the near term, and might become overfunded if assets amortize faster. The interest rate view refers to how the bank captures the maturity spread of interest rates and to its sensitivity to interest rate changes.

Interest Rate Exposure The exposure to interest rate shifts depends on the variable rate gap or, equivalently, on the gap between the average reset period of assets and the average reset period of liabilities. The larger the variable rate gap, the higher the favourable sensitivity to a parallel upward shift of the yield curve. Stable deposits do not earn long-term interest rates because their legal maturity is zero, or remains short for term deposits. Many demand deposits have a zero rate. Regulated savings accounts earn a fixed regulated rate in some countries. Demand deposits, subject to certain constraints, earn a short-term rate or a lower rate. In all cases, the stable deposit base has a short-term rate, a fixed rate close to short-term rates or a zero rate. In terms of the yield curve, they behave as if they were on the short end of the curve, rather than on the long end corresponding to their effective maturity. Hence, banks earn the market spread between the zero rate and the average maturity rate of assets, or the spread between the short-term rate of demand deposits and the average maturity rate of assets. For an upward sloping curve, the maturity spread contributes positively to the interest income, and even more so for zero-rate demand deposits.

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However, the interest rate gap, or the gap between reset dates of deposits and assets, depends on the rate of deposits. When the rate is a short-term rate, deposits are ratesensitive liabilities. When the rate is fixed or zero, they are rate-insensitive liabilities. Given their mass in the balance sheet, this changes the interest rate gap of the bank. Banks with zero-rate (or fixed-rate) deposits tend to have a positive variable rate gap, while banks with interest rate-sensitive deposits tend to have a negative variable rate gap. The zero-rate, or fixed-rate, deposits make banks favourably sensitive to a parallel upward shift of the yield curve, while the interest rate-sensitive deposits make banks adversely sensitive to the same parallel upward shift of the yield curve. In both cases, banks benefit from the positive maturity spread of an upward sloping curve (Figure 16.5). FRA

FRL

Deposit base: Deposit zero ratebase at zero rate

VRA

VRA VRL

FRA

FRL

Deposit base: Deposit base: short-term rate short-term rate VRA VRL

VRL VRA

VRL

VR Gap > 0 i L

FIGURE 16.5

A

VR Gap < 0

IM

i L

IM

A

Interest rate exposure of commercial banks

HEDGING POLICIES This section makes explicit some general rules driving the hedging policies. Then, we discuss the case of banks behaving as net lenders in continental Europe during the period of declining rates. This is a period of continuous downward shifts of upward sloping yield curves. For several players, it was a period of missed opportunities to take the right exposures and of failures to timely hedge exposures to unfavourable market movements. The context also illustrates why the trade-off between profitability and risks is not the same when rates are low and when rates are high. In times of low rates, hedging locks in a low profitability and increases the opportunity cost of giving up a possible upward move of rates. In times of high rates, the issue might shift more towards reducing the volatility of the margin rather than the level of the profitability.

General Principles Banks with a large zero-rate deposit base are net lenders, have positive variable rate gaps, and their interest income improves with upward shifts of the yield curve and with an

HEDGING ISSUES

207

increasing slope of the yield curve. Banks with a large short-term-rate deposit base are also net lenders but have negative variable rate gaps. Their interest income deteriorates with upward shifts of the yield curve and improves with an increasing slope of the yield curve. We now focus only on ‘net lender’ banks for simplicity. With the upward sloping yield curves of European countries in the nineties, forward rates were above spot rates. Banks behaving as net lenders faced a progressive decline of the yield curve. They needed to lock in the current forward rates for future investments only if they believed that future spot rates would be below current forward rates. If they expected rates to rise again, the issue would be by how much, since maintaining exposure implies betting on a rise beyond the current forward rates.

Implication for On-balance Sheet Hedging A net lender bank with a large base of zero-rate deposits tends to have a positive variable rate gap, a negative liquidity gap (excess of funds) and an average reset date of assets shorter than that of liabilities, but they benefit from a positive slope of the yield curve: • In times of increasing rates, they benefit both from the positive variable rate gap and the asset rate higher than liabilities. • In times of declining rates, they tend to close the gap, implying looking for fixed rates, lending longer or swapping the received variable rate of assets against the fixed rate. Note that declining rates right be an opportunity to embed higher margins over market rates in the price of the variable rate loans. In this case, variable rate loans would not reduce the bank’s risk but would improve its profitability. Upward sloping curves theoretically imply that future interest rates will increase to the level of forward rates and that these are the predictors of future rates because of arbitrage. In practice, this is not necessarily true. The euro term structure remained upward sloping for several years while continuously moving down for years in the late nineties, until short-term rates hit repeatedly historical lows. Rates declined while the yield curve got flatter. Going early towards fixed rates on the asset side was beneficial. After the decline, such hedges became useless because banks could only lock in low rates.

Hedging and the Level of Interest Rates Hedging policies typically vary when interest rates are low and when they are high. Starting with high interest rates, commercial banks faced a continuous decline of the yield curve, and finally the bottom level, before new hikes in interest levels began to take place in the year 2000. With high interest rates, interest rate volatility might be higher. This is when hedging policies are most effective. Derivatives reduce the volatility by closing gaps. There are timing issues, since locking fixed rates in revenues is a bet that rates will not keep increasing, otherwise the bank with favourable exposures to interest rate increases would suffer an opportunity cost. Before reaching low levels, rates decline. Banks acting as net lenders prefer to receive more fixed rates when rates decline and pay variable rates on the liabilities side. Both on-balance sheet and off-balance sheet actions become difficult:

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• Banks are reluctant to hedge their current exposures because that would lock in their revenues at the current rates, for instance through swaps, when there is still a chance of hitting a turning point beyond which rates could increase again. This temptation postpones the willingness to set up hedges in time. • Simultaneously, customers might be reluctant to accept long fixed rates in line with the profitability targets of the bank if they expect rates to keep declining. Once rates get low, not having set up hedges on time makes them useless later, shifting the pressure from off-balance sheet actions to on-balance sheet actions again. The likelihood of a rise increases, but crystallizing the low rates in the revenues does not make much sense, because the profitability reaches such a low level that it might not suffice any more to absorb the operating costs. There is a break-even value of the interest revenue such that profits net of operating costs hit zero and turn into losses if revenues keep falling. Late forward hedges do not help. Optional hedges are too costly. The European situation illustrates well this dilemma. Many banks waited before setting up hedges because of a fear of missing the opportunity of rates bouncing back up. After a while, they were reluctant to set up forward hedges because low interest revenues from hedges were not economical. Moreover, institutions collecting regulated deposits pay a regulated rate to customers. This regulated rate remains fixed until regulations change. This makes it uneconomical to collect regulated resources, while depositors find them more attractive because of low short-term rates. The decline in rates puts pressure on deregulating these rates. Because of this pressure, regulated rates become subject to changes, at infrequent intervals, using a formula linking them to market rates.

Riding the Spot and Forward Yield Curves Riding an upward sloping yield curve offers the benefit of capturing the positive maturity spread, at the cost of interest rate risk. Riding the forward yield curve, with an upward sloping yield curve, offers the benefit of capturing the positive spread between the forward and spot yields. Steeper slopes imply higher spreads between long-term and short-term interest rates. The spot yield curve is not necessarily the best bet for institutions having recurring future excess funds. The difference between forward and spot rates is higher when the slope is steeper. This is an opportunity for forward investments to lock in rates significantly higher than the spot rates. There are ‘windows of opportunity’ for riding the forward yield curve, rather than the spot yield curve. A case in point is the transformation of the yield curve in European countries in the late nineties. The slope of the spot yield curve became steep, before decreasing when approaching the euro. When the slope was very steep, there was an opportunity to shift from lending to investing forward. Banks that faced declining interest rates for lending did not hedge their gaps if they bet that rates were close to the lowest point and would start rising again. An alternative policy to lending was to invest future excess funds at forward rates when the yield curve slope was very steep. These rates ended much higher than the spot rates prevailing later at the forward investment dates, because the yield curves kept shifting downward. Riding

HEDGING ISSUES

209

Invest forward 1 year

Rates

Forward yield curve

Invest spot 2 years Spot yield curve Invest spot 1 year Time 1 year

FIGURE 16.6

2 years

Comparing spot and forward investments

the forward yield curve was more beneficial than riding the spot yield curve. Figure 16.6 summarizes this policy. Of course, when interest rates reached the bottom line in late 1999 and early 2000, the yield curve had become lower and flatter. The spread between forward and spot rates narrowed. Consequently, the ‘window of opportunity’ for investing in forward rates higher than subsequent spot rates closed.

17 ALM Simulations

ALM simulations focus on the risk and expected return trade-off under interest rate risk. They also extend to business risk because business uncertainties alter the gap profile of the bank. Return designates the expected value of the target variables, interest income or NPV, and their distribution across scenarios, while volatility of target variables across scenarios characterizes risk. This chapter focuses on interest income. Simulations project the balance sheet at future horizons and derive the interest income under different interest rate scenarios. When the bank modifies the hedge, it trades off risk and expected profitability, eventually setting the gap to zero and locking in a specific interest income value. For each gap value, the expected interest income and risk result from the set of income values across scenarios. Altering the gap generates various combinations. Not all of them are attractive. Some combinations dominate others because they provide a higher expected income at the same risk or have a lower risk for the same value of expected income. These are the ‘efficient’ combinations. Choosing one solution within the efficient set, and targeting a particular gap value, becomes a management decision depending on the willingness to take risk. Simulations with simple examples illustrate these mechanisms. The next chapter extends the scope to business risk. Simple examples involve only one variable rate, as they all vary together. In practice, there are several interest rate scenarios. Multiple variablerate exposures generate complexity and diversification effects because not all rates vary simultaneously in the same direction. EaR (‘Earnings at Risk’), in this case ‘interest income at risk’, results from the interest rate distribution given gap and multiple interest rate scenarios. The management can set an upper bound to EaR, for limiting the interest income volatility or imposing a floor to a potential decline. Such gap limits alter the expected interest income. The first section details balance sheet projections, all necessary inputs, and the projected gaps. The second section details the direct calculations of interest income under different

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211

interest rate scenarios. The third section makes explicit the risk–return profile of the balance sheet given interest rate uncertainty and exposure. It also introduces the case of multiple variable-rate gaps. The fourth section shows how controlling the exposure through hedging modifies this profile and implies a trade-off between expected interest income and uncertainty. The last section discusses the hedging policy, from ‘immunization’ of the interest income to taking exposures depending on the bank’s views on interest and setting risk limits.

BALANCE SHEET AND GAP PROJECTIONS Projections of balance sheets are necessary to project liquidity and interest rate gap profiles and the values of the target variables. Business projections result from the business policy of the bank. Projections include both existing assets and liabilities and new business.

Projections of the Existing Portfolio and of New Transactions The projection of all existing assets and liabilities determines the liquidity gap time profile. The projection of interest rate gaps requires breaking down assets and liabilities into ratesensitive items and fixed-rate items, depending on the horizon and starting as of today. The existing portfolio determines the static gaps. The new business increases the volume of the balance sheet while existing assets and liabilities amortize. The new assets and liabilities amortize as soon as they enter the balance sheet. Projections should be net of any amortization of new assets and liabilities. Projecting new business serves to obtain the total volume of funding or excess funds for each future period, as well as the volume of fixed-rate and variable-rate balance sheet items. Projections include both commercial assets and liabilities and financial ones, such as debt and equity. Equity includes retained earnings derived from the projected income statements.

Interest Rate Sensitivity of New and Existing Assets and Liabilities Interest-sensitive assets and liabilities are those whose rate is reset during each subperiod. Real world projections typically use monthly periods up to 1 or 2 years, and less frequent ones after that. In our example, we consider only a 1-year period. An item is interest rate-sensitive if there is a rate reset before the end of the year. For new business, all items are interest rate-sensitive, no matter whether the expected rate is fixed or variable. The variable–fixed distinction is not relevant to determine which assets or liabilities are interest-sensitive. A future fixed-rate loan is interest-sensitive, even if its rate remains fixed for its entire life at origination, because origination occurs in the future. The future fixed rate will depend on prevailing market conditions. On the other hand, the fixed–variable distinction, and the breakdown of variable rates for existing assets and liabilities, are necessary to obtain fixed-rate gaps and as many variablerate gaps as distinct market references. Therefore, interest-sensitive assets and liabilities include variable-rate existing transactions plus all new transactions. Interest-insensitive items include only those existing transactions that have a fixed rate. For simplicity, all sensitivities to the single variable-rate reference are 100% for all balance sheet items in our example.

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Table 17.1 provides a sample set of data for a simplified balance sheet, projected at date 1, 1 year from now. There is no need to use the balance sheet at date 0 for gap calculations. All subsequent interest revenue and cost calculations use the end of year balance sheet. We assume there is no hedge contracted for the forthcoming year. TABLE 17.1 Balance sheet projections for the banking portfolio Dates Banking portfolio Interest rate-insensitive assets (a) Interest rate-sensitive assets (b) Total assets (c = a + b) Interest rate-insensitive resources (d) Interest rate-sensitive resources (e) Total liabilities (f = d + e)

1 19 17 36 15 9 24

Projected Gap Profiles The gaps result from the 1-year balance sheet projections. All gaps are algebraic differences between assets and liabilities. The liquidity gap shows a deficit of 12. The variablerate gap before funding is +8, but the deficit of 12 counts as a variable-rate liability as long as its rate is not locked in advance, so that the post-funding variable interest rate gap is −4 (Table 17.2). TABLE 17.2

Gap projections

Dates Banking portfolio Interest-insensitive assets (a) Interest-sensitive assets (b) Total assets (c = a + b) Interest-insensitive resources (d) Interest-sensitive resources (e) Total liabilities (f = d + e) Liquidity gap (c − f )a Variable interest rate gap (b − e) Total balance sheet Variable interest rate gap after funding (b − e) − (c − f )b Gaps Liquidity gapc Interest rate gapb

1 19 17 36 15 9 24 +12 +8 −4 +12 −4

a Liquidity gaps are as algebraic differences between assets and liabilities. The +12 value corresponds to a deficit. b Funding is assumed to be variable-rate before any hedging decision is made. c Interest rate gaps are interest-sensitive assets minus interest-sensitive liabilities, or ‘variable-rate’ interest rate gaps.

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213

INTEREST INCOME PROJECTIONS The interest margin is the target of financial policy in our example. The margins apply to different levels. The interest margin generated by the banking portfolio (operating assets and liabilities only) is the ‘commercial margin’. The interest income is after funding costs, inclusive of all interest revenues or costs from both operating and financial items. In order to calculate margins, we need to relate market rates to customers’ rates. The spread between the customers’ rates and the market rate is the percentage commercial margin. Such spreads are actually uncertain for the future. Assigning values to future percentage margins requires assumptions, or taking them as equal to the objectives of the commercial policy. Value margins are the product of volume and percentage margins. In this example, we use the market rate as an internal reference rate to calculate the commercial margins. Normally, the Funds Transfer Pricing (FTP) system defines the set of internal transfer prices (see Chapter 26).

Interest Rate Scenarios In our example, we stick to the simple case of only two yield curve scenarios. The example uses two scenarios for the flat yield curve, at the 8% and 11% levels (Table 17.3). TABLE 17.3

Interest rate scenarios

Scenarios 1 (stability) 2 (increase)

Rate (%) 8 11

Note: Flat term structure of interest rates.

Commercial and Accounting Margins The assumptions are simplified. The commercial margins are 3% for assets and −3% for liabilities. This means that the average customer rate for assets is 3% above the market rate, and that the customer rate for demand and term deposits is, on average, 3% less than the market rate. When the market rate is 8%, these rates are 11% and 5%1 . In practice, percentage margins differ according to the type of asset or liability, but the calculations will be identical. The commercial margin, before the funding cost, results from the customer rates and the outstanding balances of assets and liabilities in the banking portfolio. It is: 36 × 11% − 24 × 5% = 3.96 − 1.20 = 2.76 The accounting interest margin is after cost of funding. The cost of funding is the market rate2 . Since the yield curve is flat, the cost of funds does not depend on maturity in 1 These

figures are used to simplify the example. The actual margins obviously differ for various items of assets and liabilities. 2 Plus any credit spread that applies to the bank, assumed to be included in the rates scenarios.

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this example. The interest margin after funding is the commercial margin minus the cost of funding a deficit of 12. This cost is 12 × 8% = 0.96. The net accounting margin is therefore 2.76 − 0.96 = 1.80.

The Sensitivity of Margins In this example, we assume that the percentage commercial margins remain constant when the interest rate changes. This is not a very realistic assumption since there are many reasons to relate percentage margins to interest rate levels or competition. For instance, high rates might not be acceptable to customers, and imply percentage margin reductions. For all interest-insensitive items, the customers’ rates remain unchanged when market rates move. For interest-sensitive items, the customers’ rate variation is identical to the market rate variation because of the assumption of constant commercial margins. The value of the margins after the change in interest rate from 8% to 11% results from the new customers’ rates once the market rate rose. It should be consistent with the interest rate gap calculated previously. Detailed calculations are given in Table 17.4. TABLE 17.4

Margins and rate sensitivity

Interest-insensitive assets Interest-sensitive assets Revenues Interest-insensitive resources Interest-sensitive resources Costs Commercial margin Liquidity gap Net interest margin

Volume

Initial rate

Revenues/ costs

Final rate

Revenues/ costs

19 17

11% 11%

11% 14%

15 9

5% 5%

12

8%

2.09 1.87 3.96 0.75 0.45 1.20 2.76 0.96 1.80

2.09 2.38 4.47 0.75 0.72 1.47 3.00 1.32 1.68

5% 8% 11%

The values of the commercial margins, before and after the interest rate rise, are 2.76 and 3.00. This variation is consistent with the gap model. The change in commercial margin is 0.24 for a rate increase of 3%. According to the gap model, the change is also equal to the interest rate gap times the change in interest rate. The interest rate gap of the commercial portfolio is +8 and the variation of the margin is 8 × 3% = 0.24, in line with the direct calculation. The net interest margin, after funding costs, decreases by 1.80 − 1.68 = 0.12. This is because the funding cost of the liquidity gap is indexed to the market rate and increases by 12 × 3% = 0.36. The commercial margin increase of 0.24 minus the cost increase results in the −0.12 change. Alternatively, the interest gap after funding is that of the commercial portfolio alone minus the amount of funding, or +8 − 12 = −4. This gap, multiplied by 3%, also results in a −0.12 change in margin. This assumes that the funding cost is fully variable.

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THE RISK–RETURN PROFILE The gaps summarize the balance sheet image and provide a simple technique to derive all possible variations of interest margins. If we know the original interest margin, gaps and interest rate scenarios provide all information necessary to have the risk–return profile of each future time point. The risk–return profile of the banking portfolio is the image of all attainable combinations of risk and expected return given all possible outcomes. The process requires selecting a target variable, whose set of possible values serves to characterize risk. In the example of this chapter, the interest income is the target variable, but the same approach could use the Net Present Value (NPV) as well. The interest rate changes drive the margin at a given gap. The full distribution of the interest margin values when rates vary characterizes the risk–return profile. Making explicit this trade-off serves to set limits, such as maximum gap values, and assess the consequences on expected net income.

The Risk–Return Profile of the Portfolio given Gaps Characterizing the risk–return profile is straightforward using the basic relationship: IM = gap × i The margin at date t is random, IMt , the period is from 0 to t, and the relationship between the variation of the margin and that of interest rates is: IM = gap × i = gap × (it − i0 ) To project the interest margin, we need only to combine the original IM0 at date 0 with the above relations: IMt = IM0 + IM = IM0 + gap × i The expected variation of margin depends on the expected interest rate variation. The final interest rate is a random variable following a probability distribution whose mean and standard deviation are measurable. With a given gap value, the probability distribution of the margin results from that of the interest rate3 . The change of margin is a linear function of the interest rate change with a preset gap. The expected value of the interest margin at date t, E(IMt ), is the summation of the fixed original margin plus the expected value of its variation between 0 and t. The volatility of the final margin is simply that of the variation. They are: E(IM) = gap × E(i)

and E(IMt ) = IM0 + gap × E(i)

σ (IM) = |gap| × σ (i) and σ (IMt ) = |gap| × σ (i) The vertical bars stand for absolute value, since the volatility is always positive. Consequently, the maximum deviation of the margin at a preset confidence level results directly 3 With the usual notation, the expected rate is E(i ) and its volatility is σ (i ). When X is a random variable with t t expectation E(X) and standard deviation σ (X), any variable Y = aX, a being a constant, follows a distribution with expectation a × E(X) and volatility a × σ (X). The above formula follows: E(IM) = gap × E(it ) and σ (IM) = |gap| × σ (it ).

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e rg La

p ga

Sm

Interest rate variation Confidence Level

FIGURE 17.1 preset gap

IM IM Distribution Distribution

Interest margin variation all

gap

Probability

Probability

Interest Rate Interest Rate Distribution Distribution

Probability

from the maximum deviation of the interest rate at the same confidence level. However, if the gap changes, the interest income distribution also changes. This is the case when the bank modifies the gap through its hedging policy. Figure 17.1 shows the distribution of the interest margin with two different values of the gap. The probabilities of exceeding the upper bound variations are shaded.

IM Distribution

Interest margin variation

Distribution of Interest Margin (IM) with random interest rates and a

Interest Income at Risk and VaR The usual Value at Risk (VaR), or EaR, framework applies to the interest margin. However, it is possible that no straight loss occurs if we start with a positive interest margin, because the downside deviation might not be large enough to trigger any loss. The margin volatility results from that of interest rates: σ (IM) = |gap| × σ (i). The historical volatility of interest rates is observable from time series. In this example, we assume that the yearly volatility is 1.5%. Using a normal distribution, we can define confidence levels in terms of multiples of the interest rate volatility. The interest rate will be in the range defined by ±2.33 standard deviations from the mean in 1% of all cases. Since the interest margin follows a distribution curve derived from the distribution of interest rates multiplied by the gap, similar intervals apply to the margin distribution of values. With a gap equal to 2000, the margin volatility is: σ (IM) = 2000 × 1.5% = 30. The upper bound of the one-sided deviation is 2.33 times this amount at the 1% confidence level, or 2.33 × 30 = 70 (rounded). The EaR should be the unexpected loss only. This unexpected loss depends on the expected value of the margin at the future time point. Starting from an expected margin at 20, for example, the unexpected loss is only 20 − 70 = −50.

THE RISK–RETURN TRADE-OFF WHEN HEDGING GAPS With a given interest rate distribution, the bank alters its risk–return profile by changing its gap with derivatives. Unlike the previous section, the gap becomes variable rather than preset. When altering its exposure, the bank changes both the expected margin and the

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risk, characterized by margin volatility or a downside deviation at a preset confidence level. In what follows, we ignore the cost of derivatives used to modify the gap and assume that they generate interest payments or revenues with the current flat rate.

The Risk–Return Profile The basic relationships are σ (IM) = |gap| × σ (i) and E(IM) = IM0 + gap × E(i). The gap value, controlled by the bank, appears in both equations and is now variable. Since the volatility is an absolute value, we need to specify the sign of the gap before deriving the relationship between σ (IM) and E(IM) because we eliminate the gap between the two equations to find the relationship between expected margin and margin volatility. With a positive gap, the relationship between E(IM) and σ (IM) is linear, since gap = |gap| = σ (IM)/σ (i) and: E(IM) = IM0 + [E(i)/σ (i)] × σ (IM) This relation shows that the expectation of the margin increases linearly with the positive gap. When the gap is negative, |gap| = −gap and we have a symmetric straight line with respect to the σ (IM) axis. Therefore, with a given gap value, there is a linear relationship between the expected interest margin and its volatility. The risk–return profile in the ‘expected margin–volatility of margin’ space is a set of two straight lines. If the bank selects a gap value, it also selects an expected interest income. If the bank sets the gap to zero, it fully neutralizes the margin risk. We illustrate such risk–return trade-offs with a simple example when there are only two interest rate scenarios.

Altering the Risk–Return Profile of the Margin through Gaps In general, there are a large number of interest rate scenarios. In the following example, two possible values of a single rate summarize expectations. The rate increases in the first scenario and remains stable in the second scenario. The probability of each scenario is 0.5. If the upward variation of the interest rate is +3% in the first scenario, the expected value is 1.5% (Figure 17.2). Variation Variation Initial rate Initial = 8% rate = 8%

FIGURE 17.2

Probability Probability

Final rate Final rate

+ 3%

0.5

11%

+ 0%

0.5

10%

Expected variation = +1.5%

Expected rate = 11.5%

Interest rate expectations

Swaps serve to adjust the gap to any desired value. For each value of the gap, there are two values of the margin, one for each of the interest rate scenarios, from which

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the average and the volatility of the margin derive. Gap changes generate combinations of average margin and volatility. The variations of the margins, when the interest rate changes, are equal to the gap times the variation of interest rate: IM = gap × i. If the gap is −4, as in the above example, the variations of the margin are −4 × 3% = −0.12 and −4 × 0% = 0, since +3% and 0% are the only possible deviations of interest rate. Their probabilities are both 0.5. The expected value and the variance are respectively: 0.5 × (−0.12) + 0.5 × 0 = −0.06 and 0.5 × (−0.12 − 0.06)2 + 0.5 × (−0.12 + 0.06)2 = 0.0036. The volatility is the square root, or 0.06. With various values of the gap, between +10 and −10, the average and the volatility change according to Table 17.5. TABLE 17.5 Interest rate gap −10 −8 −6 −4 −2 0 +2 +4 +6 +8 +10

Expected margin and margin volatility Volatility of the margin variation

Expected variation of margin

0.15 0.12 0.09 0.06 0.03 0.00 0.03 0.06 0.09 0.12 0.15

−0.15 −0.12 −0.09 −0.06 −0.03 0.00 +0.03 +0.06 +0.09 +0.12 +0.15

By plotting the expected margin against volatility, we obtain all combinations in the familiar risk–return space. The combinations fall along two straight lines that intersect at zero volatility when the gap is zero. If the gap differs from zero, there are two attainable values for the expected margin at any given level of volatility. This is because two gaps of opposite sign and the same absolute value, for instance +6 and −6, result in the same volatility (0.09) and in symmetrical expected variations of the margin (+0.09 and −0.09). The expected variation of the margin is positive when the gap is positive. This is intuitive since the expectation of interest rate, the average of the two scenarios, is above the current rate. A rise in interest rate results in an increase of the expected margin when the gap is positive. Negative gaps are irrational choices with this view on interest rates. The expected interest rate would make the margin decrease with such gaps. When, at a given risk, the expected margin is above another combination, the risk–return combination is ‘efficient’. In this case, only positive values of the gap generate efficient combinations that dominate others, and negative values of the gap generate inefficient combinations. Therefore, only the upper straight line represents the set of efficient combinations. It corresponds to positive gaps only (Figure 17.3). The optimum depends on the bank’s preference. One way to deal with this issue is to set limits on the margin volatility or its maximum downside variation. Such limits set the gap and, therefore, the expected margin. This result extends over multiple interest rate scenarios since the risk–return profile depends only on the expectation and volatility of interest rate changes. However, the

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Expected variation of margin

0.20 0.10

Gap = +10 Gap

0.05 0.00 −0.05

0.00

0.05

0.10

0.15

0.20

−0.10 Gap = −10

−0.15 −0.20

FIGURE 17.3

Efficient combinations

0.15

Margin volatility

Risk–return relationships

above calculation applies only with a preset gap controlled by the bank. It does not hold when the gap is random. This is precisely what happens when considering business risk, because it makes future balances of assets and liabilities uncertain. To deal with this double uncertainty, interest rate and business risks, we need a different technique. The multiple interest rate and business scenarios approach addresses this issue (Chapter 18).

Risk–Return Profiles with Multiple Interest Rates In general, there are multiple gaps for different interest rates. Since all rates vary together, adding up the interest income volatilities due to each exposure, or the worst-case exposures for each gap, would overestimate the risk. This issue is the same for market risk and credit risk when dealing with diversification. The correlation4 between risk drivers determines the extent of diversification. The two rates might not vary mechanically together, so the chances are that one varies less than the other or in the opposite direction. This reduces the overall exposure of the bank to an exposure lower than the sum of the two gaps. Section 11 of this book provides the extended framework for capturing correlation effects. In the case of interest rates, the analysis relies on gaps. The following example illustrates these issues. The balance sheet includes two subportfolios that depend on different interest indexes. These indexes are i1 and i2 . Their yearly volatilities are 3% and 2%. The correlation between these two interest rates can take several values (0, +1, +0.3). The sensitivities of the interest income are the gaps, 100 and 200, relative to those indexes (Table 17.6). The interest margin variation IM becomes a function of both gaps and interest rate changes: IM = 100 × i1 + 200 × i2 It is the summation of two random variables, with constant coefficients. The expected variation is simply the weighted summation of the expected variation of both rates, using the gaps as weights. The volatility of the interest margin results from both interest rate volatilities and the correlation. The standalone risk for the first gap is the margin volatility generated by the variation of the first index only, or 100 × 3% = 3. The standalone risk 4 The

correlation is a statistical measure of association between two random variables, which is extensively discussed in Chapter 28.

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TABLE 17.6

Overall volatility of two interest rate exposures Market volatilities

Rate i1 Rate i2 Exposures Exposure 1 Exposure 2 Portfolio volatilitya Correlation = 0 Correlation = 1 Correlation = +0.3

σ (i1 ) = 3% σ (i2 ) = 2% Gap1 = 100 Gap2 = 200 5.00 7.00 5.67

values are calculated with the general formula: [(100 × 3%)2 + (200 × 2%)2 + 2ρ(100 × 3%)(200 × 2%)] where ρ takes the values 0.1 and 0.3 (see Chapter 28).

a These

for the second exposure is 200 × 2% = 4. The sum of these two volatilities is 7, which overestimates the true risk unless rates are perfectly correlated. The actual volatility is that of a weighted summation of two random variables: [gap1 × σ (i1 )]2 + [gap2 × σ (i2 )]2 + 2ρ[gap1 × σ (i1 )][gap2 × σ (i2 )] σ (IM) = (100 × 3%)2 + (200 × 2%)2 + 2ρ(100 × 3%)(200 × 2%)

σ (IM) =

With a zero correlation the volatility is 5, and with a correlation of 0.3 it becomes 5.67. It reaches the maximum value of 7 when the correlation is 1. The margin volatility is always lower than in the perfect correlation case (7). The difference is the effect of diversification of exposures on less than perfectly correlated rates. The extreme case of a perfect correlation of 1 implies that the two rates are the same. In such a case, the gap collapses to the sum of the two gaps. When characterizing the risk–return profile of the portfolio of two exposures by the expected margin and its volatility, the latter depends on the correlation between rates. Using a normal distribution as a proxy for rates is acceptable if rates are not too close to zero. In this case, it is a simple matter to generate the distribution of the correlated rates to obtain that of the margin5 . When considering the hedging issue, one possibility is to neutralize all gaps for full immunization of interest income. Another is to use techniques taking advantage of the correlation between interest rates. In any case, when rates correlation is lower than 1, variable-rate gaps do not add together. Note that using fixed-rate gaps is equivalent to an arithmetic addition of all variable-rate gaps, as if they referred to the same index. This corresponds to a correlation of 1 and overestimates the exposure. 5 Alternatively,

interest rates follow a lognormal, more realistic distribution. The changes in interest rates still follow a stochastic process with a random normal factor. It is possible to correlate the rates by correlating their random factor. See Chapter 30 for a description of the main stochastic processes driving rates. Chapter 28 discusses techniques for generating the distribution of a sum of correlated normal variables.

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INTEREST RATE POLICY The ALM unit controls the interest rate exposure by adjusting the gap after funding. This section discusses, through simple examples, various hedging issues: immunization of the interest income; trade-off of the hedging policy when the bank has ‘views’ on future interest rates; usage of multiple gap exposures; setting limits for interest rate risk.

Hedging In order to obtain immunization, the post-funding gap should be zero. In other words, the funding and hedging policy should generate a gap offsetting the commercial portfolio positive gap. The funding variable-rate gap is equal to the fraction of debt remaining at variable rate after hedging, with a minus sign since it is a liability. Since the gap is +8 before funding, the variable-rate debt should be set at 8 to fully offset this gap. This means that the remaining fraction of debt, 12 − 8 = 4, should have an interest rate locked in as of today. An alternative and equivalent approach is to start directly from the post-funding gap. This gap is +8 − 12 = −4. Setting this gap to zero requires reducing the amount of floating rate debt from 12 to 8. Therefore, we find again that we need to lock in a rate today for the same fraction of the debt, or 4, leaving the remaining fraction of the debt, 8, with a floating rate. In the end, the interest rate gap of the funding solution should be the mirror image of the interest rate gap of the commercial portfolio. Any other solution makes the net margin volatile. The solution ‘floating rate for 8 and locked rate for 4’ neutralizes both the liquidity gap and the interest rate gap (Table 17.7). TABLE 17.7 rate gaps

Hedging the liquidity and interest

Liquidity gap Interest rate gap of the banking portfolio Fixed-rate debt Floating-rate debt Total funding Liquidity gap after funding Interest rate gap after funding and hedging

+12 +8 4 8 12 0 0

In order to lock in the rate for an amount of debt of 4, various hedges apply. In this case, we have too many variable-rate liabilities, or not enough variable-rate assets. A forward swap converting 4 of the floating-rate debt into a fixed-rate debt is a solution. The swap would therefore pay the fixed rate and receive the floating rate. Alternatively, we could increase the variable-rate fraction of assets by 4. A swap paying the fixed rate and receiving the variable rate would convert 4 of fixed-rate assets into 4 of variable-rate assets. The same swap applies to both sides.

Hedging Policy If the amount of floating-rate debt differs from 8, there is an interest rate exposure. The new gap differs from zero. For instance, locking in the rate for an amount of debt of 2

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results in a gap which is +2. The margin volatility results directly from the remaining gap after funding. Any increase in rate increases the margin since the variable-rate gap is positive. However, the price to pay is the risk that rates decrease, thereby reducing the margin. Maintaining open a gap makes sense only if it is consistent with an expected gain. In this example, the expectation embedded in the scenarios is a rise of rate, since rates are either stable or increase. A positive variable-rate gap makes sense, but a negative variable-rate gap would be irrational. The level of interest rates is an important consideration before making a decision to close the gap. A swap would lock in a forward rate. In this case, the locked rate is on the cost side, since the swaps pay the fixed rate. If the forward rate is low, the hedge is acceptable. If the forward rate is high, it will reduce the margin. Using a full hedge for a short horizon only (1 year) provides added flexibility, while later maturities remain open. This allows for the option to hedge at later periods depending on the interest rate moves. Of course, if interest rates keep decreasing, the late hedge would become ineffective. The exposure to interest rate risks has to be consistent with risk limits. The limits usually imply that adverse deviations of the interest margin remain within given bounds. This sets the volume of the necessary hedge for the corresponding period. Setting the maximum adverse deviation of the target variable, given a confidence level, is equivalent to setting a maximum gap since the margin volatility is proportional to the gap. This maximum gap is the limit.

Setting Limits for Interest Rate Risk Setting limits involves choosing the maximum volatility, or the maximum downward variation of the margin at a given confidence level. Limits would effectively cap the gap, thereby also influencing the expected income, as demonstrated earlier. With a given volatility of the margin, the maximum acceptable gap is immediately derived: σ (IM) = |gap| × σ (i). If the management caps the volatility of the margin at 20, and the interest rate volatility is 1.5% yearly, the maximum gap is 20/1.5% = 1333. Alternatively, it is common to set a maximum downward variation of the interest margin, say −100. Assuming operating costs independent of interest rates, the percentage variation of pre-tax and post-operating costs income is higher than the percentage change in interest margin at the top of the income statement. For instance, starting with a margin of 200, a maximum downside of 100 represents a 50% decrease. If operating costs are 80, the original pre-tax income is 200 − 80 = 120. Once the margin declines to 100, the remaining pre-tax income becomes 100 − 80 = 20. The percentage variation of pre-tax income is −100/120 = −80%, while that of margin is −50%. If interest rate volatility is 1.5% and confidence level is 2.5%, the upper bound of the interest rate deviation, using the normal distribution proxy for rates, is 1.96 × 1.5%, or approximately 3%. Starting from a maximum decline of the margin set at 100, at the 2.5% confidence level, the maximum sustainable gap is 100/3% = 3333. If the interest rate decreases by 3%, the margin declines by 3333 × 3% = 100, which is the maximum.

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Hedging Multiple Variable-rate Gaps For hedging purposes, the simplest way to obtain margin immunization is to neutralize both gaps with respect to the two interest rates. Another option would be to hedge both gaps at the same time using the interest rate correlation. Since rates correlate, a proxy hedge would be to take a mirror exposure to only one of the rates used as a unique driver of the interest rate risk, the size of the hedge being the algebraic summation of all gaps. This is the well-known problem of hedging two correlated exposures, when there is a residual risk (basis risk). When considering the two variable-rate gaps, respectively 100 and 200 for two different and correlated rates, the basic relationship for the interest margin is: IM = 100 × i1 + 200 × i2 Since the two rates correlate, with ρ12 being the correlation coefficient, there is a statistical relationship between the two variations of rates, such that: i1 = βi2 + ε12 IM = 100 × (βi2 + ε12 ) + 200 × i2 The value of the second gap becomes a variable G, since the bank can control it in order to minimize the margin volatility: IM = 100 × (βi2 + ε12 ) + G × i2 IM = (100 × β + G) × i2 + ε12 The variance of IM is the sum of the variance of the interest rate term and the residual term since they are independent: σ 2 (IM) = (100 × β + G)2 × σ 2 (i2 ) + σ 2 (ε12 ) There is a value X of the gap G such that this variance is minimum. Using β = 0.5 and σ 2 (ε12 ) = 1%, we find that X = −50 results in a variance of 1. This also has a direct application for setting limits to ALM risk, since the volatility is an important parameter for doing so. Note that hedging two exposures using a single interest rate results in a residual volatility. On the other hand, neutralizing each variable rate gap separately with perfect hedges (same reference rates) neutralizes basis risk. This is a rationale for managing gaps separately.

18 ALM and Business Risk

Since both interest rates and future business volumes and margins are random, banks face the issue of how to jointly optimize, through financial hedges, the resulting uncertainty from these multiple sources of risk. There are practical simulation-based solutions to this joint optimization problem, using financial hedges. The starting point of the method requires defining a number of interest rate scenarios and business scenarios. The principle consists of simulating all values of the target variable, the interest income or the Net Present Value (NPV) of the balance sheet, across all combinations of interest rate and business scenarios. The resulting set of values is organized in a matrix cross-tabulating business and interest rate scenarios. Such sets of target variable values depend on any hedge that affects interest income or NPV. Since gaps become subject to business risk, it is not feasible any more to fully hedge the interest income. When hedging solutions change, the entire set of target variable values changes. For turning around the complexity of handling too many combinations of scenarios, it is possible to summarize any set of values of the target of variables within the matrix, by a couple of values. The first is the expected value of the target variable, and the second is its volatility, or any statistics representative of its risk, both calculated across the matrix cells. For each hedging solution, there is such a couple of expected profitability and risk across scenarios. When hedging changes, both expectation and risk change, and move in the ‘risk–return’ space. The last step consists of selecting the hedging solutions that best suit the goals of the Asset–Liability Management Committee (ALCO), such as minimizing the volatility or targeting a higher expected profitability by increasing the exposure to interest rate risk. Those solutions that maximize expected profitability at constant risk or minimize risk at constant expected profitability make up a set called the ‘efficient frontier’. All other hedging solutions are discarded. It is up to the management to decide

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what level of risk is acceptable. This methodology is flexible and accommodates a large number of simulations. The technique allows us to investigate the impact on the risk–return profile of the balance sheet under a variety of assumptions, for example: What is the impact on the risk–return profile of assumptions on volumes of demand deposits, loans with prepayment risk or committed lines whose usage depends on customer initiatives? Which funding and hedging solutions minimize the risk when only interest rate risk exists, when there is business risk only and when both interact? How can the hedging solutions help to optimize the risk–return combination? The Asset–Liability Management (ALM) simulations also extend to optional risks because the direct calculation of interest income or NPV values allows us to consider the effect of caps and floors on interest rates, as explained in Section 7 of this book, which further details these risks. Finally, the ALM simulations provide a unique example of joint control of interest rate and business risks, compared to market risk and credit risk measures, which both rely on a ‘crystallized’ portfolio as of today. The first section of this chapter illustrates the methodology in a simple example combining two simple interest rate scenarios with two business scenarios only. It provides the calculation of interest income, and of its summary statistics, expectation and volatility, across the matrix of scenarios. The second section details the calculations when changing the hedging solution, considering only two different hedging solutions for simplicity. The third section summarizes the findings in the risk–return space, demonstrating that extending simulations to a larger number of scenarios does not raise any particular difficulty. It defines the efficient frontier and draws some general conclusions.

MULTIPLE SCENARIOS WITH BUSINESS AND INTEREST RATE RISKS The previous methodology applies to a unique gap value for a given time point. This is equivalent to considering a unique business scenario as if it were certain, since the unique value of the gap results directly from asset and liability volumes. Ignoring business risk is not realistic, and amounts to hiding risks rather than revealing them when defining an ALM policy. Moreover, the ALM view is medium-term, so that business uncertainty is not negligible. To capture business risk, it is necessary to define multiple business scenarios. Multiple scenarios result in several projections of balance sheets and liquidity and interest rate gap profiles. For hedging purposes, the risk management issue becomes more complex, with interest income volatility resulting from both interest rate risk and business risk. Still, it is possible to characterize the risk–return profile and the hedging policies when both risks influence the interest income and the NPV. Business risk influences the choice of the best hedging policy.

The Risk–Return Profile of the Balance Sheet When using discrete scenarios for interest rates and for business projections, it is possible to summarize all of them in a matrix cross-tabulating all interest rate scenarios with all business scenarios. For each cell, and for each time point, it is feasible to calculate the

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margin or the NPV. The process is not fully mechanical, however, because there is a need to make assumptions on funding and interest rate management. Constructing a risk–return profile and hedging are two different steps. For the first step, we consider that there are no new hedges other than the existing ones. Hence, all future liquidity gaps result in funding or investing at prevailing rates. The ultimate goal of the simulations remains to define the ‘best’ hedge. If we calculate an interest income value or NPV for each cell of the matrix, we still face the issue of summarizing the findings in a meaningful way in terms of return and risk. A simple way to summarize all values resulting from all combinations of scenarios is to convert the matrix into a couple of parameters: the expected interest income (or NPV) and the interest income (or NPV) volatility across scenarios. This process summarizes the entire matrix, whatever its dimensions, into a single pair of values (Figure 18.1). The volatility measures the risk and the expectation measures the return. This technique accommodates any number of scenarios. It is possible to assign probabilities to each scenario instead of considering that all are equally probable, and both expected margin and margin volatility would depend on such probabilities. If all combinations have the same probability, the arithmetic average and the volatility are sufficient. Business / Interest rate

FIGURE 18.1 scenarios

A

B

Rate 1

IM & NPV IM & NPV

Rate 2

IM & NPV IM & NPV

Expected IM & NPV Expected + IM & NPV + Volatility Volatility

Matrix of scenarios: cross-tabulating business scenarios and interest rate

However, since the interest income or NPV depends on the funding, investing and hedging choices, there are as many matrices as there are ways of managing the liquidity and the interest rate gaps. We address this issue in a second step, when looking for, and defining, the ‘best’ policies.

