Arvanitis a., Gregory J. Credit.. the Complete Guide to Pricing, Hedging and Risk Management (Risk...

Credit:THE COMPLETE GUIDE TO PRICING,HEDGING AND RISK MANAGEMENT

Credit:THE COMPLETE GUIDE TO PRICING,HEDGING AND RISK MANAGEMENT

Angelo Arvanitis and Jon Gregory

Published by Risk Books, a division of the Risk Waters Group.

Haymarket House2829 HaymarketLondon SW1Y 4RXTel: +44 (0)20 7484 9700Fax: +44 (0)20 7484 9758E-mail: [email protected]: www.riskbooks.com

Every effort has been made to secure the permission of individual copyrightholders for inclusion.

Risk Waters Group Ltd 2001

ISBN 1 899 332 731

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library

Risk Books Commissioning Editor: Emma Elvy Project Editor: Sarah Jenkins

Typeset by Special Edition Pre-press Services

Printed and bound in Great Britain by Bookcraft (Bath) Ltd, Somerset

Conditions of saleAll rights reserved. No part of this publication may be reproduced in any material form whetherby photocopying or storing in any medium by electronic means whether or not transiently orincidentally to some other use for this publication without the prior written consent of the copyrightowner except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 orunder the terms of a licence issued by the Copyright Licensing Agency Limited of 90, TottenhamCourt Road, London W1P 0LP.

Warning: the doing of any unauthorised act in relation to this work may result in both civil andcriminal liability.

Every effort has been made to ensure the accuracy of the text at the time of publication. However,no responsibility for loss occasioned to any person acting or refraining from acting as a result of thematerial contained in this publication will be accepted by Risk Waters Group Ltd.

Many of the product names contained in this publication are registered trade marks, and Risk Bookshas made every effort to print them with the capitalisation and punctuation used by the trademarkowner. For reasons of textual clarity, it is not our house style