The Matrix Approach A simple example details the methodology. When using two scenarios, there are only two columns and two rows. It is possible to summarize each business scenario by a gap value. Referring to the example detailed, we use two such gap values: −4 and −8 crosstabulated with two interest rate scenarios, with +8% and +11% as flat rates. The original value of the margin with rate at 8% is 1.80 with a gap equal to −4. When rates vary, it is easy to derive the new margin value within each column, since the gap is constant along a column. The matrix in this very simple case is as given in Table 18.1, with the corresponding expectation and volatility on the right-hand side, for a single time point 1 year from now. Using two business scenarios, with net gaps of −4 and −8, and two interest rate scenarios, at +8% and +11%, we find the expectations of the margin change and of the margin volatility in Table 18.1. The variations are respectively −0.12 and −0.24 when the rate moves up from 8% to 11%. Starting from a margin of 1.80, the corresponding final

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TABLE 18.1 Matrix of margins: two interest rate scenarios and a single business scenario Rate

8% 11%

Scenario Gap = −4 Gap = −8 +1.80 +1.68

1.80 1.56

Risk–return profile

E(IM) σ (IM)

1.7100 0.1149

margins are 1.68 and 1.56. The expected margin and its volatility across the four cells are respectively 1.7100 and 0.11491 . In the range of scenarios considered, the worst-case value of the margin is 1.56. By multiplying the number of scenarios, we could get many more margin values, and find a wider distribution than the three values above. In the current simple set of four cases, we have a 2 × 2 matrix of margins for each financing solution. If we change the financing solution, we obtain a new matrix. There are as many matrices as there are financing solutions.

Handling Multiple Simulations and Business Risk The matrix approach serves best for handling multiple business scenarios as well as multiple yield curve scenarios. There is no limitation on the number of discrete scenarios. The drawback of the matrix approach is that it summarizes the risk–return profile of the portfolio using two parameters only, which is attractive but incomplete. To characterize the full portfolio risk, we would prefer to have a large distribution of values for each future time point. In addition, probabilities are subjective for business scenarios. This approach contrasts with the market risk ‘Value at Risk’ (VaR) and the credit risk VaR techniques, because they use only a ‘crystallized’ portfolio as of today. With a single business scenario, we could proceed with the same technique to derive an ALM VaR with full probability distributions of all yield curves. This would allow us to generate distributions of margins, or of the NPV, at all forward time points fully complying with observed rates. The adverse deviations at a preset confidence level derive from such simulations. We use this technique when detailing later the specifics of NPV VaR with given asset and liability structures. However, ignoring business risk is not realistic with ALM. It does not make much sense to generate a very large number of interest rate scenarios if we ignore business risk, since there is no point in being comprehensive with rates if we ignore another significant risk. The next chapter details how the matrix methodology helps in selecting the ‘best’ hedging solutions.

HEDGING BOTH INTEREST RATE AND BUSINESS RISKS For any given scenario, there is a unique hedging solution immunizing the margin against changes in interest rates. This hedging solution is the one that offsets the interest rate gap. With several scenarios, it is not possible to lock in the margin for all scenarios by 1 The

standard deviation uses the formula for a sample, not over the entire population of outcomes (i.e. it divides the squared deviations from the mean by 3, not 4).

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hedges, because gaps differ from one business scenario to another. To determine hedging solutions, we need to simulate how changing the hedge alters the entire risk–return profile corresponding to each business scenario.

The Matrix Technique, Business Risk and Hedging When ignoring new hedges, we generate a number of values for the target variables identical to the number of combinations of interest rate and business scenarios, conveniently summarized into a couple of statistics, the ‘expected value’ and ‘volatility’. When modifying the hedge, we change the entire matrix of values. There are as many matrices of values of the target variables, interest income or NPV, as there are hedging solutions. Summarizing each set of new values in a risk–return space allows us to determine which hedges result in better risk–return profiles than others. Figure 18.2 illustrates the process. Hedging 2 Hedging 2 Hedging 3

Business / A Interest rate Business / 1 Taux A Marge B Interest rate 2 Marge Business / Taux Taux 1 A Marge B Interest rate Taux 2 Marge IM & NPV IM & NPV Rate 1 Rate 2

IM & NPV IM & NPV

B

Expected IM & NPV / Volatility Expected IM & NPV / Volatility Expected IM & NPV / Volatility

FIGURE 18.2 scenarios

Matrix of scenarios: cross-tabulating business scenarios and interest rate

Simulations offer the maximum flexibility for testing all possible combinations of assumptions. Multiplying the number of scenarios leads to a large number of simulations. Two business scenarios and two interest rate scenarios generate four combinations. With two hedging scenarios, there are eight combinations. The number of simulations could increase drastically, making the interpretation of the results too complex. To handle this complexity, summarizing each matrix using a pair of values helps greatly. Only the risk–return combinations generated for each hedging solution serve, independently of the size of the matrix. The approach facilitates the identification of the best solutions within this array of combinations. In what follows, we ignore probabilities assigned to business scenarios. Using probabilities would require assigning probabilities to each interest rate scenario and to each business scenario. Therefore, each cell of the matrix has a probability of occurrence that becomes the joint probability of the particular pair of business and interest rate scenarios. In the case of uniform probabilities, there is no need to worry about different likelihoods of occurrence when using the matrix technique. Otherwise, we should derive these joint

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probabilities. Under independence between business scenarios and interest rate scenarios, these would be the product of the probabilities assigned to the particular business scenario and the particular interest rate scenario, as explained in Chapter 28.

Business Scenarios The interest rate scenarios are those of the first analysis with only one base case for business projections (interest scenarios 1 and 2). One business scenario (A) is identical to that used in the previous chapter. A second business scenario (B) is considered. Each business scenario corresponds to one set of projected balance sheet, liquidity gap and interest rate gap. The scenario B is shown in Table 18.2. The starting points are the liquidity gaps and the interest rate gaps of the banking portfolio for scenarios A and B. TABLE 18.2

Two business scenarios

Projected balance sheet Interest-insensitive assets Interest-sensitive assets Interest-insensitive resources Interest-sensitive resources Total assets Total liabilities Gaps Liquidity gaps Interest rate gap—banking portfolio Interest rate gap—total balance sheet

Scenario A

Scenario B

19 17 15 9 36 24

22 17 14 13 39 27

12 8 −4

12 4 −8

We use the same 3% commercial margins to obtain customer rates and margins in value. For instance, with scenario B, at the 8% rate, the value of the commercial margin is 39 × 11% − 27 × 5% = 4.29 − 1.35 = 2.94, and the net margin, after funding costs, is 2.94 − 0.96 = 1.98. The margins are different between scenarios A and B because the volumes of assets and liabilities generating these margins are different (Table 18.3). TABLE 18.3 scenario 1

Interest margins: interest rate

Interest rate gap (banking portfolio) Initial commercial margina Liquidity gap (a) Cost of funding (a) × 8% Net interest margin after funding a The

A

B

+8 2.76 +12 0.96 1.80

+4 2.94 +12 0.96 1.98

initial margins correspond to an interest rate of 8%.

When the interest rate changes, the variations of margins result from the interest rate gaps. The cost of funding depends on the hedging solution. The fraction of the total funding whose rate is locked in through swaps defines a hedging solution.

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Hedging Scenarios There are two hedging scenarios. The first solution locks in the interest rate for a debt of 4. The remaining debt, 12 − 4 = 8, required to bridge the liquidity gap is a floating rate debt. The second solution locks in the interest rate for an amount of 8, the remaining 4 being floating rate debt. These are the H1 and H2 hedging solutions. It is possible to think of the first hedging scenario as the existing hedge in place and of H2 as an alternative solution (Table 18.4). TABLE 18.4

Funding scenarios

Scenario Interest rate gap (banking portfolio) Liquidity gap Hedginga Hedging H1 Hedging H2 Interest rate gap after hedgingb Hedging H1 Hedging H2

A

B

+8 −12

+8 −12

8vr + 4fr 4vr + 8fr 0 +4

+4 0

a The

hedging solution is the fraction of total funding whose rate is locked in. ‘fr’ and ‘vr’ designate respectively the fractions of debt with fixed rate and variable rate. b The interest rate gap is that of the banking portfolio less the floating rate debt.

The interest rate gap after funding is the difference between the commercial portfolio gap and the gap resulting from the funding solution. This gap is simply the floating rate debt with a minus sign. With the hedging solution H1, the floating rate debt is 8. The interest rate gap post-funding is therefore +8 − 8 = 0. In scenario B, the solution H1 results in an interest rate gap post-funding of +8 − 4 = +4.

The Matrices of Net Interest Margins The banking portfolio margins and the net interest margins derive from the initial values of margins using the gaps and the variation of the interest rate. For scenario A, the initial value of the net margin, with interest of 8%, is 1.80. For scenario B, the net margin with initial interest rate of 8% is 1.98. The margin variation with a change in interest rates of i = +3% is: (Net interest margin) = gap × i It is easier to derive the matrix of net margins from initial margins and gaps than to recalculate all margins directly. A first matrix cross-tabulates the business scenarios A and B with the interest rate values of 8% and 11%, given the hedging solution H1. The second matrix uses the hedging solution H2. For instance, with the business scenario A and the hedging solution H1, both the initial net margin and the final margin, after an interest rate increase of 3%, are equal. With the hedging solution H2 and the scenario B, the initial margin is 1.98. Since the interest rate gap after hedging is −4, the final margin is 1.98 − 4 × 3% = 1.86 when the rate increases by 3% (Table 18.5).

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TABLE 18.5

The net margin matrix after hedging

Business scenario

A

B

Fixed rate amount: 4 Net margin Rate 1: 8% Rate 2: 11% Business scenario

Hedging H1 1.80 + 0 × i 1.80 1.80 A

Hedging H2 1.80 − 4 × i 1.98 1.86 B

Fixed rate amount: 8 Net margin Rate 1: 8% Rate 2: 11%

Hedging H1 1.80 + 4 × i 1.80 1.92

Hedging H2 1.98 + 0 × i 1.98 1.98

In order to define the best hedging solutions, we need to compare the risk–return combinations.

RISK–RETURN COMBINATIONS First, the risk–return profiles generated by the two hedging solutions are calculated. By comparing them, we see how several funding and hedging solutions compare, and how to optimize the solution given a target risk level. For each matrix, the average value of the margins and the volatility2 across the cells of each matrix are calculated. The Sharpe ratio, the ratio of the expected margin to the margin volatility, is also calculated. This ratio is an example of a risk-adjusted measure of profitability3 . The results are as given in Table 18.6. TABLE 18.6 Net combinations Hedging H1 Rate 1 Rate 2 Hedging H2 Rate 1 Rate 2

A 1.80 1.80 A 1.80 1.92

margin B 1.98 1.86 B 1.98 1.98

matrix

and

risk–return

Mean Volatility Mean/volatility Mean Volatility Mean/volatility

1.860 0.085 21.920 1.920 0.085 22.627

The first solution generates a lower average margin with the same risk. The second generates a higher margin, with the same risk. The best solution is therefore H2. The Sharpe ratio is greater with H2. 2 The

volatility is the square root of the sum of the squared deviations of the margin from the mean. All values in the matrix have the same probability. The expected value is the arithmetic mean. 3 The Sharpe ratio is a convenient measure of the risk-adjusted performance of a portfolio. It also serves when modelling the credit risk of the portfolio.

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In general, a large number of hedging solutions are considered. When graphed in the risk–return space, they generate a cloud of points4 . Each dot summarizes a matrix combining several target variable values corresponding to each pair of interest rate scenarios and business scenarios. In spite of the high number of simulations, changing hedging scenarios simply moves the dots in the risk–return space. Some solutions appear inefficient immediately. They are those with an expected return lower than others, at the same risk level or, alternatively, those with the same expected return as others, but with a higher risk. These solutions are inefficient since others dominate them. The only solutions to consider are those that dominate others. The set of these solutions is the ‘efficient frontier’ (Figure 18.3). Increasing return and constant risk

Expected margin Efficient frontier

With new hedge H2 Without new hedge H1

Decreasing risk, constant return Minimum risk solution Margin volatility

FIGURE 18.3

Risk–return combinations

A risk–return combination is efficient if there is no other better solution at any given level of risk or return. The efficiency criterion leads to several combinations rather than a single one. The optimization problem is as follows: • Minimize the volatility of the margin subject to the constraint of a constant return. • Maximize the expected margin subject to the constraint of a constant risk. For each level of profitability, or of return, there is an optimal solution. When the risk, or the return, varies, the optimum solution moves along the efficient frontier. In order to choose a solution, a risk level has to be set first. For instance, the minimum risk leads to the combination located at the left and on the efficient frontier. When both interest rate and business risks interact, solutions neutralizing the margin risk do not exist. Only a minimum risk solution exists.

4 The

graph shows the risk–return profiles in a general case. In the example, the only variable that changes the risk–return combination is the gap after funding. When this gap varies continuously, the risk–return combinations move along a curve. The upper ‘leg’ of this curve is the efficient frontier of this example.

19 ALM ‘Risk and Return’ Reporting and Policy

The Asset–Liability Management (ALM) reporting system should provide all elements for decision-making purposes to the ALM Committee (ALCO). The reporting should provide answers to such essential questions as: Why did the interest income vary in the last period? Which interest income drivers caused the variations? Funding, investment and hedging issues necessitate reporting along the two basic financial dimensions, earnings and risks. To move back and forth from a financial perspective, risk and return, to a business policy view, it is also necessary to break down risk and performance measures by transaction, product family, market segment and business line. ALM reporting links gaps, interest incomes and values of transactions to business units, products, markets and business lines. Reports slice gaps, incomes and gaps along these dimensions. To analyse the interest income variations, ALM reports should break down the variations due to the interest income drivers: changes in the structure of the existing by level of interest earned or paid, due to both amortization effects plus new business; changes of the yield curve shape. Moving back and forth from financial to business dimensions, plus the necessity of relating interest margin changes to identified drivers, creates high demands on the ALM information system, which should record all business and financial data. This chapter provides examples of reporting related to common ALCO issues. The first section lists the typical ALCO agenda for a commercial bank. The second section describes the reporting system specifications. The third and fourth sections illustrate breakdowns of gaps and interest incomes by product family and market segment. The last section focuses on explaining interest income variations.

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ALM MISSIONS AND AGENDA Table 19.1 lists the basic missions and agenda of ALCO. The next subsections discuss them further. TABLE 19.1

ALM scope and missions

Liquidity management

Funding deficits Investing excess funds Liquidity ratios

Measuring and controlling interest rate risk Recommendations off-balance sheet

Gaps and NPV reporting Hedging policy and instruments Hedging programmes

Recommendations on-balance sheet

Flows: new business Existing portfolio

Transfer pricing system

Economic prices and benchmarks Pricing benchmarks

Preparing the ALCO

Recent history, variance analysis (projections versus realizations) GAP and NPV reports ‘What if’ analyses and simulations . . .

Risk-adjusted pricing

Mark-up to transfer prices (from credit risk allocation system) Mispricing: gaps target prices versus effective prices

Reporting to general management

The ALCO Typical Agenda and Issues Without discussing further some obvious items of the list in Table 19.1, we detail some representative issues that need ALCO attention. Historical Analysis

Typical questions include the following: • How do volumes, margins and fees behave? • How do previous projections differ from realizations, in terms of interest margins, fees and volumes? • What explains the actual–projected volume gaps, based on actual and projected interest rate gaps, and the product mix changes?

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A review of commercial actions and their results is necessary to see which corrective actions or new actions are required. These involve promotions and incentives for customers, or all development policies in terms of product mix. Without this historical, or ex post, analysis, it remains difficult to look forward. On-balance Sheet Actions

There is an important distinction between existing portfolios, which generate the bulk of revenues, and new business for future developments. Actions on the volume–product mix and pricing have an impact on new business. All future actions need analysis both in business terms and in risk–return terms. New development policies require projections of risk exposure and revenue levels. Conversely, interest rate and liquidity projections help define new business policies. This is a two-way interaction. Interest rate risk and liquidity risk are not the only factors influencing these decisions. However, the new transactions typically represent only a small fraction of the total balance sheet. Other actions on the existing portfolio might be more effective. They include commercial actions, such as incentives for customers to convert some products into others, for example fixed to floating rate loans and vice versa, or increasing the service ‘intensity’, i.e. the number of services, per customer. Off-balance Sheet Actions

Off-balance sheet actions are, mainly, hedging policies through derivatives and setting up a hedging programme for both the short and long-term. Hedges crystallize in the revenues the current market conditions, so that there are no substitutes to ‘on-balance sheet’ actions. In low interest rate environments, forward hedges (swaps) embed current low rates in the future revenues. In high interest rate environments, high interest rate volatility becomes the issue, rather than the interest rate level. The hedging policy arbitrages between revenues level, interest rate risk volatility and the costs of hedging (both opportunity costs and direct costs). The financial policy relates to funding, hedging and investing. However, it depends on business policy guidelines. Defining and revising the hedging programme depends on expectations with respect to interest rates, on the trade-off between hedging risk versus level of revenues, and on the ‘risk appetite’ of the bank, some willing to neutralize risk and others betting on favourable market movements to various degrees.

OVERVIEW OF THE ALM REPORTING SYSTEM The ALM reporting system moves from the ALM risk data warehouse down to final reports, some of them being purely financial and others cross-tabulating risk and return with business dimensions. The business perspective requires in fact any type of breakdown of the bank aggregates, by product, market segment and business unit. In addition, reporting the sources of gaps, along any of these relevant dimensions, raises the issue of what contributes to risk and return. Therefore, the system should provide functionalities to ‘drill down’ into aggregates to zoom in on the detailed data. Moreover, front-ends should include ‘what if’ and simulation capabilities, notably for decision-making purposes (Figure 19.1).

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Front-ends and Reporting

ALM Data Warehouse Products & Markets

Aid to Decisionmaking: Pricing Gaps, Simulations & ALM Reports

Volumes Balance Amortization Renewal & expected usage Implicit options Currency ...

Actual / Projected Variance Analysis Mispricing to Benchmarks

Reference rate Rate level, guaranteed historical rate, historical target rate Nature of rate (calculation mode). Interest rate profile Sensitivity to reference rates Options characteristics ...

FIGURE 19.1

OLAP Analysis

Rates

Business Business Reports: Reports: Markets & Markets & Products: Products: 'Risk−Return' "Risk - Return" ...

ALM data warehouse, front-ends and reporting

BREAKING DOWN GAPS BY SEGMENTS In this section, we illustrate the breakdown of gaps and ‘all-in revenues’ (interest plus fees) along the product and market dimensions. We show gaps and volumes first, and gaps and volumes in conjunction with ‘all-in margins’ in a second set of figures. All sample charts focus on risks measured by gaps. Since gaps are differences in volumes of assets and liabilities, there is no difficulty in splitting these aggregates into specific product or market segment contributions measured by their outstanding balances of assets and liabilities. The time profile of existing assets and liabilities is the static gap profile. Figure 19.2 shows the declining outstanding balances of assets and liabilities. The static gap, measured by assets minus liabilities, tends to be positive in the first periods, which implies a need for new funding. The picture is very different from the dynamic profiles because of the volume of new transactions during each of the first three periods. Figure 19.3 shows total asset and liability values, together with the liquidity gaps and the variable interest rate gaps. Since the total volumes of assets and liabilities are used, these gaps are dynamic. The positive liquidity gaps represent a permanent need for new funds at each period, and the negative variable interest rate gaps show that the bank interest income is adversely influenced by rising interest rates. The dynamic time profile of gaps increases up to period 3 and then declines because we assume that no projections are feasible beyond

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1500 Existing liabilities (total) 1000

Existing assets (total)

Amounts

500 0 1

2

3

4

5

6

−500 −1000 Time

FIGURE 19.2

Static gaps time profile 2000

Amounts & Gaps

1500 1000 500 0 −500 −1000 1

2

3

4

5

6

−1500 Time Total Assets

FIGURE 19.3

Total Liabilities

Interest Rate Gap

Assets and liabilities and the liquidity and variable interest rate gaps

period 3. After period 3, the time profile becomes static and total assets and liabilities decline.

Breakdown of Dynamic Gaps by Segments Figures 19.4 and 19.5 show the dynamic time profiles of assets and liabilities and their breakdown into six hypothetical segments, at each date. Bars represent volumes of assets or liabilities. The differences are the gaps. Each bar is broken down into the six segments of interest. The bar charts show which assets and liabilities contribute more or less to the gap. This type of reporting links the gap and asset and liability time profiles to the commercial policy. For instance, we see that the top segment of each time bar is among the highest volumes of assets contributing to the positive liquidity gap, or the need for

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2000 1500 Assets

Amount

1000 500 0

Time

−500 −1000

1

2

−1500

FIGURE 19.4

3 Liabilities

4

5

6 Bars broken down by product−market segments

Breakdown of the liquidity gaps by segments 2000 Bars broken down by product−market segments

Assets

1500

Amount

1000 500 0 −500 −1000 −1500 −2000

1 2

3

4

5

6

Liabilities

Time Liquidity Gap New Liabilities (cum.) Variable Rate Existing Liabilities Fixed Rate Existing Liabilities New Assets (cum.) Variable Rate Existing Assets Fixed Rate Existing Assets

FIGURE 19.5

Breakdown of interest rate gaps and volumes by segments

new funds. Some assets and liabilities are variable rate and others fixed rate. Figure 19.5 shows a similar breakdown using the type of rate, fixed or variable. These figures are not sufficient, however, since we have only the contributions to gaps without the levels of margins.

BREAKDOWN OF REVENUES AND GAPS, AND THE PRODUCTS–MARKETS MIX The revenues might be: • All-in margins, meaning that they combine various sources of revenues.

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• Pure interest rate spreads, such as customers’ rate minus any internal transfer rate used as the reference for defining commercial margins. • Pre-operating expenses, and in some cases, post-direct operating expenses.

Breakdown of Interest Income by Segments Figure 19.6 shows the contributions to the ‘all-in margin’ (interest margin and fees) of the different types of products or markets across time. The vertical axis shows the all-in revenue divided by the asset volume, and the horizontal axis shows time. The profitability measure is a ‘Return On Assets’ (ROA). This is not a risk-adjusted measure, since that would require capital allocation. In this case, the total cumulated margin, as a percentage, seems more or less constant at 10%. Hence, the interest incomes in dollars are increasing with volume since the percentage remains constant while volumes grow in the first 3 years. 12%

All-in Margin

10% 8% 6% 4%

Bars broken down by product−market segments

2% 0% 1

2

3

4

5

6

Time

FIGURE 19.6

Breakdown of percentages of ‘all-in margins’ by segments

This chart shows only interest income contributions, without comparing them with contributions to gaps. Therefore, it is incomplete.

Contributions of Segments to Revenues and Gaps Figure 19.7 compares both interest income and volumes. The chart brings together contributions to gaps and margins in percentages (of volumes) at a given date. The six segments of the example correspond to the six bars along the horizontal axis. The comparison shows contrasting combinations. Low volume and low margin for the fourth segment make it a loser (relatively), high volume and high margin for the first segment make it a winner (relatively).

‘Gap Risk’–Return Profiles A common financial view plots volumes (gap risk) against incomes (in value) in the same chart. The traditional business view suggests that more volume with high income is

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3.5%

Volumes ($)

300

3.0%

250 Volume & Margin

Margins (%)

2.5%

200

2.0% 150 1.5% 100

1.0%

50

0.5%

0

0.0% Segments

FIGURE 19.7

Comparing margins and volumes by products and market lines

better, and conversely. Under this view, the volume–income space is split into four basic quadrants, and the game is to try to reach the area above the diagonal making volume at least in line with margins: more volume with higher margins and less volume with lower incomes. An alternative view highlights volumes as contributions to gaps. It turns out that this is the same as the risk–income view when volume measures risk, since volume is the contribution to gaps. Hence, the two views are consistent. In Figure 19.8, each dot represents one of the six segments. Some contribute little to gaps and have higher income, 3.5% 3.0%

Higher income

100

Return On Asset

2.5% 2.0%

200

1.5%

200

100

1.0%

150

Lower 'risk'

250

0.5% 0.0% 0

50

100

150

200

250

300

Volume (Value)

FIGURE 19.8

Margins versus volumes (contribution to liquidity and interest rate risks)

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some contribute to both with similar magnitude, and others contribute to gaps and little to income. ‘Losers’ are the segments combining high volumes with low incomes. The low volume–high income segments are dilemmas, since it might be worth taking on more risk for these. The first diagonal dots are acceptable, with the ‘winners’ being the high volume–high income combinations. Another general comment is that the view on risks is truncated. There is not yet any adjustment for credit risk in incomes since gap risk is not credit risk. Hence, credit risk versus income, or the view of Risk-adjusted Return on Capital (RaRoC), provides a global view on the complete risk–return profiles. The conclusion is that the reports of this chapter provide useful views on the sources of gap risk, but they do not serve the purpose of optimizing the portfolio considering all risks. This issue relates more to the credit risk and to the related RaRoC of segments or transactions.

ANALYSIS OF THE VARIATIONS OF INTEREST INCOME Often, it is necessary to explain the changes in the interest margin between two dates. In addition, with projections and budgets, there are deviations between actual interest income and projected values from the previous projections. The sources of variations of interest income are: the change in portfolio structure by level of interest rate; the new business of the elapsed period; the changes in the yield curve. • The amortization of the existing portfolio creates variations of interest income because assets and liabilities earn or pay different interest rates. When they amortize the corresponding revenues or costs disappear. This effect is unrelated to the variations of interest rates. Tracking this variation between any two dates requires having all information on flows resulting from amortization and on the level of rates of each individual asset and liability. To isolate this effect, it is necessary to track the portfolio structure by interest rate level. • The new business creates variations as well, both because of the pure volume effect on income and costs and because of the new rates of these assets and liabilities. New assets and liabilities substitute progressively for those that amortize, but they earn or pay different rates. • The changes in the yield curve create variations of the interest income independent of volume variations of assets and liabilities. Note that the gap provides only the change due to a parallel shift of the yield curve, while commercial banks are sensitive to the slope of the yield curve because of the maturity gaps between interest rates of liabilities and assets. Moreover, the level of interest income depends on the portfolio structure by products and market segments since each of these generates different levels of spread. It becomes necessary to trace back such deviations to the actual versus projected differences between all interest margin drivers, volumes, interest rates and commercial percentage margins. This

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allows us to allocate the predicted–actual margin variance to errors in past interest expectations and to the actual–predicted variances between volumes and percentage margins. This analysis is required to understand the changes in interest margins between two ALM committees. Corrective actions depend on which drivers explain the variations of the interest margin. Using gaps to explain the changes in interest income is not sufficient in general. First, gaps show the sensitivity of the interest income due to a parallel shift of the interest rate used as reference. If we use a unique gap, we ignore non-parallel shifts of the yield curve. With as many gaps as there are interest rates, it is feasible to track the change in interest income due to changes of all interest rates used as references. If we assume no portfolio structure change between two dates, implying that the interest margin as a percentage is independent of volume, the change in interest income between any two dates depends on the shifts of interest rates, the change in volumes and the interaction effect of volume–interest rate. This is the usual problem of explaining variances of costs or revenues, resulting from variances of prices, volumes and product mix. In the case of interest income, there is variation due to the pure volume change, the gap effect due to the variations of the interest rates between the two dates, and an interaction effect. The interaction effect when both price (interest rate) and volume vary results from the simplified equation using a single interest rate: IM = [i × (A − L)] = (A − L) × i + i × (A − L) + i × (A − L) In the equation, IM is the interest margin, (A − L) is the difference between interest rate-sensitive assets and liabilities, or the variable rate gap, and i is the unique interest rate driver. The interaction term results from the change in product between the gap and the interest rate. Table 19.2 shows how these effects combine when there is no portfolio structure effect. TABLE 19.2 The analysis of volume and price effects on the variation of interest margin Actual–predicted variance IM = IM1 − IM0 IM due to interest rate variations at constant margins and volume IM due to volume variations at constant interest rate and margins IM due to price variations (interest rate and margins) combined with volume variations (interaction effect) IM

2.80 −0.90 3.00 0.70

2.80

Note that the calculations depend on the numerical values of the start and end values of rates and volumes. In this example, the 2.80 increase of margin is due to: 1. A decrease of −0.90 due to a negative variable rate interest gap at initial date combined with an interest rate increase. 2. An increase of +3.00 due to the expansion of assets and liabilities between initial and final dates.

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3. An increase of +0.70 due to the increased customer rates of new assets and liabilities, resulting from the increase in interest rates and from any variation of commercial margins. In general, there is an additional effect portfolio structure, depending on the amortization of existing assets and liabilities plus the new business. For instance, there are only two assets and a single debt. The two assets earn the rates 10% and 12%, while the debt pays 8%. Total assets are 100, divided into two assets of 50 each, and the debt is 100. At the end of period 1 (between dates 0 and 1), the asset earning 12% amortizes and the bank originates a new loan earning 11%. During period 1, the revenue is 50 × 10% + 50 × 12% − 100 × 8% = 0.050 + 0.060 − 0.080 = 0.030. Even if there is no change in interest rate and overall volume, the interest income of the next period 2 changes due to the substitution of the old loan by the new loan. The revenue becomes 50 × 10% + 50 × 11% − 100 × 8% = 0.050 + 0.055 − 0.080 = 0.025. This change is due to the change in portfolio structure by level of interest rates. This change might be independent of interest rate change if, for instance, the new loan earns less than the first one due to increased competition between lenders.

SECTION 7 Options and Convexity Risk in Banking

20 Implicit Options Risk

This chapter explains the nature of embedded, or implicit, options and details their payoff, in the event of exercise for the individual borrower. Implicit options exist essentially in retail banking and for individuals. A typical example of an implicit option is the facility of prepaying, or renegotiating, a fixed rate loan when rates decline. The borrower’s benefit is the interest costs saving, increasing with the gap between current rates and the loan fixed rate, and with the residual maturity of the loan. The cost for the bank is the mirror image of the borrower’s benefit. The bank’s margin over the fixed rate of debt matching such loans narrows, or eventually turns out negative. Other banking options relate to variable rate loans with caps on interest rates, or to transfers of resources from demand deposits to interest-earning products, when interest rates increase, raising the financial cost to the bank. Embedded options in banking balance sheets raise a number of issues: What is the cost for the lender and the pricing mark-up that would compensate this cost? What is the risk–return profile of implicit options for the lender? What is the portfolio downside risk for options, or downside ‘convexity risk’? How can banks hedge such risks? Because the outstanding balances of loans with embedded options are quite large, several models based on historical data help in assessing the prepayment rates of borrowers. Prepayment models make the option exercise a function of the differential between the rate of loans and the prevailing rates, and of demographic and behavioural factors, which also influence the attitude of individuals. From such prepayments, the payoffs for borrowers and the costs for lenders follow. Payoffs under exercise differ from the value of the option. The value of the option is higher than the payoff as long as there is a chance that future payoffs increase beyond their current value if rates drift further away. This chapter assesses the benefit to the borrower, calculated as the payoff of the option under various interest rate levels. For a long-term fixed rate loan, the option payoff is the time profile of the differential savings (annuities) after and before exercising the

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renegotiation option. Alternatively, it is the present value of these cash savings at the date of renegotiation. The payoff increases with the differential between the initial and the new interest rates and the residual maturity of the loan. The next chapter addresses the valuation of options based on the simulation of representative interest rate scenarios. The first section details the nature of options embedded in various banking and insurance contracts. Subsequent sections illustrate prepayment risk. The second section refers briefly to prepayment models. The last section calculates the payoff of the option under various levels of interest rates, using a simple fixed rate loan as an example.

OPTIONAL RISK: TWO EXAMPLES The most well known options are the prepayment options embedded in mortgage loans. These represent a large amount of the outstanding balances, and their maturity is usually very long. The prepayment option is quite valuable for fixed rate loans because it usually has a very long horizon and significant chances of being in-the-money during the life of the loan. There are other examples of embedded options. Guaranteed rate insurance contracts are similar for insurance companies, except that they are liabilities instead of assets. The prepayment option has more value (for a fixed rate loan) when the interest rate decreases. Prepaying the loan and contracting a new loan at a lower rate, eventually with the same bank (renegotiation), might be valuable to the customer. The prepayment usually generates a cost to the borrower, but that cost can be lower than the expected gain from exercise. For instance, a prepayment penalty of 3% of the outstanding balance might be imposed on the borrower. Nevertheless, if the rate declines substantially, this cost does not offset the gain of borrowing at a lower rate. Prepayments are an issue for the lender if fixed rate borrowings back the existing loans. Initially, there is no interest rate risk and the loan margin remains the original one. However, if the rates decline, customers might substitute a new loan for the old one at a lower price, while the debt stays at the same rate, thereby deteriorating the margin. For insurance contracts, the issue is symmetrical. The insurance companies provide guaranteed return contracts. In these contracts, the customer benefits from a minimum rate of return over the life of the contract, plus the benefit of a higher return if interest rates increase. In order to have that return, the insurance company invests at a fixed rate to match the interest cost of the contract. If the rate increases, the beneficiary can renew the contract in order to obtain a higher rate. Nevertheless, the insurance company still earns the fixed rate investment in its balance sheet. Hence, the margin declines. The obvious solution to prepayments of fixed rate loans is to make variable rate loans. However, these do not offer much protection against a rise in interest rates for the borrower, unless covenants are included to cap the rate. Therefore, fixed rate loans or variable rate loans with a cap on the borrower’s rate are still common in many countries. A theoretical way of limiting losses would be to make the customer pay for the option. This implies valuing the option and that competition allows pricing it to customers. Another way would be to hedge the option risk, and pay the cost of hedging. Caps and floors are adapted for such protections. The lender can hedge the risk of a decrease in the rate of the loan by using a floor that guarantees a minimum fixed return, even though the rate of the new loan is lower. An insurance company can use a cap to set a maximum value on the rate guaranteed to customers in the case of a rise in interest rates. However,

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such hedges have a cost for banks. The underlying issue is to determine the value that customers should pay to compensate the additional cost of options for banks.

MODELLING PREPAYMENTS The modelling of prepayments is necessary to project the gap profiles. There is a significant difference between contractual flows and effective maturity. The purpose of models is to relate prepayment rates to those factors that influence them1 . Prepayment models can also help to value the option. If such models capture the relationship between prepayments and interest rate, it becomes easier to value options by simulating a large number of prepayment scenarios. When prepayments are considered, the expected return of a loan will differ from the original return due to the rollover of the loan at a lower rate at future dates. Models can help to determine the amount subject to renewal, and the actual cost of these loans, as a function of interest scenarios. If multiple scenarios are used, the expected return of the loan is the average over many possible outcomes. The valuation of options follows such lines, except that models are not always available or accurate enough. Models specify how the prepayment rate, the ratio of prepayment to outstanding balances, changes over time, based on several factors. The simplest model is the Constant Prepayment Rate (CPR) model that simply states that the prepayment at any date is the product of a prepayment rate with the outstanding balances of loans (Figure 20.1). The prepayment rate usually depends on the age of the loans. During the early stages of the loans, the prepayment rate increases. Then, it tends to have a more or less stable value for mature loans. Modifying the model parameters to fit the specifics of a loan portfolio is feasible. The simple prepayment rate model captures the basic features of the renegotiation behaviour: renegotiation does not occur in the early stage of a loan because the rates do not drift suddenly away and because the rate negotiation is still recent. When the loan gets closer to maturity, the renegotiation has fewer benefits for the borrower because savings occur only in the residual time to maturity. Therefore, renegotiation rates increase steadily, reach a level stage and then stay there or eventually level off at periods close to maturity. Hence, the models highlight the role of ageing of loans, as a major input of Prepayment rate

First period

FIGURE 20.1 1 See,

Time

‘Constant Prepayment Rate’ models

for example, Bartlett (1989) for a review of prepayment issues and models serving for securitizations of residential mortgages.

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prepayment models, in addition to the rate differentials. This makes it necessary to isolate cohorts of loans, defined by period of origination, to capture the ‘seasoning effect’, or ‘ageing effect’, as well as the rate differential, which varies across cohorts. Other models use such factors as the level of interest rate, the time path of interest rates, the ageing of the loan, the economic conditions and the specifics of customers. These factors serve to fit the modelled prepayment rates to the observed ones. However, such models require a sufficient amount of historical data. An additional complexity is that the time path of interest rates is relevant for options. If the rates decline substantially at the early stage of a loan, they trigger an early prepayment. If the decline occurs later, the prepayment will also occur later. Therefore, the current level of interest rates is not sufficient to capture timing effects. The entire path of interest rates is relevant. This is one reason why the prepayment rate does not relate easily to the current interest level, as intuition might suggest. The valuation of the option uses models of the time path of rates and of the optimum prepayment or renegotiation behaviour (see Chapter 21).

GAINS AND LOSSES FROM THE PREPAYMENT OPTION In this section, an example serves to make explicit the gains for the borrower and the losses for the lender. The gain is valued under the assumption of immediate exercise once the option is in-the-money. In reality, the option is not exercised as soon as it becomes in-the-money. The exercise behaviour depends on future expected gains, which can be higher or lower than the current gain from immediate exercise. This is the time value of the option. The next chapter captures this feature using a valuation methodology of options. The example used is that of a fixed rate and amortizing loan, repaid with constant annuities including interest and capital. The customer renews the loan at a new rate if he exercises the prepayment option. The loan might be representative of a generation of loan portfolios. In actual portfolios, the behaviour of different generations of loans (differing by age) will differ. The gain from prepayment results from the characteristics of the new loan. The new loan has a maturity equal to the residual maturity of the original loan at the exercise date. The decision rule is to prepay and renew the debt as soon as the combined operation ‘prepayment plus new loan’ becomes profitable for the borrower. If the borrower repays the loan, he might have penalty costs, for instance 3% of the outstanding balance. The gain from prepayment results from the difference between the annuities of the original and the new loan plus the penalty cost. It spreads over the entire time profile of differential annuities between the old and the new loan until maturity. Either we measure these differences or we calculate their present value. The original loan has a 12% fixed rate, with an amount of 1000, and with a maturity of 8 years. Table 20.1 details the repayment schedule of the loan. The repayment or renegotiation occurs at date 4, when the interest rate decreases to 10%. The new annuities result from the equality between their present value, calculated as of date 4, and the amount borrowed. The amount of the new loan is the outstanding balance at date 4 of the old loan plus the penalty, which is 3% of this principal balance. It is identical to the present value of the new annuities calculated at the date of renewal at the 10% rate until maturity: Present value of future annuities at t = 4, at 10% = outstanding balance at date 4 × (1 + 3%)

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TABLE 20.1

251

Repayment schedule of the original loan

Amount 1000 Date

0

Maturity 8 years 1

Discount rate 12% 3 4 5

2

6

Annuity 201.30 7

8

a

PV at 12% 1000 Annuity (rounded) 201.30 201.30 201.30 201.30 201.30 201.30 201.30 201.30 Principal repayment 81.30 91.06 101.99 114.22 127.93 143.28 160.48 179.73 Outstanding balance 918.70 827.64 725.65 611.43 483.50 340.21 179.73 0.00 Discount rateb 12.00% a PV

is the present value of future annuities. discount rate is the rate that makes the present value of future annuities equal to the amount borrowed. It is equal in this case to the nominal rate (12%) used to calculate interest payments. b The

The gain for the borrower is the present value, as of date 4, of the difference between the old and new annuities. The present value of the old annuities at the new rate is the market value of the outstanding debt at date 4. The value of the old debt becomes higher than the outstanding balance if the rate decreases. This change is a gain for the lender and a loss for the borrower. The economic gain from prepayment, for the borrower, as of the rate reset date T , is equal to the difference between the present values of the old and the new debts: PVT ,new rate (old debt) − PVT ,new rate (new debt) = PVT ,new rate (old debt) − outstanding balance × (1 + 3%) PVT ,new rate is the present value at the prepayment date (T = 4) and at the new rate (10%). The calculation of both market values of debt and annuities is shown in Table 20.2. The new annuity is 198.75, lower than the former annuity of 201.30, in spite of the additional TABLE 20.2

Gains of prepayment for the borrower

Amount Initial rate Original maturity in years Annuity Schedule Date

1000 12.00% 8 201 1

2

Principal repayment 81.30 91.06 Outstanding balance 918.70 827.64 Penalty (3%) 0 0 New debt 0 0 Old annuity 201.30 201.30 New annuity 0 0 Annuity profile 201.30 201.30 Differential flows (new–old) 0 0 Gain for the borrower Present value at date 4 at the 10% rate PV old debt PV new date Present value of gain at 4

Penalty Date of prepayment New rate

3.00% 4 10.00%

3

4

5

6

7

8

101.99 725.65 0 0 201.30 0 201.30 0

114.22 611.43 18.34 629.77 201.30 0 201.30 0

127.93 483.50 0 0 201.30 198.75 198.75 2.63

143.28 340.21 0 0 201.30 198.75 198.75 2.63

160.48 179.73 0 0 201.30 198.75 198.75 2.63

179.73 0.00 0 0 201.30 198.75 198.75 2.63

638.10 629.77 8.33

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cost due to the 3% penalty. The annual gain in value is 2.63. The present value of this gain, at date 4, and at the new rate (10%), is 8.33. The borrower gain increases when the new rate decreases. Above 12%, there is no gain. Below 12%, the gain depends on the difference between the penalty in case of prepayment and the differential gain between the old and the new loans. The gain can be valued at various levels of the new interest rate (Table 20.3). Table 20.3 gives the value of the old debt at the current new rate. The outstanding balance at date 4 is 611.43. The market value of the debt is equal to 611.43 only when the rate is 12%. The exercise price of the option is equal to the prepayment amount, which is the outstanding balance plus the 3% penalty, or 611.43(1 + 3%) = 629.77, rounded to 630. This price is determined by the prepayment date since the amortization schedule is contractual. The present value of gains, at date 4, is the difference between the market value of the old debt and 630, which is the present value of the new debt. The exercise value of the option is the maximum of 0 and the market value of the old debt minus 630. It is given in the last column of Table 20.3. TABLE 20.3 New rate

Borrower gains from prepayment at date 4 PV(old debt) at 4 12% (a)

PV(new debt) at 4 (b)

698 682 667 652 638 625 611 599 587

630 630 630 630 630 630 630 630 630

6% 7% 8% 9% 10% 11% 12% 13% 14%

PV of gains at 4 (c = b − a)

Exercise value of the option at 4 (d = max[0, c])

68 52 37 22 8 −5 −18 −31 −43

68 52 37 22 8 0 0 0 0

Gain from the option 80 60 40 20 0 6%

FIGURE 20.2

7%

8%

9%

10% 11% 12% 13% 14% Interest rate

Payoff profile of the prepayment option

The borrower gains are higher when the interest rate decreases. If the rate increases, the option has a zero exercise value. The profile of payoffs as a function of interest rate has the usual shape of the payoff profile of an option. Positive gains appear when the decline in interest rate below 12% is sufficient to offset the penalty (Figure 20.2).

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The same profile is the cost of the exercise of the option for the lender. This cost becomes very important when the decline in rates is significant. If the rate moves down to 8%, the present value of the loss for the bank reaches 37. This is 3.7% of the original capital, or 37/667 = 5.5% of the outstanding capital at date 4. This example is simplified, but it demonstrates the importance of valuing options. The next chapter provides an example of the valuation of a prepayment option.

21 The Value of Implicit Options

The value of an option combines its liquidation value, or the payoff under exercise, plus the value of waiting further for larger payoffs. Valuing an option requires simulating all future outcomes for interest rates, at various periods, to determine when the option is in-the-money and what are the gains under exercise. The option value discounts the expected future gains, using the simulated rates as discount rates. Various interest rate models potentially apply to the valuation of options1 . The purpose of this chapter is to illustrate how they apply for valuing prepayment options. The value of options depends on the entire time path of the interest rates from origination to maturity, because interest rate changes can trigger early or late exercise of the option. This is a significant departure from classical balance sheet Asset–Liability Management (ALM) models, which use in general a small number of scenarios. The valuation of options necessitates simulating the future time paths of interest rates and deriving all future corresponding payoffs. The interest simulation technique applies because it generates the entire spectrum of payoffs of the option up to a certain horizon, including all intermediate values. The valuation is an average of all future payoffs weighted by probabilities of occurrences. In this chapter, we use the simple technique based on ‘binomial trees’. Other examples of time path simulations are given in Section 11 of this book. From a risk perspective, the issue is to find the expected and worst-case values of the option from the distribution of its future values at various time points. From a pricing perspective, the option is an asset for the borrower and it generates an expected cost for the bank, which the bank should charge to borrowers. The worst-case value makes sense in a ‘Value at Risk’ (VaR) framework, when considering worst-case losses for determining 1 The

bibliography is rather exhaustive and technical. A good introduction to the valuation of options is Hull (2000).

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economic capital. The sections on Net Present Value (NPV) and optional risk develop this view further. The expected value of options relates to pricing. Pricing optional risk to borrowers requires converting a single present value as of today into a percentage mark-up. The mark-up reflects the difference in values between a straight fixed rate debt, that the borrower cannot repay2 , and a ‘callable debt’, which is a straight debt combined with the prepayment option. The callable loan is an asset for the lender that has a lower value than the straight debt for the bank, the difference being the option value. Conversely, the callable loan is a liability for the borrower that has a lower value for him because he can sell it back to the lender (he has a put option). The percentage mark-up is the ‘Option-Adjusted Spread’ (OAS). The OAS is the spread added to the loan rate making the value of the ‘callable’ loan identical to the value of a straight loan. The callable debt, viewed as an asset held by the bank, has a lower value than the straight debt for the lender. To bring the value of the straight debt in line with the value of the debt minus the value of the option, it is necessary to increase the discount rates applied to the future cash flows of the debt. The additional spread is the OAS. For pricing the option, banks add the OAS to the customer rate, if competition allows. The first section summarizes some basic principles and findings on option valuation. The second section uses the binomial tree technique to simulate the time path of interest rates. The third section derives the value of the prepayment option from the simulated ‘tree’ of interest rate values. The last section calculates the corresponding OAS.

RISK AND PRICING ISSUES WITH IMPLICIT OPTIONS The valuation process of implicit options addresses both issues of pricing and of risk simultaneously. From a pricing perspective, the expected value of the option should be a cost transferred to customers. From a risk perspective, the issue is to find the worst-case values of options. In fact, the same technique applies to both issues, since valuation requires simulation of the entire time paths of interest rates to find all possible payoffs. The valuation issue shows the limitations of the discrete yield curve scenarios used in ALM. The ALCO would like to concentrate on a few major assumptions, while the option risk requires considering a comprehensive set of interest rate time paths. Pricing an option as the expected value of all outcomes requires finding all its potential values under a full range of variations of interest rates. This is why the two blocks of ‘ALCO simulations’ and ‘options valuation’ often do not rely on the same technique. The value of an option is an expected value over all future outcomes. The interest rate risk is that of an increase in that value, since it is a loss for the bank, measured as the loss percentile at a forward horizon. Simulating all interest rate scenarios up to the forward horizon and after provides both. 2 The

terminology is confusing. In fact the ‘callable loan’ is a straight fixed rate loan plus a put option to sell it at a lower than market value when interest rates decline. The value of the loan for the bank (lender) is that of the straight debt minus the put value, which increases when rates decline. It is more like a ‘putable loan’.

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The value of an interest rate option depends on several parameters. They include the interest rate volatility, the maturity of the option and the exercise price, the risk-free rate. There are many option valuation models. Black’s (1976) model allows pricing interest rate options. The model applies to European options, and assumes that the underlying asset has a lognormal distribution at maturity. In the case of loan prepayments, the borrower has the right to sell to the bank the loan for its face value, which is a put option on a bond. Black’s model assumes that the bond price is lognormal. This allows us to derive formulas for the option price. In this chapter, we concentrate on numerical techniques, which allow us to show the entire process of simulations, rather than using the closed-form formula. For long-term options, such as mortgage options, the Heath, Jarrow and Morton model (Heath et al., 1992) allows more freedom than previous models in customizing the volatility term structure, and addressing the need to periodically revalue long-term options such as those of mortgage loans. They require using Monte Carlo simulations. However, such advanced models are more difficult to implement. Simple techniques provide reasonable estimates, as shown in the example. We concentrate on ‘American’ options allowing exercise at any time between current date and maturity. As soon as the option provides a positive payoff, the option holder faces a dilemma between waiting for a higher payoff and the risk of losing the current option positive liquidation value because of future adverse deviations. The choice of early or late exercise depends on interest rate expectations. The risk drivers of options are the entire time paths of interest rates up to maturity. The basic valuation process includes several steps, which are identical to those used for measuring market risk VaR, because this implies revaluing a portfolio of market instruments at a forward date. In the current case, we need both the current price for valuing the additional cost charged to borrowers and the distribution of future values for valuing downside risk. The four main steps addressing both valuation as of today and forward downside risk are: • Simulate the stochastic process of interest rates, through models or Monte Carlo simulations or an intermediate technique, such as ‘binomial trees’, explained below. • Revalue the option at each time point between current date and horizon. • Derive the ‘as of date value’ from the entire spectrum of future values and their probabilities. The current valuation addresses the pricing issue. • When the range of values is at a forward date, the process allows us to find potential deviations of values at this horizon. This technique serves for measuring the downside risk rather than the price. Intermediate techniques, such as binomial trees, are convenient and simple for valuing options3 . To illustrate the technique, the following section uses the simplest version of the common ‘binomial’ methodology. The technique of interest rate trees is a discrete time representation of the stochastic process of a short-term rate, such as those mentioned in Chapter 30. 3 There

are variations around the basic original binomial model. Hull and White (1994, 1996) provided techniques for building ‘general trees’ which comply with various modelling constraints on interest rates for incorporating mean-reversion. Mean-reversion designates the fact that interest rates tend to revert to their long-term mean when they drift along time. Some additional details on interest rate models are given in Chapter 31.

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A SIMPLE ‘BINOMIAL TREE’ TECHNIQUE APPLIED TO INTEREST RATES We use the simplest possible model of interest rates4 for illustration purposes. We limit the modelling of interest rates to the simple and attractive ‘binomial’ model. The ‘binomial’ name of the technique refers to the fact that the interest rate can take only two values at each step, starting from an initial value.

The Binomial Process Each ‘step’ is a small interval of time between two consecutive dates t and t + 15 . Given a current value of the rate at date t, there are only two possible values at t + 1. The rate can only move up or down by a fixed amount. After the first step (period), we repeat the process over all periods dividing the horizon of the simulations. The shorter this period, the higher the number of steps6 . Starting from a given initial value, the interest rate moves up or down at each step. The magnitudes of these movements are u and d. They are percentage coefficients applied to the starting value of the rate to obtain the final values after one step. If it is the interest rate at date t: it+1 = u × it

or

d × it

It is convenient to choose d = 1/u. Since u × d = 1, an up step followed by a down step results in a previous value, minimizing the number of nodes. The rate at t + 1 is u × rate(t) or d × rate(t). With several steps, the tree looks like the chart in Figure 21.1. In addition to u and d, we also need to specify what is the probability of an up and of a down move.

i(0) u x u i(0) u i(0) u x d

i(0) i(0) d

i(0) d x d

FIGURE 21.1 4 See

The binomial tree of rates

Cornyn and Mays (1997) for a review of interest rate models and their applications to financial institutions. literature uses a small interval t, which tends towards zero. 6 The binomial approach to options is expanded in Cox et al. (1979) and Cox and Rubinstein (1985).

5 The

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Natural and ‘Risk-neutral’ Probabilities The ‘binomial tree’ of rates requires defining the values of the ‘up’ and ‘down’ moves, u and d, and of their probabilities consistent with the market. Finding the appropriate values requires referring to risk-neutrality. This allows us to derive u and d such that the return and volatility are in line with the real world behaviour of interest rates. The financial theory makes a distinction between actual, or ‘natural’, probabilities and the so-called ‘risk-neutral’ probabilities. Risk-neutrality

Risk-neutrality implies indifference between the expected value of an uncertain gain and a certain gain equal to this expected value. Risk aversion implies that the value of the uncertain gain is lower than its expected value. Risk-neutrality implies identity between the expected outcome and the certain outcome (see Chapter 8 for brief definitions). The implication is that, under risk-neutrality, all securities should have the same risk-free return. There exists a set of risk-neutral probabilities such that the expected value equals the value under risk aversion. Intuitively, the risk-neutral probabilities are higher for downside deviations than actual probabilities. For instance, if an investor values a bet providing 150 and 50 with the same probability, his expected gain is 100. Under risk aversion, the value might fall to 90. This value under risk aversion is the expected value under riskneutral probabilities. This implies that the risk-neutral probability of the down move is higher than the actual natural downside probability, 0.5. The risk-neutral probability of the down move is p ∗ such that p ∗ × 50 + (1 − p ∗ ) × 150 = 90, or p ∗ = 0.6. The Parameters of the Binomial Model

Risk-neutrality implies that there exists a set of risk-neutral probabilities, p ∗ and 1 − p ∗ , of up and down moves such that: • The return is the risk-free rate r, or the expectation of the value after one period is 1 + r. • The volatility of distribution of the ‘up’ value and the ‘down’ value is identical to the √ market volatility σ t for a small period t. The corresponding equations are: p ∗ u + (1 − p ∗ )d = 1 + r

Resulting in:

[p ∗ u2 + (1 − p ∗ )d 2 ] − [p ∗ u + (1 − p ∗ )d]2 = σ 2 t √ u = exp[σ t]

and

√ d = exp[−σ t]

The derivation of these equations is given in the appendix to this chapter.

Using Numerical Data In the example used in this section, the yearly volatility is 15% and the steps correspond to a 1-year period. The numerical values selected for u and d are:

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259

u = exp(15%) = 1.1618

and

d = exp(15%) = 0.8607

With the above numerical values, the sequence of rates is that of Table 21.1. The riskneutral probabilities are simply 50% and 50% to make calculations more tractable. This implies that the real world probability of a down jump in interest rate is lower than 50% since the actual risky return should be higher than the risk-free rate in the real market. TABLE 21.1 values

The tree of numerical

Dates

0

1

2

Rates

10.00%

11.62% 8.61%

13.50% 10.00% 7.41%

After two up and down steps, the value at date 2 is the initial value, 10%, since u = 1/d.

VALUING DEBT AND OPTIONS WITH SIMULATED RATES In this section, we apply the binomial tree technique to value a risky debt. This implies ensuring that the binomial tree is consistent with market prices. To meet this specification, we need to proceed with various practical steps.

Value of Debt under Uncertainty Any asset generates future flows and its current value is the discounted value of those flows. The simulated rates serve for discounting. For instance, a zero-coupon bond generates a terminal flow of 100 at date 2. Its current value is equal to the discounted value of this flow of 100 using the rates generated by the simulation. There are as many possible values as there are interest rate paths to arrive at date 2. In a no-uncertainty world, the time path of rates is unique. When rates are stochastic, the volatility is positive and there are numerous interest rate paths diverging from each other. Each sequence of rates is a realization of uncertain paths of rates over time. For each sequence of rates, there is a discounted value of this stream of flows. With all sequences, there are as many discounted values as there are time paths of rate values. The value of the asset is the average of these values assuming that all time paths of rates are equally probable7 . The simple zero-coupon bond above illustrates the calculation process. The terminal flow of 100 is certain. Its present value as of date 1 uses the rate at period 2. Its present value as of date 0 uses the rate of period 1, which is the current rate. There are only two possible sequences of rates in this example, and hence two possible values of the same asset: 100/[(1 + 10.00%)(1 + 11.62%)] = 81.445 100/[(1 + 10.00%)(1 + 8.61%)] = 83.702 7 The maximum number of time paths is 2n where n is the number of periods, since at each date there are twice as many nodes as at the preceding date.

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The average value is (81.445 + 83.702)/2 = 82.574, which is the expected value given the uncertainty on interest rates (Table 21.2). TABLE 21.2 Values of the bond at various dates Dates

0

1

2

Values

82.574

89.590 92.073

100 100 100

Another technique for calculating this value serves later. At date 2, the value of the flow is 100 for all time paths of rates. However, this flow has two possible values at date 1 that correspond to the two possible values of the rates: 100/(1 + 11.62%) = 89.590 and 100/(1 + 8.61%) = 92.073. The current value results from discounting the average value at date 1, using the current rate, which gives [(89.590 + 92.073)/2]/(1 + 10.00%) = 82.574. The result is the same as above, which is obvious in this case. The benefit of this second method lies in calculating a current value using a recursive process starting from the end. From final values, we derive values at preceding dates until the current date. This second technique illustrates the mark-to-future process at date 1. In this case, there are two possible states only at date 1. From this simple distribution, we derive the expected value and the worst-case value. The expected value at date 1 is the average, or (89.590 + 92.073)/2 = 91.012. The worst-case value is 89.590. The binomial model is general. The only difference between a straight debt and an option is that the discounted flows of the option change with rates, instead of being independent of interest rate scenarios. Since it is possible to use flows defined conditionally upon the value of the interest rates, the model applies.

The Global Calibration of the Binomial Tree The value of listed assets, calculated with all the time paths of interest rates, should replicate those observed on the market. In order to link observed prices and calculated prices, the rates of each node of the tree need an adjustment, which is another piece of the calibration process. The binomial tree above does not yet include all available information. The rate volatility sets the upward and downward moves at each step. However, this unique adjustment does not capture the trend of rates, any liquidity premium or credit spread observed in markets, and the risk aversion making expected values less than actual values. All factors should show up in simulated rates as they do in the market. Instead of including progressively all factors that influence rates, it is easier to make a global calibration. This captures all missing factors in the model. In the above example, the zero-coupon value is 82.574. Its actual price might diverge from this. For instance, let us assume that its observed price is 81.5. The simulated rates need adjustment to reconcile the two values. In this example, the simulated rates should be above those used in the example, so that the simulated price decreases until it reaches the observed price. There is no need to change the volatility, since the up and down moves already replicate the market volatility for all rates individually. In the case of zero-coupon maturity at date 2, the initial rate value is 10%, and the rates at period 2 do not count.

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261

Hence, only rates as of date 1 need adjustment. The easiest way to modify them is to add a common constant to both rates at date 1. The constant has an empirical value such that it matches the calculated and observed prices. This constant is the ‘drift’. In the example, the drift should be positive to increase the rate and decrease the discounted value. Numerical calculation shows that the required drift is around 1.40% to obtain a value of 81.5. The calibrated rates at 1 become: i1u = 11.62% + 1.40% = 13.02% and i1d = 8.61% + 1.40% = 10.01%. Once the rates at date 1 are adjusted, the calibration extends to subsequent periods using the prices of listed assets having longer maturity. It is necessary to repeat this process to extend the adjustment of the whole tree. This technique enforces the ‘external’ consistency with market data.

The Valuation of Options In order to obtain an option value, we start from its terminal values given the interest rate values simulated at this date. Once all possible terminal values are determined from the terminal values of rates, the recursive process serves to derive the values at other dates. For American options, whose exercise is feasible at any time between now and maturity, there is a choice at any date when the option is in-the-money. Exercising the option depends on whether the borrower is willing to wait for opportunities that are more profitable or not. An American option has two values at each node (date t) of the tree: a value under no exercise and the exercise value. The value when waiting, rather than exercising immediately, is simply the average of the two possible values at the next date t + 1, just as for any other asset. Under immediate exercise, the value is the difference between the strike price and the gain. The optimum rule is to use the maximum of these two values. With this additional rule, the value of the options results from the same backward process along the tree, starting from terminal values.

THE CURRENT VALUATION OF THE PREPAYMENT OPTION This methodology applies to a loan amortized with constant annuities, whose maturity is 5 years and fixed rate 10%. The current rate is also 10%. The yearly volatility of rates is 15%. The borrower considers a prepayment when the decline in rates generates future savings whose present value exceeds the penalty of 3%. However, being in-the-money does not necessarily trigger exercise, since immediate prepayment may be less profitable than deferred prepayment. The borrower makes the optimal decision by comparing the immediate gain with the expected gains of later periods. The expected gain is the expected value of all discounted gains one period later, while the immediate gain is the exercise value. The process first simulates the rate values at all dates, then revalues the debt accordingly, calculates the strike price as 1.03 times the outstanding principal, and calculates the payoff under immediate exercise. The last step is to calculate the optimum payoff at each node and value the present value of these optimum payoffs to get the option value.

Simulating Rates Equal yearly periods divide the future. The binomial tree starts at 10%. However, there is a need to adjust the rates with a uniform drift of −0.10% applied to all rates to find the exact value 1000 of the straight debt under interest rate uncertainty. The values of u

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TABLE 21.3

The binomial tree of rates (5 years, 1-year rate, initial rate 10%)

Dates (Eoy) Ratesa

0

1

2

3

4

5

9.90%

11.52% 8.51%

13.40% 9.90% 7.31%

15.59% 11.52% 8.51% 6.28%

18.12% 13.40% 9.90% 7.31% 5.39%

21.07% 15.59% 11.52% 8.51% 6.28% 4.63%

a

In order to find exactly the value of 1000 when moving along the ‘tree’ of rates, it is necessary to adjust all rates by deducting a constant drift of −0.10%. Using the original rates starting at 10% results in a value of the debt at date 0 higher than 1000 because we average all debt values which are not a linear function of rates. Instead of starting at 10% we start at 10% − 0.10% = 9.90%. The same drift applies to all rates of the tree. This uniform drift results from a numerical calculation.

and d correspond to a yearly volatility of 15%. The binomial tree is the same as above, and extends over five periods. Dates are at ‘end of year’ (Eoy), so that date 1 is the end of period 1 (Table 21.3).

The Original Loan Since dates refer to the end of each year, we also need to specify when exactly the annuity flow occurs. At any date, end of period, the borrower pays immediately the annuity after the date. Hence, the outstanding debt includes this annuity without discounting. The date 0 is the date of origination. The terminal date 5 is the end of the last period, the date of the last annuity, 263.80, completing the amortization. Hence, the last flow, 263.80, occurs at date 5. The value of existing debt is exactly 263.80 before the annuity is paid. Table 21.4 shows the original repayment schedule of the loan. The constant annuity is 263.80 with an original rate of 10%. TABLE 21.4

Repayment schedule of loan

Dates (Eoy) Loan Annuities Capital reimbursement Outstanding balance

0

1

2

3

4

5

263.80 163.80 836.20

263.80 180.18 656.03

263.80 198.19 457.83

263.80 218.01 239.82

263.80 239.82 0.00

1000

The present value of all annuities at a discount rate of 10% is exactly 10008 .

The Market Value of a Loan, the Exercise Price and the Payoff of the Option Given the above strike price, the borrower can repurchase the debt and contract a new debt at the new rate. The market value of the old debt varies with interest rates. This market 8 The

1000 values discount all annuities of 263.80, starting at Eoy 1, so that: 1000 =

5

t=1 263.80/(1

+ i)t .

THE VALUE OF IMPLICIT OPTIONS

263

value is the underlying of the prepayment option. It is the present value of all subsequent annuities, given the market rate. The rate values of each node serve to calculate this time profile of the present value. The gain from prepayment is the difference between the current value of the debt and the current strike price (Table 21.5). TABLE 21.5

Market value of loan, exercise price and payoff from immediate exercise

Dates

0

1

Value at origination

1000.00

1071.72 1126.34

Principal + 3%

1030.00

836.20

Gains

0.00

2

3

Valuation: 881.73 689.25 920.29 712.25 951.60 730.78 745.41

Exercise price: 656.03 457.83

Payoff from immediate exercise:a 0.00 0.00 0.00 26.34 0.47 0.00 31.78 9.15 23.79

4

5

487.12 496.42 503.82 509.62 514.10

263.80 263.80 263.80 263.80 263.80 263.80

239.82

0.00

0.00 0.00 0.21 6.01 10.49

0.00 0.00 0.00 0.00 0.00

a Value

= max(market value of debt − strike, 0). The strike is the outstanding balance plus a penalty. The above values are rounded at the second digit: the payoff 26.34 at date 1 is 1126.34 − 263.80 − 836.20 = 26.34.

As an example, the following calculations are detailed at date 1, when the rate is 8.51% (line 2 and column 1 of the binomial tree and of the debt value table): the value of the debt; the exercise price; the payoff from exercise; the value of the option. Value of Debt

The recursive process starting from the terminal values of the debt applies to the calculation of the debt value along any time path of rates. At date 4, the debt value discounts this 263.80 flow with the interest rates of the binomial tree. The same process applies to all previous dates until date 0. At any intermediate date, the value of the debt includes the annuity of the period, without discounting, plus the average discounted value of the debt at date 2 resulting from previous calculations. This value is: 1126.34 = 263.80 + 0.5(920.29 + 951.60)/(1 + 8.51%) The Payoff from Exercise of the Option

The outstanding balance is the basis for calculating the penalty, set at 3% at any date. The strike price of the option is 1.03 times the outstanding balance at all dates before maturity. The gain, when exercising the prepayment option, is the difference between the

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market value of existing debt and the exercise price, which is the value of new debt plus penalty. The time profile of the exercise price starts at 1030 and declines until maturity. This difference between market value of debt and strike price is a gain. Otherwise, the exercise value of the option is zero. A first calculation provides the gain with immediate exercise, without considering further opportunities of exercise. For instance, at date 1 the value of debt, after payment of the annuity, is 1126.34 − 263.80 = 862.54. The exercise price is 836.20. The liquidation value of the option is 862.54 − 836.20 = 26.34. In many instances, there is no gain, the option is ‘out-of-the-money’ and the gain is zero.

Valuation of the Option under Deferred Exercise The immediate gain is not the optimum gain for the borrower since future rate deviations might generate even bigger gains. The optimum behaviour rule applies. The expected value of deferred gains at the next period is the average of their present values. Finally, the value of the option at t is the maximum between immediate exercise and the present value of expected gains of the next period, or zero. This behavioural rule changes the time profile of gains. The rounded present value of the option as of today is 12.85 (Table 21.6). TABLE 21.6 Dates Value of option

a Option

The expected gains from the prepayment optiona 0

1

2

3

4

5

12.85

1.90 26.34

0.04 4.21 31.78

0.00 0.09 9.15 23.79

0.00 0.00 0.21 6.01 10.49

0.00 0.00 0.00 0.00 0.00 0.00

value = max(exercise value at t, expected exercise values at t + 1 discounted to t, 0).

The value of the prepayment option is 12.85, for a loan of 1000, or 1.285% of the original value of the loan.

CURRENT VALUE AND RISK-BASED PRICING Given the value of the option, pricing requires the determination of an additional markup corresponding to the expected value to the borrower. This mark-up, equivalent to the current value of the option, is the OAS. The original loan has a value of 1000 at 10%, equal to its face value when considering a straight debt without the prepayment option. The value of the debt with the prepayment option is equal to the value of the straight debt less the value of the option given away, or 1000 − 12.85 = 987.15. For the lender, the loan is an asset whose value is less than that of a straight debt of 1000 at 10%, the difference being the value of the prepayment option. For the borrower, the loan is a liability whose value is also lower than the 1000 at 10%. The lender gave away a right whose value is 12.85.

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265

This cost is 1.285% of the current loan principal. With a current rate of 10%, the ‘debt plus option’ package has a value of 987.15, or only 98.715% of the value of the debt without option. This decline in loan value by the amount of the option value should appear under ‘fair value’ in the balance sheet. It represents the economic value of the debt, lower than the book value by that of the option given away by the bank. In this case, fair value is a better image than book value, because it makes explicit the losses due to prepayments rather than accruing losses later on when borrowers actually repay or renegotiate their loan rate. For practical purposes, banks lend 1000 when customer rates are 10%. In fact, the customer rate should be above this value since the 10% does not account for the value of the option. Pricing this prepayment risk necessitates a mark-up over the straight debt price. This periodical mark-up is an additional margin equivalent to the value of the option. The spread required to make the value of the debt alone identical to that of the debt with the option is the OAS. To determine the OAS, we need to find the loan rate making the loan value less the option value equal to 1000. The straight loan value is 1000 + 12.85 = 1012.85. In order to decrease this value to 1000, we need to increase the yield of the loan. This yield is 10.23%. At this new yield, the annuity is 265.32 instead of 263.80. This is a proxy calculation. The OAS is 23 basis points (Table 21.7). TABLE 21.7

Debt 1000 Current rate 10.00%

Summary of characteristics of the loan A: Loan without prepayment option Rate 10.00% B: Loan with prepayment option Value of option Value of debt without option 12.85 1000 Maturity 5

Annuity 263.80 Value of debt with option 1012.85

OAS 0.23%

The methodology used to determine the OAS is similar to that of calibration. However, the goal differs. Calibration serves to link the model to the market. OAS serves to determine the cost of the option embedded in a loan and the required mark-up for pricing it to the borrower.

THE RISK PROFILE OF THE OPTION To determine the risk profile of the option at a given horizon, we need its distribution. The simulation of interest rate paths provides this distribution. In this simple case, the up and down probabilities are 0.5 each. From these up and down probabilities, we can derive the probabilities of reaching each node of the tree. At any future horizon, the losses are simply the payoffs to the option holder, under exercise. Using the end of period 3 as a horizon, the probabilities of each payoff depend on how many paths end up with the same value of the debt, remembering that several paths can end up at the same value, because each ‘up and down’ combination results in the same final value as a ‘down and up’ combination. At date 3, there are four values of the option payoff. There are more than four time paths leading to these four different values of rates at date 3. The probability of a single time path for three consecutive periods is

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0.53 = 12.5% because we use a constant up and down probability of 0.5. When three time paths lead to the same node, the probability is 3 × 12.5% = 37.5%. At date 3, we have one single path leading to each extreme node and three time paths leading to each of the two intermediate nodes. The probability distribution of reaching each of the four nodes follows. For each node, we have the payoff of the option, which equates the loss for the bank. The distribution of the payoffs, or the bank losses, follows (Table 21.8). TABLE 21.8 option

Payoff, or bank’s cost, distribution from an

Date

Probabilities 0.53 3 × 0.53 3 × 0.53 0.53

Total

Bank’s cost = option payoff

12.5% 37.5% 37.5% 12.5%

0.00 0.09 9.15 23.79

100%

The lowest value is 23.79 and the worst-case value at the 75% confidence interval is 9.15. Detailed simulations of interest rates would lead to a more continuous distribution. The loss distribution is highly skewed to the left, as for any option, with significant losses for two nodes (9.15 and 23.79) (Figure 21.2).

Probability

40% 30% 20% 10% 0%

Loss 0

FIGURE 21.2

10

20

30

Standalone risk of the implicit option

Extending the simulation to the entire banking portfolio implicit options would provide the VaR due to option risk. The NPV simulations derive such an ALM VaR for optional risk, as illustrated in Chapter 25.

APPENDIX: THE BINOMIAL MODEL The natural probability p is that of an up move, while 1 − p is the probability of a down move: p = 1 − p = 50%. Similar definitions apply to p∗ and 1 − p ∗ in a risk-neutral world. There is a set of risk-neutral probabilities p ∗ such that the expectation becomes equal to 90: p ∗ × 150 + (1 − p∗ ) × 50 = 90 or: p ∗ = (90 − 50)/(150 − 50) = 40%

THE VALUE OF IMPLICIT OPTIONS

267

The risk-neutral probability p ∗ differs from the natural probability p = 50% of the real world because of risk aversion. Generalizing, the expected value of any security using risk-neutral probabilities is E ∗ (V ), where E is the expectation operator and ∗ stands for the risk-neutral probability in calculating E ∗ (V ). This risk-neutral expected value is such that it provides the risk-free return by definition. Since we know the risk-free return r, we have a first equation specifying that u and d are such that E ∗ (V ) = 1 + r for a unit value. The expectation of the value after one time interval is: p × u + (1 − p) × d = 1 + r. This is not enough to specify u and d. We need a second equation. A second specification of the problem is that the volatility should mimic the actual volatility of rates. The volatility in a risk-neutral world is identical to the volatility in the real world. Given natural or risk-neutral probabilities, the variance over a small time interval t of the values is9 : [p ∗ u2 + (1 − p ∗ )d 2 ] − [p ∗ u + (1 − p ∗ )d]2 = σ 2 t This equality holds with both p and p ∗ . Under risk-neutrality, the expected return is the risk-free return r. Under risk aversion, it has the higher expected value µ. Therefore, p and p ∗ are such that: p = (eµt − d)/(u − d)

p ∗ = (ert − d)/(u − d)

Using p ∗ × u + (1 − p ∗ ) × d = 1 + r and p ∗ = (ert − d)/(u − d), we find the values of u and d: √ √ u = exp[σ t] and d = exp[−σ t] The yearly volatility is σ and t is the unit time for one step, say in years. Hence, u and d are such that the volatility is the same in both real and risk-neutral worlds. However, the return has to be the risk-free return in the risk-neutral world and the expected return, above r, in the real world.

9 The

variance of a random variable X is V (X) = E(X2 ) − [E(X)]2 .

SECTION 8 Mark-to-Market Management in Banking

22 Market Value and NPV of the Balance Sheet

The interest income is a popular target of interest rate risk management because it is a simple measure of current profitability that shows up directly in the accounting reports. It has, however, several drawbacks. Interest income characterizes only specific periods, ignoring any income beyond the horizon. Measuring short-term and long-term profitability provides a more comprehensive view of the future. The present values of assets and liabilities capture all future flows generated by assets and liabilities, from today up to the longest maturity. The mark-to-market value of the balance sheet discounts all future flows using market discount rates. It is the Net Present Value (NPV) of the balance sheet, since it nets liability values from asset values. The calculation of an NPV does not imply any assumption about liquidity of loans. It is a ‘fair’ or ‘economic’ value calculation, not the calculation of a price at which assets could trade: • It differs from the ‘fair values’ of assets by the value of credit risk of borrowers, because it uses market rates, without differentiating them according to the risk of individual borrowers. • It also differs from the value of equity because it considers the balance sheet as a portfolio of fixed income assets, long in loans and short in the bank’s liabilities, exposed to interest rate risk only, while the stock price depends on equity market parameters and is subject to all bank risks. To put it more simply, NPV can be negative, while stock prices cannot. • The NPV at market rates represents the summation of all future interest income of assets and liabilities, plus the present value of equity positioned as of the latest date of calculations.

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This last property provides a major link between the accounting income measures and the NPV, under some specific assumptions. It makes NPV a relevant long-term target of ALM policy, since controlling the risk of the NPV is equivalent to controlling the risk of the entire stream of interest incomes up to the latest maturity. The presentation of the NPV comprises two steps. This chapter discusses the interpretation of the NPV and its relationship with net interest margins and accounting profitability. The first section discusses alternative views of NPV. The second section demonstrates, through examples, the link between NPV at market rates and the discounted value of the future stream of periodical interest incomes up to the longest maturity, plus an adjustment for equity. Chapter 23 deals with the issue of controlling the interest rate risk using the NPV, rather than interest income, as the target variable of ALM. Duration gaps substitute the classical interest rate gaps for this purpose.

VIEWS ON THE NPV The NPV of the balance sheet is the present value of assets minus the present value of liabilities, excluding equity. This present value can use different discount rates. When using the market rates, we have an ‘ALM’ NPV that has the properties described below. ‘Fair value’ calculation would use the yields required by the market dependent on credit spreads of individual assets. This fair value differs from the value of assets under ‘ALM’ NPV, which is a mark-to-model calculation. There are various possible views on NPV. We review some of them, focusing on what they imply and eventual inconsistencies. Possible views of NPV are: a mark-to-market value of a portfolio; a measure of performance defined by the gaps between the asset and liability yields and the market rates; a measure of the future stream of interest incomes up to the latest maturity in the banking portfolio.

Definition The present value of the stream of future flows Ft is: Ft /(1 + yt )t V = t

The market rates are the zero-coupon rates yt derived from yield curves, using the cost of the debts of the bank. The formula provides a value of any asset that generates contractual flows. The NPV of the balance sheet is the value of assets minus that of debts. Both result from the Discounted Cash Flow (DCF) model. The calculation of NPV uses market rates with, eventually, the credit spread of the bank compensating its credit risk. The NPV using market rates, with or without the bank’s credit spread, differs from the ‘fair value’ of the balance sheet. Hence, NPV is more a ‘mark-to-model’ value whose calculation rules derive from the interpretations of its calculation, as explained below.

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273

NPV as a Portfolio The NPV is the theoretical value of a portfolio long in assets and short in liabilities. The NPV can be either positive or negative, since the NPV is the net value of assets and liabilities. The change in NPV depends entirely on the sensitivities of these two bonds. A subsequent chapter (Chapter 23) discusses the behaviour of NPV when market rates change. The NPV ‘looks like’ a proxy for equity value since it nets the values of assets and debts, so that equity is the initial investment of an investor who buys the assets and borrows the liabilities. However, NPV and market value of equity are not at all identical. First, NPV can be negative whereas equity value is always positive. Second, this interpretation assimilates equity to a bond, which it is not. Using market rates as discount rates is equivalent to considering the investor as a ‘net’ lender, instead of an equity holder. Since equity investors take on all risks of the bank, the required rate of return should be above the cost of debt. Therefore, it does not make sense to consider the equity investor as a ‘pure lender’. In addition, the NPV at market rates represents the market value of the balance sheet even though assets and liabilities are not actually liquid. If assets and liabilities were marketable, it would be true that an investor could replicate the bank’s portfolio with a net bond. In such a case, the market value of equity would be identical to the NPV at market rates because the investor would not be an equity investor. He would actually hold a ‘leveraged’ portfolio of bonds. The problem is that the market assets do not truly replicate operating assets and liabilities because they are neither liquid nor negotiable. Hence, the market value of equity is actually different from the theoretical NPV at market rates. The market value of equity is the discounted value of the flows that compensate equity, with a risk-adjusted discount rate. These flows to equity are dividends and capital gains or losses. In addition, the relevant discount rate is not the interest rate, but the required return on equity, given the risk of equity.

NPV and Assets and Liabilities Yields The NPV at market rates is an economic measure of performance. If the value of assets is above face value, it means that their return is above the market rates, a normal situation if loans generate a margin above market rates. If the bank pays depositors a rate lower than market rates, deposits cost less than market rates and the market value of liabilities is below the face value. This results in a positive NPV, even if there were no equity, because assets are valued above face value and liabilities below face value. Accordingly, when the NPV increases, margins between customer rates and market rates widen, the economic performance improves, and vice versa. When the bank lends at 10% for 1 year, and borrows in the market at 8%, it gets 1100 as proceeds from the loan in 1 year and repays 1080 for what it borrows. The difference is the margin, 20. In addition, the present value of the asset is 1100/(1 + 8%) = 1018.52, above 1000. The gap of 18.52 over face value is the present value of the margin at the

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prevailing market rate. If asset values at this rate are higher than book values, they provide an excess spread over the bank’s cost of debt. If not, they provide a spread that does not compensate the bank’s cost of debt. If assets yield exactly the cost of debt for the bank, their value is at par. The same process applies to bank deposits. When using the market rates for the bank’s deposits, their value is under face value. If the bank collects deposits of 1000 for 1 year at 5% when the market rate is 10%, the value of the deposits is 1050/(1 + 8%) = 972.22, lower than the face value. The difference is the present value of the margin at the market rate. The interpretation is that lending deposits costing 5% in the market at 10% would provide a margin of 1000 × 5% = 50. This simple reasoning is the foundation of the interpretation of the NPV as the present value of future periodical interest incomes of the bank.

NPV and the Future Stream of Interest Incomes Since we include all flows in discounting, the NPV should relate to the margins of the various future periods calculated using market rates as references. We show in the next section that the NPV is equal to the discounted value of the future stream of interest incomes, with an adjustment for equity. Since this NPV represents the discounted value of a stream of future interest incomes, all sensitivity measures for the NPV apply to this discounted stream of future margins. This is an important conclusion since the major shortcoming of the interest margin as a target variable is the limited horizon over which it is calculated. Using NPV instead as a measure of all future margins, the NPV sensitivity applies to the entire stream of margins up to the longest horizon. The next chapter expands the NPV sensitivity as a function of the durations of assets and liabilities.

NPV AND INTEREST INCOME FOR A BANK WITHOUT CAPITAL The relationship between NPV and interest income provides the link between market and accounting measures. It is easier to discuss this relation in two steps: the case of a bank without equity, which is a pure portfolio of bonds; a bank with equity capital. This section considers only the zero equity case and the next the case where equity differs from zero. The discussion uses an example to make explicit the calculations and assumptions. The discount rate used is a flat market rate of 10% applying to the bank, which can fluctuate.

Sample Bank Balance Sheet The bank funding is a 1-year debt at the current rate of 9%. This rate is fixed. The asset has a longer maturity than debt. It is a single bullet loan, with 3-year maturity, having a contractual fixed rate of 11%. Hence, the bank charges to customers the current market rate plus a 1% margin. The current market rate is 10%. This margin is immune to interest

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275

TABLE 22.1 Example of a simplified balance sheet

Amount Fixed rate Maturity

Assets

Liabilities

1000 11% 3 years

1000 9% 1 year

rate changes for the first year only, since the debt rate reset occurs only after 1 year. The balance sheet is given in Table 22.1. The cash flows are not identical to the accounting margins (Table 22.2). The differences are the principal repayments. For instance, at year 1, two cash flows occur: the interest revenue of 110 and the repayment of the bank’s debt plus interest cost, or 1110. However, the accounting margin depends only on interest revenues and costs. It is equal to 110 − 90 = 20. For subsequent years, the projection of margins requires assumptions, since the debt for years 2 and 3 has to be renewed. TABLE 22.2 The stream of cash flows generated by assets and liabilities Dates Assets Liabilities

0

1

2

3

1000 1000

110 1090

110

1110

The profile of cash flows remains unchanged even though the interest rate varies. This is not so for margins, after the first period, because they depend on the cost of debt, which can change after the first year. Therefore, projecting margins requires assumptions with respect to the renewal of debt. We assume that the debt rolls over yearly at the market rate prevailing at renewal dates. The horizon is the longest maturity, or 3 years. The cost of debt is 80 if the market rate does not change. If the market rate changes, so does the cost of debt beyond 1 year. The rollover assumption implies that the cash flows change, since there is no repayment at the end of period 1, and until the end of period 3, where the debt ceases to roll over. If the market rate increases by 1%, up to 11%, the cost of debt follows and rises to 10%. The stream of cash flows for the rollover debt becomes (90, 90, 1090) and the margin decreases starting from the second year (Table 22.3). TABLE 22.3 The future stream of cash flows and margins generated by assets and liabilities Dates

0

1

2

3

Assets 1000 110 110 1110 Liabilities 1000 90 100 1100 Interest revenues and costs when debt is rolled over at the current rate Revenues — 110 110 110 Costs — 90 100 100 Margins — 20 10 10

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NPV and Projected Interest Income with Constant Market Rate The values of assets and liabilities, using the original rate, and the NPV are: PV(asset) = 110/(1 + 10%) + 110/(1 + 10%)2 + 1110/(1 + 10%)3 = 1024.869 PV(debt) = 90/(1 + 10%) + 90/(1 + 10%)2 + 1090/(1 + 10%)3 = 975.131 The NPV is the difference: NPV = PV(asset) − PV(debt) = 49.737 Next, we calculate the present value of all future interest incomes at the prevailing market rate of 10%: Present value of interest income = 20/(1 + 10%) + 20/(1 + 10%)2 + 20/(1 + 10%)3 = 49.737 These two calculations provide evidence that the NPV from cash flows is exactly equal to the present value of interest incomes at the same rate: NPV = present value of margins = 49.737

NPV and Projected Interest Income with Market Rate Changes When the market rate changes, the cash flows do not change, but the projected margins change since the costs of debt are subject to resets at the market rates. We assume now parallel shifts of the yield curve. The flat market rate takes the new value of 9% from the end of year 1 and up to maturity. The cost of debt changes accordingly to 8% assuming a constant margin between the cost of debt and the market rate. The flows are as given in Table 22.4. TABLE 22.4 years Periods

Flows generated over three 0

1

2

3

Assets 1000 110 110 1110 Liabilities 1000 1090 Rollover of debt at 8%, market rate at 9% Revenues — 110 110 110 Costs — 90 80 80 Margins — 20 30 30

The NPV calculation remains based on the same flows, but the discount rate changes. For simplicity, we assume that the rate changes the very first day of the first period. The margins also change with the rate, and the present value discounts them with the new market rate. With a 9% market rate, the margins increase after the first year. The calculations of the NPV and the discounted value of margins are: PV(asset) = 110/(1 + 9%) + 110/(1 + 9%)2 + 1110/(1 + 9%)3 = 1050.626 PV(debt) = 90/(1 + 9%) + 90/(1 + 9%)2 + 1090/(1 + 9%)3 = 983.861

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277

The NPV is the difference: NPV = PV(asset) − PV(debt) = 66.765 The discounted value of periodical interest incomes is: Discounted margins = 20/(1 + 9%) + 30/(1 + 9%)2 + 30/(1 + 9%)3 = 66.765 When the rate changes, the equality still holds. Using the 9% flat rate, or any other rate, shows that this equality holds for any level of market rates.

The Interpretation of the Relationship between Discounted Margins and NPV For the above equality to hold, we need to reset both the discount rate and the margin values according to rate changes. Taking an extreme example with a sudden large jump of interest rate, we see what a negative NPV means. The negative NPV results from projected margins becoming negative after the first year. Both asset and liability values decline. With a 15% market rate, the projected margins beyond year 1 become 110 − 140 = −30. The two calculations, discounted flows and discounted margins, provide the same value: NPV = discounted interest incomes = 20/(1 + 15%) − 30/(1 + 15%)2 − 30/(1 + 15%)3 = −25.018 The change from a positive NPV to a negative NPV is the image of the future negative margins given the rising cost of debt. This results from the discrepancies of the asset and liability maturities and the resulting mismatch between interest rate resets. The short-term debt rolls over sooner than assets, which generate a constant 11% return. The variable interest rate gap is negative, calculated as the difference between interest-sensitive assets and liabilities. This implies that the discounted margins decrease when the rate increases. In the example, discounted margins change from 49.737 to −25.018. The change in NPV is the difference between variations of the asset and liability values. When rates increase, both decrease. Nevertheless, the asset value decreases more than the debt value. This is why the NPV declines and becomes negative. The sensitivity of the asset value is higher than the sensitivity of the debt value. The following chapter discusses the sensitivity of the mark-to-market value, which is proportional to duration. In the current example, the duration of assets is above that of debt. Accordingly, assets are more sensitive to rate changes than debt. The above results are obtained for a balance sheet without equity. Similar results are obtained when equity differs from zero, but they require an adjustment for the equity term.

DISCOUNTED MARGINS AND NPV WITH CAPITAL With capital, the debt is lower than assets and the interest income increases because of the smaller volume of debt. In the example below, equity is 100, debt is 900 and assets are 1000. All other assumptions are unchanged. At the 10% market rate, the first year

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cost of debt becomes 9% × 900 = 81 instead of 90 because the amount of debt is lower (Table 22.5). TABLE 22.5 Example of a simplified balance sheet Assets Loans at 11% Capital Debt at 9%

Liabilities

1000 100 900

We can duplicate the same calculations as above with the new set of data. Cash flows and projected margins correspond to the new level of debt: PV(asset) = 110/(1 + 10%) + 110/(1 + 10%)2 + 1110/(1 + 10%)3 = 1024.869 PV(debt) = 81/(1 + 10%) + 81/(1 + 10%)2 + 1081/(1 + 10%)3 = 877.618 The NPV is the difference: NPV = PV(asset) − PV(debt) = 147.250 The stream of future margins becomes 29 for all 3 years. Its discounted value at the market rate of 10% becomes: Discounted margins = 29/(1 + 10%) + 29/(1 + 10%)2 + 29/(1 + 10%)3 = 72.119 The present value of margins becomes lower than the balance sheet NPV. The gap results from equity. The difference between the NPV and the discounted margins is 147.250 − 72.119 = 75.131. We see that this difference is the present value of the equity, at the market rate, positioned at the latest date of the calculation, at date 3: 100/(1 + 10%)3 = 75.131 Therefore, the NPV is still equivalent to the discounted value of future interest income plus the present value of equity positioned at the latest date of the calculation. Table 22.6 illustrates what happens when the market rate varies, under the same assumptions as above, with a constant margin of −1% for the rollover debt. The discount rate being equal to the market rate, it changes as well. TABLE 22.6 Market rate 9% 10% 11%

Discounted values of cash flows and margins at market rates Present value of margins

Present value of 100 at date 3

Present value of margins + capital

NPV

87.932 72.119 56.982

77.218 75.131 73.119

165.151 147.250 130.102

165.151 147.250 130.102

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Conclusion All the above calculations use a very simple balance sheet. However, since all assets and liabilities can be broken down into simple zero-coupon items, the calculations still hold for real balance sheets. In the end, the assumptions required for the equivalence of discounted margins plus capital and NPV are the following: • Projections of interest income should extend up to the longest maturity of balance sheet items. • Both the stream of projected margins and the discount rate need adjustment when rates change: future margin resets occur when the current market rate changes until maturity1 . • The discount rates are the market rates. • The NPV at market rates, or at the bank’s market rates, differs from the ‘fair value’ which uses rates embedding the credit spreads of the borrower. • The difference between NPV and discounted margins at market rates is the present value of equity positioned at a date that is the longest maturity of all assets and liabilities.

1 Rate

deviations from 8% require adjusting both margins and discount rate.

23 NPV and Interest Rate Risk

The Net Present Value (NPV) is an alternative target variable to interest income. The sensitivity of interest rate assets and liabilities is the ‘duration’. Since the NPV is the difference between mark-to-market values of loans and debts, its sensitivity to rate changes depends on their durations. The duration is the percentage change of a market value for a unit ‘parallel’ shift of the yield curve (up and down moves of all rates). The duration of assets and liabilities is readily available from their time profile of cash flows and the current rates. Intuitively, the interest rate sensitivity of the NPV depends on mismatches between the duration of assets and the duration of liabilities. Such mismatches are ‘duration gaps’. Controlling duration gaps is similar to controlling interest rate gaps. Derivatives and futures contracts alter the gap. Simple duration formulas allow us to define which gaps are relevant for controlling the sensitivity of several NPV derived target variables. These include the NPV of the balance sheet, the leverage ratio of debt to asset in mark-tomarket values, or the duration of ‘equity’ as the net portfolio of loans minus bank debts. Maintaining adequate duration gaps within bounds through derivatives or futures allows us to control the interest rate sensitivities of mark-to-market target variables. Duration-based sensitivities do not apply when the yield curve shape changes, or when the interest shock is not small. Durations are, notably, poor proxies of changes with embedded options (for further details see Chapter 24). This chapter focuses on duration gaps between assets and liabilities, and how to use them. The first section describes duration and its properties. The second section lists NPV derived target variables of the Asset–Liability Management (ALM) policy, and defines the relevant duration gaps to which they are sensitive. Controlling these duration gaps allows us to monitor the risk on such target variables. The last section explains why derivatives and futures modify durations and help to hedge or control the risk of the NPV.

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THE SENSITIVITY OF MARKET VALUES AND DURATION The sensitivity is the change in market value generated by a parallel shift of the yield curve. In technical terms, it is the derivative of the value of the asset with respect to interest rate. The sensitivity is the modified duration of an asset. The duration is the ratio of the present value of future flows, weighted by dates, to the market value of the asset. The modified duration applies when the yield curve is flat, since it is the ratio of the duration to (1 + y), where y is the flat rate. However, it is always possible to derive a sensitivity formula when the yield curve is not flat, by taking the first derivative with respect to a parallel shift of the entire yield curve. In this case, we consider that a common shock y applies to all rates yt and take the derivative with respect to y rather that y. The appendix to this chapter uses this technique to find the condition of immunization of the interest income over a given period when flows do not occur at the same date within the period. The cash flows are Ft for the different dates t, and yt are the market rates. Then the duration is:

N N tFt /(1 + yt )t Duration = Ft /(1 + yt )t t=1

t=1

The duration formula seems complex. In fact, it represents the average maturity of future flows, using the ratio of the present value of each flow to the present value of all flows as weights for the different dates1 . For a zero-coupon, the duration is identical to maturity. In this case, all intermediate flows are zero. The formula for the duration shows that D = M, where M is the zero-coupon maturity. The duration has several important properties that are explained in specialized texts2 . Subsequent sections list the properties of duration relevant for ALM. The duration is a ‘local’ measure. When the rate or the dates change, the duration drifts as well. This makes it less convenient to use with important changes because duration ceases to be constant.

Sensitivity of Market Values to Changes in Interest Rates The modified duration is the duration multiplied by 1/(1 + y). The modified duration is the sensitivity to interest rate changes of the market value and applies when the yield curve is flat. The general formula of sensitivity is: V /V = −[D/(1 + y)] × y The relative change in value, as a percentage, is equal to the modified duration times the absolute percentage interest rate change. This formula results from the differentiation of the value with respect to the interest rate. The sensitivity of the value is the change in value of the asset generated by a unit change in interest rate. This value is equal to the market value of the asset multiplied by the modified duration and by the change in interest rate: V = −[D/(1 + y)] × V × y 1 This

is different from the simple time weighted average of flows because of discounting. (1987) book is entirely dedicated to duration definitions and properties.

2 Bierwag’s

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The duration of a portfolio is the average of the durations of assets weighted by their market values. This property is convenient to derive the durations of a portfolio as a simple function of the durations of its individual components.

Duration and Return Immunization The return of an asset calculated over a horizon equal to its duration is immune to any interest rate variation. With fixed rate assets, obtaining the yield to maturity requires holding the asset until maturity. The yield to maturity is the discount rate making the present value of the future cash flow identical to its price. When selling the asset before maturity, the return is uncertain because the price at the date of sale is unknown. This price depends on the prevailing interest rate at this date. The Discounted Cash Flow (DCF) model, using the current market rates as discount rates, provides its value. If the rates increase, the prices decrease, and conversely. The holding period return combines the current yield (the interest paid) and the capital gain or loss at the end of the period. The total return from holding an asset depends on the usage of intermediate interest flows. Investors might reinvest intermediate interest payments at the prevailing market rate. The return, for a given horizon, results from the capital gain or loss, plus the interest payments, and plus the proceeds of the reinvestments of the intermediate flows up to this horizon. If the interest rate increases during the holding period, there is a capital loss due to the decline in price. Simultaneously, intermediate flows benefit from a higher reinvestment rate up to the horizon at a higher rate. If the interest rate decreases, there is a capital gain at the horizon. At the same time, all intermediate reinvestment rates get lower. These two effects tend to offset each other. There is some horizon such that the net effect on the future value of the proceeds from holding the asset, the reinvestment of the intermediate flows, plus the capital gain or loss cancel out. When this happens, the future value at the horizon is immune to interest rate changes. This horizon is the duration of the asset3 .

Duration and Maturity The duration increases with maturity, but less than proportionately. It is a ‘convex’ function of maturity. The change in duration with maturity is the slope of the tangent to the curve that relates the duration to the maturity for a given asset (Figure 23.1). When time passes, from t to t + 1, the residual life of the asset decreases by 1. Nevertheless, the duration decreases by less than 1 because of convexity. If the duration is 2 years in January 1999, it will be more than 1 year after 1 year, because it diminishes less than residual maturity. This phenomenon is the ‘duration drift’ over time. Due to duration drift, any constraint on duration that holds at a given date does not after a while. For instance, an investor who wants to lock in a return over a horizon of 5 years will set the duration of the portfolio to 5 at the start date. After 1 year, the residual life decreases by 1 year, but the duration decreases by less than 1 year. The residual time to the horizon is 4 years, but the duration will be somewhere between 5 and 4 years. The portfolio duration needs readjustment to make it equal to the residual time of 4 years. This adjustment is continuous, or frequent, because of duration drift. 3 This

is demonstrated in dedicated texts and discussed extensively by Bierwag (1987).

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Duration

t

FIGURE 23.1

t+1

Maturity

The relationship between duration and maturity

THE DURATION GAP AND THE TARGETS OF INTEREST RATE POLICY The target variables of ALM policies include: the interest income over specified periods, the NPV of the balance sheet; the leverage ratio expressed with market values; the market return of the portfolio of assets and liabilities. The immunization conditions for all variables hold under the assumption of parallel shifts in the yield curve.

Immunization of the Interest Margin With the gap model, all flows within periods are supposed to occur at the same date. This generates errors due to reinvestments or borrowings within the period. The accurate condition of immunization of the interest margin over a given horizon depends on the duration of the flows of assets and liabilities over the period. The condition that makes the margin immune is: VA(1 − DA ) = VL(1 − DL ) where VA and VL are the market values of assets and liabilities, and DA and DL are the durations of assets and liabilities. The demonstration of this formula is in the appendix to this chapter. The formula summarizes the streams of flows for both assets and liabilities with two parameters: their market values and their durations. It says that the streams of flows generated by both assets and liabilities match when their durations, weighted by their market values, are equal. Figure 23.2 illustrates the immunization condition that makes market values of assets and liabilities, weighted by duration, equal.

NPV of the Balance Sheet To neutralize the sensitivity of the market value, the variations in values of assets and liabilities should be identical. This condition implies a relationship between the market values and the duration of assets and liabilities. With a parallel shift of the yield curve equal to i, the condition is: NPV/i = (VA − VL)/i

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VA 1−DA

DL DA

VL

0

FIGURE 23.2

1−DL

1

Summarizing a stream of flows into a single ‘equivalent flow’

The changes in values of assets and liabilities result from their durations, DA and DL . They are (−DA × VA × i) and (−DL × VL × i). If we divide by i, we obtain: NPV/i = [1/(1 + i)][−DA × VA + DL × VL] The immunization condition is: VA × DA = VL × DL This condition states that the changes in the market value of assets and liabilities are equal. When considering the net portfolio of assets minus liabilities, the condition stipulates that the change in asset values matches that of liabilities. Note that this duration gap is in value, not in years. When adjusting durations, it is more convenient to manipulate durations in years. The immunization condition is: DA /DL = VL/VA The formula stipulates that the ratio of the duration and liabilities should be equal to the ratio of market values of liabilities to assets. The NPV sensitivity is the sensitivity of the net portfolio of assets minus liabilities. Its duration is a linear function of these, using market value weights: NPV/NPV = [1/(1 + i)]{[−DA × VA + DL × VL]/NPV} × i The sensitivity of the NPV has a duration equal to the term in brackets. An alternative designation for the duration of NPV is the ‘duration of equity’. It implies dividing the duration gap in value by the NPV. This formula expresses the duration of NPV in years rather than in value. However, it leverages the unweighted duration gap (DA − DL ) by dividing by the NPV, a lower value than those of assets and liabilities. For instance, using the orders of magnitudes of book values, we can consider approximately that VL = 96% × VA since equity is 4% of assets (weighted according to regulatory forfeits). Assuming that the NPV is at par with the book value, it represents around 4% of the balance sheet. If we have a simple duration gap DA − DL = 1, with DA = 2 while DL = 1, the weighted duration gap is around: −2 × 100% + 1 × 96% = −1.04 The ‘equity’ or NPV duration is −1.04/4% = −26, which is much higher than the unweighted duration gap because of the leverage effect of equity on debt. On the other

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hand, the ratio of the durations of assets and liabilities is around DA /DL = 0.96/1, or 96%. These conditions are extremely simple. The durations of assets and liabilities are averages of durations of individual lines weighted with their market values. Therefore, the duration of any portfolio is easy to obtain from the durations of the assets. This makes it easy to find out which assets or liabilities create the major imbalances in the portfolio.

Leverage The leverage is the ratio of debt to equity. With market values, the change of market leverage results from the durations of assets and liabilities. The condition of immunization of the market leverage is very simple, once calculations are done4 : DA = DL This condition says that the durations of assets and liabilities should match, or that the duration gap should be set at zero. By contrast, immunization of the NPV would imply different durations of assets and liabilities, proportional to their relative values, which necessarily differ in general because of equity. The intuition behind this condition is simple. When durations match, any change in rates generates the same percentage change of asset and debt values. Since the change of the debt to asset ratio is the difference between these two percentage changes, it becomes zero. If the debt to asset ratio is constant, the debt to equity ratio is also constant. Hence, leverage is immune to changes in interest rates.

The Market Return on Equity for a Given Horizon The market return on equity is the return on the net value of assets and liabilities. Its duration is that of the NPV. Setting the duration to the given horizon locks in the return of the portfolio over that horizon: DE = DNPV = H

CONTROLLING DURATION WITH DERIVATIVES The adjustment of duration necessitates changing the weights of durations of the various items in the balance sheet. Unfortunately, customers determine what they want, which 4 The

change in asset to equity ratio is: [VA/VL]/i = [−1/E 2 ]{E × VA/i − VA × E/i} = −[1/E 2 ]{[−E × VA × DA /(1 + i)] + [VA × E × DE /(1 + i)]}

This can be simplified to: [VA/E]/i = [−VA × E/E 2 ][1/(1 + i)](DA − DL ) The sensitivity of the ratio is zero when DA = DL .

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sets the duration value. Hence, modifying directly the durations of assets and liabilities is not the easiest way to adjust the portfolio duration. An alternative method is to use derivatives. Hedges, such as interest rate swaps, modify duration to the extent that they modify the interest flows. When converting a fixed rate asset into a variable rate asset through an interest rate swap, the value becomes insensitive to interest rates, except for the discounted value of the excess margin over the reference rate. In the case where the excess margin is zero, we have a ‘floater’. The value of a floater is constant because the discounted value of future cash flows, both variable interest revenues and capital repayments, remains equal to face value. Its duration is zero. When the floater provides an excess margin over the discount rates, the value becomes sensitive to rate because its value is that of a floater plus the discounted value of the constant margin. But its duration becomes much lower than a fixed rate asset with the same face value and the same amortizing profile because the sensitivity relates to the margin amount only, which is much smaller than the face value of the asset. Future contracts have durations identical to the underlying asset. Any transaction on the futures market is therefore equivalent to lending or borrowing the underlying asset. Hence, futures provide a flexible way to adjust durations through off-balance sheet transactions rather than on-balance sheet adjustments.

APPENDIX: THE IMMUNIZATION OF THE NET MARGIN OVER A PERIOD The horizon is 1 year. The rate-sensitive assets within this period are Aj for date j and the rate-sensitive liabilities are Lk for date k. Before assets and liabilities mature, it is assumed that the assets generate a fixed return rj until date j and that the liabilities generate a fixed cost ij until date j . When they roll over, the rates take the new values prevailing at dates j and k (Figure 23.3). Aj 1−tj

tj 0

1

tk tj

1−tk

Lk

FIGURE 23.3

Time profile of cash flows within a period

The reinvestment of the inflows Aj occurs over the residual period 1 − tj . The refunding of the outflows Lk occurs during the residual period 1 − tk . Until the revision dates, the

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287

return rj and the costs rk are fixed. At the time of renewal, rate resets occur according to market conditions. The reinvestment, or the new funding, uses the rates ij or ik , assumed constant until the end of the period. The Interest Margin (IM) of the period results from both historical rates and the new rates used for rolling over the flows. The interests and costs accrue from the dates of the flows, j or k, up to 1 year. The residual periods are 1 − tj and 1 − tk . The net margin is: {Aj [(1 + rj )tj (1 + ij )1−tj − (1 + rj )tj ]} IM = j

−

k

{Lk [(1 + rk )tk (1 + ik )1−tk − (1 + rk )tk ]}

This expression combines revenues and costs calculated at the historical rates with those calculated after the rollover of transactions at the new rates. For instance, an inflow of 100 occurs at tj = 30, with a historical rate of 10% before renewal. The rate jumps to 12% after reinvestment. The total interest flow up to the end of the period is: 100(1 + 10%)30/360 [(1 + 12%)330/360 − 1] The amount 100(1 + 10%)30/360 is the future value of the capital at the renewal date. We reinvest this amount up to 1 year at 12%. We deduct the amount of capital from the future value to obtain the interest flows only. To find out when the net margin is immune to change in interest rates, we calculate the derivative with respect to a change in interest rate and make it equal to zero. The change in rate is a parallel shift λ of the yield curve: ij −→ ij + λ ik −→ ik + λ The derivative of the margin with respect to λ is: [Aj (1 + rj )tj (1 + ij ) − tj (1 − tj )] δIM/δλ = j

−

k

[Lk (1 + rk )tk (1 + ik ) − tk (1 − tk )]

The market values of all assets and liabilities using the historical rates prevailing at the beginning of the period are: VAj = Aj (1 + rj )tj /(1 + ij )tj

and VLk = Lk (1 + rk )tk /(1 + ij )tk

Using these market values, the expression of the derivative simplifies and becomes: VLk (1 − tk ) VAj (1 − tj ) − δIM/δλ = j

k

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This condition becomes, using VA and VL for the market values of all assets and all liabilities: VA × 1− tk VLk /VL = 0 1− tj VAj /VA − VL × j

k

VA and VL are weighted by 1 minus the present values of all assets and liabilities times the reset dates. The weights are equal to 1 minus the duration of assets and liabilities DA and DL . This formula is in the text.

24 NPV and Convexity Risks

Using NPV as target variable, its expected value measures profitability, while risk results from the distribution of NPV around its mean resulting from interest rate variations. Market values and interest rates vary inversely because market values discount the future flows with interest rates. The NPV–interest rate profile is an intermediate step visualizing how interest rate deviations alter the NPV. It is convenient for making explicit ‘convexity risk’. The NPV risk results from its probability distribution, which can serve for calculating an Asset–Liability Management (ALM) ‘Value at Risk’ (VaR). Convexity risk arises from the shape of the relationship between asset and liability values to interest rates. Both values vary inversely with rates, but the slopes as well as the curvatures of the asset–rate curve and the liability–rate curve generally differ. When slopes only differ, the NPV is sensitive to interest rate deviations because there is a mismatch between asset value changes and liability value changes. Controlling the duration gap between assets and liabilities allows us to maintain the overall sensitivities of the NPV within desired bounds. When interest rate drifts get wider, the difference in curvatures, or ‘convexities’, of the asset–rate and the liability–rate profiles becomes significant, resulting in significant variations of the NPV even when slopes are similar at the current rate. This is ‘convexity risk’, which becomes relevant in times of high interest rate volatility. Options magnify convexity risk. ‘Gapping’ techniques ignore implicit options turning a fixed rate loan into a variable rate one because of renegotiation, or a variable rate loan into a fixed rate one because of a contractual cap on the variable rate. With such options, the NPV variation ceases to be approximately proportional to interest rate deviations, because the NPV durations are not constant when rates vary widely and when options change duration. ‘Convexity risk’ becomes relevant in the presence of interest rate options even when interest rate volatility remains limited. Finally, NPV sensitivity results from both duration and convexity mismatches of assets and liabilities. Convexity risk remains hidden unless interest rates vary significantly.

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NPV–interest rate profiles visualize this risk. Interest rate volatility and implicit options make a strong case for measuring the downside risk of the balance sheet. NPV is an adequate target variable for revealing such risk because it captures long-term profitability when interest rate shifts become more important. Even though there is no capital requirement for ALM risk, VaR techniques applied to NPV capture this downside risk and value it. This chapter focuses on the specifics of ‘convexity’ risk for the banking portfolio starting from the ‘market value–interest rate’ profiles of assets and liabilities. The first section details the relationship between the market value of the balance sheet and the interest rate. The second section visualizes duration and convexity effects. The third section illustrates how various asset and liability duration and convexity mismatches alter the shape of the NPV–interest rate profile, eventually leading to strong adverse deviations of NPV when interest rates deviate both upwards and downwards. The last section addresses hedging issues in the context of low and high volatility of interest rates. Chapter 25 discusses the valuation of downside NPV risk with VaR techniques.

THE MARKET VALUE OF THE BALANCE SHEET AND THE INTEREST RATE The sensitivity to interest rate variations of the NPV is interesting because the NPV captures the entire stream of future flows of margins. The easiest way to visualize the interest risk exposure of the NPV is to use the ‘market value–interest rate’ profile. For any asset, the value varies inversely with interest rates because the value discounts future cash flows with a discount rate that varies. The relationship is not linear (Figure 24.1). The curvature of the shape looks upwards. This profile is a complete representation of the sensitivity of the market value. Market value

V

i

FIGURE 24.1

Rate

The market value–interest rate profile

For the NPV, the profile results from the difference of two profiles: the asset profile and the liability profile. If the NPV calculation uses the costs of liabilities (wacc), it is equal to the present value of all streams of margins up to the longest maturity, except for equity. The shape of the NPV profile results from those of the asset and liability profiles, as shown in Figure 24.2.

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291

Value

Assets

Liabilities

Rate NPV

Rate

FIGURE 24.2

The NPV–interest rate profile

The two profiles, for assets and liabilities, have the usual shape with an upward looking convexity. However, the difference, the NPV profile, has a shape that depends on the relative curvatures of the profiles of assets and liabilities. Since the market values of assets and liabilities can vary over a large range, and since their curvatures are generally different, the NPV profile can have very different shapes. In Figure 24.2, the NPV is positive, and the curvature of liabilities exceeds that of assets. Nevertheless, this is only an example among many possibilities. In this example, the NPV is almost insensitive to interest rate variations. The sensitivity analysis of NPV leads to duration gaps and suggests matching the durations of assets and liabilities to make the NPV immune to variations of interest rates. However, this is a ‘local’ (context-dependent) rule. Sensitivities of both assets and liabilities and, therefore, the sensitivity of the NPV depend on the level of interest rates. When the variation of interest rates becomes important, the overall profile provides a better picture of interest rate risk. Neutralizing the duration gap is acceptable when the variations of interest rates are small. If they get larger, the NPV again becomes sensitive to market movements. The next section discusses duration and convexity.

DURATION AND CONVEXITY The duration is a good criterion whenever interest rate variations remain small and when the convexity of asset and liability profiles with rates remains small. If one of these two conditions does not hold, matching durations do not make the NPV immune to rate changes and the duration gaps cease to be a good measure of sensitivity. Since the duration is the first derivative of value with respect to interest rate, the second derivative shows how it changes with rates, and there are techniques to overcome the duration limitations and provide a better proxy for the real ‘value–rate’ profile.

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Convexity The sensitivity of a financial asset is the variation of its value when the market rate moves. Graphically, the slope of the ‘market value–interest rate’ profile relates to duration. When the interest rate moves, so does the duration. Graphically, the curvature of the profile shows how the slope changes. The curvature means that the sensitivity to downward moves of the interest rate is higher than the sensitivity to upward moves. The effect of a decrease in rates, from 9% to 8%, is bigger than the change generated by a move from 4% to 3%. ‘Convexity’ measures the change in duration when the rates move. Because of convexity, the market value changes are not linear. For small variations of rates, the duration does capture the change in asset value. Nevertheless, for bigger variations, the convexity alters the sensitivity significantly (Figure 24.3). Market values

Sensitivity

Rate

FIGURE 24.3

Sensitivity and the level of interest rates

Visually, convexity is the change in slope that measures duration at various levels of interest rates.

The Sources of Convexity The first source of convexity is that the relationship between market value and discount rates is not linear. This is obvious since the calculation of a present value uses discount factors, such as (1 + i)t . A fraction of convexity comes from this mathematical formula. Because of discounting, convexity depends on the dates of flows. The sensitivity is proportional to duration, which represents a market value weighted average maturity of future flows. Convexity, on the other hand, is more a function of the dispersion of flows around that average. Financially speaking, modifying convexity is equivalent to changing the time dispersion of flows. A third source of convexity is the existence of options. The relationship between the market value of an option and the underlying parameter looks like a broken line. When the option is out-of-the-money, the sensitivity is very low. When the option is in-themoney, the sensitivity is close to 1. Therefore, the ‘market value–interest rate’ profile is quite different for an option. It shows a very strong curvature when the options are at-the-money. Figure 24.4 shows the profile of a floor that gains value when the interest rate decreases.

NPV AND CONVEXITY RISKS

Value

293

Floor in-the-money, high sensitivity Floor out-of-the-money, low sensitivity

Rate Strike = Guaranteed rate

FIGURE 24.4

Market value–interest rate profile: example of a floor

The implication is that, with options embedded in the balance sheet of banks, the convexity effects increase greatly.

The Measure of Convexity Controlling convexity implies measuring convexity. Mathematically, convexity results, like duration, from the formula that gives the price as the discounted value of future cash flows. The duration is the first-order derivative with respect to the interest rate. The convexity is the second-order derivative. This formula of convexity allows us to calculate it as a function of the cash flows and the interest rate. For options, the convexity results from the equation that relates the price to all parameters that influence its value. Just as for duration, convexity is also a local measure. It depends on the level of interest rate. Therefore, building the ‘market value–interest rate’ profile requires changing the yield curve and revaluing the portfolio of assets and liabilities for all values of the interest rates. Simulations allow us to obtain a complete view of the entire profile. The convexity has value in the market because it provides an additional return when the volatility of interest rates becomes significant. The higher the volatility of interest rates, the higher the value of convexity. The implication is that it is better to have a higher convexity of assets when the interest rate volatility increases, because this would increase the NPV. This intuition results from Figure 24.5. The convexity of assets makes Market value of assets

V

i

FIGURE 24.5

The value of convexity

Rate

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the average value of assets between two distant interest rates higher than the current value. Hence, interest rate volatility increases the expected value of assets. The converse is also true. When convexity is negative, a high volatility of rates results in a lower value of assets, everything else held constant. The value of convexity alters the NPV–rate profile when large deviations of rates occur. The appendix to this chapter expands the visualization with both assets and liabilities having a common duration but different convexities.

THE SENSITIVITY OF NPV The value–interest profiles of NPV have various shapes depending on the relative durations and relative convexities of the assets and liabilities. Some are desirable profiles and others are unfavourable. What follows describes various situations.

Duration Mismatch A duration mismatch—weighted by asset and liability values—between assets and liabilities makes the NPV sensitive to rates. The change in NPV is proportional to the change in interest rates since the values of assets and liabilities are approximately linear functions of the rate change. The NPV changes if the durations do not match. Figure 24.6 shows that there is a value of interest rates such that the NPV is zero, reached when rates decrease. When the interest rate increases, the NPV increases. Value

Liabilities

Assets

Rate NPV

Rate

FIGURE 24.6

Value–rate profile and NPV–duration mismatch

By modifying and neutralizing the duration gap, it is possible to have a more favourable case. In Figure 24.7, the convexities are not very important, and the durations of assets

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Small variations of rates

Value

Liabilities Assets

NPV

Rate

Rate

FIGURE 24.7

Variations in NPV with matched durations and different convexities

and liabilities match at the level where the NPV reaches its maximum. With significant changes in interest rates, the NPV remains positive. However, the different convexities weaken the protection of matching durations when interest rates move significantly away from the current value. With options embedded in the balance sheet, the NPV becomes unstable, because options increase convexity effects. In such cases, the simulation of the entire profile of the NPV becomes important.

The Influence of Implicit Options Options are the major source of convexity in balance sheets, and the main source of errors when using local measures of sensitivity to determine risk. They cap the values of assets and impose floors on the values of liabilities. For instance, the value of a loan with a prepayment option cannot go above some maximum value since the borrower can repay the loan at a fixed price (outstanding capital plus penalty). Even though the borrower might not react immediately, he will do so beyond some downward variation of the interest rate. In this case, a decline in interest rate generates an increase in the value of the asset, until we reach the cap when renegotiation occurs. The value of a straight debt (without prepayment option) would increase beyond that cap. The difference between the cap and the value of the straight debt is the value of the option. On the liability side, options generate a floor for the value of some liabilities when the interest rates rise. This can happen with a deposit that earns no return at all, or provides a fixed rate to depositors. Beyond some upper level of market rates, the value of the deposit hits a floor instead of decreasing with the rise in interest rates. This is when depositors shift their funds towards interest-bearing assets. In such a case, even if the resources stay within the bank, they keep from now on a constant value when rates increase since they become variable rate assets.

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Everything happens as if fixed rate assets and liabilities behaved as variable rate assets and liabilities beyond some variations of interest rates. The shape of the ‘value–interest rate profile’ flattens and becomes horizontal when hitting such caps and floors, as shown in Figure 24.8. Profile of the asset without option

Value Value of the option

Loan with prepayment Deposits

Rate

FIGURE 24.8

Values of assets and liabilities under large interest rate shifts

NPV and Optional Risks The options generate a worst-case situation for banks. The combination of cap values of assets with floor values of liabilities generates a ‘scissor effect’ on the NPV (Figure 24.9). Value Capped value of assets

Floor value of liabilities

Rate NPV Rate

NPV < 0

FIGURE 24.9

NPV > 0 NPV > 0

The ‘scissor effect’ on NPV due to embedded options

This is because the convexity of the assets is the opposite of the convexity of the liabilities beyond some variations. When the interest rate is in the normal range, the NPV remains

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positive. However, if it deviates beyond upper and lower bounds, the options gain value. The change in rate generates a negative NPV beyond those values that trigger the caps or floors. This happens for both upward and downward variations of the interest rates. Since there are multiple embedded options with varying exercise prices, the change is progressive.

CONTROLLING AND OPTIMIZING THE NPV RISK–RETURN PROFILE The goal of interest rate policies is to control the risk of the NPV through both duration and convexity gaps. The common practice is to limit the NPV sensitivity by closing down the duration gap. However, this is not sufficient to narrow the convexity gap. Theoretically, it is, however, possible to eliminate the risk of negative NPV and turn the NPV convexity upside down.

Duration and Convexity Gaps To immunize the NPV to changes in interest rates, the asset-weighted duration and the liability-weighted duration should match. This policy works for limited variations of rates. In order to keep the NPV immune to interest rates for larger variations, the convexities of assets and liabilities should also match. There is a better solution. When the convexity of assets is higher than the convexity of liabilities, any variation of interest rate increases the NPV as shown in Figure 24.10. The NPV then has a higher expected value when the volatility of rates increases. Value

Rate NPV

Rate

FIGURE 24.10

Convexities and NPV sensitivity to interest rate optimization

Controlling Convexity Risk The theoretical answer to protect the NPV against duration and convexity risks is to match duration and increase the convexity of assets. Optional hedges allow us to modify

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the NPV–interest rate profile, to obtain the desired shape. Theoretically, floors protect the lender against a decrease in return from loans. Caps protect the bank against a rise in cost of resources. Floor values will decrease significantly when interest rates decrease significantly. Cap values will increase the NPV when interest rates increase significantly. Both caps and floors correct convexity effects and flatten the asset and liability profiles when rates change. Since optional hedges are expensive, we face the usual trade-off between the cost of hedging and the gain from hedging. Any technique to reduce the cost of such hedges helps. The simplest is to set up an early hedge out-of-the-money. Doing otherwise might imply a prohibitive decrease in NPV.

APPENDIX: WHERE DOES CONVEXITY VALUE COME FROM? The value of convexity appears visually when considering the ‘market value–interest rate’ profile. To make the comparison between two assets of different convexities visible, it is preferable to start with assets of equal value and equal duration, but with different convexities (Figure 24.11).

High convexity

Low convexity

Value High convexity

Max = 17 Max = 15 Mean = 14

Low convexity

Mean = 12.5 Min = 11

Min = 10 Min = 6%

Value of convexity = 14 − 12.5

FIGURE 24.11

Max = 10%

Rate

Expected rate = 8%

Expected values of assets of different convexities

It is not obvious how to define assets with the same value and the same duration. One way is to use a zero-coupon and a composite asset called a ‘dumbbell’. The zero-coupon duration is equal to its maturity. The dumbbell combines two zero-coupons of different durations. The dumbbell duration is the weighted average of the duration of the two zerocoupons. If the average duration of this portfolio matches that of the zero-coupon, we have portfolios of identical market value and identical duration. A property of zero-coupons is to have the lowest convexity, everything else being equal1 . The ‘market value–interest rate’ profiles of these assets, the single zero-coupon and the dumbbell, look like those of Figure 24.11. They intersect at the current price and have 1 What

drives convexity is not always intuitive. This property holds only when everything else is equal. Otherwise, it might not be true.

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the same tangent at this price, since they have the same sensitivity. However, when the interest rate moves away from its original value, the market values diverge because of the differences in convexity. Under certainty, the two assets would have the same value. Under uncertainty, they have an expected value. Uncertainty begins with two possible values of interest rate. Two interest rates (6% and 10%) with equal probabilities result in a common expected value of interest rate identical to the current value (8%). Even though the volatility of interest rates increases when the variations from the mean grow bigger, the average value of the interest rate stays the same. However, the expected value of the asset changes when the rate volatility increases. The reason is that the increase in value when the interest rate declines is higher than the decrease in value when the interest rate moves upwards by the same amount. This asymmetry in the deviation of market values is higher when the convexity (the curvature of the profile) is higher. The expected value of the assets under the two equal probability rate scenarios is the average of the corresponding two values of the assets. The least convex has a value of (15 + 10)/2 = 12.5, and the most convex has an expected value of (17 + 11)/2 = 14. Therefore, the expected value of the most convex asset (14) is higher than the expected value of the least convex asset (12.5). This difference is the value of convexity (1.5). The value of convexity starts at zero, under certainty, and increases with the volatility of rates. Convexity explains the behaviour of the NPV when the volatility of rates increases. With the same variations, it depends essentially on the duration gap. With larger deviations of rates, it depends also on the convexity gap.

25 NPV Distribution and VaR

Multiple simulations serve to optimize hedges and determine interest income at risk, or ‘Earnings at Risk’ (EaR) for interest rate risk as the worst-case deviation with a preset confidence level of interest income. The same technique transposes readily to Net Present Value (NPV) and values downside risk when both duration and convexity mismatches magnify the NPV risk. The current regulations do not impose capital requirements for Asset–Liability Management (ALM), making the ALM ‘Value at Risk’ (VaR) of lower priority than market or credit risk VaR. However, the important adverse effects of convexity risk suggest that it is worthwhile, if not necessary, to work out NPV VaR when considering long-term horizons. The construction of a distribution of NPV at future time points follows the same basic steps as other VaR calculations. The specific of ALM VaR is that interest rate variations beyond some upper and lower bounds trigger adverse effects whatever the direction of interest rate moves, because of convexity risk. Such adverse effects make it necessary both to value this risk and to find these bounds. The same remarks apply for interest income as well, except that NPV crystallizes in a single figure the long-term risk, which short-term simulations of interest income might not reveal or underestimate. The first section discusses the specifics of NPV downside risk. The second section shows that modelling the NPV VaR uses similar techniques as market risk VaR. The third section illustrates the techniques with a simplified example, in two steps: using the Delta VaR technique designed for market risk (see Section 11 of this book); showing how convexity risk alters the shape of the NPV distribution and the NPV VaR.

OVERVIEW AND SPECIFICS OF NPV RISK AND DOWNSIDE RISK VaR applies to NPV as it does for other risks, but there are a number of basic differences. They result in an NPV distribution whose profile is highly skewed when the balance sheet embeds a significant convexity risk.

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301

Interest Income and NPV Distributions The interest income VaR approach looks similar to the EaR approach. In fact, it differs because the interest income results from explicit calculations, whereas EaR concentrates on observed earnings rather than modelling them. In addition, EaR does not use simulations, whereas ALM models do. When transposing the technique to NPV, it is sufficient to identify the main risk drivers and their distributions. These are a small number of correlated interest rates. Characterizing the NPV risk traditionally uses the NPV sensitivity. The higher the sensitivity, the higher the risk. In addition, setting limits implies bounding this sensitivity. To get downside risk requires going one step further. This is not current practice, but its value is to demonstrate that the same technique applies to various risks and to demonstrate the value of the NPV downside risk. The next step consists of calculating a distribution of NPV values at future time points. The time points need to fit the needs of ALM to have medium-term visibility. A 1 to 2-year horizon is adequate, as for credit risk. Determining a forward distribution of NPV at the 1-year time point requires considering all cash flows accruing from assets and liabilities up to maturity. ALM models consider business risk over some short to medium-term horizon. Since we need probability distributions, we ignore business risk here, thereby getting closer to the ‘crystallized’ portfolio approach of market and credit risks. The usual steps are generating correlated distributions of the main interest rate drivers and revaluing the NPV for each scenario. In what follows, we perform the exercise using only a flat yield curve and a single rate. Using correlated interest rates would require the techniques to be expanded for market and credit risks. Because of the relatively high correlations of interest rates for a single currency, parallel shifts of yield curve assuming perfect correlations are acceptable as a first proxy. They do not suffice, however, because a fraction of the interest income results from the maturity gap between assets and liabilities, making both interest income and NPV sensitive to the steepness of the yield curve. Going further would entail using correlations or factor models of the yield curve. For illustration purposes, there is no need to do so here. Deriving the NPV value from its current sensitivities is not adequate for a medium-term horizon and in the presence of options. Full simulations of the NPV values, requiring revaluations of all assets and liabilities, and of options, are necessary to capture the extreme deviations when discussing VaR. Without option risk, the NPV distribution would be similar to that of interest rates, since interest margin variations remain more or less proportional to interest rate deviations. With options, the NPV distribution drifts away from the interest rate distribution because options break down this simple relationship and introduce abrupt changes in the sensitivity to interest rates. Because of convexity risk, the reason for fat tails in the NPV distributions lies with the adverse effect of large deviations of interest rates on both sides of current interest rates. The fat tails of the NPV distribution result from convexity risk. Once the NPV distribution is identified, it becomes easy to derive the expected value and the worst-case deviations at preset confidence levels. Since the NPV is usually positive, losses occur once deviations exceed the expected NPV value. The VaR is this adverse deviation minus any positive expected NPV.

Downside Risk and Convexity Downside risk appears when interest rates deviate both upwards and downwards. From a VaR standpoint, this is a specific situation. It stems from the nature of implicit options,

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some of them contributing to the decrease in NPV when rates increase, and others when rates decrease. This ‘double adverse exposure’ results from the implicit options in loans and deposits. As a result, downside risk appears in two worst-case scenarios of rising and decreasing rates. Since it is possible to model interest rate deviations, it is easy to assign probabilities to such deviations and to assign probabilities to the NPV adverse changes. Note, however, that the ‘convexity’ risk is not mechanically interest risk-driven. It depends also on behavioural patterns of customers, and demographic or economic factors for prepayments or renegotiations of loans, or on legal and tax factors (for deposits). In addition, we ignore business risk here. We could use business scenarios with assigned probabilities. This would result in several NPV distributions that we should combine into one. For such reasons, the probability distribution of adverse variations of NPV is not purely interest rate-driven. Figure 25.1 shows the VaR of the NPV at a preset confidence level. Since the NPV is positive, losses occur only when the NPV decreases beyond the positive expected value. Probability Interest rate distribution

NPV

Interest rate

NPV distribution Probability + 0 NPV −

FIGURE 25.1

Interest rate distribution NPV at given confidence level

NPV, implicit options and VaR

NPV VAR In order to derive the NPV distribution from interest rate distributions, ignoring other risk factors, a first technique mimics the technique for market risk and, notably, Delta VaR as expanded in Section 11 of this book. It relies on the sensitivity, which is the NPV duration. Current durations do not account for changes in sensitivity due to time drift of duration and convexity risk. Forward calculations of duration correct the first drawback. They imply positioning the portfolio at the future time points and recalculating the durations for each value of the future interest rate. We simplify somewhat the technique and assume them given. High interest rate volatility is dealt with through convexity later on. The alternative technique uses multiple simulations of risk drivers plus revaluation of the NPV for each set of trial values of rates, to obtain the NPV distribution, similarly to the simulation technique used for market risk. We summarize here the main steps for both techniques. The process replicates the techniques expanded in Chapter 29 and serves as a transition. The sensitivity of the NPV is the net duration of assets and liabilities, weighted by the mark-to-market value of assets and liabilities. This allows us to convert any drift of rates into a drift of NPVs. Proceeding along such lines makes the NPV random, as the interest rate is. We use bold characters in what follows for random variables. As usual,

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VaR requires looking for the maximum variations of the NPV resulting from the shift of interest rates. Let Si be the sensitivity of the NPV to interest rate i, this sensitivity resulting from the above duration formula. The volatility of the NPV is: σ (NPV) = Si × σ (i) With a given interest rate volatility, the maximum change, at a given confidence level, is a multiple of the volatility. The multiple results from the shape of the distribution of interest rates. With a normal curve, as a proxy for simplification, a multiple of 2.33 provides the maximum change at 1% confidence level: VaR = 2.33 × Si × σ (i) When there are several interest rates to consider, and when a common shift is not an acceptable assumption, the proxy for the variation of the NPV is a linear combination of the variations due to a change in each interest rate selected as a risk driver: NPV = S1 × i1 + S2 × i2 + S3 × i3 + · · · The indices 1, 2, 3 refer to different interest rates. Since all interest rate changes are uncertain, the volatility of the NPV is the volatility of a sum of random variables. Deriving the volatility of this sum requires assumptions on the correlations between interest rates. The issue is the same for market risk except that interest rates correlate more than other market parameters do. This simple formula is the basic relationship of the simple Delta VaR model for market risk. The distribution of the NPV changes is the sum of the random changes of the mark-to-market values of the individual transactions, which depend on those of underlying rates. Using the normal approximation makes the linear NPV changes normal as well. Figure 25.2 represents the distribution and VaR for NPV at a single time point in the future. V NP

Probabilities

Lower bound of NPV at a given confidence level

e

Tim R

Va

FIGURE 25.2

Projecting NPV at a future horizon (no convexity risk)

In fact, convexity risk plus the long-term horizon makes this technique inappropriate. When time passes, duration drifts. In addition, interest rate volatility grows and activates options. The Delta VaR relationship ceases to apply. Multiple revaluations of NPV for various sets of rates are necessary to capture the changes in durations and NPV. In order

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to illustrate the effect of convexity risk, we develop in the next section a simplified example.

MULTIPLE SIMULATIONS AND NPV VAR This section provides simulations of NPV with varying rates and uses Monte Carlo simulation, with a flat yield curve, to calculate the NPV VaR. In order to proceed through the various steps, we need to simulate random values of a single risk driver, which is a flat market rate. The forward revaluation of the NPV is straightforward with only two assets, the loan portfolio and the debt. Assets and debt are both fixed rate in a first stage. We introduce convexity risk later on in the example using a variable rate loan with a cap on the interest rate of the loan. Using ‘with and without’ cap simulations illustrates how the cap changes the NPV distribution. Revaluation depends on the interest rates of assets and liabilities. The simulation without cap uses a fixed rate, while the simulation with cap on the variable rate loan necessitates that the loan be variable up to the cap. The first subsection provides a set of simplified data. The next two subsections provide the two simulations.

The Bank’s Balance Sheet Data The bank sample portfolio is as given in Table 25.1. TABLE 25.1

Market Asset Liability Date Asset Liability

Sample bank portfolio Rate 8.0% Rate

Amount

9.0% 8.0% 0

1000.00 960.00 1 90.00 1036.80

2

3

4

5

90.00

90.00

90.00

1090.00

The bank NPV at the flat yield curve, with interest rates at 8%, is the present value at 8% of all asset and liability cash flows, as in Table 25.2. TABLE 25.2 Sample bank portfolio NPV Present value Asset Liability NPV

1123.12 1036.80 86.32

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When the interest rate increases, both values of assets and liabilities decrease. There are two cases: without significant convexity risk and with convexity risk for assets due to a prepayment option.

NPV VaR with Fixed Rate Asset and Liability In the simpler case, without convexity, both asset and debt have fixed rates, the loan yielding a 9% rate and the liability rate being 8%. The NPV calculation is a forward calculation at date 1. Since the liability matures at date 1, its value remains constant whatever the rate value because there is no discounting at this date. Both asset and liability rates are fixed rates. The asset and liability value–rate profiles and the NPV profile when the market rate changes are given in Figure 25.3. 1600 Asset value

1400 1200 Values & NPV

1000 800

Debt value

600 NPV

400 200 0 −200 0%

2%

4%

6%

8%

10% 12% 14% 16%

−400 Market Rate

FIGURE 25.3

Asset, debt and NPV values when the market rate changes

The NPV decreases continuously with rates because the asset value decreases, while the debt value remains constant. To calculate the NPV VaR, we vary the interest rate randomly, using a normal curve for the flat rate, with an expected value of 8% and a yearly volatility of 2%1 . With 1000 simulations, we generate the interest rate values and the corresponding NPV values. With the distribution of NPV values, we derive the NPV at various confidence levels and the VaR at these confidence levels. The break-even rate making the NPV negative is around 11% (Figure 25.4). The NPV reaches negative values, triggering losses. The VaR requires setting confidence levels. The VaR at various NPV percentiles results from the distribution (Figure 25.5). The graph shows that the NPV VaR at 1% confidence level is close to −60. The NPV expected value is 86.32, and the NPV (1%) shows that the loss from the expected value is close to 146 (86 + 60). 1 We

need to ensure that the interest rate does not become negative. We could also use a lognormal curve for the market rate. In this simulation, the rate remains well above 0.

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12% 10%

NPV

8% 6% 4% 2% 0% −20

−10

0

0

0

100

200

300

400

Frequency (%)

FIGURE 25.4

NPV distribution with market rate changes 20

NPV

0 −20 0%

2%

4%

6%

8%

10%

−40 −60 −80 Confidence Level

FIGURE 25.5

NPV values and confidence levels

NPV VaR with Convexity Risk in Assets In the second case, we simulate convexity for the asset by introducing a formula that caps the asset rate: asset rate = max(market rate + 1%, 9%). The asset rate is either the market rate plus a margin of 1% above the market rate, or 9%. It is a floater when the market rate is lower than 8%. When the market rate hits 8%, the loan has a fixed rate equal to 9%. This simulates a prepayment option triggered at 9% by borrowers. The asset value varies when the market rate changes, with a change of slope at 8%. The increase with rate when rate values are lower than 8% results from the discounting flows indexed to rates, and when the asset rate hits the 9% cap the asset value starts decreasing. The liability value remains constant because we have a unique flow at date 1, which is the date for the forward valuation. Consequently, the NPV shows a bump at 8%, and becomes negative when the market rate exceeds some upper bound close to 11% (Figure 25.6). Figure 25.7 shows the NPV bump. When proceeding with similar simulations of market rate as above, we obtain the NPV distribution and the NPV percentiles for deriving the VaR (Figure 25.8).

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Asset value

1200 1000

Liability value Values & NPV

800 600 400

NPV

200 0 −200 0%

2%

4%

6%

8%

10%

12%

14%

16%

Market Rate

FIGURE 25.6

Asset, debt and NPV values when the market rate changes 100 50

NPV

0 −50

0%

2%

4%

6%

8%

10%

12%

−100 −150 Market Rate

FIGURE 25.7

NPV ‘bump’ when the market rate changes 45% 40% 35% Frequency

30% 25% 20% 15% 10% 5% 0% −150

−100

−50

0

50

NPV

FIGURE 25.8

Distribution of NPV with market rate changes

14%

16%

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20

NPV

0 −20 0%

2%

4%

6%

8%

10%

12%

−40 −60 −80 Confidence Level

FIGURE 25.9

NPV values and confidence levels

The NPV distribution becomes highly skewed, illustrating convexity effects. The bump effectively flattens the NPV–rate profile, with a concentration of value around the bump which results in a much higher frequency around the maximum value. On the other hand, both increases and decreases in market rate have an adverse effect due to convexity risk. Hence, there are higher frequencies of adverse deviations and negative values (Figure 25.9).

SECTION 9 Funds Transfer Pricing

26 FTP Systems

The two main tools for integrating global risk management with decision-making are the Funds Transfer Pricing (FTP) system and the capital allocation system. As a reminder, FTP serves to allocate interest income, while the capital allocation system serves to allocate risks. Transfer prices serve as reference rates for calculating interest income of transactions, product lines, market segments and business units. They also transfer the liquidity and the interest rate risks from the ‘business sphere’ to Asset–Liability Management (ALM). The capital allocation system is the complement to the FTP system, since it allocates capital, a necessary step for the calculation of risk-based pricing and risk-based performance (ex post). Chapters 51 and 52, discussing ‘risk contributions’, address the capital allocation issue. The current chapter addresses three major FTP issues for commercial banking: • The goals of the transfer pricing system. • The transfer of funds across business units and with the ALM units. • The measurement of performance given the transfer price. It does not address the issue of defining economic transfer prices, deferred to the next chapter, and assumes them given at this first stage. The FTP system specifications are: • Transferring funds between units. • Breaking down interest income by transaction or client, or any subportfolio such as business units, product families or market segments. • Setting target profitability for business units. • Transferring interest rate risk, which is beyond the control of business units, to ALM. ALM missions are to maintain interest rate risk within limits while minimizing the cost of funding or maximizing the return of investments.

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• Pricing funds to business units with economic benchmarks, using economic transfer prices. • Eventually combining economic prices with commercial incentives. An internal system exchanges capital between units. Transfers go through ALM. The ALM unit is the central pole buying all resources from business lines, collecting them through deposits and selling funds to the lending business lines. Transfer prices allow the calculation of interest income based on transfer prices, in such a way that the algebraic addition of interest income of all units, including ALM, sums up to the accounting interest income of the overall banking portfolio (because all internal sales and purchases of funds compensate). The first section introduces the specifications of the transfer pricing system. The second section discusses the netting mechanisms for excesses and deficits of funds within the bank. The third section details the calculation of performance, for business lines as well as for the entire bank, and its breakdown by business units. The fourth section provides simple calculations of interest income, and shows how the interest incomes allocated to business units or individual transactions add up exactly to the bank’s overall margin. The fifth section addresses the choice of target profitability and risk limits for ALM and business lines. The last section contrasts the commercial and financial views of transfer prices, and raises the issue of defining ‘economic’ transfer prices, providing a transition to the next chapter.

THE GOALS OF THE TRANSFER PRICING SYSTEM The FTP system serves several major purposes, to: • Allocate funds within the banks. • Calculate the performance margins of a transaction or any subportfolio of transactions and its contributions to the overall margin of the bank (revenues allocation). • Define economic benchmarks for pricing and performance measurement purposes. This implies choosing the right reference for economic transfer prices. The ‘all-in’ cost of funds to the bank provides this reference. • Define pricing policies: risk-based pricing is the pricing that would compensate the risks of the bank, independent of whether this pricing is effective or not, because of competition, in line with the overall profitability target of the banks. • Provide incentives or penalties, differentiating the transfer prices to bring them in line with the commercial policy, which may or may not be in line with the target risk-based prices. • Provide mispricing reports, making explicit the differences between the effective prices and what they should be, that is the target risk-based pricing. • Transfer liquidity and interest rate risk to the ALM unit, making the performance of business lines independent of market movements that are beyond their control. The list demonstrates that the FTP system is a strategic tool, and that it is the main interface between the commercial sphere and the financial sphere of the bank. Any malfunctioning or inconsistency in the system interferes with commercial and financial management, and

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313

might create a gap between global policies and operations management. Moreover, it is not feasible to implement a risk management system without setting up a consistent and comprehensive FTP system. All business units of a financial institution share a common resource: liquidity. The first function of FTP is to exchange funds between business units with ALM, since business units do not have balanced uses and resources. The FTP nets the balances of sources and uses of funds within the bank. The exchange of funds between units with ALM requires a pricing system. The FTP system serves the other major purpose of setting up internal transfer prices. Transfer prices serve to calculate revenues as spreads between customers’ prices and internal references. Without them, there is no way to calculate internal margins of transactions, product lines, customer or market segments, and business units. Thus, transfer prices provide a major link between the global bank earnings and individual subportfolios or individual transactions. The presentation arranges the above issues in two main groups (Figure 26.1): Allocate funds Measure performance

FTP system

Define economic benchmarks Pricing

Economic transfer prices

Transfer risks to ALM

FIGURE 26.1

The ‘Funds Transfer Pricing’ system and its applications

• The organization of the FTP. • The definition of economic transfer prices.

THE INTERNAL MANAGEMENT OF FUNDS AND NETTING Since uses and resources of funds are generally unbalanced for business units, the FTP system allows netting the differences and allocating funds to those having liquidity deficits, or purchasing excesses where they appear. There are several solutions for organizing the system. An important choice is to decide which balances are ‘netted’ and how.

ALM, Treasury and Management Control The organization varies across banks although, putting together all interested parties, it should allow us to perform the necessary functions of the FTP. Various entities are

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potentially interested in an FTP scheme. Since the Treasury is the unit which, in the end, raises debts or invests excesses in the market, it is obviously interested in the netting of cash flows. Since internal prices also serve to monitor commercial margins, management control is also involved. Since ALM is the unit in charge of managing the liquidity and interest rate exposures of the bank, the internal prices should be consistent with the choices of ALM. These various units participate in transfer pricing from various angles. They might have overlapping missions. The organization might change according to specific management choices. The definition of the scope of each unit should avoid any overlapping. In this chapter, we consider that ALM is in charge of the system, and that other management units are end-users of the system.

Internal Pools of Funds Pools of funds are virtual locations where all funds, excesses and deficits, are centralized. The concept of pools of funds is more analytical than real. It allocates funds and defines the prices of these transfers. Netting

The FTP nets the excesses of some business units with the deficits of others. This necessitates a central pool of resources to group excesses and deficits and net them. The central pool lends to deficit units and purchases the excesses of others. Starting from scratch, the first thought is to use this simplest system. The simplest solution is to use a unique price for such transfers. The system is ‘passive’, since it simply records excesses and deficits and nets them. Some systems use several pools of funds, for instance grouping them according to maturity and setting prices by maturity. Moreover, at first sight, it sounds simple to exchange only the net balances of business units (Figure 26.2). Market

Sale of resources to A

Purchase of net excess of B

Central pooling of net balances Business unit A Deficit of funds

FIGURE 26.2

Business unit B Excess of funds

Transfers of net balances only

Since assets and liabilities are netted before transfer to the central pool, all assets and liabilities, generated by the operations of A and B, do not transit through the system.

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Therefore, the transfer prices apply only to net balances. Their impact is limited to netted balances. The system is simple. The flows are consequences of the operations of business units and the fund transfer system does not influence them. A more active management requires a different system. Pricing all Outstanding Balances

Instead of exchanging only the net excesses or deficits, an alternative is that the ALM purchases all resources and sets a price for all uses of funds of business units, without prior local netting of assets and liabilities. The full amounts of assets and liabilities transit through the central pool. This central pool is the ALM unit. This scheme creates a full internal capital market with internal prices. Post-transfers, the ALM either invests any global excess or funds any global deficit in the capital markets. This system is an active management tool. It does more than ‘record’ the balances resulting from the decisions of business units. The major difference with the system exchanging net local balances only is that the internal prices hit all assets and liabilities of each business unit. By setting transfer prices, the ALM has a powerful leverage on business units. The decision-making process, customer pricing and commercial policy of these units become highly dependent on transfer prices. Transfer prices might also serve as incentives to promote some product lines or market segments through lower prices, or to discourage, through higher prices, the development of others. Such decisions depend on the bank’s business policy. Using such a system in a very active manner remains a management decision. In any case, this scheme makes the FTP a very powerful tool to influence the commercial policies of all business units (Figure 26.3). Market

Sale of all uses of funds

Purchase of all resources

Central pool of all assets and liabilities

Purchase of all resources Sale of all uses of funds

Business unit A

FIGURE 26.3

Business unit B

The central pool of all assets and liabilities

This system serves as a reference in what follows because it ‘hits’ all assets and liabilities. For instance, it was impossible to calculate a margin of an individual transaction in the previous system, since there was no way to ‘hit’ that transaction. Now, the FTP system makes it possible, since all transactions transfer through the central pool. This is a critical property to develop the full functions of the FTP system. Given the powerful potential of this solution, the issue of defining relevant economic prices is critical.

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MEASURING PERFORMANCE Any transfer pricing system separates margins into business, or commercial, margins and financial margins. The commercial margin is the spread between customer prices and internal prices. The financial margin is that of ALM, which results from the volumes exchanged plus the spreads between internal prices and the market prices used to borrow or invest. In addition, calculating the spread is feasible for any individual transactions as well as for any subportfolio. This allows the FTP to isolate the contribution to the overall bank margin of all business units and of ALM. In the FTP, the banking portfolio is the mirror image of the ALM portfolio since ALM buys all liabilities and sells all assets. The sum of the margins generated by the business units and those generated by the ALM balance sheet should be equal to the actual interest margin of the bank (Figure 26.4). Business balance Business sheet balance sheet

Commercial margins Customer prices Transfer prices

ALM balance sheet ALM balance sheet

ALM margins on internal prices Revenues and costs from investing or borrowing in the market

Accounting margin of the bank Accounting margin of the bank

FIGURE 26.4

The bank’s balance sheet and the ALM balance sheet

In general, the summation of all business line internal margins calculated over the transfer prices will differ from the accounting margin since it ignores the ALM margin. Without the latter, there is a missing link between the commercial margins and the accounting earnings of the bank. On the other hand, calculating all margins properly, including the ALM margin, makes them sum the accounting margin of the bank because internal transfers neutralize over the entire bank. The FTP implies an internal analytical recording of all revenues to allow reconciliation with the bank’s income statement. The next paragraph illustrates this reconciliation process. The calculations use a simple example. The commercial margin calculation scope is the entire balance sheet of all business units.

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• For the bank, we simply sum all revenues and costs from lending and borrowing, including any funding on the market. • For the business units, revenues result from customer prices minus the cost of any internal purchase of resources by the central unit (ALM). Costs result from the interest paid to customers (depositors) and the revenues of selling these resources to the ALM at transfer prices charged by the ALM. • For the ALM unit, revenues result from charging the lending units the cost of their funds, and the costs result from purchasing from them their collected resources. In addition, ALM gets a market cost when funding a deficit and a market revenue from investing any global excess of funds.

THE FTP SYSTEM AND MARGIN CALCULATIONS OF BUSINESS UNITS The margin calculations use a simplified balance sheet. After calculating the bank’s accounting margin, we proceed by breaking it down across business units and ALM using the transfer prices. The system accommodates cost allocations for completing the analytical income statements, and any differentiation of multiple transfer prices, down to individual transactions. As long as the ALM records properly internal transactions, the overall accounting profitability reconciles with the interest income breakdown across business lines and transactions.

The Accounting Margin: Example The business balance sheet generates a deficit funded by ALM. The average customer price for borrowers is 12% and the average customer rate paid to depositors is 6%. The ALM borrows in the market at the 9% current market rate. There is a unique transfer price. It applies to both assets sold by ALM to business units and resources purchased by ALM from business units. This unique value is 9.20%. Note that this value differs from the market rate deliberately. There could be several transfer prices, but the principle would be the same. Table 26.1 shows the balance sheet and the averaged customer rates. TABLE 26.1

Assets Loans Total Resources Deposits Funding Total

Bank’s balance sheet Volume

Rate

2000 2000

12.00%

1200 800 2000

6.00% 9.00%

The full cost of resources is the financial cost plus the operating cost, either direct or indirect. The analytical profitability statements, using both transfer prices and cost allocations, follows. The transfer price mechanism ensures the consistency of all analytical

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income statements of the business units. If transfer prices change, they simply transfer margins from one unit to another, but they do not change the overall bank margin. Cost allocations add up to the overall costs as well. To illustrate the process, we subdivide the bank into three business units: • Collecting resources. • Originating loans. • ALM, managing financial risks. Table 26.1 shows the aggregated balance sheet. The volumes of loans, deposits and external funding are under the control of each of the three business units. The direct calculation of the accounting margin is straightforward since it depends only on customer rates and the funding cost by the ALM: 2000 × 12% − 1200 × 6% − 800 × 9% = 96

Breaking Down the Bank Margin into Contributions of Business Units From the average customer rates, the transfer price set by the ALM unit and the market rate (assuming a flat yield curve), we calculate the margins for each commercial unit purchasing or selling funds to the ALM. The ALM unit buys and sells funds internally and funds externally the bank’s deficit. The corresponding margins are determined using the usual set of calculations detailed in Table 26.2. TABLE 26.2 Market rate Transfer price Margin

The calculation of margins 9.00% 9.20% Calculation

Direct calculation of margin Accounting margin 2000 × 12.00% − 1200 × 6.00% − 800 × 9.00% Commercial margins Loans 2000 × (12.00% − 9.20%) Deposits 1200 × (9.20% − 6.00%) Total commercial margin 2000 × (12.00% − 9.20%) + 1200 × (9.20% − 6.00%) ALM margin 2000 × 9.20% − 1200 × 9.20% − 800 × 9.00% Bank margin Commercial margin + ALM margin

Value 96.0 56.0 38.4 94.4 1.6 96.0

With a unique internal price set at 9.20%, the commercial margin as a percentage is 12% − 9.20% = 2.80% for uses of funds and 9.20% − 6% = 2.80% for resources. ALM charges the lending activities at the transfer price and purchases the resources collected at this same price. The total commercial margin is 94.4, and is lower than the bank’s accounting margin. This total commercial margin adds the contributions of the lending activity and the collection of resources. The lending activity generates a margin of 56.0, while the collection of resources generates 38.4. These two activities could be different business units. The system allocates the business margin to the different business units.

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The ALM has symmetrical revenues and costs, plus the cost of funding the deficit of 800 in the market. Its margin is 1.6. The commercial margin is 94.4, but the accounting margin of the bank is 96.0. This is due to the positive ALM margin, because ALM actually overcharges the market rate to the business units with a transfer price of 9.20%, higher than the market rate (9%). If we add the contributions of business units and the ALM margin, the result is 96.0, the accounting margin. The mechanism reconciles the contribution calculations with the bank’s accounting margin. It is easy to verify that, if the ALM margin is zero, the entire bank’s margin becomes equal to the commercial margin. This is a reference case: ALM is neutral and generates neither profit nor loss. Then the entire bank’s margin is ‘in’ the commercial units, as in the calculations below. Note that we keep the customers’ rates constant, so that the internal spreads change between customers’ rates and transfer prices. They become 9% − 8% = 1% for lending and 8% − 3% = 5% for collecting deposits. Evidently, since all customers’ rates are constant as well as the cost of market funds, the accounting margin remains at 4. Nevertheless, the allocation of this same accounting margin between business units and ALM differs (Table 26.3). TABLE 26.3

The calculation of margins

Market rate Transfer price Margin

9.00% 9.00% Calculation

Direct calculation of margin Accounting margin 2000 × 12.00% − 1200 × 6.00% − 800 × 9.00% Commercial margins Loans 2000 × (12.00% − 9.00%) Deposits 1200 × (9.00% − 6.00%) Total commercial margin 2000 × (12.00% − 9.00%) + 1200 × (9.00% − 6.00%) ALM margin 2000 × 9.20% − 1200 × 9.20% − 800 × 9.00% Bank margin Commercial margin + ALM margin

Value 96.0 60.0 36.0 96.0 0.0 96.0

The above examples show that: • Transfer prices allocate the bank’s margin between the business units and ALM. • The overall net interest margin is always equal to the sum of the commercial margin and the ALM margin.

Analytical Income Statements The cost allocation is extremely simplified. We use three different percentages for lending, collecting resources and the ALM units. The percentages apply to volumes (Table 26.4). Combining the cost allocations with the interest margins, we check that all interest margins add up to the overall bank margin and that the aggregated operating income is the bank operating income (Table 26.5).

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TABLE 26.4 units

Operating costs allocated to business

Operating cost

Volume

Loans Collection of resources ALM Total operating cost

2.00% 1.80% 0.08% 3.88%

Cost 2000 1200 3200

40.00 21.60 2.56 64.00

TABLE 26.5 Business unit income statements and bank consolidated income statements Income statement Interest margin Operating cost (2%) Operating income

‘Loans’

‘Deposits’

‘ALM’

‘Bank’

56.0 40.0 16.0

38.4 26.7 11.7

1.6 2.6 −1.0

96.0 69.3 26.7

Differentiation of Transfer Prices The above calculations use a unique transfer price. In practice, it is possible to differentiate the transfer prices by product lines, market segments or individual transactions. If we use a different transfer price for lending and collecting deposits, the customer rates being 12% for lending and 6% for deposits, the margin is: Commercial margin = assets(12% − TPassets ) + liabilities(TPliabilities − 6%) Commercial margin = assets × assets contribution (%) + resources × resources contribution (%) The commercial margin depends directly on the transfer prices and it changes if we use different ones. But whatever transfer prices we use, the ALM uses the same. When consolidating business transaction margins and ALM internal transactions, we end up with the accounting margin of the bank. This provides degrees of freedom in setting transfer prices. It is possible to use economic transfer prices differentiated by transaction, as recommended in the next chapter, and still reconcile all transaction margins with the bank’s accounting margin through the ALM margin. The facility of differentiating prices according to activities, product lines or markets, while maintaining reconciliation through the ALM margin, allows us to choose whatever criteria we want for transfer prices. The next chapter shows that economic prices reflecting the ‘true’ cost of funding are the best ones. However, it might be legitimate to also use transfer prices as commercial signals for developing some markets and products while restricting business on others. As long as we record the transfer prices and their gaps with economic prices, we know the cost of such a commercial policy. Transfer of revenues and costs neutralizes within the bank, whether or not there is a unique transfer price or several. In fact, no matter what the transfer prices are, they allocate the revenues. Setting economic benchmarks for these prices is an issue addressed in the next chapter.

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ALM AND BUSINESS UNIT PROFITABILITY GOALS The overall profitability target should translate into consistent objectives across business units and the ALM unit. We discuss setting objectives for the ALM unit and move on to the definition of business line target profitability levels, from an overall profitability goal. The overall profitability relates to the overall risk of the bank, as target profitability on economic capital. We assume it given, and examine how to break it down.

ALM Profitability and Risks ALM is a special business unit since it can have various missions. If its target profit is set to zero, it behaves as a cost centre whose responsibility is to minimize the cost of funding and hedge the bank against interest rate risk. If the bank has excess funds, the mission of ALM is still to keep the global interest rate risk within limits, while maximizing the return on excess funds. This is consistent with a strategy that makes commercial banking the core business, and where the ALM policy is to hedge liquidity and interest rate risks with cost-minimizing, or revenue-maximizing, objectives. However, minimizing the cost of funding could be an incentive for ALM to maintain some exposure to interest rate risk depending on expectations with respect to interest rates. In general, banking portfolios generate either deficits or excesses of funds. Nevertheless, a policy of hedging systematically all exposures, for all future periods, has a cost. This is the opportunity cost of neutralizing any potential gain resulting from market movements. However, without a full hedge, there is an interest rate risk. The ALM unit should be responsible for this risk and benefit from any saving in the cost of funds obtained thanks to maintaining risk exposure. This saving is the difference between the funding cost under full hedging policy and the effective funding cost of the actual policy conducted by ALM. This cost saving is its Profit and Loss (P&L). Since it is not reasonable to speculate on rates, interest rate risk limits should apply to ALM exposures. Giving such flexibility to ALM turns it into a profit centre, which is in charge of optimizing the funding policy within specified limits on gaps, earnings volatility or ‘Value at Risk’ (VaR). Making such an organization viable requires several conditions. All liquidity and interest rate risks should actually be under ALM control, and the commercial margins should not have any exposure to interest rate risk. This imposes on the FTP system specifications other than above, developed subsequently. In addition, making the ALM a profit centre subject to limits requires proper monitoring of what the funding costs would be under full hedging over a period. Otherwise, it is not possible to compare the effective funding costs with a benchmark (Figure 26.5).

Setting Target Commercial Margins Banks have an overall profitability goal. They need to send signals to business units and allocate target contributions to the various business units consistent with this overall goal. These contributions are the spreads over transfer prices times the volume. Assuming transfer prices given, the issue is to define commercial contributions of the various units such that they sum exactly to the target accounting net margin.

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Risks Risks

ALM ALM P&L P&L Maintain risk within limits

FIGURE 26.5

Return Return

Minimize funding cost Maximize investment return

Policies and profitability of ALM

The accounting net interest margin is the summation of commercial margins, or contributions, and of the ALM margin. If ALM is a pure hedging entity, business units generate the entire margin. In such a case, the sum of commercial margins becomes identical to the target accounting margin. If ALM is a profit centre, any projected profit or loss of ALM plus the commercial margins is necessarily equal to the consolidated margin. Taking the example of lending only, we have only one business unit. The commercial margin, as a percentage of assets, is the spread between the unknown customer rate and the transfer price, or X − 8%. The value of the margin is the product of this spread with asset volume. The accounting margin is, by definition, the total of the commercial margin of business units and of the ALM margin: Mbank = Mcommercial + MALM Therefore, the commercial margin is the difference between the accounting margin and the internal ALM margin. The process requires defining what the ALM margin is. The ALM margin might be zero, making the sum of commercial contributions to the margin identical to the accounting margin. This would be the benchmark case, since any P&L of the ALM is a ‘speculative’ cost saving ‘on top’ of the net margin of the banking book. The simple example below shows how to move from a target overall accounting interest margin towards the commercial contributions (margins) of business units. Once these margins are defined, the target customer prices follow, since they are equal to the transfer prices plus the percentage margin on average. We use the following data. The volume of assets is 100, the cost of borrowing in the market is 10% and the internal transfer price is 8%. There is a difference between the market rate and the internal transfer rate to make the argument general. The volume of equity is 4% of assets, or 40. The target net interest margin is 25% of equity, or 10. The outstanding debt is 1000 − 40 = 960. The cost of debt is 8% × 960 = 76.8. Table 26.6 summarizes the data. The issue is to find the appropriate target commercial margin over transfer price. The ALM margin is: MALM = 8% × 1000 − 10% × 960 = −16 It is negative because the ALM prices internal funds at a rate lower than the market rate and subsidizes the commercial units. The target commercial margin is the difference

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TABLE 26.6

Setting a target economic margin

Target accounting margin Cost of debt Transfer price Volume of assets Equity Debt Target commercial margin over transfer price

10 10% 8% 1000 40 960 ?

between the targeted accounting margin and that of the ALM: Target IMcommercial = target IMoverall − IMALM = 10 − (−16) = 26 Given the volume of assets, it is easy to derive the average target customer rate X from this target margin. The margin value 26 represents 2.6% of outstanding assets. The required spread as a percentage of assets is: X − 8% = 2.6% The average rate X charged to customers should be 10.6%. It is sufficient to set a target commercial margin at 26, setting some minimum customer rate to avoid selling below cost, so that commercial entities might have flexibility in combining the percentage margin and the volume from one transaction to another as long as they reach the overall 26 goal. We check that the calculation ensures that the net interest margin is 10. It is the total revenue less the cost of debt: 10.6% × 1000 − 96 = 10. This equation allows us to calculate the average customer rate directly since the target accounting margin, the cost of debt and the size of assets are given. It simply says: (X − 8%) × 1000 − 96 = 10. The same overall summations hold with multiple transfer prices. Therefore, transfer prices allow us to define the global commercial margin above transfer prices consistent with a given target net accounting margin. Setting the actual business line target contributions to this consolidated margin implies adjusting the profitability to the risk originated by business units. For lending, credit risk is the main one. Within an integrated system, the modulation of target income depends on the risk contributions of business lines to the overall risk of the portfolio (as defined in Chapters 51 and 52).

THE FINANCIAL AND COMMERCIAL RATIONALE OF TRANSFER PRICES The FTP system is the interface between the commercial universe and the financial universe. In order to serve this purpose, the transfer prices should be in line with both commercial and financial constraints. From a financial standpoint, the intuition is that transfer prices should reflect market conditions. From a commercial standpoint, customer prices should follow business policy guidelines subject to constraints from competition. In other words, transfer prices should be consistent with two different types of benchmarks: those derived from the financial universe and those derived from the commercial policy. Transfer prices should also be consistent with market rates. The next chapter develops this rationale further. However, the intuition is simple. If transfer prices lead to much higher customer rates than the market offers, it would be impossible to sustain competition

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and customers would arbitrage them with the market. For buying resources, a similar rationale applies. The alternative uses of resources are lending or investing. If lending provides much less than the market, the bank is better off investing. If the opposite holds, resources priced internally at more than market prices would hurt the lending margin. Hence, the market provides a benchmark. On the other hand, ignoring competitors’ prices does not make much sense. This leads to the usual trade-off between profitability and market share. Banks are reluctant to give up market share, because it would drive them out of business. Since competition and market rates are not easy to reconcile, there has to be some mispricing. Mispricing is the difference between ‘economic prices’ and effective pricing. Mispricing is not an error since it is business-driven. Nevertheless, it deserves monitoring for profitability and business management, and preventing mispricing from being too important to be sustainable. Monitoring mispricing implies keeping track of target prices and effective prices, to report any discrepancy between the two. At this stage, the mispricing concept applies to economic transfer prices limited to such benchmarks as the cost of funds for lending and benchmarks of investment return for collecting resources. It extends to riskbased pricing, when target prices charge the credit risk to customers. This has to do with capital allocation. A logical conclusion is that using two sets of internal prices makes sense. One set of transfer prices should refer to economic benchmarks, such as market rates. The other set of transfer prices serves as commercial signals. Any discrepancy between the two prices is the cost of the commercial policy. These discrepancies are the penalties (mark-up) or subsidies (mark-down) consistent with the commercial policy. This scheme reconciles diverging functions of the transfer prices and makes explicit the cost of enforcing commercial policies that are not in line with market interest rates.

27 Economic Transfer Prices

This chapter focuses on economic benchmarks for transfer prices that comply with all specifications of the transfer pricing scheme. The basic principle for defining such ‘economic’ transfer prices is to refer to market prices because of arbitrage opportunities between bank rates and market rates whenever discrepancies appear. Economic benchmarks derive from market prices. Mark-ups or mark-downs over the economic benchmarks serve for pricing. There are two types of such add-ons. Some serve for defining risk-based pricing. Others are commercial incentives or penalties resulting from the business policy, driving the product–market segments mix. Note that implementing this second set of commercial mark-ups or mark-downs requires tracking the discrepancies with economic prices which, once aggregated, represent the cost of the commercial policy. Economic benchmarks for transfer prices are ‘all-in’ costs of funds. The ‘all-in’ cost of funds applies to lending activities and represents the cost of obtaining these funds, while complying with all banks’ balance sheet constraints, such as liquidity ratio. It is a market-based cost with add-ons for liquidity risk and other constraints. It is the cost of a ‘notional’ debt mimicking the loans. It is notional because no rule imposes that Asset–Liability Management (ALM) continuously raises such debts. The rationale of this scheme is to make sure that lending provides at least the return the bank could have on the market, making it worthwhile to lend. The target price adds up the cost of funding with mark-ups and mark-downs, plus a target margin in line with the overall profitability goal. To ensure effective transfer of interest rate risk from business units to ALM, economic prices of individual transactions are historical, calculated from prevailing market rates at origination, and guaranteed over the life of the transaction. The guaranteed prices make the interest incomes of transactions and business lines insensitive to interest rates over subsequent periods until maturity of the transaction. The cost of funds view applies to lending activities. For excess resources, market rates are the obvious references. The difference is that investments generate interest rate risk

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and exposure to the credit risk of issuers, while lending generates exposure to the credit risk of borrowers, ALM taking care of interest rate risk. Investments of excess funds raise the issue of defining meaningful risk and return benchmarks. These result, in general, from investment portfolios serving as ‘notional’ references. The first section expands the arbitrage argument to define transfer prices referring to market rates. The second section provides an overview of the transfer pricing scheme with all inputs, from the cost of funds to the final customer price, using a sample set of calculations. The third section defines the economic cost of funds, the foundation on which all additional mark-ups and mark-downs pile up. The fourth section makes explicit the conditions for ensuring effective transfer of interest rate risk to the ALM unit.

COMMERCIAL MARGIN AND MATURITY SPREAD Discrepancies of banks’ prices with market prices lead to arbitrage by customers. The drawback of a unique transfer price, equal to the average cost of funds of the bank, is that it would serve as a reference for both long and short loans. If the term structure of market rates is upward sloping, customer prices are below market rates on the long end of the curve and above market rates in the short maturity range (Figure 27.1). The unique transfer subsidizes long-term borrowers and penalizes short-term borrowers. A first conclusion is that transfer prices should differ according to maturities. Rate Rate for borrowers Commercial margins

Transfer price Rate for depositors Maturity

FIGURE 27.1

Drawbacks of a single transfer price

It is sensible to remove the maturity spread from the commercial margins, since this spread is beyond the control of the business lines. The maturity spread depends on market conditions. On the other hand, it is under the control of ALM, which can swap longterm rates against short-term rates, and the spread is in the accounting margin of the bank. Most commercial banks ride the yield curve by lending long and borrowing short, benefiting from this spread. On the other hand, riding the yield curve implies the risk of ‘twists’ in the curve. The risk of shifts and twists should be under the responsibility of ALM, rather than affecting commercial margins. Otherwise, these would embed interest rate risk. Business lines would appear responsible for financial conditions beyond their control. Making commercial margins subject to shifts and twists of the yield curve would ultimately lead to closing and opening offices according to what happens to the yield curve. The implication is that ‘commercial’ margins do not include the contribution of the market spread between long and short rates, although it contributes to the accounting margin, and this contribution should be under the control of ALM.

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327

In order to ensure that the bank makes a positive margin whatever the rates, and to isolate the maturity spread from the commercial margin, it is necessary to relate transfer prices to maturity. Bullet loan prices should be above market rates of the same maturity. Term deposit rates should be below the market rate of the same maturity. Demand deposit rates should be below the short-term rate. When lending above the market rate, the bank can borrow the money on the market and have a positive margin whatever the maturity. When collecting resources below market rates, the bank can invest them in the market and make a profit. Figure 27.2 illustrates this pricing scheme. Customer rate Rate Market spread

Commercial spread Customer rate

Time

FIGURE 27.2

Commercial and market spreads

This suggests that market rates are the relevant benchmarks. It also illustrates the arbitrage argument making discrepancies irrelevant for both customers and the bank. This basic pricing scheme is the foundation of economic transfer prices, although the relevant benchmarks require further specifications.

PRICING SCHEMES Transfer prices differ for lending and calculating margins on resources. For assets, transfer prices include all financial costs: they are ‘all-in’ costs of funds, with all factors influencing this cost. For deposits, the transfer prices should reflect the market rates on investing opportunities concurrent with lending.

Lending Activities There are several pricing rationales in practice, mixing economic criteria and commercial criteria. It is possible to combine them if we can fully trace the components of pricing and combine them consistently for reporting purposes. • Risk-based pricing is the benchmark, and should be purely economic. It implies two basic ingredients: the cost of funds plus mark-ups, notably for credit risk in lending. The mark-ups for risk result from the risk allocation system, and derive from capital allocations. We consider them as given here and concentrate on the economic costs of funds.

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• Commercial pricing refers to mark-ups and mark-downs over economic benchmarks to drive the business policies through incentives and penalties differentiated by product and market. Such mark-ups and mark-downs are purely business-driven. A comprehensive pricing scheme might include risk-based references plus such commercial mark-ups and mark-downs. ‘Effective pricing’ refers to actual prices used by banks. They might differ from the target risk-based prices simply because pricing is subject to competitive pressures. Mispricing is the difference between effective prices and target prices. The ‘all-in’ cost of funds serves as the foundation for transfer prices. To get the economic transfer price, other economic costs should add up. They include the expected loss on credit risk. Risk-based pricing implies a mark-up related to the credit standing of the borrower. This all-in cost of funding, plus these economic mark-ups, is the foundation of the transfer price. Note that, at this level, the bank does not earn any profit yet since it simply balances the overall cost of funding and risk. Conventions are necessary at this stage. If the cost of allocated risk includes the target profitability, the bank actually defines a transfer price that provides adequate profitability. An alternative presentation would be to add a margin, which is the target profitability on allocated capital. Such a target margin, or contribution, should be in line with the overall profitability goal of the bank, and absorb the allocated operating costs. We adopt this second presentation below, because it makes a distinction between the economic transfer price and the risk-based price, the difference being precisely the target return on allocated capital. Table 27.1 summarizes the format of the economic income statement. TABLE 27.1 incentives

Risk-based pricing and commercial

Component

%

Cost of funding +Cost of liquidity =‘All-in’ cost of funding −Expected loss from credit risk =Economic transfer price +Operating allocated costs +Risk-based margin for compensating credit risk capital =Target risk-based price +Business mark-ups or mark-downs =Customer price

Such transfer prices are before any business mark-ups or mark-downs. Additional mark-ups or mark-downs, which result from deliberate commercial policies of providing incentives and penalties, could also affect the prices. Our convention is to consider only the economic transfer price plus a profitability contribution. The rationale is that we need to track these to isolate the cost of the business policy, while other mark-ups or mark-downs are purely business-driven. Moreover, we also need to track mispricing or the gap between effective prices and target risk-based prices for reporting purposes, and take corrective action. Figure 27.3 summarizes all schemes and mispricing.

ECONOMIC TRANSFER PRICES

'Pure' risk "Pure" based Risk Based pricing Pricing

All-in cost of funds

329

Expected loss

Mark-up for risk: capital

Risk Risk based based price price

Mispricing Effective Effective pricing pricing

FIGURE 27.3

All-in cost of funds

Effective mark-up/ down

Effective Effective pricing pricing

Pricing schemes and mispricing

At this stage, we have not yet defined the ‘all-in’ cost of funds. The next section does this.

Transaction versus Client Revenues and Pricing Risk-based pricing might not be competitive at the individual transaction level simply because market spreads are not high enough to price all costs to a large corporate which has access to markets directly. This is a normal situation, because there is no reason why local markets should price credit risk, as seen in credit spreads, in line with the target of banks based on capital allocation for credit risk. Section 16 of this book addresses again the ‘consistency’ issue between internal pricing and external pricing of risk. However, this does not imply that the overall client revenue cannot meet the bank’s target profitability. Banks provide products and services and obtain as compensation interest spreads and fees. The overall client revenue is the relevant measure for calculating profitability, because it groups all forms of revenues from all transactions, plus all services resulting from the bank’s relationship with its clients. Loans and services are a bundle. Using risk-based pricing at the transaction level might simply drive away business that would be profitable enough at the client level. The client is a better base for assessing profitability than standalone transactions. This is a strong argument for developing economic income statements at the client level.

Target Risk-based Pricing Calculations Economic pricing for loans includes the cost of funds plus any mark-up for risk plus a target margin in line with the overall profitability goal. The cost of funding depends on rules used to define the funding associated with various loans. The pure economic benchmark is the one notional economic cost of funds, described below, which is the cost of funding that exactly mirrors the loan. We assume here that this cost of funds is 7%. The ‘all-in’ funding cost of the loan adds up the cost of maintaining the liquidity ratio and the balance sheet structure at their required level. We use here an add-on of 0.2% to obtain an ‘all-in’ cost of funds of 7.2%. Other relevant items are expected loss and allocated cost. Note that the overall profitability of the bank should also consider an additional contribution for non-allocated operating costs1 . 1 Chapters

53 and 54 provide details on full-blown economic statements pre- or post-operating costs.

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The required margin results from the required capital and the target return on that capital. For an outstanding balance of 1000 that requires no capital, the price needs to absorb only the all-in cost of funding. When there is credit risk, the rationale for defining the target required margin based on allocated capital is as follows. If capital replaces a fraction of debt because the loan is risky, there is an additional cost because capital costs more than debt. The additional cost is the differential cost between equity and debt weighted by the amount of capital required. The same calculations apply to regulatory capital or economic capital. The pre-tax cost of equity is 25% in our example. The additional cost is the amount of equity times the differential between the cost of equity and the cost of debt: (25% − cost of debt) × equity. In order to transform this value into a percentage of the loan outstanding balance, we divide this cost in value by the loan size, assuming that the ratio of equity to assets is 4% for example. This gives the general formula for the risk premium: (25% − cost of debt) × 4%. In this example, the cost of capital is (25% − 7%) × 4% = 0.72%. All components of transfer prices are given in Table 27.2. To obtain a pure economic price, we set commercial incentives to zero. TABLE 27.2 prices

Components of transfer

Component Cost of debt +Cost of liquidity +Expected losses +Operating costs =Transfer price +Risk-based margin =Target risk-based price +Commercial incentives =Customer rate

% 7.00 0.20 0.50 0.50 8.20 0.72 8.92 0 8.92

The cost of debt, set to 7%, is the most important item. The next section discusses the all-in cost of funds.

THE COST OF FUNDS FOR LOANS The transfer price depends on the definition of the funds backing the assets. There are two views on the issue of defining the transaction that best matches the assets or the liabilities. The first is to refer to existing assets and resources. The underlying assumption is that existing resources fund assets. The second view is to define a ‘best’ funding solution and use the cost of this funding solution as the pure cost of funding.

The Cost of Existing Resources Using the existing resources sounds intuitively appealing. These resources actually fund assets, so their cost has to be relevant. However, this solution raises conceptual inconsistencies. There are several types of assets and liabilities with different characteristics.

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Each type of resource has a different cost. Therefore, we use either the ‘weighted average cost of capital’ (wacc), the average cost of resources or several costs. We pointed out earlier the shortcomings of a single transfer price. Another solution could be to match assets and resources based on their similar characteristics. For instance, we can try to match long-term resources with long-term assets, and so on for other maturities. This is the rationale for multiple pools of funds based on maturity buckets. This principle raises several issues: • Since the volumes of assets and resources of a given category, for instance long-term items, are in general different, the match cannot be exact. If long-term assets have a larger volume than long-term resources, other resources have to fill up the deficit. Rules are required to define which other resources will fund the mismatch, and the resulting cost of funds becomes dependent on such conventions. • For business units, there is a similar problem. Generally, the balance sheet under the control of a business unit is not balanced. Any deficit needs matching by resources of other business units. However, it does not sound right to assign the cost of funds of another business unit, for instance a unit collecting a lot of ‘cheap’ deposits, to one that generates a deficit. The collection of cheap resources should rather increase the profitability of the unit getting such resources. • Any cheap resource, such as deposits, subsidizes the profitability of assets. Matching a long-term loan with the core fraction of demand deposits might be acceptable in terms of maturity. However, this matching actually transfers the low cost of deposits to the margin allocated to loans. Economically, this assumes that new deposits fund each new dollar of loans, which is unrealistic. This view is inconsistent from both organizational and economic standpoints. It implies a transfer of income from demand deposits to lending. In general, transferring the low cost of some resources to lending does not leave any compensation for collecting such resources. Low cost resources subsidize lending and lose their margin in the process! Given inconsistencies and unrealistic assumptions, the matching of assets with existing resources is not economic. Transfer prices require other benchmarks.

The ‘Notional’ Funding of Assets The unique funding solution that neutralizes both liquidity and interest rate risk of a specific asset is the funding that ‘mirrors’ the time profile of flows and the interest rate nature. For instance, with a fixed rate term loan, the funding should replicate exactly the amortization profile and carry a fixed rate. Such funding is more ‘notional’ than real. It does not depend on the existing resources. It does not imply either that ALM actually decides to implement full cash flow matching. It serves as a benchmark for determining the cost of funds backing any given asset. In some cases, the replication is obvious and the cost of funds also. A bullet debt matches a bullet loan. The relevant rate is the market rate corresponding to the maturity of the transaction. In other cases, the notional funding is different. For an amortizing loan, the outstanding balance varies over time until maturity. Therefore, the market rate of this maturity is not adequate. Using it would assume that the loan does not amortize over

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time. The funding that actually replicates the time profile of the loan is a combination of debts of various maturities. In the example below, a loan of 100 amortizes in 2 years, the repayments of capital being 40 and 60. Figure 27.4 shows the profile of the reference debt. The funding solution combines two spot bullet debts, of 1 and 2-year maturity, contracted at the market rates2 . This example demonstrates that it is possible to find a unique notional funding for any fixed rate asset whose amortizing schedule is known. Outstanding balance

40

60

Debt spot 1 year

Debt spot 2 years

1 year

FIGURE 27.4

2 years

Time

A 2-year amortizing loan

The solution is even simpler with floating rates than with fixed rates. For floating rate transactions, the replicating funding is a floating rate debt with reset dates matching those of the assets. The amortization profile should mirror that of the floating rate asset. However, for those assets that have no maturity, conventions are necessary.

The Cost of Funds For a fixed rate loan, the funding solution used as reference has a cost that depends on the combinations of volumes borrowed and maturities. The cost is not a single market rate, but a combination of market rates. In the above example, there are two layers of debt: 60 for 2 years and 40 for 1 year. The relevant rates are the spot rates for these maturities. However, there should be a unique transfer price for a given loan. It is the average cost of funds of the two debts. Its exact definition is that of an actuarial rate. This rate is the discount rate making the present value of the future flows generated by the two debts equal to the amount borrowed. The future outflows are the capital repayments and interest. The interest flows cumulate those of the 1-year and 2-year market rates. If the debts are zero-coupon, the interest payments are at maturity. The flows are 40(1 + r1 ) in 1 year and 60(1 + r2 )2 in 2 years. 2 There

is another solution for backing the loan. For instance, a spot debt for 1 year could be contracted for the full amount of the loan, that is 100 for 1 year at the spot market rate, followed by another 1-year debt, for an amount of 60, starting 1 year from now. Nevertheless, such funding does not replicate the loan, since a fraction of the debt needs renewal for another year at a rate unknown today.

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The cost r of this composite funding is a discount rate such that: 100 = 40 × (1 + r1 )/(1 + r) + 60 × (1 + r2 )2 /(1 + r)2 The discount rate is somewhere between the two market rates. An approximate solution uses a linear approximation of the exact formula: 100 = 40(1 + r1 − r) + 60(1 + 2r2 − 2r) r = (40 × r1 + 60 × 2 × r2 )/(40 + 2 × 60) The rate r is the weighted average of the spot rates for 1 and 2 years, using weights combining the size of each debt and its maturity. For instance, r1 counts roughly twice and r2 only once. With rates r1 and r2 equal to 8% and 9%, r = 8.75%. The rate is closer to 9%, because the 2-year debt is the one whose amount and maturity are the highest. In practice, transfer price tables provide the composite market rates immediately, given the time profile of loans and the current market rates.

The Benefits of Notional Funding Using the cost of funds of a debt that fully replicates the assets offers numerous economic benefits. Perfect matching implies that: • The margin of the asset is immune to interest rate movements. • There is no need for conventions to assign existing resources to usages of funds. • There is no transfer of income generated by collecting resources to the income of lending activities. • The calculation of a transfer price is mechanical and easy. This objective choice avoids possible debates generated by conventions about transfer prices. In addition, this reference actually separates the income of the banking portfolio from those of ALM. It also separates business risk from financial risk and locks in commercial margins at origination of the loan, if we use as transfer price the cost of this notional funding mimicking the loan. The ALM unit does not have to use a funding policy that actually immunizes the interest margin of the bank. The debt replicating the asset characteristics is a ‘notional debt’. The ALM policy generally deviates from perfect matching of the flows of individual assets depending on interest rate views for example. On the other hand, a perfect match is feasible at the consolidated level of the bank for neutralizing both liquidity and interest rate risk. Doing so at the individual transaction level would result in over-hedging because it would ignore the natural offsets between assets and liabilities, resulting in gap profiles. Comparing the perfect matching cost with the effective cost resulting from the ALM policy determines the performance of the ALM. If ALM policy results in a cost saving compared to the cost of a global perfect match, this saving is the profit of ALM. If the effective policy results in a higher cost, ex post, the cost differential over the benchmark is a loss. This requires keeping track of the cost of funding under perfect matching and the effective cost of funding for the same period. The differential is the ALM contribution.

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Transfer Prices for Resources Assigning transfer prices to resources should follow the same principles, except that some resources have no tangible maturity. The arbitrage rationale suffices to define economic transfer prices to ALM. It is feasible to invest resources with a contractual maturity up to that maturity, so that the corresponding market rate serves as reference. For resources without maturity, such as the volatile fraction of demand deposits, the maturity is conventional, such as a 1-year maturity, or modelled to some extent according to the effective time profile of resources.

TRANSFERRING LIQUIDITY AND INTEREST RATE RISK TO ALM Two factors help to fully separate commercial and financial risks. First, the commercial margins become independent of the market maturity spread of interest rates and of the ‘twist’ of the yield curve. ALM is responsible for managing the spread risk. Second, referring to a debt replicating the asset removes the liquidity and the market risks from the commercial margin. This takes care of the transfer of interest rate risk to ALM. However, guaranteeing this risk transfer requires another piece of the Funds Transfer Pricing (FTP) to actually protect business margins against financial risks. For instance, if a fixed rate asset generates 11% over a transfer price of 9%, the margin is 2%. Recalculating the margin of the same asset with a new rate at the next period would result in a change, unless the calculation follows a rule consistent with the system. For instance, resetting the transfer price after one period at 10%, because of an upward shift of 1% in the yield curve, results in a decrease of the margin to 1%. This decrease is not acceptable, because it is not under the control of business lines. The philosophy of the system is to draw a clear-cut frontier between commercial and financial risks. An additional component of the FTP scheme allows us to achieve this goal. The calculation of transfer price as the cost of the mirror debt occurs at the origination date of the asset. This rules out transfer price resets because of market rate changes. The transfer price assigned to a transaction over successive periods is the historical transfer price. Doing otherwise puts the commercial margin at risk, instead of locking it over the life of the transaction. Margins calculated over reset transfer prices are apparent margins. They match current market conditions and change with them. The FTP system should ensure that transfer prices are assigned to individual transactions for the life of the transactions. These prices are historical prices ‘guaranteed’ for transactions and business units. Using such historical prices for each transaction puts an additional burden on the information system. Nevertheless, without doing so, the margins lose economic meaning. In addition, the consistency between what business units control by and their performance measures breaks down. Table 27.3 shows the time profile of both apparent and locked in margins calculated, respectively, over reset prices and guaranteed historical transfer prices. The pricing remains in line with a target margin of 2% over the transfer price. Meeting this goal at origination does not lock in the apparent margin over time. Only the margin over guaranteed prices remains constant. This last addition makes the transfer pricing system comprehensive and consistent.

ECONOMIC TRANSFER PRICES

TABLE 27.3

335

Margins on apparent transfer prices and historical transfer prices

Transactions

Period 1

Period 2

Period 3

A B C

100

100 100

100 100 100

Current rate Target % margin Customer rate

10% 2% 12%

11% 2% 13%

12% 2% 14%

Apparent margin over current rate Apparent margin (value) Apparent margin (%)

100 × (12 − 10)

100 × (12 − 11) +100 × (13 − 11)

100 × (12 − 12) + 100 ×(13 − 12) + 100 × (14 − 12)

2

3

3

2/100 = 2%

3/200 = 1.5%

3/300 = 1%

Margin over guaranteed historical price Margin over guaranteed price (value) Margin over guaranteed price (%)

100 × (12 − 10)

100 × (12 − 10) +100 × (13 − 11)

100 × (12 − 10) + 100 ×(13 − 11) + 100 × (14 − 12)

2

4

6

2%

2%

2%

BENCHMARKS FOR EXCESS RESOURCES Some banks have structurally excess resources or dedicate deliberately some resources to an investment portfolio, more or less independently of the lending opportunities. Under a global view, the bank considers global management of both loan and investment portfolios. The management of invested funds integrates with ALM policy. Alternatively, it is common to set up an investment policy independently of the loan portfolio, because there are excess funds structurally, or because the bank wants to have a permanent investment portfolio, independent of lending opportunities. This policy applies for example when investing equity into a segregated risk-free portfolio rather than putting these funds at risk with loans. The transfer price for the portfolio becomes irrelevant because the issue is to maximize the portfolio return under risk limits, and to evaluate performance in relation to a benchmark return. The benchmark return depends on the risk. The bank needs to define guidelines for investing funds. When minimizing risk is the goal, a common practice consists of smoothing out market rate variations by structuring adequately the investment portfolio. It implies breaking down the investment into several fractions invested over several maturities and rolling them over, like a ‘tractor’, when each tranche matures. This is the ‘ladder’ policy. It avoids crystallizing the current yield curve in large blocks of investments. The ‘tractor’ averages the time series of all yield curves up to the management horizon. Segregation of assets in different subportfolios creates potential conflicts with the global ALM view of the balance sheet. Any segmentation of the portfolio of assets might

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result in overall undesired mismatches not in line with global guidelines. For instance requiring a long-term maturity for an investment portfolio, while the loan maturities are business-driven, could increase the gap between the average reset date of assets and that of liabilities. Hedging this gap with derivatives is inconsistent with the long-term goal for the portfolio, because it is equivalent to closing the gap by synthetically creating a short-term rate exposure rather than the long-term one. In general, segmentation of the balance sheet creates mismatch and does not comply with the basic ALM philosophy. If controlling the overall remains the goal, gaps over all compartments should add up. If they do not, the bank maintains an undesired global exposure.

SECTION 10 Portfolio Analysis: Correlations

28 Correlations and Portfolio Effects

Combining risks does not follow the usual arithmetic rules, unlike income. The summation of two risks, each equal to 1, is not 2. It is usually lower because of diversification. Quantification based on correlations shows that the sum is in the range of 0 to 2. This is the essence of diversification and correlations. Diversification reduces the volatility of aggregated portfolio income, or Profit and Loss (P&L) of market values, because these go up for certain transactions and down for others, thereby compensating each other to some extent. Moreover, credit risk losses do not occur simultaneously. In what follows, we simply consider changes of values of transactions, whatever the nature of risk that triggers them. Portfolio risk results from individual risks. The random future portfolio values, from which potential losses derive, are the algebraic sum of individual random future transaction values or returns. Returns are the relative changes in values between the current date and the horizon selected for measuring risk. The distribution of the portfolio values is the distribution of this sum. The sum of random individual transaction values or returns follows a distribution dependent on the interdependencies, or correlations, between these individual random values. The expected value, or loss, does not depend on correlations. The dispersion around the mean of the portfolio value, or volatility, and the downside risk do. Individual risks and their correlations drive the portfolio risk and the extent of the diversification effect, which is the difference between the (arithmetic) sum of these individual transaction risks and the risk of the sum. Because portfolio risk depends so much on interdependencies between individual transaction risks, they play a key role in portfolio risk models. Various techniques serve for capturing the correlation, or diversification, effect:

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• Correlations quantify interdependencies in statistical terms, and are critical parameters for valuing the random changes in portfolio values. Basic statistical rules allow us to understand how to deal with sums of correlated random individual values of facilities. • When dealing with discrete default events, the credit standing of individual obligors depend on each other. Conditional probabilities are an alternative view of correlations between individual risks that facilitates the modelling of some random events. • When looking forward, the ‘portfolio risk building block’ of models relies on simulations of random future changes of value, or ‘returns’, of transactions, triggered by risk events. Forward-looking simulations require generating random values of ‘risk factors’ in order to explore all future scenarios and determine the forward distribution of the portfolio values. Such random scenarios should comply with variance and correlation constraints between risk factors. Special techniques serve to construct such simulations. Several examples use them later on for constructing portfolio loss distributions. The first section explains why correlations are critical parameters for determining portfolio risk. The second section defines correlations and summarizes the essential formulas applying to sums of correlated random variables. The third section introduces conditional probabilities. The fourth section presents some simple basic techniques, relying on factor models, for generating correlated values of risk drivers.

WHY ARE CORRELATIONS IMPORTANT? Correlations are measures of the association between changes in any pair of random variables. In the field of risk management, the random variables are individual values of facilities that the random portfolio value aggregates. The subsequent sections expand the essentials for dealing with correlated value changes.

Correlations and Interdependencies between Individual Losses within a Portfolio For the market portfolio, the P&L is the variation of market values. Market values are market-driven. The issue for measuring downside risk is to quantify the worst-case adverse value deviations of a portfolio of instruments whose individual values are market-driven. It does not make sense to assume that all individual values deviate simultaneously on the ‘wrong’ side because they depend on market parameter movements, which are interdependent. To measure market risk realistically, we need to assess the magnitudes of the deviations of the market parameters and how they relate to each other. The interdependencies, or correlations, between parameters, plus their volatilities, drive the individual values. This is the major feature of market Value at Risk (VaR) calculations. The modelling relates individual value deviations, or returns, to the market parameter co-movements. For credit risk, the same phenomenon occurs. It does not make sense to assume that all obligors are going to default at the same time. The principle for modelling simultaneous changes in credit states, including the migrations to the default state, is to relate the credit standing of firms to factors, such as economic–industry conditions, that influence all of them. If these get worse, defaults and credit deteriorations increase, if they improve,

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the reverse happens. Such dependence creates some association between credit states of all obligors and their possible defaults. This implies that we need to model correlations between credit events, both default and migration events. In all cases, correlations are key parameters for combining risks within portfolios of transactions, both for market and credit risk. This chapter has the limited goal of summarizing the basics of correlations and the main formulas often used for measuring portfolio statistics, notably volatility of losses. Such definitions are a prerequisite whenever risk aggregation or risk allocation are considered. Throughout the chapter, we use bold characters for random variables and italics or regular characters for their particular values1 .

The Portfolio Values as the Aggregation of Individual Facility Values In order to proceed, we need the basic definitions, notations and statistical formulas applying to the summation of random variables. As a preliminary step, we show that the portfolio value is the summation of all individual transaction values. The presentation does not depend on whether we deal with market risk or credit risk. In both cases, we have changes of facility values between now (date 0) and a future horizon H . For market risk, the horizon is the liquidation period. For credit risk, it is the horizon necessary to manage the portfolio risk or the capital, typically 1 year in credit portfolio models. For market risk, we link instrument values to the underlying market parameters that drive them. For credit risk, the value changes depend on credit risk migrations, eventually related to factors common to all borrowers such as the economic–industry conditions. The value change of any transaction i between current date 0 and horizon H is: Vi = (Vi , H − Vi,0 ) = Vi,0 × yi The unit value change yi is a random variable independent of the size of exposure i. It is simply the discrete time return of asset i. A loss is a negative variation of value, whether due to default or a downside risk migration. For market risk, the changes in values of each individual transaction are either gains or losses. The algebraic summation of all gains and losses is the portfolio loss. Since the individual value changes are random, so is the algebraic summation. The distribution of negative changes, or adverse changes, is the loss distribution. The random change portfolio value Vp is the sum of the random individual losses Vi , and the change in the portfolio value Vp is the sum of the changes in all individual transactions Vi : Vp = Vi i

Vp = 1 For

Vi

i

instance X is a random variable, and X or X designates a particular numerical value.

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The variations of values are Vi = (Vi ,H − Vi,0 ) = Vi,0 × yi so that the portfolio return yp is: wi,0 × yi Vi,0 × yi /Vp,0 = Vi /Vp,0 = yp = Vp /Vp,0 = i

i

i

In this formula, the portfolio return is the weighted sum of the individual asset returns, the weights wi,0 being the ratios of initial exposures to the total portfolio value. By definition, they sum to 1. The same definitions apply for credit risk, except that credit risk results from credit risk migrations, including the migration to the default state. Whenever there is a risk migration, there is a change in value. Whenever the migration ends in the default state, the value moves down to the loss given default, that is the loss net of recoveries. Models ignoring migrations are default only. In default mode, all changes in values are negative, or losses, and there are only two credit states at the horizon.

PORTFOLIO RISK AS A FUNCTION OF INDIVIDUAL RISKS This section uses the classical concepts of expectations and variance applied to the portfolio value or loss viewed as the algebraic sum of individual value changes. We provide here only a reminder of basic, but essential, definitions.

Expectation of Portfolio Value or Loss The expectation of the sum of random variables is the sum of their expectations. This relation holds for any number of random variables. In general, using compact notations for the summation: ai E(Xi ) E ai Xi = i

i

The immediate application is that the expectation of the change in value of a portfolio is simply the sum of all individual value changes of each individual instrument within the portfolio: E(Vp ) = i Vi . This result does not depend on any correlation between the individual values or losses.

Correlations The value or loss volatility of the portfolio measures the dispersion of the distribution of values. The volatility of value is identical to that of value variations because the initial value is certain. The loss volatility, although distinct from loss percentiles, is an intermediate proxy measure to the unexpected loss concept2 . The portfolio value or loss volatility depends on the correlation between individual values or losses and on their variance–covariance matrix. The correlation measures the extent to which random variables change together or not, in the same direction or in opposite directions. Two statistics characterize this association: the 2 Loss

percentiles are often expressed as a multiple of loss volatility. See Chapters 7 and 50 for specific details.

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correlation coefficient and the covariance. A correlation and a covariance characterize pairs of random variables. The covariance is the weighted sum of the products of the deviations from the mean of two variables X and Y, the weights being the joint probabilities of occurrence of each pair of values. The coefficient of correlation is simpler to interpret because it is in the range −1 to +1. It is calculated as the ratio of the covariance by the product of the volatilities (which are the square roots of variances) of X and Y. The value +1 means that the two variables change together. The correlation −1 means that they always vary in opposite directions. The zero correlation means that they are independent. The formulas are as follows: σXY = ρXY σX σY where σXY is the covariance between variables X and Y; ρXY is the correlation between variables X and Y; σX and σY are the standard deviations, or volatilities, of variables X and Y.

Correlation and Volatility of a Sum of Random Variables The volatility of a sum depends on the correlations between variables. It is the square root of the variance. The variance of a sum is not, in general, the sum of the variances. It is the sum of the variances of each random variable plus all covariance terms for each pair of variables. Therefore, we start with the simple case of a pair of random variables and proceed towards the extension to any number of random variables. The general formula applies for determining the portfolio loss volatility and the risk contributions to the portfolio volatility of each individual exposure. Hence, it is most important. Two Variables

For two variables, the formulas are as follows: V (X + Y) = σ 2 (X + Y) = V (X) + V (Y) + 2 Cov(X, Y)

V (X + Y) = σ 2 (X) + σ 2 (Y) + 2ρXY σ (X)σ (Y)

The covariance between X and Y is Cov(X, Y), V (X) is the variance of X, σ (X) is the standard deviation and ρXY the correlation coefficient. Since the covariance is a function of the correlation coefficient, the two above formulas are identical. If the covariances are not zero, the variance of the sum differs from the sum of variances. The correlation term drops out of the formulas when the two variables are independent: V (X + Y) = σ 2 (X) + σ 2 (Y) σ (X + Y) = σ 2 (X) + σ 2 (Y)

The variance of the sum becomes the sum of variances only when all covariances are equal to zero. This result is valid only when all variables move independently of each other. The volatility is the square root of the variance. Since the variance of the sum is the sum of variances and the volatility is less than the sum of volatilities. It is the square root of the sum of the squared values of volatilities.

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As an example, the volatilities of two equity returns are 2.601% and 1.826%, and the correlation between returns is 27.901%. The sum of volatilities is 2.601% + 1.826% = 4.427%. The variance of the sum is: V (X + Y) = 2.601%2 + 1.826%2 + 2 × 37.901% × 1.826% × 2.601% = 0.137% The volatility is: σ (X + Y) =

√

0.137% = 3.7014% < 4.427%

This volatility is lower than the sum of volatilities. This is the essence of the diversification effect, since it shows that deviations of X and Y compensate to some extent when they vary. Combining risks implies that the risk of the sum, measured by volatility, is less than the sum of risks. Risks do not add arithmetically, except in the extreme case where correlation is perfect (value 1). Extension to Any Number of Variables

This formula extends the calculation of the variance and of the volatility to any number N of variables Zi . This is the single most important formula for calculating the portfolio return or value volatility. The portfolio volatility formula also serves for calculating the risk contribution of individual facilities to the total portfolio loss volatility because it adds the contributions of each facility to this volatility. The formula also illustrates why the sum of individual risks is not the risk of the sum, due to the correlation effects embedded in the covariance terms. The general formula is: N N N 2 σ Zi = σij = σi2 + ρij σi σi σi2 + i=1

i =j

i=j =1

i=j =1

i =j

In this formula, σi2 is the variance of parameter i, equal to the square of the standard deviation. σi and σij are the covariance between variables Zi and Zj . The summation corresponds to a generalized summation over all pairs of random variables Zi and Zj . It collapses to the above simple formulas for two variables. The summation i =j is equiva lent to the more explicit formula i =j with i and j =1,N , a summation over all values of i and j , being allowed to vary from 1 to N , but with i not equal to j since, in this case, we have the first variance terms in the above formula. A similar, and more compact, notation writes that the variance of the random variables summing N random variables is the summation of all σij using the convention that, whenever i = j , we obtain the variance σi2 = σj2 : N 2 σ Zi = σij i,j with i and j =1,N

i=1

Reverting to the portfolio return, we need to include the weights of each individual exposure in the formulas. If we use the portfolio return as the random variable characterizing the relative change in portfolio value, using the simple properties of variances and covariances, we find that: N N N 2 σ ρij wi wj σi σi wi wj σij = wi2 σi2 + wi2 σi2 + Zi = i=1

i=j =1

i =j

i=j =1

i =j

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The variance σi and the covariance σij now refer to the individual asset returns. The weights are constant in the formula, which implies a crystallized portfolio, with constant asset structure. This is the basic formula providing the relative portfolio value over a short period. It applies as long as the weights are approximately constant.

Visual Representation of the Diversification Effect The diversification effect is the gap between the sum of volatilities and the volatility of a sum. In practice, the random variables are the individual values of returns of assets in the portfolio. The diversification effect is the gap between the volatility of the portfolio and the sum of the volatilities of each individual loss. A simple image visualizes the formula of the standard deviation of a sum of two variables. The visualization shows the impact of correlation on the volatility of a sum. A vector whose length is the volatility represents each variable. The angle between the vectors varies inversely with correlation. The vectors are parallel whenever the correlation is zero and opposed when the correlation is −1. With such conventions, the vector equal to the geometric summation of the two vectors representing each variable represents the overall risk. The length of this vector is identical to the volatility of the sum of the two variables3 . Visually, its length is less than the sum of the lengths of the two vectors representing volatilities, except when the correlation is equal to +1. The geometric visualization shows how the volatility of a sum changes when the correlation changes (Figure 28.1). Variable 2

1+2

Angle = correlation −0 if ρ = +1 −90° if ρ = 0 −180° if ρ = −1 Variable 1 Length of vectors = volatility

FIGURE 28.1 variables

Geometric representation of the volatility of the sum of two random

Figure 28.2 groups different cases. The volatilities of the two variables are set to 1. The only change is that of their correlation changes. The volatility of the sum is the length of the geometric summation of vectors 1 and 2. It varies between 0 when the correlation is −1, up to 2 when the correlation is +1. The intermediate case, when correlation is zero, √ shows that the volatility is 2. 3 This

result uses the formula for the variance of a sum, expressing the length of the diagonal as a function of the sides of the square of which it is a diagonal.

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Correlation +1

σ1+2 = 2

FIGURE 28.2

Correlation > 0

Independence Correlation 0

Correlation < 0 Correlation − 1

σ1+2 = √2

σ1+2 = 0

The change in volatility of a sum when the correlation changes

CONDITIONAL AND UNCONDITIONAL PROBABILITIES Conditional probabilities provide an alternative view to correlations that serves in several instances. The unconditional probabilities are the ‘normal’ probabilities, when we have no information on any event that influences the random events, such as defaults. Conditional probabilities, on the other hand, embed the effects of information correlated with the random event.

Conditional Probabilities and Correlations In the market universe, we have time series of prices and market parameter values, making it easy to see whether they vary together, in the same direction, in opposite directions, or when there is no association at all. The universe of credit risk is less visible. We have information on the assessment of the risk of obligors, but the scarcity of data makes measures of correlations from historical data hazardous. In addition, we cannot observe the correlation between default events of two different firms in general because the joint default of two firms is a very rare event. Correlation still exists. Typically, the default probabilities vary across economic cycles. Since the credit standing of all firms depends on the state of the economy, the variations of economic conditions create correlations between default events. If economic or industry conditions worsen, all default probabilities tend to increase and vice versa. Therefore, the same default probability should differ when there is no information on future economic conditions and if it is possible to assess how likely the future states of the economy will be, better or worse than average. Probabilities depending on random economic scenarios are conditional probabilities on the state of the economy. In general, when common factors influence various risks, the risks are conditional on the values of these factors. Correlations influence the probability that two obligors default together. Default probabilities depending on the default events of the other firm are conditional probabilities on the status, default or non-default, of the other firm. The purpose of the next sections is to explain the properties of joint and conditional probabilities, and to illustrate how they relate to correlations. Applications include the definitions of joint default or migration events for a pair of obligors, the derivation of default probabilities conditional on ‘scoring models’ providing information on the credit standing of borrowers, or making risk events dependent on some common factor, such as the state of the economy, that influences them.

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Conditional Probabilities Conditional probabilities are probabilities conditional on some information that influences this probability. If X is a random event whose probability of occurrence depends on some random scenario S, then the probability of X occurring given S is the conditional probability P (X|S). We characterize default events by a random variable X taking only two values, default or non-default, or 1 and 0 respectively for these two states. The unconditional default probability of X taking the default value 1 is P (X). This is an unconditional default probability that applies when we have no particular information on whatever influences this probability. The random state of the economy S influences the likelihood of X values. The probability of X defaulting once S is known, or P (X|S), is the conditional default probability of default of X given S. The conditional probability depends on the correlation between the two events. For example, if economic conditions worsen, the default probability of all firms increases. In such a case, there is a correlation between the default event X and the occurrence of worsening economic conditions, a value of S. If there is no relation, the conditional probability P (X|S) collapses to the unconditional probability P (X). If there is a positive relation, the conditional probability increases above the unconditional probability, and if there is a negative correlation, it decreases. For instance, if the unconditional default probability is 1% for a firm, its conditional probability on S is lower than 1% if the economy improves and higher when it deteriorates.

Joint Probabilities A joint probability is the probability that two events occur simultaneously. It applies to both continuous and discrete variables. The two random events are X and Y. The relationship between the joint probability that both X and Y occur and the conditional probabilities of X given occurrence of Y, or Y given occurrence of X, is: P (X, Y) = P (X) × P (Y|X) = P (Y) × P (X|Y) A first example relates the default probability to the state of the economy. A firm has an ‘unconditional’ default probability of 1%. The random value F, taking the values ‘default’ and ‘non-default’, characterizes the status of the firm. The firm’s unconditional default probability P (F = default) = 1%. The unconditional probability represents the average across all possible states of the economy. Let us now consider three states of the economy: base, optimistic and worst-case. The probability of observing the worst-case state is 20%. Let us assume that the corresponding default probability increases to 2%. This probability is conditional on S. The formula is: P (F = default|S = worst-case) = 2%. The joint probability of having simultaneously a default of the firm and a worst-case state of the economy is, by definition: P (F = default, S = worst-case) = P (S = worst-case) × P (F = default|S = worst-case) = 20% × 2% = 0.4% As a summary: P (F = default) = 1%

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P (F = default|S = worst-case) = 2% P (F = default, S = worst-case) = 0.4% When X and Y are independent, the joint probability collapses to the product of their standard probabilities, and the conditional probabilities are equal to the standard probabilities: P (X, Y) = P (X) × P (Y) P (X|Y) = P (X)

and P (Y|X) = P (Y)

If the state of the economy does not influence the credit standing of the firm, the conditional probability of default becomes equal to the unconditional default probability, or 1%. This implies that P (F = default|S = worst-case) = P (F = default) = 1%. As a consequence, the joint probability of having a worst-case situation plus default drops to P (F = default, S = worst-case) = 1% × 20% = 0.2% < 0.4%. The equality of conditional to unconditional probabilities, under independence, is consistent with a joint probability equal to the product of the unconditional probabilities: P (X, Y) = P (X) × P (Y). With correlated variables X and Y, the joint probability depends on the correlation. If the correlation is positive, the joint probability increases above the product of the unconditional probabilities. This implies that P (X|Y) > P (X) and P (Y|X) > P (Y). With positive correlation between X and Y, conditional probabilities are higher than unconditional probabilities. This is consistent with P (X, Y) being higher than the product of the unconditional probabilities P (X) and P (Y) since: P (Y|X) = P (X, Y)/P (X) and P (X|Y) = P (X, Y)/P (Y). In the universe of credit risk, another typical joint probability of interest is the joint default of two obligors X and Y, or P (X, Y). If the chance that X defaults increases when Y does, there is a correlation between the two defaults. This correlation increases the joint probability of default, the conditional probability of X defaulting given Y defaults, or P (X|Y), and P (Y|X) as well. If X and Y are positively correlated events, the occurrence of Y increases the probability of X occurring. This is equivalent to increasing the joint default probabilities of X and Y. Correlation relates to conditional probabilities. It is equivalent to say that there is a correlation between events A and B, and to consider that the probability that B occurs increases above B’s ‘unconditional’ probability if A occurs. For instance, the unconditional probabilities of default of A and B are 1% and 2% respectively. If the default events A and B are independent, the Joint Default Probability (JDP) is 2% × 1% = 0.02%. However, if the probability that B defaults when A does becomes 10%, the conditional probability of B given A’s default P (B|A) is 10%, much higher than the unconditional probability of B’s default, which is P (B) = 2%. This implies a strong correlation between A and B default events. Then, the joint default probability becomes: JDP(A, B) = P (A|B) × P (B) = 10% × 1% = 0.1% This is much higher than 2% × 1% = 0.02%, the joint default probability if the defaults were independent.

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Averaging Conditional Probabilities and Bayes’ Rule When inferring probabilities, the rule for averaging conditional probabilities is helpful. Let’s assume there are three (or more) possible values for S, representing the state of the economy. The default probabilities of firms vary with the state of the economy. If it worsens, the default probabilities increase and vice versa. Let’s assume we have only three possible states of the economy, with unconditional probabilities P (S = S1), P (S = S2) and P (S = S3) respectively equal to 50%, 30% and 20%, summing up to 1 since there is no other possible state. The third case is the worst-case scenario of the above. The averaging rule stipulates that the default probability of a firm F is such that: P (F) = P (F|S1) × P (S = S1) + P (F|S2) × P (S = S2) + P (F|S3) × P (S = S3) The probability of X defaulting under a particular state of the economy varies with the scenario since we assume a correlation between the two events, default and state of the economy. Taking 0.5%, 1% and 1.5% as conditional default probabilities for the three states respectively, the unconditional default probability has to be: P (F) = 0.5% × 50% + 1% × 30% + 1.5% × 20% = 0.85% The unconditional (or standard) default probability is the weighted average of the conditional default probabilities under various states of the economy, the weights being the probabilities of occurrence of each state. Its interpretation is that of an average over several years where all states of the economy occurred. The unconditional probability is the weighted average of all conditional probabilities over all possible states of the economy: K P (F) = P (F|Si) × P (Si) i=1

If S is a continuous variable, we transform the summation symbol into an integration symbol over all values of S using P (S) as the probability of S being in the small interval dS: P (X) = P (X|S) × P (S) dS

Conditioning and Correlation Correlation implies that conditional probabilities differ from unconditional probabilities, and the converse is true. Correlation and conditional probabilities relate to each other, but they are not identical. Conditional probabilities are within the (0, 1) range, while correlations are within the (−1, +1) range. Therefore, it is not simple to define the exact relationship between conditionality and correlation. As an illustration of the relation, the joint probability of default of two obligors depends on the correlation and the unconditional default probabilities of the obligors. The relation results from a simple analysis, developed in Chapter 46, explaining the loss distribution for

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a portfolio of two obligors X and Y, whose random defaults are variables X and Y taking the value 1 for default and 0 for non-default. Such ‘binary’ variables are called Bernoulli variables. For discrete Bernoulli variables, the relationship between the unconditional probabilities and correlation is: P (Y, Z) = P (Y) × P (Z) + ρYZ × P (Y)[1 − P (Y)] × P (Z)[1 − P (Z)]

The joint default probability increases with a positive correlation between default events. However, there are boundary values for correlations and for unconditional default probabilities. Intuitively, a perfect correlation of +1 would imply that X and Y default together. This is possible only to the extent that the default probabilities are identical. Different unconditional probabilities are inconsistent with such perfect correlation.

Conditioning, Expectation and Variance Some additional formulas serve in applications. The conditional probability distribution of X subject to Y is the probability distribution of X given a value of Y. The conditional expectation of X results from setting Y first, then cumulating (‘integrating’) over all values of X: E(X) = E[E(X|Y)]

E(X|Y) = E(X) implies that X does not depend on Y. The expectation of a variable can be calculated by taking the expectation of X when Y is set, then taking the expectation of all expected values of X given Y when Y changes. E(X|Y) = E(X) implies that X does not depend on Y. Conditional expectations are additive, like unconditional expectations. These formulas serve to define the expectation when a loss distribution of a portfolio is conditional on the value of some external factors, such as the state of the economy. The variance of a random event X depending on another event Y results from both the variance of X given Y and the variance of the conditioning factor Y: Var(X) = Var[E(X|Y)] + E[Var(X|Y)]

It is convenient to use this formula in conjunction with: Var(X) = E(X2 ) − [E(X)]2 . The variance of X is the expectation of the square of X minus the square of the expectation. This allows decomposition of the variance into the fraction of variance due to the conditioning factor Y and the variance due to the event X.

Applications of Joint and Conditional Probabilities for Credit Events Joint probabilities apply to a wide spectrum of applications. They serve in credit risk analysis when dealing with credit events. Such applications include the following. The joint default probability of a pair of obligors depends on the correlation between their default events, characterized by probabilities of default conditional on the default or non-default of the second obligor. The two-obligor portfolio is the simplest of all portfolios. Increasing the default probability of an obligor conditional on default of the other increases correlation between defaults. There is a one-to-one correspondence between default correlation and conditional probability, once the unconditional default probabilities are defined.

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The same two-obligor portfolio characterizes the borrower–guarantor relation, since default occurs only when both default together. This is the ‘double’ default event when there are third-party guarantees. The technique allows us to value third-party guarantees in terms of a lower joint default probability. Modelling default probability from scoring models relies on conditional probabilities given the value of the score, because the score value provides information on the credit standing of a borrower. The score serves for estimating ‘posterior’ default probabilities, differing from ‘prior’ probabilities, which are the unconditional probabilities, applying when we do not have the score information. When defaults are dependent on some common factors, they become conditional on the factor values. This is a simple way to determine the shape of the distribution of correlated portfolio defaults by varying the common factors that influence the credit standing of all obligors within the portfolio.

GENERATING CORRELATED RISK DRIVERS WITH SINGLE FACTOR MODELS This section shows how to generate, through multiple simulations, two correlated variables, with a given correlation coefficient. The correlation coefficient ρij is given. The variables can be credit risk drivers such as asset values of firms or economic factors influencing the default rates of segments. We first show how to generate two random normal variables having a preset correlation coefficient. Then, we proceed by showing how to generate N random variables having a uniform correlation coefficient. This is a specific extension of the preset pair correlation case. It applies when simulating values of credit risk drivers for a portfolio of obligors with known average correlation. We implement this technique to demonstrate the sensitivity of the loss distribution to the average correlation in Chapters 45–49. Next, we drop the uniform correlation and discuss the case of multiple random variables with different pair correlations. To generate random values complying with a variance–covariance structure, Cholevsky’s decomposition is convenient. This technique serves to generate the distribution of correlated variables when conducting Monte Carlo simulations.

Generating Two Correlated Variables Each random variable Zi is a function of a common factor Z and a specific factor εi . The specific factor is independent of the common factor. All cross-correlations between Z and εi or εj are 0. All random variables are normal standardized, meaning that they have zero mean and variance equal to 1. The correlation is between each Zi variable and the common factor is ρi , while the correlation between the two random variables Zi and Zj ρij . We use the equations for two different variables Zi and Zj : Zi = ρ × Z + 1 − ρij2 × εi Zj = ρ × Z + 1 − ρij2 × εj

The variance of Zi is:

Var(Zi ) = ρi2 Var(Z) + (1 − ρi2 ) Var(εi ) + 2ρi 1 − ρi2 × Cov(Zi , εi )

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Given that Var(Zi ) = Var(Z) = Var(εi ) = 1 and Cov(Zi , εi ) = 0, we have: Var(Zi ) = ρi2 + (1 − ρi2 ) + 0 = 1

Risk models make a distinction between the general risk, related to a common factor influencing the values of all transactions, and the specific risk, which depends on the transaction only. If Zi represents the transaction ‘i’ and if Z represents a factor influencing its risk and that of other transactions, the general risk is the variance of Zi due to the factor Z. From above, the general risk is simply equal to the correlation coefficient ρi2 and the specific risk to its complement to 1, or (1 − ρi2 ). In this very simple case, we have a straightforward decomposition of total risk. Another correlation of interest is the pair correlations between any pair of variables, Zi and Zj . The covariance and their correlation between these variables derive from their common dependence on the factor Z. All cross-covariances between the factor Z and the residual terms εi and εj are zero by definition. Therefore: Cov(Zi , Zj ) = Cov(ρi × Z + 1 − ρij2 × εi , ρj × Z + 1 − ρij2 × εj ) Cov(Zi , Zj ) = Cov(ρi Z, ρj Z) = ρi ρj Var(Z) = ρi ρj

Hence, all pairs such as Zi and Zj have a predetermined correlation ρij = ρi ρj Var(Z). This technique shows how to generate two correlated variables having a predetermined correlation coefficient ρij by generating independent normal variables: • Generate a standardized normal variable εi . • Generate a standardized normal variable εj . • Generate a standardized normal variable Z. Then Zi and Zj , calculated with the above linear function using ρij as the correlation coefficient, have a correlation equal to this ρij . The technique applies for generating N correlated variables with various pair correlations. To use the above technique, we generate independently K random draws of the common factor Z and K random draws of each of the specific risks εi (N variables) following standardized normal distributions. K is the number of trials or runs. Combining the random

values of Z and εi with the above equation Zi = ρ × Z + 1 − ρi2 × εi , we get K values of the N Zi . The K values of all Zi are such that all cross-correlations are equal to ρ. To generate K = 1000 trials for N variables, we need to generate 1000 × (N + 1) values since there are N residuals εi plus the unique common factor. In the case of a uniform correlation, all pair correlations and covariances of Zi and Zj are equal to a common ρ. The technique generates random values with standardized normal distributions and all pair correlations equal to a given value ρ. The technique serves for generating samples of N uniformly correlated normal variables. This procedure is implemented in Chapter 48.

Generating Random Values Complying with a Correlation Structure The correlation structure results from multi-factor modelling of credit drivers and credit events, or any proxy such as the equity return correlations of Credit Metrics. Simulation

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requires generating risk drivers, such as asset values, complying with this correlation structure. Of course, in such a case, variances and covariances differ across pairs of variables. Cholevsky’s decomposition allows us to impose directly a variance–covariance structure on N random normal variables4 . It allows us to derive from the generation of N independent variables N other random variables that comply with the given correlation structure, and that are linear functions of the independent variables. The difference with the previous technique is that we start from a full variance–covariance matrix. First, we generate K independent samples of values of N variables Xi . It is convenient to use normal standardized variables. K is the number of simulation runs. Then, we convert the N Xi variables into another set of N Zj variables. The Zj variables are linear combinations of the independent Xi . The Cholevsky decomposition provides the coefficients of these linear combinations. The coefficients are such that the new Zj variables comply with the imposed variance–covariance matrix. The Decomposition Technique

We use the simple case of two random variables. We start from X1 , a normal standardized variable. Once we have X1 , we derive the α12 and α22 for X2 : Z1 = α11 X1 Z2 = α12 X1 + α22 X2 These numbers are such that: 2 Var(Z1 ) = α11 =1

This requires α11 = 1, or Z1 = X1 :

2 2 Var(Z2 ) = α12 + α22 =1

Cov(Z1 , Z2 ) = Cov(α11 X1 , α12 X1 + α22 X2 ) = α11 × α12 = ρ12 Since α11 = 1, we have two equations with two unknown values α12 and α22 : 2 2 α12 + α22 =1

α12 = ρ12 2 α22 = 1 − ρ12 Matrix Format with Two Random Variables

Therefore, the two variables Z1 , Z2 are linear functions of standardized independent normal variables X1 , X2 according to: Z 1 = X1 Z2 = ρX2 + 4 If

1 − ρ 2 X2

all variables are normal standardized, the variance–covariance matrix collapses to the correlation matrix, with correlation coefficients varying for each pair of variables and all diagonal terms equal to 1.

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The above equations show that ZT = LXT using L as the lower triangular matrix of the αij coefficients, where XT is the column vector of Xi variables, and ZT is also a column vector of Zj variables. Using the matrix format of Cholevsky’s decomposition, the process starts from the variance–covariance matrix of two normal standardized variables, which is extremely simple. If Zk are standardized normal variables, they have covariance matrix: 1 ρ

= ρ 1 The Cholevsky decomposition of is: 1 1 0 1 ρ ×

= = ρ 1 0 ρ 1 − ρ2

ρ 1 − ρ2

Therefore, the variables Z1 and Z2 having correlation ρ are linear functions of independent standardized normal variables X1 and X2 according to the relation: 1 0 X1 Z1 × = 2 X2 Z2 ρ 1−ρ

Then:

Z1 Z2

=

X 1 ρX2 + 1 − ρ 2 X2

These are the relations used in preceding sections for generating pairs of correlated variables. The appendix to this chapter provides general formulas with N variables.

APPENDIX: CHOLEVSKY DECOMPOSITION This appendix shows how to generate any number of correlated normal variables, starting with a set of independent variables and using Cholevsky’s distribution to obtain linear combinations of these independent variables complying with a given correlation structure.

The Matrix Format for N Variables Generalizing to N variables requires an N -dimensional squared variance–covariance matrix . In general, L and U stand respectively for ‘lower’ triangular matrix and ‘upper’ triangular matrix. The U matrix is the transpose of the L matrix: U = LT . By construction of L = ZT (XT ) − 1, LU = LLT is the original variance–covariance matrix . The above equations show that ZT = LXT using L as the lower triangular matrix of the αij coefficients, where XT is the column vector of Xi variables, and ZT is also a column vector of Zj variables. Solving the matrix equation ZT = LXT determines L: Z j = L × Xi α11 Z1 Z2 α12 = .. .. . .

0 α22 .. .

X1 ... X2 ... × .. .. . .

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The solution uses an iterative process as indicated below for determining the coefficients, as illustrated with the two-variable case.

Determining the Set of Coefficients of the Cholevsky Decomposition The Cholevsky decomposition determines the set of coefficients αim of the linear combinations of the Xj such that the Zj comply with the given variance–covariance structure: Zj =

j

αim Xi

m=1

This set of coefficients transforms the independent variables Xi into correlated variables Zj . The coefficients result from a set of equations. The variance of Zj is the sum of the variances of the Xi weighted by the squares of αij , for i varying from 1 to j . It is equal to 1. This results in a first set of equations, one for each j , with j varying from 1 to N : j j 2 αim =1 Var(Zj ) = Var αim Xi = m=1

m=1

The index m varies from 1 to j , j being the index of Zj varying from 1 to N because there are as many correlated variables as independent variables Xi . Then, for all i different from j , we impose the correlation structure ρij . All variances are 1 and the covariances are equal to the correlation coefficients. The covariance of Zj and Zk is: j k αj m Xi , Cov(Zj , Zk ) = Cov αj k Xi = ρij m=1

m=1

This imposes a constraint on the coefficients: αj m αj k = ρij j =m

The problem is to find the set of αj m for each value of j and m, both varying from 1 to N , transforming the independent variables Xi into correlated Zj that are linear functions of the Xi . These values are the solution of the above set of equations for all αim . Determining these values follows an iterative process.

Example of Cholevsky’s Decomposition The original variance–covariance matrix is . The Cholevsky decomposition of a symmetrical matrix is such that = LU. The following is an example of such a decomposition using a variance–covariance matrix:

4 2 14

=L×U 2 14 2 0 0 2 17 5 = 1 4 0 × 0 5 83 7 3 5 0

1 7 4 3 0 5

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If ZT is the column vector of the correlated normal Zj , and XT is the column vector of the independent normal Xi variables, ZT derives from the XT vector of independent variables through: ZT = LXT If we proceed for all Zi , we find k values for each Zi , and the values comply with the correlation structure. Table 28.1 makes two sample calculations of two sets of three values for the three variables Zi starting from two sets of three values for each of the three independent variables Xm , of which the Zi are linear combinations. The construction of the coefficients of the L matrix ensures that the values of the Zi comply with the original variance–covariance matrix. TABLE 28.1 The Z values from the values of the independent variable using L L

X1

Column vector X: X2 X3

1 2 ··· ···

3 1 ··· ···

2 3 ··· ···

2 0 0 1 4 0 7 3 5 Column vector Z = XL: Z1 Z2 Z3 19 26 ··· ···

18 13 ··· ···

10 15 ··· ···

The first row of sample values 19, 18, 10 for Z1 , Z2 , Z3 results from the first row of X values and the matrix L coefficients: 1 × 2 + 3 × 1 + 2 × 7 = 19 1 × 0 + 3 × 4 + 2 × 3 = 18 1 × 0 + 3 × 0 + 2 × 5 = 10 We can generate as many values as we need from random trials of normally independent X. The above Z values comply with the variance–covariance matrix = LLT = LU.

SECTION 11 Market Risk

29 Market Risk Building Blocks

Market risk is the potential downside deviation of the market value of transactions and of the trading portfolio during the liquidation period. Therefore, market risk focuses on market value deviations and market parameters are the main risk drivers. Figure 29.1 summarizes the specifics of the market risk building blocks using the common structure followed throughout the book for all risks. We focus on the first two main blocks: standalone risk of individual transactions and portfolio risk. The capital allocation and risk–return building blocks are technically identical to that of credit risk.

BLOCK I: STANDALONE RISK Market risk results from the distribution of the value variations between current date and horizon of individual assets. In percentage terms, these deviations are the random asset returns. Consequently, all we need for measuring risk are the distributions of the returns between now and the future short-term horizon. The risk drivers are all market parameters to which mark-to-market values are sensitive. They include all interest rates by currency, equity indexes and foreign exchange rates. The exposures are mark-to-market values. Exposure values map to different market parameters, depending on the type of transaction. Simple instruments map to few main parameters, such as equity indexes for stocks. Derivatives are sensitive to several market parameters, such as underlying asset value, underlying asset volatility, interest rate and time to maturity. Mapping exposures to risk drivers is a prerequisite for determining the distribution of random future asset returns. Valuing risk implies defining the downside deviations between current value and the random market value at the horizon of the liquidation period. These variations of

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Building Blocks: Market Risk Building Blocks: Market Risk I.

I.

II. II.

1

Risk drivers

Market parameters (interest rates, equity indexes, foreign exchange rates)

2

Risk exposures

Mark-to-market values. Mapped to market parameters

3

Standalone risk

Valuation of adverse deviations of market returns over liquidation period

Correlations

Correlations between selected market parameters mapped to individual exposures

4

5

6

FIGURE 29.1

Portfolio risk

Capital

Loss distributions aggregate algebraically the individual asset returns. The 'Delta− normal' or Monte Carlo simulation techniques serve for deriving the individual and correlated asset returns VaR is a loss percentile

Major building blocks of market risk

values result directly from the distribution of the future asset returns. Financial models traditionally view small period returns as following specific random processes. Using such processes, it is possible to generate a random set of time paths of market parameter changes or of individual asset returns to obtain their final values at horizon. Then, we can revalue each asset for each final outcome. These are the basics of the full revaluation technique, which is resource intensive. The Delta ‘Value at Risk’ (VaR) technique uses several simplifications to bypass full revaluations: mapping individual asset returns to market parameter returns; constant sensitivities of asset returns to underlying value drivers; linear relationship between asset returns and market parameter returns; normal distributions of market parameter returns as a proxy for actual distributions over short horizons. This solution is evidently more economical than full revaluation because there are many fewer market parameters than individual assets. Alternatively, when it is not possible to ignore the changes in sensitivity with market movements, full revaluations at horizon under various scenarios serve for improving the accuracy of the distribution of the simulated values (Figure 29.2). All loss statistics and loss percentiles derive from the distribution of horizon returns of each individual exposure. The distribution of returns characterizes the standalone risk. When moving on to portfolios, diversification eliminates a large fraction of the sum of individual risks. The portfolio risk building block needs to model correlations and use them to model the distribution of the portfolio return.

MARKET RISK BUILDING BLOCKS

361

Distribution of forward market values

Loss

Current value

FIGURE 29.2

Forward valuation and loss valuation

BLOCK II: PORTFOLIO RISK Portfolio risk results from the distributions of all individual and correlated asset returns. In practice, it is more economical to derive asset returns from the common factors that influence them, using sensitivities. These factors, or risk drivers, are the market parameters. Correlations between the returns of individual exposures are derived from those of market parameters. Sometimes, it is sufficient to observe directly the market parameters for measuring historical correlations. Sometimes, it is necessary to use ‘factor models’ of market parameters for simulating possible scenarios. Factor models make the market parameters dependent on a common set of factors, thereby correlating them. The most well known single-factor model relates individual equity returns to the single equity index return. Some interest rate models and yield curve models can also serve to capture the distributions of the various rates. Because it is complex to model all rates, other models serve for modelling the random yield curve scenarios from factors that represent the main changes in shape of the yield curve. This is a two-stage modelling process, from factors to market parameters, and from these risk drivers to individual market values of exposures. Since factors correlate, risk drivers also do, as well as the returns of individual exposures (Figure 29.3).

FIGURE 29.3

Factors

Market parameters

Correlated factors

Correlated market parameters

Correlated asset values & returns

Modelling risk driver correlations

Portfolio risk results from the forward revaluation of portfolio returns, as the algebraic summation of all individual asset value changes, under various market parameter scenarios. VaR is the loss percentile at the preset confidence level. Generating the various

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Variance covariance structure

Market risk drivers

Simulate new 'risk driver' values

−

Loss Distribution VaR

Revalue portfolio

New 'trials'

FIGURE 29.4 distribution

Current MTM values

Gain / loss

0

+

The multiple simulation process for generating the market risk value

scenarios requires techniques for obtaining sets of market parameter values complying with the observed volatilities and correlations. The Delta VaR methodology, also known as the ‘linear–normal model’, relates individual transaction returns to market parameter changes through sensitivities. It suffers from limitations due to the assumption of constant sensitivities. Under these simplifying assumptions, the distribution of the portfolio return between current date and horizon is a normal distribution. Loss statistics and loss percentiles, which measure VaR, are obtained from analytical formulas. Nevertheless, the sensitivity-based approach is not appropriate with optional instruments because their sensitivities are not stable. The multiple simulation methodology (Monte Carlo simulation) remedies the drawback of the attractive, but too simple, linear model. It reverts to the direct modelling of asset values from the underlying random and correlated market parameters. The process implies generating correlated random values of market parameters and deriving, for each set, the values of individual assets. The random risk drivers need to comply with the actual variance–covariance structure observed in the market. Multiplying the number of trials results in a distribution of market parameter values. A distribution of the portfolio returns and value changes is then derived from these runs. This allows us to determine the VaR. The drawback of the Monte Carlo simulation is that it is calculation intensive, so that banks prefer the Delta VaR technique or intermediate techniques whenever possible (Figure 29.4).

30 Standalone Market Risk

This chapter reviews the techniques for measuring the market risk of individual transactions. Market risk results from the distribution of the value variations between current date and horizon of individual assets. In percentage terms, these deviations are the random asset returns. Consequently, all we need for measuring risk are the distributions of the returns of individual assets between the current date and the short-term liquidation horizon. Losses are downside moves from the current value, when ignoring the expected return for the short liquidation period. The main drivers influencing the returns of market instruments are the market parameters. Individual asset values also serve for options, such as stock options, of which values depend on that of the underlying stock. The literature designates them as ‘risk factors’: they are ‘risk drivers’ as well as ‘value drivers’ because risk materializes as changes of values or of market parameters. The risk measures are loss statistics, such as value volatility and downside variations of values at a preset confidence level. All statistics result from the distribution of the random asset values at the liquidation horizon. Pricing models in finance consider current prices as the present value of all future outcomes for individual assets. The VaR perspective is the reverse of the pricing perspective. We start from current prices and need to derive all potential adverse deviations of asset values or returns until a future time point. There are two basic techniques for doing so. Full revaluation determines the distribution of all asset values and returns after having modelled the full time path until horizon of those parameters that influence each individual asset price. Partial revaluation relies only on asset returns over the small time period until the liquidation horizon and uses shortcuts to relate these variations directly to market parameter returns over the period through sensitivities. Full revaluation at a future date for VaR purposes is intensive in terms of resources and often impractical at the full scale of the entire portfolio. Pricing models provide

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closed-form formulas facilitating revaluation at the horizon once we have the distribution of the parameters that drive values and returns of individual assets. The theoretical process comprises three steps: choosing intermediate time points as ‘vertices’ for intermediate calculations; generating random time paths of asset and market parameter returns from now to the horizon; revaluing from these time paths the assets at the liquidation horizon. The Delta VaR technique uses several simplifications for bypassing full revaluations: mapping individual asset returns to market parameter returns; constant sensitivities of asset returns to underlying value drivers; linear relationships between asset returns and market parameter returns; normal distributions of market parameter returns as a proxy for actual distributions over short horizons. This solution is more economical than full revaluation because there are many fewer market parameters than individual assets. The Delta–normal VaR technique that provides an analytically tractable distribution of asset returns and values at the horizon makes it simple to derive value statistics, volatility and value percentiles for a single period. When complex assets, such as exotic options, make this process excessively simplistic, we need to revert to full revaluation. However, the full power of the Delta–normal technique appears when dealing with portfolios of assets, rather than a single asset, because the portfolio return depends in a linear way on all market parameter returns mapped to the various assets in the portfolio. The values of all instruments depend on a common set of value drivers that comply with the variance–covariance structure. Therefore, we need to mimic this correlation–volatility structure before revaluation. The portfolio return, or final value, sums algebraically all individual asset returns and depends on offsetting effects between positive and negative variations and on correlations between individual asset returns, as subsequent chapters explain. The current chapter shows the process in the case of a single asset, as an intermediate step. The first section provides an overview of VaR and modelling of returns, and discusses the full valuation technique and the Delta VaR technique. The second section describes well-known stochastic processes for returns, stock prices and interest rates. The third section addresses the volatility measurement issue, which is a major ingredient for modelling variations of returns. The fourth section explains how to derive standalone risk measures from the normal and lognormal distribution of prices. The last section addresses the mapping issues and describes some common sensitivities, which are the foundations of the Delta VaR technique.

VAR AND FUTURE ASSET RETURNS The market risk loss is the adverse deviation of value due to market movements. This variation is Vi = (Vi ,H − Vi,0 ) = Vi,0 × ri . It is the deviation from current value or, equivalently, the return times the current value. The return measures the unit deviation of value. Hence, measuring value deviations over short time periods is equivalent to measuring the random returns from the current value in percentage terms. Therefore, to measure market risk VaR, it is sufficient to model random returns between the current date and the horizon. In what follows, we use both value and cumulative returns equivalently since they map on a one-to-one basis at the horizon. The valuation building block of market risk models relies on two basic techniques:

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• Full revaluation at a future horizon, set as the liquidation period. • Partial revaluation of individual asset returns from market movements, also known as the Delta VaR technique, which applies to both single assets and portfolios.

Full Revaluation The finance literature relies extensively on modelling returns for pricing models. Theoretical finance derives prices from modelled time paths of returns. Returns follow stochastic processes. Stochastic processes specify the distribution of asset returns, or of market parameters, over each time interval, as a function of their expected values and volatilities. Implementing these stochastic processes allows us to model future outcomes, and to find all future asset payoffs, notably when these payoffs are contingent on outcomes, such as for options. For common assets, including standard options, modelling the stochastic processes of returns results in closed-form pricing formulas. For complex options, the modelling of return processes allows us to conduct simulations of all future outcomes and derive the current price as the expected present value of all option payoffs over this range. Derivative pricing models derive the return of derivatives from the return process of the underlying asset. The most well known example is the pricing of European stock options. The Black–Scholes equation prices a European equity option by modelling the return of the option as a function of the underlying stock return using Itˆo’s lemma. The principle applies to all derivatives1 . Accordingly, the literature on stochastic processes expanded considerably. Stochastic processes serve for modelling all asset returns and prices. They apply to stock prices, interest rates and derivatives, whose returns depend on the underlying asset return process2 . The VaR perspective is a sort of ‘reverse’ pricing perspective. Instead of finding the current price from the present value of all possible future outcomes, we derive all possible asset values for all future outcomes from the current data. There is a duality between cumulative returns and values. The time paths of returns map on a one-to-one basis to future values. This is obvious using the single discrete time formula for the random return y(0, t) = (Vt − V0 )/V0 , with Vt and V0 being random values at date t and the current certain value at date 0. However, discrete time returns have drawbacks, as illustrated in the section on returns below, making it necessary to use small time intervals and construct the time path of returns, over a sequence of small periods. The random cumulative return links final value to initial value. Continuous finance makes extensive use of instantaneous ‘logarithmic’ returns over small time intervals, rather than arithmetic returns, because they are additive, as demonstrated in the appendix to this chapter. There are important variations across asset classes for modelling future outcomes. For stocks, we need to model the time path of stock returns and derive the distribution of future prices, given the expected return and its volatility as measured from historical observations. We find, as shown below, that stock prices follow a lognormal distribution whose parameters depend on the expected return and its volatility. For bonds, we need 1 See

Hull (2000) and Merton (1990) for comprehensive reviews. Neftci (1996) for an introduction to the mathematics of stochastic processes, Smith (1976) and Cox and Ross (1976) for option pricing.

2 See

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to derive all future interest scenarios at the future date from their stochastic processes to revalue the bond for each of these scenarios by discounting at appropriate market rates the future contractual cash flows. For options, we need to derive all future outcomes for all their value drivers, which include the underlying prices, such as stock prices for equity options, and revalue the options at the future date. For complex derivatives, the process is the same except that we do not have closedform pricing formulas. Therefore, we need to revert to simulations of future option payoffs. The difference is that the price as of a given date results from all possible future outcomes. A forward revaluation requires determining all possible outcomes as of this future date. Unfortunately, there are many starting points at future dates, so we need to replicate the process for each of them. This mechanism is a ‘simulation within a simulation’. The first simulation provides all outcomes at the horizon. A second set of simulations replicates the process for all dates beyond the horizon starting from each horizon state. Evidently, such a process consumes many resources, making it unpractical. Derivatives looking backward, such as barrier options and ‘look back’3 options, raise different obstacles. The values of these options depend on all intermediate outcomes between the current date and a future date. Revaluation at a future date depends on these intermediate states. The ‘full revaluation’ technique requires all outcomes for all ingredients of future prices and recalculates future prices for each scenario. The distribution of future cumulative returns between today and the horizon, provides all that is needed for VaR calculations. Accordingly, the next section details some basic stochastic processes, those applying to stock prices and those applying to interest rates, to show the implications of using the full revaluation technique. Stochastic processes specify the distribution of asset returns, or of market parameters, over each time interval, as a function of the expected value of the return and its volatility. For stocks, the process allows stock prices to drift away from initial value without imposing any bound on prices. However, the time paths of interest rates follow specific stochastic processes because, unlike stock prices, they tend to revert towards a long-term average value. The next section provides an overview of these basic stochastic processes. Once stochastic processes of value–risk drivers are specified, the technique implies: • Choosing intermediate time points as ‘vertices’ for intermediate calculations. • Generating random time paths of assets and market parameter returns from now to each time point. • Revaluing from these time paths the asset at the liquidation horizon.

Partial Revaluation and Delta VaR It is obvious from the above that the process is resource intensive and impracticable in most cases. Additional shortcuts help resolve such complexities. Partial revaluation uses much simpler techniques. It starts from the observation that all we need for VaR purposes 3A

‘look back’ option has a payoff linking to past values of the underlying. For instance, the payoff of a look back equity option could be the maximum of the stock price between inception and the horizon. Barrier options disappear when the underlying hits a barrier, and also depend on the time path of past values.

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are variations of values or returns for each individual asset, and that value drivers for assets are market parameters. To be more specific, the drivers of stock returns are the equity indexes. The drivers of interest-bearing assets are the interest rates along the yield curve. The drivers of option values are these market parameters plus the underlying asset prices, for instance an interest rate or a stock price. There are many fewer market parameters than individual assets in a portfolio. Therefore, starting from market parameters to obtain asset returns is evidently more economical than modelling each asset return. The first prerequisite is to map asset returns to market parameters. Stock returns map to equity indexes, interest-bearing assets map to interest rates, stock options map to the market parameters of interest rates, stock return volatilities and stock prices, and so on for other options. The next step is to relate asset returns to the selected underlying market parameters. Sensitivities provide a simple proxy relationship as long as the variations of market parameters are not too important. This is acceptable in many cases for short-term horizons, except for options when market movements trigger abrupt shifts in sensitivities. Using constant sensitivities makes the relationship between market parameter changes and asset returns linear. This is a major simplification compared to using the pricing formulas of individual assets. The next step is to model the distribution of the market parameter returns. A normal distribution has very attractive properties: the linear combination of random normal variables is also a normal variable. Combining these shortcuts results in a simple technique. The asset returns follow normal distributions derived from a linear combination of random normal parameter returns. The derivation of all loss statistics and VaR becomes quite easy since we know all percentiles of the normal distribution. For single assets, a single value driver is sufficient. For options, there are several value drivers, but the option returns remain a linear combination of these. For a portfolio, various assets depend on different market parameters, so that we always have a linear function of all of them. These are the foundations of the ‘Delta–normal VaR’ technique. It provides normal distributions of asset returns from which VaR derives directly. The Delta VaR technique faces some complexities with interest rates. Modelling future outcomes of the yield curve is not an easy task. Simple interest rate models might suffice, for instance when there are only parallel shifts in the yield curve. In general, however, interest-bearing asset values depend on the entire yield curve. It is not practical to use full-blown yield curve models. One technique consists of selecting only some interest rates along the yield curve for references. An alternative technique consists of using factor models of the yield curve. These make the yield curve a function of a few factors that behave randomly, with specific volatilities, subject to prior fitting to historical data. They allow us to simulate random shocks on the yield curve, including those that alter its shape (level, slope and ‘bump’). Converting these factors into yield curves provides the required distribution of selected interest rates. The partial revaluation process requires the following steps, that substitute for full revaluation: • • • •

Mapping asset returns to value–risk drivers. Simple modelling of the market parameter returns. Measuring the sensitivities of asset returns to market parameters. Deriving the asset return distribution as a linear function of these distributions.

Limitations of the Delta–normal technique are:

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Constant sensitivities. Proxy mapping to market parameters, generating basis risk. Proxy of actual distributions of value–risk drivers. Ignoring ‘fat tails’ of actual distributions.

Notably, the Delta VaR technique collapses when sensitivities cannot be considered constant, as for options and look back derivatives. Moreover, only on sensitivities for stocks ignores risk unrelated to market parameters, or the specific risk of stocks. The diagonal model of stock returns remedies this drawback, as explained in the next chapter. Nevertheless, the Delta VaR technique shines for portfolios. Portfolios of assets make the overall return dependent on the entire set of market parameters. Their random movements depend on their correlations. Since the Delta VaR technique makes individual asset returns a linear function of random parameters, it also applies to the portfolio return. The difference when dealing with multiple market parameters resides in correlations affecting their co-movements. Since it is a relatively simple matter to model correlated random variables, using techniques detailed in Chapter 28, the Delta VaR technique allows us to handle correlations without difficulty.

STOCHASTIC PROCESSES OF RETURNS To fully revaluate asset values at the horizon, the common technique consists of modelling the individual asset returns to derive the value at the horizon by cumulating intermediate returns. This section explains the basic mechanisms of common stochastic processes and how time paths of returns relate to final values. Pricing models view returns as following ‘stochastic’ processes across successive short time intervals. Measuring risk requires the distribution of cumulative returns to the horizon given the current value. The first subsection illustrates the drawbacks of a single-period discrete return. The next subsections discuss two types of processes serving to model asset returns: the Wiener process for stock prices and processes applicable to interest rates. The last subsection discusses volatility measures, which are critical inputs to risk measures and stochastic models of returns.

Single-period Return The return over a discrete period is the relative variation of value between two dates. For a short time interval, the single-period return over that period is sufficient to obtain the final value because there is a one-to-one correspondence between the random return value and the final value, according to the above equation. Random returns between dates 0 and H are y(0, H ), values indexed by date are V0 for the current (certain) value and VH for the random value at horizon H . The return y, between any two dates, and the value at the horizon, are: y(0, H ) = (VH − V0 )/V0 VH = V0 (1 + y) With a single-period return, the value distribution at H derives from that of the return because VH = V0 (1 + y). The obvious limitation of this technique for modelling returns

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and final values is that a normal distribution allows the random return to hit values lower than −100%, resulting in non-acceptable negative values of the asset. For a sufficiently short horizon, this formulation is convenient because normal distributions are easier to handle. When the period gets very small, the return becomes y = dV/V , where dV is the small change in value. This makes it more acceptable to use normal distributions. On the other hand, it makes it necessary to use cumulative returns, which are not normally distributed, between discrete dates, current date and horizon.

Stock Return Processes Under the efficient markets hypothesis, all stock values reflect all past information available. Consequently, variations of values in successive periods are independent from one period to another because the flow of new information, or ‘innovations’, is random. In addition, small period returns follow approximately normal distributions with a positive mean. This assumption applies notably to stock prices, but it is also a proxy for short-term interest rates. The mean is the expected return, such as the risk-free return for a risk-free asset and the risky expected market yield for risky assets. Over very small intervals, returns follow stochastic processes. The Wiener process is a common stochastic process, adequate for stock prices: dVt = µVt dt + σ Vt dzt This equation stipulates that the small change in value of a stock price Vt is the sum of a term µVt dt, which reflects the expected return µ per unit time, times the initial stock value, and proportional to the time interval dt, plus a random term. This first term is the expected value of the relative increase dVt /Vt of the asset price, because the expected value of the random term dzt is zero. The random component of the return is σ Vt dzt , proportional to the return volatility σ , the price Vt and a random ‘noise’, dzt , following a standardized normal variable (mean 0 and volatility 1). This simple process models the behaviour of the price through time. dVt follows a ‘generalized Wiener process’, with drift µ and volatility σ . The process makes the instantaneous return dVt /Vt a direct function of its mean and volatility σ : y = dVt /Vt = µdt + σ dzt This equation shows that the return follows a normal distribution around the time drift defined by the expected return µ dt. Since dx/x is the derivative of ln(x), this equation shows that the logarithm of stock price at date t follows a normal distribution4 . Therefore, the stock price at date t follows a lognormal distribution whose equation is5 : Vt = V0 exp[(µ − 21 σ 2 )t + σ zt ] 4 This

necessitates integration from date 0 to t of the small value increments over each dt. lognormal distribution is the distribution of a variable whose logarithm follows a normal distribution. The summation of all dzt over time results in a normal variable for zt . The equation of the final value of the stock results from expressing the logarithm of the stock price as following the process d[ln(S)] = [(µ − 12 σ 2 )dt + σ dz]. 5A

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For stocks, the random process is consistent with the efficient markets hypothesis stating that, if all prices fully reflect the available information, they fluctuate randomly with the flow of new information. Therefore, new prices are dependent only on ‘innovations’ dzt , not on past prices. The dzt are uncorrelated through time and follow the same distribution at all dates. Under this assumption, the process is ‘stationary’, meaning that it remains the same through time. Innovations are identically independently distributed (iid). Because the innovations dzt are independent variables of variance (σ dt)2 , their sum zt has a variance equal to the sum of variances. Using dt = 1, and t time√intervals, the variance of the cumulated innovations is t × σ 2 , and the volatility is σ t. It increases as the square root of time. This is the ‘square root of time’ rule for the volatility. It results in an ever-increasing volatility of the random price. The formula makes it easy to generate time paths of the St by generating normal standardized dzt and cumulating them until date t, with preset values of the instantaneous expected return µ and its volatility σ . This simple process is very popular. It applies to asset prices and market parameters whose variance increases over time.

Generating Time Paths of Stock Values: Sample Simulation The simulation of time paths of value drivers allows us to model the forward values of any asset. We illustrate here the mechanism with the simple Wiener process, which results in a lognormal distribution. The technique comprises three steps: • Choosing intermediate time points as ‘vertices’ for intermediate calculations of the returns. • Generating random time paths of market parameter returns from now to the horizon based on the specific parameters characterizing the asset process. The final asset value distribution at the horizon results from cumulative returns along each time path. To generate a discrete path, we use a unit value for t = t − (t − 1) and the standard N(0, 1) normal variable for the innovations z. With several time intervals, we obtain the intermediate values of the price. The process uses the following inputs: • • • • •

Initial price V0 = 10. Time interval t = 0.1 year. Expected return 10% yearly, or 10%/10 = 1% per 1/10 of a year. Yearly volatility of the random component 20%, or 20%/10 = 2% per 1/10 of a year. The innovation at each time interval z as a standardized normal distribution.

The price after one step is: Vt − Vt−1 = Vt−1 (µ + z × t) = Vt−1 (1% × 0.1 + z × 0.2 ×

√

0.1)

Simulating a time path over 1 year consists of generating a time series of 10 random draws of 10 innovation values. Simulating several time paths consists of repeating the process as many times as necessary. This allows us to calculate any final value that depends on past values, such as the value of a look back option on the stock. Figure 30.1 shows various time paths and the final price at date 10.

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13 12 11

Value

10 9 8 7 6 5 4 0

3

4

5 6 Time

7

8

9

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22.5

18.8

15

11.3

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20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% 0

FIGURE 30.2

2

Time paths of value

Probability

FIGURE 30.1

1

Distribution of final value

Figure 30.2 shows the simulation of the final prices after 1000 time path simulations, which is a lognormal distribution. The appendix to this chapter shows why logarithmic returns provide good proxies of instantaneous returns and facilitate handling cumulative returns when the number of subperiods makes them small enough. The formulas provide useful shortcuts when modelling cumulative returns.

Interest Rate Processes Interest rate processes are more complex than the basic stock price processes for two reasons: • There are several interest rates, so we need to model the behaviour of several parameters rather than only one. • Interest rates tend to revert to some long-term average over long periods. To capture the behaviour of interest rates, we need to model the entire time structure of rates, with models using ‘mean reverting’ stochastic processes. Interest rate models view

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interest rates as following a stochastic process through time. A stochastic process specifies the change in the rate as a function of the time interval and a random ‘noise’ mimicking the volatility of rates. Several stochastic processes apply to interest rates. Mean reversion implies that rates revert to some long-term average and avoid extreme variations of the interest rate to mimic the actual behaviour of rates. Mean reverting processes prevent rates from drifting too far away from a central tendency materialized by a long-term rate. This contrasts with stock prices, which can drift away from initial values without bounds. A ‘term structure model’ describes the evolution of the yield curve through time. Models generate the entire yield curve using one or two factors. When using models to generate interest rate scenarios, the technique consists of generating random values of the factor(s) driving the rate and the error term. Random interest rate scenarios follow. To find all possible values of a bond at horizon, we need to simulate yield curve scenarios between now and the horizon. For each scenario, there is a unique value of the bond price resulting from its discounting contractual cash flows at maturity with rates of simulated yield curves. Since the same yield curve applies to all interest-driven assets, ignoring the fluctuation of credit spreads, we use the same yield curve scenarios for all bonds. For derivatives, the final values result from the underlying asset values, as for equity derivatives. One-factor models involve only one source of uncertainty, usually the short-term rate. They assume that all rates move in the same direction. However, the magnitude of the change is not the same for all rates. The following equation represents a one-factor model of short-term rates rt . Using such a model implies that all rates fully correlate with this short-term rate. The random dzt term follows a standardized normal distribution and makes the rate changes stochastic: drt = κ(θ − rt )dt + σ (rt )γ dzt The parameter κ < 1 determines the speed of mean reversion to a long-term mean θ . When the rate is high, the term in parentheses becomes negative, and contributes to a downward move of the rate. When the parameter γ = 0, the change of rate is normally distributed as the random term. When γ = 1, the rate rt follows a lognormal distribution, as the stock prices do with the standard Wiener process. When γ = 0.5, the variance of the rate is proportional to its level, implying that low interest rates are less volatile than high interest rates. One-factor models use the following processes: Rendleman and Bartter: dr = κrdt + σ rdz Vasicek’s model: Cox, Ingersoll and Ross model:

dr = κ(θ − r)dt + σ dz √ dr = κ(θ − r)dt + σ r dz

Two-factor models assume two sources of uncertainty (Brennan and Schwartz, 1979; Cox et al., 1985; Longstaff and Schwarz, 1992). The ‘no-arbitrage’ models start by fitting the current term structure of rates, which provides the benefit of starting from actual rates that can serve for pricing purposes or defining interest rate scenarios from the current state. These models comply with the principle of no risk-free arbitrage across maturities of the spot yield curve. However, this constraint implies that the current shape of the yield curve drives to a certain extent the future short rates. The Ho and Lee (1986) model uses a binomial tree of bond prices. Hull

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and White extended the original technique6 . Advanced models allow long-term modelling of the yield curves (Heath et al., 1992). For VaR purposes, it is not practical to use elaborate interest rate models. Simpler techniques are necessary. Selecting some interest rates as references and allowing them to vary, while complying with given variance–covariance structure, is the simplest technique. Principal component analyses use historical data on market parameters to define components that explain the movements7 . The application of this technique to term structure modelling shows that the most important factors represent the parallel shifts, twists (slope changes) and curvature of the term structure of yield. Principal component analysis has many attractive properties. The rates of different maturities are a linear function of the factors. The technique provides the sensitivities of rates to each factor and the factor volatilities. Since factors are independent, it is easy to derive the volatility of a rate given sensitivities and factor volatilities. For simulation, allowing factors to vary randomly, as well as the random error, generates a full distribution of yield curves.

MEASURING VOLATILITIES The volatility is a basic ingredient of all processes leading to the horizon value distribution. There are several techniques to obtain volatilities. The first is to derive them from historical observations of prices and market parameters. More elaborate approaches try to capture the instability of volatility over time. They use moving average, with or without assigning higher weights to the most recent observations, or ARCH–GARCH models of the time behaviour of observed volatilities. A third technique is to use the implicit volatilities embedded in options prices, by reverse engineering pricing models of options to derive from the price the volatility parameter. Implicit volatilities look forward by definition, as market prices do, but they might be very unstable.

Square Root of Time Rule Historical volatility measures require us to define the horizon of the period of observations and the frequency of observations. The frequency stipulates whether we measure a daily, monthly or yearly volatility. It is easier to use daily measures because there is more information. Then, we move to other periods using the ‘square root of time rule’ for volatilities. The square root of time rule applies only when there are no bounds to cumulative returns, as mentioned above for stock prices, and if the underlying process is stationary. The rule stems from the independence assumption across time of consecutive value changes. For instance, the change in values between dates 0 and 2, V (0, 2), is the sum of the change between dates 0 and 1 and between dates 1 and 2 (end of period dates), V (0, 1) + V (1, 2). The issue is to find the volatility of the change σ02 of V (0, 2) between 0 and 2 from the volatilities between 0 and 1, σ01 , and between 1 and 2, σ12 , which are supposed identical, σ01 = σ12 = σ1 , the unit period volatility. A simple rule of summation indicates the volatility of the sum is the square 6 See 7 See

Hull (2000) for a review of all models and details on the Hull and White extensions. Frye (1997) for factor models of interest rates, also Litterman and Scheinkman (1988).

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√ root of the sum of the squared volatilities, or σ02 = σ1 2. The rule extends easily to any number of units of time, if variations across time periods remain independent and have a constant √ volatility. Using the rule, with a 1% daily volatility, the yearly volatility is σ1year = σ1day 250 = 1% × 15.811 = 15.811%. This simple rule allows us to convert market volatilities over various lengths of time. The relationship between the volatility σt √ over t periods and the volatility over a unit σ1 period is σt = σ1 t (Figure 30.3). σ

σ √t

Time t

FIGURE 30.3

Volatility time profile of a market parameter over various horizons

Practical rules are helpful, even though reality does not comply with the underlying assumptions. For long-term horizons, the above volatility would become infinite, which is unacceptable. Many phenomena are ‘mean reverting’, such as interest rates. The above formula provides a practical way to infer short-term volatilities only. In addition, real phenomena are not ‘stationary’, implying that volatilities are not stable over time, and the assumption does not hold.

Non-stationarity Various techniques deal with non-stationarity, which designates the fact that the random process from which we sample the volatility observations is not stable over time. Since the VaR is highly sensitive to volatilities of market parameters, it is of major importance to use relevant and conservative values. They extend from simple conservative rules to elaborate models of the time behaviour of volatility, ARCH–GARCH models (Engle, 1993; Nelson, 1990a). Volatilities are highly unstable. Depending on the period, they can be high or very low. There are a number of techniques for dealing with this issue8 . Some are very simple, such as taking the highest between a short-term volatility (3 months) and a long-term volatility (say 2 years). If volatility increases, the most recent one serves as a reference. If there are past periods of high volatility, the longer period volatility serves as a reference. Other techniques consist of using moving averages of volatilities. A moving average is the average of a determined number k of successive observations. The average moves with time since the sample of the last k observations moves when time passes. For example, we could have a time series of values such as: 2, 3, 5, 6, 4, 3. A three-period moving average would calculate the average over three consecutive observations. At the third date it is 8 For

updating volatilities in simulations, see Hull (2000).

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(2 + 3 + 5)/3, then it becomes (3 + 5 + 6)/3 at the fourth date, (5 + 6 + 4)/3 at the fifth date, and finally (6 + 4 + 3)/3 at the last date. Moving averages minimize deviations from the long-term mean. Arithmetic moving averages place equal weights on all observations, ignoring the fact that the latest ones convey more information. Exponential smoothing places higher weights on the most recent observations and mimics better the most recent changes. The Exponentially Weighted Moving Average (EWMA) model uses weights decreasing exponentially when moving back in time. Risk Metrics uses a variation of these ‘averaging’ techniques9 . The GARCH family of models aims to model the time behaviour of volatility. Volatility seems to follow a mean reverting process, so that high values tend to smooth out after a while. The models attempt to capture these patterns assuming that the variance calculated over a given period as of t depends on the variance as of t − 1, over the same horizon, plus the latest available observation. The model equation, using ht as the variance and rt as the return, is: 2 ht = α0 + α1 rt−1 + α2 ht−1 The variance ht is a conditional variance, dependent on past observations. By contrast, the long-term unconditional mean is h, obtained by setting h = ht = ht−1 and observing that E(r2t−1 ) = h : h = α0 + α1 h + α2 h implies h = α0 /(1 − α1 − α2 ). The model results in a series of variance shocks followed by smoothed values of variance.

VALUE DISTRIBUTIONS AT THE HORIZON AND STANDALONE MARKET RISK The measures of standalone market risk of single instruments are loss statistics, loss volatility and loss percentiles, all resulting from the distribution of final values at horizon. Applied to a single asset, we obtain the standalone VaR at a preset confidence level. This section uses the normal and lognormal distributions to derive these measures. The normal distribution has the drawbacks mentioned above, but serves for the Delta–normal technique for a short-term horizon. When reverting to continuous stochastic processes of the time paths of asset returns, we obtain lognormal distributions at the horizon for stock prices. The value percentiles derive from this lognormal distribution.

Normal Values Distribution at Horizon Return yt and value Vt are such that Vt = V0 (1 + yt ) and both follow a normal distribution. The confidence level for the value percentiles is such that: Prob[Vt ≤ V (α)] = α Converting this equation into a condition on the random return between 0 and t: Prob[V0 (1 + yt ) ≤ V (α)] = α Prob{yt ≤ [V (α) − V0 ]/V0 } = α 9 J.

P. Morgan, Risk Metrics Monitor, 1995.

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yt follows N(µ, σ ) and zt = (yt − µ)/σ follows N(0, 1), the normal standardized distribution. Since yt = σ zt + µ, the preceding inequality becomes: Prob{zt ≤ [V (α)/V0 ] − (1 + µ)/σ } = α By definition, Prob[zt ≤ z(α)] = α. The probability of a standardized normal variable being lower than a given threshold z(α) is by definition (α), where is the cumulative standard distribution. By definition, z(α) = −1 (α), with −1 being the standardized normal inverse. This inequality defines z(α) and the value percentile V (α) derives from the linear relationship between z(α) and V (α): z(α) = −1 (α) = [V (α)/V0 − (1 + µ)]/σ V (α)/V0 = 1 + µ + −1 (α)σ In this formula, −1 (α) is negative, so that the final value can be lower than the initial value as long as the downside deviation exceeds the upside variation due to the asset return. As an example, let’s assume that the expected yearly return µ is 10%, the yearly return volatility σ is 15%, the horizon t is 1 year and the value at date 0, V0 , is 1. This simplifies the formula. If the confidence level is 1%, Prob[zt ≤ z(α)] = α implies that z(α) = −1 (α) = −2.3263. Hence: V (α)/V0 = 1 + 10% − 2.3263 × 15% = 75.11% The downside deviation of the value at the preset confidence level of 1% is 1 − 75.11% = 24.89% in absolute value.

Lognormal Values Distribution at Horizon When using normal distributions of continuous returns, the distribution at the horizon is a lognormal distribution. The lognormal distribution applies notably to stock prices. Defining the distribution requires the return volatility. Deriving value percentiles from a lognormal distribution is less straightforward than for a normal distribution. If the return follows a Wiener process, the value at date t is: √ Vt = V0 exp[(µ − 21 σ 2 )t + σ t zt ] The random component zt follows a standardized normal distribution N(0, 1). The parameters µ and σ are the mean and volatility of the instantaneous rate of return of the firm. The value percentile V (α) at the preset confidence level α is such that: Prob[Vt ≤ V (α)] = α From the distribution of the value at t, we derive the value percentile V (α) using the inequality: √ Prob{exp[(µ − 12 σ 2 )t + σ t zt ] ≤ V (α)/V0 } = α This inequality requires that zt be such that: √ Prob{(µ − 21 σ 2 )t + σ t zt ≤ ln[V (α)/V0 ]} = α √ Prob{zt ≤ ln[V (α)/V0 ] − (µ − 21 σ 2 )t/σ t} = α

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The probability of a standardized normal variable being lower than a given threshold z(α) is by definition (α), where is the cumulative standard distribution, and z(α) = −1 (α), with −1 being the standardized normal inverse. The equality Prob[zt ≤ z(α)] = α defines z(α) and, then, the value percentile V (α). This requires determining z(α) first, and moving from there to V (α): √ z(α) = −1 (α) = {ln[V (α)/V0 ] − (µ − 12 σ 2 )t}/σ t √ ln[V (α)/V0 ] = −1 (α)σ t + (µ − 12 σ 2 )t √ V (α)/V0 = exp[−1 (α)σ t + (µ − 21 σ 2 )t] In this formula, −1 (α) is negative, so that the final value is lower than the initial value as long as the downside deviation exceeds the upside variation due to the asset return adjusted for the volatility term. As an example, we assume that the expected yearly return µ is 10%, the yearly return volatility σ is 15%, the horizon t is 1 year and the value at date 0 is 1, so that we have the final value as a percentage of the initial value. This simplifies the formula. If the confidence level is 1%, z(α) = −1 (α) = −2.3263. The argument of the exponential above is: √ −1 (α)σ t + (µ − 12 σ 2 )t = −2.3263 × 15% + (10% − 12 × 15%2 ) = −0.260195 The exponential of this term is 77.09%. The downside deviation of the value at the preset confidence level of 1% is 1 − 77.09% = 22.91% in absolute value. We observe that this downside deviation is lower than under the normal approximation, because the normal approximation has a ‘fatter’ downside tail than the lognormal distribution.

MAPPING INSTRUMENTS TO RISK FACTORS Modelling all individual asset returns within a portfolio is overly complex to handle. Since the number of market indexes is much smaller than that of individual assets, it is more efficient to derive asset returns from the time paths of market parameters that influence their values. On the other hand, it necessitates modelling the relation between asset returns and market parameters. The process requires two steps: identifying those market parameters which influence values, a process called ‘mapping’; modelling the sensitivity of individual asset values to market parameters. Mapping and using sensitivities to underlying market parameters is the foundation of the Delta VaR technique. The first subsection describes the mapping process, the second one explains how to derive standalone market risk under the Delta VaR technique with a single asset. The next subsections specify the sensitivities of various types of assets.

The Mapping Process The mapping process results from pricing models in order to identify the main value drivers of asset values. Mapping individual asset returns to market parameters often uses only a subset of all market parameters for simplification purposes. The subset of interest

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rates corresponds to selected maturities. For interest rate references that do not correspond to these maturities, it is possible to interpolate a proxy of the corresponding rate using the references. The exclusion of some value drivers results in ‘basis risk’, since the modelled values do not track exactly the actual asset prices any more. The variations of value of any market instrument result from its sensitivities to market parameters and the volatility of these parameters. The mapping of individual asset values to market parameters (also designated as risk drivers or risk factors) results from pricing models, showing which parameters influence values. Most relations between asset values and value drivers have a linear approximation. In some cases, the relationship is statistical, and implies always a significant error, such as for stocks. In such cases, it becomes necessary to account for the error in estimating the risk. In other cases, the relation results from a closed-form formula such as for bonds or simple derivatives. When dealing with derivatives, such as interest rate swaps or foreign exchange swaps, or forward rate agreements, it is necessary to break down the derivative into simpler components to reduce complexity and find direct relations with market parameters driving their values. For example, a forward transaction is a combination of long and short transactions, such as a forward interest rate (lend long until final date and borrow short until start date). An interest rate swap is a combination of fixed and variable flows indexed to the term structure of rates. A forward swap is a combination of a long swap and a short swap. Sensitivities are unstable because they are local measures. This is a major limitation, notably for options. Their sensitivity with respect to the underlying parameter is the delta of the option. An ‘out-of-the-money’ option has a relatively stable and low delta. When the option is ‘at-the-money’, its delta changes significantly with the underlying price. When it is ‘in-the-money’, the value increases almost proportionally with the market parameter and the delta tends progressively towards 1. With out-of-the-money options, the portfolio behaves as if no options exist. When options are in-the-money, the sensitivity gets close to 1. In other words, the portfolio behaves as if its sensitivity structure changes with variations in the underlying assets. This is the convexity risk, encountered with implicit options in Asset–Liability Management (ALM) models. Whenever convexity is significant within a portfolio, the simple Delta VaR model becomes unreliable10 . Extensions, as described in the next chapter, partially correct such deficiencies. Sensitivities are ‘local’ measures that change with the context. Accordingly, calculating the value change of an instrument as a linear function of market parameter changes is only an approximation. When a model relates the price of an asset to several parameters, the ‘Taylor’ expansion11 of the formula is a proxy relationship of the value change resulting from a change in value drivers. It relates the value change to changes in all parameters using all derivatives of the actual function. A Taylor series makes the change of the value of instruments as a function of the first, second, third-order derivatives, and so on. For large changes, it is preferable to consider additional terms of Taylor expansion of the equation beyond 10 Another

case in point is the measure of credit risk for options. Since the horizon gets much longer than for market risk, the assumption of constant sensitivities collapses. 11 See the Taylor expansion formula in Chapter 32.

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the first one. Additional terms better proxy the curvature of the relationship between the asset returns and their drivers.

Standalone Risk from Market Returns The return sensitivity change in asset i is, in general, a function of several market parameters k: K sik × mk /mk0 yi = Vi /Vi0 = 1

The index k refers to any of the K relevant market parameters, with k varying from 1 to K. The index i refers to any one of N transactions, with i varying from 1 to N . Since one transaction often depends on only one market parameter, and in general on only a small number of them, most of the sensitivities are zero. The standalone risk of an instrument results directly from the sensitivities and volatilities of market parameters. The equity index return has a volatility of 20%. The volatility of the stock return results from the above relationship: σ (yi ) = σ (Vi /Vi0 ) = sik × σ (mk /mk0 ) σ (Vi ) = σ (yi ) × Vi0 = sik × σ (mk /mk0 ) × Vi0 Let’s assume that the stock return has a sensitivity of 2 to the equity index. Then: σ (yi ) = σ (Vi )/Vi0 = sik × σ (mk /mk0 ) = 2 × 20% = 40% If the initial asset value is 1000, the resulting value volatility is 400. If the value follows approximately a normal distribution, the loss percentile at the 1% confidence level is 2.33 × 400 in value, or 932. Calculations are very simple because the linear approximation implies that the product of a random normal return by a constant follows also a normal distribution.

Stocks For stocks, the ‘beta’ β relates the change in stock price to the change in market index for the same period. It is the sensitivity of the equity return (a percentage change) to the index return (a percentage change). β is the risk measure in the Capital Asset Pricing Model (CAPM), the well-known model12 that provides the theoretical price of a stock and the required return by the market as a function of risk. It is the coefficient of the regression of the historical equity periodical returns against the similar index return providing β as the coefficient of the regression. The return is the ratio of the variation in stock price, or 12 The

ments.

original presentation is in Sharpe (1964). See also Sharpe and Alexander (1990) for subsequent develop-

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index value, to the original value, for any period. There are as many return observations as periods (daily, weekly, monthly, and so on). β is higher or lower than 1, and it is 1 by definition for the equity index. The return of the stock i and of the index Im are: ri = Pi /Pi rm = Im /Im β results from a regression on historical returns fitting the equation below to time series of observations: r i = βi r m + α + ε i In the case of stocks, there is no deterministic relationship between the equity index and the stock return. Rather, there is a general risk resulting from general market index changes plus an error term. Ignoring the error term: ri = βi rm β relates the relative price changes of a stock and an equity index, which is the market parameter in this case. If a stock has a β of 1.2, a change in index return rm of 1% results, on average, in a 1.2% increase of the price return ri . Hence ri = Pi /Pi increases by ri = 1.2%. Considering the initial stock price as given, the variation Pi is 1.2% × Pi when the variation of the index Im is 1% × Im . Hence, β relates the relative and the absolute changes in stock price and equity index. Using statistical fits implies an error term. The stock return depends on the equity index return plus a random ‘innovation’ term, which is independent of the equity index return by definition. The error term is the fraction of return of the stock unexplained by general index variations. This fraction is the specific risk of the stocks, as opposed to the general risk shared by all stocks related to the equity index. The variance of stock return is: σ 2 (ri ) = βi σ 2 (rm ) + σ 2 (εi ). It sums the variance due to general risk and specific risk. In practice, since error terms offset to a significant extent, simplified techniques ignore the sum of the specific risks for all stock prices, which is the variance of a sum of independent variables. For portfolio models, the ‘diagonal model’ of stock returns allows us to consider the specific risk from the innovation term for all stocks.

Bonds and Loans The sensitivity of bond prices to interest rate shocks, shocks being parallel shifts of the entire spectrum of rates, is the ‘duration’. If the duration of a bond is 5, it means the bond value will change by 5% when all rates deviate by 1%. A common measure of sensitivity for bonds is the basis point measure. A basis point (bp) is 1% of 1%, or 0.0001. It is the deviation, expressed in basis points, for a unit basis point change of interest rates. In the previous example, the basis point change of the bond value is 5 bp for a 1 bp change in all interest rates. This is the ‘DV01’ measure of bond sensitivity.

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Sensitivities also apply to credit risk, because bond prices depend on ‘credit spreads’, the difference between the market rate applying to a risky bond and the risk-free market rate of government bonds. Since bond values are sensitive to credit spread, any widening of spreads reduces the value. The sensitivity of bonds with respect to credit spreads is similar to that of interest rates. For non-tradable assets, such as banking loans, sensitivities to interest rates also apply to their mark-to-market values and are also durations. These sensitivities serve for measuring the risks of the mark-to-market value of the balance sheet, or Net Present Value (NPV). Durations help when considering only parallel shifts of the yield curves. They do not capture changes in the shape of the curve. Considering all changes in interest rates over all maturities requires the sensitivities to all market rates to obtain proxy asset returns. For the NPV calculations, for ALM, assets and liabilities are fully revalued for each yield curve scenario, without relying on sensitivities. Technically, the change of a single rate along the curve implies a change of the discount factor relative to that rate. For VaR purposes, it is necessary to simulate yield curve changes, either from correlations across rates or from principal components factor models.

Foreign Exchange Exposures The sensitivity of the dollar value of any exposure labelled in foreign currency to the exchange rate is the variation of this dollar value due to a unit variation of the exchange rate. For instance, the dollar value of 1000 euros, with an exchange rate of 1 EUR/USD, is 1000 USD. If the exchange rate becomes 8 EUR/USD, the dollar value becomes 800 USD. The variation is −20%, which is in value −1000 × 20%. The relative change in dollar value of exposure and that of the market parameter value are identical to −20%. The value sensitivity of the exposure in USD with respect to the exchange rate is simply 1.

Options Options are the right, but not the obligation, to buy or sell some underlying parameter. Derivative, and option, sensitivities result from models, or ‘pricers’, which relate their values to their drivers. The value of options depends on a number of parameters, as shown originally in the Black–Scholes model: the value of the underlying, the horizon to maturity, the volatility of the underlying, the risk-free interest rate. The model is in line with intuition. The option value increases with the underlying asset value, its volatility and maturity, and decreases with interest rate. Option sensitivities are known as the ‘Greek letters’. The sensitivity with respect to the underlying asset is the ‘delta’ δ. Intuitively, the δ is low if the option is ‘out-of-the-money’ (asset price below strike) because we do not get any money by exercising unless the asset value changes significantly. However, when exercising provides a positive payoff, the δ gets closer to 1: if strike is 100 and asset price is 120, the payoff is 120 − 100 = 20; if the asset price increases by 1 to 121, the payoff increases by 1. The sensitivity can be anywhere in the entire range between 0 and 1, when

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the sensitivity increases from near zero values to values close to one when the option is in-the-money. The variation of δ is the ‘convexity’ of the option. Gamma (γ ) is the change in δ when the underlying changes. It is the change in slope of the curve representing the option value as a function of the underlying. The longer the horizon, the higher the chances that the stock moves above the strike price. Hence, the option is sensitive to the time to maturity, the sensitivity being theta (θ ). The shorter the time to maturity, the lower is the value of the option. This value change when time passes is the ‘time decay’ of the option value. The higher the volatility of the underlying asset, the higher the chances that the value moves above the strike during a given period. Hence, the option has a positive sensitivity to the underlying asset volatility, which is the ‘vega’ (ν). Since any payoff appears only in the future, its value today requires discounting, and so that it varies inversely with the level of interest rates. Rho (ρ) is the change due to a variation of the risk-free rate. The option has several sensitivities, one for each relevant parameter that influences its value.

APPENDIX 1: CUMULATIVE RETURNS OVER TIME PERIODS When using a single terminal point, we have a straightforward correspondence between the discrete one-period return and the final value. This is not so for cumulative returns over a large number of intervals. Terminal values result from cumulated returns over intermediate periods. In practice, we simulate returns for each set of subperiods, for instance 10 single-period returns, as in the example below, by drawing randomly 10 values from the distribution fitting the stochastic process of returns. Each of the 10 endof-period values results from the value at the start date times the random percentage return plus 1 (Vt /Vt−1 = 1 + y). Calculating the 10 end-of-period values directly provides the final one at horizon. The example above illustrates the process. There are shortcuts for obtaining cumulative returns if returns are ‘logarithmic’ and when using small intervals. The arithmetic return is yt = (Vt − Vt−1 )/Vt−1 . The logarithm of the price ratio, ln(Vt /Vt−1 ), is identical to ln[1 + y(t)]13 . It is approximately equal to the arithmetic return when the return is small because ln(1 + y) is approximately identical to y. From the logarithmic definition, the value after a time interval t is such that: Vt+t = Vt exp(yt). With arithmetic return yt = (Vt − Vt−1 )/Vt−1 , the final value Vt becomes negative for values of y lower than −100%. With logarithmic returns, the final value cannot be negative even with negative returns. This makes the normal distribution for the return y acceptable. If the logarithm of Vt /V0 follows a normal distribution of y, Vt follows a lognormal distribution, by definition. Finally, logarithmic returns are additive across periods. When combining returns across consecutive periods, the return between dates 0 and t is y(0, t) = ln(Vt /V0 ). The logarithmic returns are additive across periods. With two consecutive periods, we have: y(0, 2) = ln(V2 /V0 ) = ln[(V2 /V1 ) × (V1 /V0 )] y(0, 2) = ln(V2 /V1 ) + ln(V1 /V0 ) = y(0, 1) + y(1, 2) If, for instance, V1 /V0 = 120% and V2 /V1 = 90%, with V0 = 1, V1 = 1.2 and V2 = 90% × 1.2 = 1.08, the overall return is 8%, which differs from the arithmetic summation 20% − 10% = 10% because the percentages apply to different initial values. In the 13 The

return is yt = (Vt − Vt−1 )/Vt−1 and 1 + yt = 1 + (Vt − Vt−1 )/Vt−1 = 1 + (Vt /Vt−1 − 1) = Vt /Vt−1 .

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example of Table 30.1, the exact cumulative return between initial and final values is 8% from the actual values. This is very close to the logarithm cumulative return of 7.696%, which is the exact summation of single-period logarithmic returns. The arithmetic summation of the single-period arithmetic returns is 10%, significantly above the actual 8%. TABLE 30.1

Cumulating arithmetic and logarithm returns

Vt V0 V1 V2

1.00 1.20 1.08

Cumulative return

Vt /Vt−1 = 1 + y

ln(Vt /Vt−1 )

Arithmetic return yi

120.00% 90.00% V2 /V0 − 1

18.232% −10.536% ln(V2 /V0 )

20.00% −10.00% y1 + y2

8.00%

7.696%

10.00%

Since the sum of random normal variables is also a normal variable, so is y(0, t) for any horizon t. Since y(0, t) = ln(Vt /V0 ) follows a normal distribution, the final value follows a lognormal distribution.

31 Modelling Correlations and Multi-factor Models for Market Risk

To model correlations, the common principle for all portfolio models is to relate the individual risks of each transaction to a set of common factors. For market risk, the market values of individual transactions are sensitive to risk drivers that alter their values. When all risk factors vary, the dependence of the individual asset returns on this set of common factors generates correlations between them. Because common factors influence simultaneously all transaction risks, they create a ‘general’ risk, defined as the risk common to all assets. The fraction of individual risk unrelated to common factors is ‘specific’ risk. Factor models are two-sided, depending on the initial purpose. The first purpose is to model the correlations between risk drivers on which individual risk correlations depend. The second purpose is to construct loss distributions for portfolios. The next chapter explains how to proceed with factor models. We address only the first issue in this chapter. It is necessary to capture the interdependencies between market parameters to correlate future values of market parameters and assets. Individual asset value variations depend on these correlations, some of them being positive and others negative. They sum algebraically to obtain portfolio values. The diversification effect results from offsetting effects between individual variations. For market risk, correlations derive directly from observed market prices or market parameters. Pricing models of derivatives or bonds help because they relate in a deterministic way the prices of assets to the market parameters that derive them. In such a

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case, it is sufficient to correlate the market parameters to obtain correlated individual asset returns (or values) using the pricing models with the correlated market parameter values as inputs. When the relationship to market parameters is stochastic, for example between stock returns and equity indexes, it includes an error term, measuring the random component of individual asset returns unrelated to market parameters. When using factor models, asset return variations result from the volatility of the factors plus that of the error term, measuring the volatility of the asset return unrelated to factors. The error term contributes to the risk. The volatility of the common factors is the ‘general risk’, and the volatility of the error term is the ‘specific risk’. Specific risk is important for stock returns with general risk driven by equity indexes and specific risk related to each individual stock. These principles underlie the ‘diagonal’ model for stocks. Fitting factor models to returns allows us to determine both the return variances plus all their covariance terms from the coefficients of the factor models. In addition, the contribution of common factors to the return volatility is the general risk, while the volatility of the error term is the specific risk. The one-factor model of stocks illustrates these properties. Multi-factor models provide the same information. ‘Orthogonal’ factor models are the simplest because the factors are independent of each other. The first section briefly summarizes why correlations between risk drivers are key inputs for capturing diversification effects through variance–covariance matrices. The second section describes the specifics of the ‘correlation building block’ of market risk. The third section explains how to derive correlations, volatilities and general plus specific risk from factor models of individual asset returns. The appendix describes various types of factor models.

WHY IS THE VARIANCE–COVARIANCE MATRIX NECESSARY FOR MODELLING PORTFOLIO LOSSES? The risk over a portfolio is the change in all mark-to-market values of individual instruments. The loss of the portfolio is the algebraic sum of all gains and losses for all individual positions. Some exposures gain values, others lose values. These changes are market parameter-driven. Since these correlate, all individual returns correlate as well. When they relate deterministically to risk drivers, the entire value variations result from sensitivities. When there exists only a statistical relationship, the risk drivers generate general risk, to which it is necessary to add specific risk, the fraction of risk not related to the common factors. A simplistic view would use simultaneous adverse changes, captured as a function of both the risk parameter volatility and the instrument sensitivities. This is a most conservative rule because it is not possible that all market parameters will change simultaneously in such a way that they trigger losses simultaneously for all individual positions. Such arithmetic addition of all risks would greatly over-estimate the portfolio risk. Sensitivities do not add. Adding them is equivalent to assuming that all parameters change simultaneously in adverse directions, which is unrealistic. The key to measuring portfolio risk lies in capturing the market parameter interdependencies. It is easy to observe that some vary together and others inversely. Sometimes the relationship is strong and sometimes it is loose. The structure of these relationships results in a variance–covariance matrix and a correlation matrix between all market parameters

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to which the instrument values map. The process for measuring portfolio risk comprises two steps: • Imposing this variance–covariance structure on market parameter deviations. • Modelling the portfolio return distribution when all market parameters vary in compliance with such variance–covariance structure. The correlation methodology aims to replicate realistic simultaneous changes in all market parameters, in line with observed changes. This is a prerequisite for revaluing the portfolio at the horizon. When the correlation structure is available, the correlations between individual asset values result from those of the stochastic processes driving the asset values. Several assets depend on the correlated market parameters, such as equity indexes. It is possible to correlate market parameters and prices, assuming a normal distribution of their values as an approximation. When implementing full valuation techniques, the issue is to correlate random time paths of returns for different assets. To generate the paths of N correlated assets, indexed i, we use several correlated random processes as follows. The stochastic process driving the market parameter or the individual asset return includes a random term, such as the process applying to stock prices: yi = dVit /Vti = µi dt + σ i dzit , where i is the index specific to an individual asset. This process results in the value distribution at√a forward date t lognormal distribution of stock prices: Sti = S0i exp[(µi − 21 σi2 )t + σi t]. When considering a pair of stocks, we apply a correlation on the random innovations dzit of the process. In order to simulate correlated normal innovations, standard procedures apply, such as the Cholevsky decomposition, explained below. The final prices of the pair of stocks correlate even though they follow lognormal distributions. The techniques for portfolio revaluation at the horizon vary from using sensitivities to value small changes of instrument values to full revaluation of all instruments given multiple correlated scenarios of time paths of the market parameters. The linear relationship between values and underlying market parameters is the foundation of the Delta VaR model. It ends up as an analytical model because the weighted summation of random normal variables is also a normal variable. Hence, only the mean and standard deviation of the portfolio value suffice to define the entire distribution. Monte Carlo simulations allow us to bypass the restrictive assumption that individual returns are linear functions of market parameters. In both cases, the prerequisite is to have the correlations and the variance–covariance matrix of all relevant market parameters.

MODELLING CORRELATIONS BETWEEN INDIVIDUAL ASSET RETURNS AND MARKET PARAMETER RETURNS Correlations and variances of individual asset returns and market parameters are observable. Therefore, the first technique for obtaining the matrix is to measure it through direct observations, usually on an historical basis. This raises several difficulties. First, the high number of assets makes it unpractical to create a matrix with all pair covariances between individual asset returns. Second, the observed variances and covariances might not comply

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with the basic property of variance–covariance matrices, that of being semi-definite positive. This happens because some observed parameters correlate strongly1 . Because of the large number of asset returns, it is more efficient to derive correlations from the correlation of the ‘factors’ driving these returns. This option requires a prior mapping of asset returns to market parameters. Risk Metrics provides correlations between the main market parameters. Risk drivers are interdependent because they depend on a common set of factors. For instance, all stock returns relate to equity index returns through statistical fitting. The technique extends to multiple factors. One example of a multiple-factor model is the Arbitrage Pricing Theory (APT) model (see Ross, 1976), which considers that more than one factor influences the equity returns. Other factor models model interest rates, making them correlated. Principal Component Analysis (PCA) uses orthogonal factors influencing all rates for generating yield curve changes. Correlations between instrument returns within each block result from their common dependence on such factors. For market risk, the risk factors are market parameters, or observable parameters that drive the risk. Factors and risk drivers are not necessarily identical however. For example, it is possible to identify a number of factors that influence the equity index returns or the interest rates. Equity indexes and interest rates are the direct drivers of asset returns. On the other hand, factors influence risk drivers without interacting directly with returns as drivers do. They serve to correlate the distributions of risk drivers as an alternative technique to direct measures of correlations. The common dependence of risk drivers, such as interest rates, on a set of factors makes them correlate. Hence, we need to distinguish three levels: factors, risk drivers and market values (Figure 31.1). From Risk Factors to Risks Market Risk

FIGURE 31.1

Factors

Risk factors

Risk drivers

Market parameters

Individual risks

Market value changes

From risk factors to correlation of individual risks

In many instances, however, risk factors are identical to the risk drivers for market risk. They are the market parameters, under the broad sense of yield curves, foreign exchange rates, equity indexes, plus their volatilities. Since market parameters are directly observable, there is no need to model them. Historical direct observations are feasible. Interest rate models allow us to generate yield curves, such that all interest rates remain consistent with arbitrage relationships. Factor models offer a convenient alternative technique, notably in two cases: • For equity returns, the Capital Asset Pricing Model (CAPM) is a single-factor model serving to calculate all variances and covariances of individual stocks, plus the specific 1 When

two variables have a correlation of 1, the matrix does not have the desired properties. Measuring errors of highly correlated variables might give a similar result.

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risk of each stock. The APT model uses ‘orthogonal’ factors to achieve the same purpose, using more than one factor to model equity returns. • For yield curves, multi-factor models help summarize in a convenient way the basic transformation of interest rates, parallel shifts, slope variations and bumps of actual yield curves. Figure 31.2 provides an overview of the correlation modelling building block of market risk models. The purpose of the ‘revaluation’ block of models is to link risk drivers to asset returns and values. As such, this revaluation block does not trigger correlations. Correlation between risks results rather from the correlations between the risk drivers. Since returns result from risk drivers through the revaluation process, correlated risk drivers generate return correlations.

Factors

Revaluation Revaluation building block building block

Factors are either observed explicit market parameters or principal components factors ...

Sensitivities, revaluations

Market risk driver correlations

Market risk drivers are market parameters driving transaction values

Market risk

Asset return correlations

Correlated MTM changes

Asset A

Asset B

Asset ...

Market risk: Correlations and volatility Market risk: Correlations and volatility

FIGURE 31.2

Individual value changes correlate due to the dependence on correlated risk drivers

From market factors to correlated transaction market risk

IMPLEMENTING FACTOR MODELS Factor models serve, notably, to simplify the modelling of correlations between individual stocks and that of interest rates. In this section we show how to derive correlations from factor models. The main example deals with stock prices. For stocks, it is important both to model correlations and to isolate specific risk, the risk unrelated to common factors. The section addresses the one-factor model of stock returns and multiple-factor models, such as those of PCA.

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Deriving the Variance–Covariance for Stocks The direct measures of the variances–covariances of stock returns would imply measuring N 2 terms, for a portfolio of N stocks. The one-factor model allows us to measure the covariances using a linear relationship between the stock return and the market index return. This is a direct application of a one-factor model, whose formulas are given in the appendix to this chapter. The market model is: ri = αi + βi rm + εi The relationships simply indicate that all stock returns co-vary with the index, although their sensitivities βi vary with each stock2 . The statistical relationship is a one-factor model, with rm being the explicit factor influencing the individual returns. For equity returns, we illustrate first the attractive property of using one factor only and proceed to describe the general ‘diagonal’ model of stock return correlations. Pairs of Stocks

By definition of the regression model, the random equity index return is independent of the residual εi . As a result, the covariance between any pair of equity returns depends only on the variance of the single factor and the coefficients β1 and β2 . The covariance and the correlation between any pair of equity returns r1 and r2 is simply3 : Cov(r1 , r2 ) = β1 β2 σ 2 (rm ) ρij = Cov(ri , rj )/σ (ri )σ (rj ) = βi βj σ 2 (rm )/σ (ri )σ (rj ) Moreover, the volatility of any individual asset return is: [σ (ri )]2 = βi2 [σ (rm )]2 + [σ (εi )]2 The return ri volatility is the sum of the systematic variance of the market, the general risk generated by rm weighted by the squared sensitivity βi2 , and of the specific, or idiosyncratic, risk of an individual stock. Let’s assume that all returns are ri standardized normal variables, with unit variance, for the index return as well as for the stocks picked. The coefficients are βi = 1 and βj = 1.5. The ri represent the equity returns of two obligors. The model fit sets the volatility of the residual, measuring specific risk: σ 2 (r1 ) = (1)2 1 + 1.5 = 2.5 = 1.5812 σ 2 (r2 ) = (1.2)2 1 + 1 = 2.94 = 1.7152 Cov(r1 , r2 ) = 1 × 1.2 × 1 = 1.2 ρ12 = 1.2/(1.581 × 1.715) = 44.25% 2 The CAPM models this empirical finding and shows that the return on any asset i is the risk-free rate r plus f a risk premium equal to the differential of the random market return rm and the risk-free rate, times the βi of the asset: ri = rf + βi (rm − rf ). 3 Cov(r , r ) = Cov(α + β r + ε , α + β r + ε ) = β β Cov(r , r ) because all cross-covariance 1 2 1 1 m 1 2 2 m 2 1 2 m m terms, Cov(r1 , ε1 ), Cov(r1 , ε2 ), Cov(ε1 , ε2 ), as well as the covariances between r2 and the residuals, are zero. In addition, Cov(rm , rm ) is the variance of the equity index return.

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In the case of the first obligor, the error variance is 1.5 and the variance of the factor is 1. Total variance is 2.5. The R 2 is the ratio of 1/2.5 = 40%. In the second case, total variance is 2.94, and the explained variance is 1.44, hence R 2 is 1.44/2.94 = 48.98%. The R 2 provides a direct estimate of the general versus specific risk. When no fit is available, for private firms for example, it is necessary to specify the R 2 using for instance the average over all firms, which is in the range of 20% to 30% for listed companies in the main stock exchanges. Deriving the Variance–Covariance for the Entire Stock Portfolio

The covariance between asset i and asset j returns depends only on βi and βj , and on the factor variance. With N assets, there are N variances and N × (N − 1) covariance terms, a total of N 2 terms in the variance–covariance matrix. Using the one-factor model allows us to calculate the N × (N − 1) covariance terms from N coefficients βi plus the factor volatility, or N + 1 terms compared to N × (N − 1). Hence, we summarize N 2 terms of the variance–covariance matrix using only N + 1. In matrix format, the variance–covariance matrix becomes: σ (ε1 )2 0 0 ··· β 1 β 1 β1 β2 β1 β 3 · · · β 2 β 1 β2 β2 β2 β 3 · · · 0 0 ··· σ (ε2 )2 2

= σm2 β3 β1 β3 β2 β3 β3 · · · + 0 0 σ (ε3 ) · · · .. .. .. .. .. .. .. .. . . . . . . . .

All the off-diagonal terms result from the βi , plus the market index volatility. All the diagonal terms depend on the βi , plus the market index volatility, plus the specific risk term. This provides the entire matrix. The first term of the matrix is simply ββ T × σm2 , where β T stands for the transpose vector of the sensitivities. This is the ‘diagonal’ model of stock returns variances and covariances.

The APT Model for Stock Returns A well-known example of multi-factor modelling is the APT model of equity returns. Contrasting with the CAPM, or the simpler ‘market model’, the APT model makes the equity returns dependent on several independent factors rather than a single one. Multiple factors generate some additional complexity. Common factors are the source of general credit risk, the risk to which each obligor is subject. The error term is the specific or idiosyncratic risk of asset i. By construction, it is independent of all common factors. The analytical form of risk results from the variance of the Yi : σ 2 (Y1 ) = (βi1 )2 Var(X1 ) + (βi2 )2 Var(X2 ) + (βi3 )2 Var(X3 ) + Var(εi ) σ 2 (Y1 ) =

K i=1

(βik )2 Var(Xk ) + Var(εi )

The general risk is the sum of the variances of the factors weighted by the model coefficients, and the specific risk is the residual variance. The appendix to this chapter explains how to obtain these formulas.

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Orthogonal Multiple-factor Models A practical model for changing randomly the shape of the yield curve is PCA. In this case, factors are linear functions of observed variables and are independent of each other. In such a case, it is easy to derive the variances and covariances, or correlations, from the factor variances and the model coefficients. We use here a simple example of such calculations, with a two-factor model of two variables. The general formulas are given in the appendix to this chapter. The models for two correlated indexes and two factors are: Y1 = β10 + 1 × X1 + 1.2 × X2 + ε1 Y2 = β20 + 0.8 × X1 + 0.5 × X2 + ε2 X1 and X2 are standardized normal independent factors (zero mean and unit variance). The variances of the residuals Var(ε1 ) and Var(ε2 ) are respectively 1.5 and 1. The crosscovariance between X1 and X2 is zero. The volatility of the residuals measures the specific risk and results from the model fit. The variance adds the general factor risk plus the specific risks Var(ε1 ) and Var(ε2 ). The variances of the indexes Y1 and Y2 follow: σ 2 (Y1 ) = (1)2 1 + (1.2)2 1 + 1.5 = 3.94 and σ (Y1 ) = 1.9852 σ 2 (Y2 ) = (0.8)2 1 + (0.5)2 1 + 1 = 0.64 + 0.25 + 1 = 1.89 and σ (Y2 ) = 1.3752 The R 2 of the regression of Y1 and Y2 (Yi ) on X1 and X2 (Xk ) is the ratio of explained variance by all factors to the error variance or, equivalently, the ratio of general to total risk. In the case of the first asset, the error variance is 1.5 and that of factors X1 and X2 is 2.44. Total variance is 3.94. The R 2 is 2.44/3.94 = 61.93%. In the second case, the total variance is 1.89 and the explained variance is 0.89, hence R 2 is 0.89/1.89 = 47.09%. The covariance between Y1 and Y2 is: Cov(Y1 , Y2 ) = 1 × 1.2 × 1 + 0.8 × 0.5 × 1 = 1.600 The corresponding correlation coefficient is: ρ12 = 1.600/(1.985 × 1.375) = 58.63%

General versus Specific Risk When relating individual returns to factors, the factors generate general risk, the risk common to all assets, and the residual risk is the specific risk. Specific risk appears only whenever there is no deterministic relationship between the factors and the risk drivers, or between the risk drivers and the returns (such as the closed-form formulas of pricing models). This is the case for stocks, as illustrated above. Specific risk also appears when modelling interest rates from underlying factors. The specific risk is the variance of the error terms in these models. When using the Delta VaR technique, ignoring specific risk underestimates the overall risk. There is always a fraction of the portfolio volatility that does not relate to common factors or risk drivers. The diagonal model described above makes explicit the specific risk component.

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APPENDIX: IMPLEMENTING FACTOR MODELS FOR MODELLING CORRELATIONS Factor models serve to model correlations between risk drivers generating correlated distributions of credit events within a portfolio. To model correlations, we face the issue of constructing a variance–covariance matrix for credit risk drivers, such as asset values or factors that drive the credit standing. From the matrix, we can infer any pair correlation. A distinct, but related, issue is that such a matrix is information intensive for a portfolio. A variance–covariance matrix is squared, and has N 2 terms for a portfolio of N obligors. Therefore, we need to reduce the information to manageable dimensions. Factor models address these issues. We provide examples of the single-factor model, or orthogonal-factor (independent factors) models obtained with PCA, and of multiple-factor models with correlated factors.

Measuring and Modelling Correlations with Single-factor Models In the case of a single factor, the decomposition is obvious because the residual term is independent, by definition of the single factor. The Single-factor Model

We start with this simple case. The one-factor model form is: Yi = βi0 + βi X1 + εi The variance of Yi is simply the sum of the variance due to the single factor and to that of the residual: σ 2 (Yi ) = (βi )2 Var(X1 ) + Var(εi ) The covariance between any two Yi and Yj is: Cov(Yi , Yj ) = Cov(βi0 + βi X1 + εi , βj 0 + βj X1 + εj ) Cov(Yi , Yj ) = βi βj Var(X1 ) This formula simplifies because all cross-correlations between factors and residuals are zero by construction of the model. All residuals εi are independent of X1 . The correlation between Yi and Yj is: ρij = Cov(Yi , Yj )/σ (Yi )σ (Yj ) = βi βj Var(X1 )/σ (Yi )σ (Yj ) When the single factor explains a large fraction of the risk driver volatility, the systematic risk is relatively high, and the specific risk is low. The opposite holds when the single factor explains only a small fraction of the volatility. Note that the R 2 of the regression of Yi on Xi is, by definition, the ratio of variance explained by the factor to the total variance or, equivalently, the ratio of general to total risk. For N assets, there are N variances plus covariances, resulting in N 2 terms. Using the single-factor model, we need only the N β i , plus the N residual variances, plus the

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single factor variance, or 2N + 1 items of information. This is the diagonal model of correlations, as shown when modelling equity risk.

Multi-factor Models for Measuring Correlations The first subsection extends the definitions to a multi-factor setting. The second subsection shows how to derive variances, covariances and correlations from models using orthogonal factors, and the third subsection is a brief extension to correlated, rather than independent, multiple factors. General and Specific Risk in Multi-factor Models

In general, a multi-factor model relates some random variable Y to common factors Xk : Yi = βi0 + βi1 X1 + βi2 X2 + βi3 X3 + · · · + εi The index i refers to asset returns, while the index k refers to the factor. In the equity universe, the variable explained by factors is the equity return of stocks. They are random just as the factors and the residual are, but they are all sensitive to each of the common factors. Multiple-factor Models with Orthogonal Factors

To illustrate the general formulas, we use a two-factor model. The two factors are independent, or ‘orthogonal’. Yi is the random asset return of asset i. The two-factor model is: Yi = βi0 + βi1 X1 + βi2 X2 + εi The general formula is: Yi =

k

βi1 Xk + εi

The index i refers to the assets, while the index k refers to the factor. The R 2 of the regression of Yi on Xk is the ratio of explained variance by all factors to the total variance. From this general formula, it is easy to derive the variance of Yi , which simplifies because we have a linear combination of independent variables. For pairs of assets, we derive both covariances and correlations. Again, the formulas simplify because of zero cross-correlations between factors and residuals. All Xk are standardized normal orthogonal factors, the variances of the residuals σ 2 (εi ) are respectively 1.5 and 1 for obligors 1 and 2. The cross-covariances between all Xk are zero. The volatility of the residuals measures the specific risk and results from the model fit. The general formula for the variance of the variable is: σ 2 (Y1 ) = Cov(β11 X1 + β12 X2 + ε1 , β11 X1 + β12 X2 + ε1 ) σ 2 (Y1 ) = (β11 )2 Var(X1 ) + (β12 )2 Var(X2 ) + Var(ε1 )

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The extension to any number K of factors is straightforward. The variance is the summation of general risk variances plus the specific risk variance of the residual. The variance is: (βi1 )2 Var(Xk ) + Var(εi ) Var(Yi ) = k

With orthogonal factors, all covariances between factors and residuals are zero and the other covariances are: Cov(Y1 , Y2 ) = Cov(β11 X1 + β12 X2 + ε1 , β21 X1 + β22 X2 + ε2 ) = β11 β21 Var(X1 ) + β12 β22 Var(X2 ) The covariance between Yi and Yj collapses to: Cov(Yi , Yj ) = βik βj k Var(Xk ) k

The correlation between Yi and Yj is: ρij = Cov(Yi , Yj )/σ (Yi )σ (Yj ) For two assets i and j , the corresponding correlation coefficient is: ρij = [β1i β1j Var(X1 ) + β2i β2j Var(X2 )]/σ (Yi )σ (Yj )

General Multi-factor Models The models are similar except that there is no simplification due to zero cross-correlations. All formulas for variances, covariances and correlations depend on all cross-correlations, making them more complex. Generally, relating asset returns to market parameters is a multi-factor setting where factors are market parameters. Therefore, we need to account for the observed cross-correlations of market parameters. When factors are not orthogonal, all cross-covariances between factors contribute to both variances and covariances of the Yi . Covariances of Risk Drivers

The covariance of two indexes Y1 and Y2 is: Cov(Y1 , Y2 ) = Cov(β10 + β11 X1 + β12 X2 + ε1 , β20 + β21 X1 + β22 X2 + ε2 ) All the covariances between constant and any random variables are zero. The covariances between any factor and the residuals are zero because the factors extract the correlations between the random asset values leaving only an uncorrelated residual. We expand all formulas to see the details: Cov(Y1 , Y2 ) = Cov(β11 X1 + β12 X2 + ε1 , β21 X1 + β22 X2 + ε2 ) Simplifying: Cov(Y1 , Y2 ) = β11 β21 Var(X1 ) + β12 β22 Var(X2 ) + β11 β22 Cov(X1 , X2 ) + β12 β21 Cov(X2 , X1 ) + β13 β21 Cov(X3 , X1 )

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All cross-covariances of factors apply for calculating the asset covariance, but all crosscovariances between factors and residuals are zero by construction of the model. Variances of Risk Drivers

The variance of an asset value results from setting Y1 = Y2 : σ 2 (Y1 ) = Cov(β11 X1 + β12 X2 + ε1 , β11 X1 + β12 X2 + ε1 ) Simplifying4 : σ 2 (Y1 ) = (β11 )2 Var(X1 ) + (β12 )2 Var(X2 ) + 2β12 β11 Cov(X1 , X2 ) + Var(ε1 ) For a single obligor, there are two factor covariance terms, plus two factor variance terms, plus one variance of the error term. The extension to K factors is straightforward. The variance depends on the variances of each factor, plus the covariance terms, plus the variance of the error term representing the specific risk. In general, for each obligor or segment, there are K factor variance terms, plus K × (K − 1) covariance terms, plus one specific risk term. Aggregating over the N obligors or segments generates N times these terms. There are N × K variance terms, plus N specific risk terms, plus N × (N − 1) × K × (K − 1) covariance terms. The N × K variance terms count less than the covariance terms. This mechanism makes the specific risk much lower than general risk when the number of obligors increases.

4 Var(Y

1)

= (β11 )2 Var(X1 ) + β11 β12 Cov(X1 , X2 ) + β12 β11 Cov(X2 , X1 ) + (β12 )2 Var(X2 ) + Var(ε1 ).

32 Portfolio Market Risk

The goal of modelling portfolio risk is to obtain the distribution of the portfolio returns at the horizon, set at the liquidation period. This implies a forward revaluation at the horizon date of all instruments once market parameters change. Since these are the value drivers of individual assets within the portfolio, the prerequisite is to model the market parameter random deviations complying with their correlation structure. The next step is to derive individual asset return distributions from market parameters to get all possible portfolio returns. Loss statistics and loss percentiles providing the market risk ‘Value at Risk’ (VaR) derive from the portfolio return distribution. To achieve this ultimate goal, techniques range from the Delta VaR technique to full-blown simulations of market parameters and portfolio values. Portfolios benefit from diversification. The portfolio return volatility decreases with the number of assets, down to a floor resulting from general risk. However, the value of risk relates to portfolio value rather than return. The portfolio value volatility increases with the number of assets, and the incremental volatility for a new asset increases with the average correlation of the portfolio. When dealing with single assets, there is no need to worry about standalone risk. For portfolios of assets, we need to include the effect of the correlation between individual asset returns and between risk drivers, or market parameters, influencing these individual returns. The Delta VaR model relates linearly asset returns to market parameter returns using instrument sensitivities. The essentials are that the portfolio return is a linear combination of random normals. Therefore it follows a normal distribution. The volatility of the portfolio return applies to any set of linear combinations of random variables. Since we can calculate the volatility of the portfolio return, and since we know that it is normally distributed, we have all that we need to measure VaR. When the assumptions get unrealistic, we need to extend the simple Delta VaR technique or rely on full revaluation at horizon.

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Full-blown simulations consist of generating risk driver returns complying with the variance–covariance structure observed in the markets, and revaluating each individual transaction. Revaluation uses pricing models, or simulation techniques for complex derivatives. The portfolio return distribution results from the full revaluation for all trials. Forward looking simulations generate random market parameter values complying with market volatilities and correlations. Intermediate techniques use sensitivities to save the time intensive calculations. Historical simulations use past values of all risk drivers, which effectively embed existing correlations. Other intermediate techniques include ‘Delta–Gamma’ techniques, or grid simulations. Because of model risk, modelled returns deviate from actual returns. This necessitates back testing and stress testing modelled VaR to ensure that tracking errors remain within acceptable bounds. The first section shows how the portfolio return and the portfolio value volatility vary with the number of assets in a simple portfolio. The second section summarizes the Delta VaR technique. The third section expands fully the calculation of the Delta–normal VaR technique using the simple example of a two-asset portfolio. The fourth section reviews intermediate techniques. The fifth section expands the simulation technique and details some intermediate stages before moving to full-blown Monte Carlo simulations. The last section addresses back and stress testing of market risk VaR.

THE EFFECT OF DIVERSIFICATION AND CORRELATION ON THE PORTFOLIO VOLATILITY The standard representation of the diversification effect applies to portfolio and asset returns. The principles date from Markowitz principles (see Markowitz, 1952). The portfolio return varies with common factors and because of specific ri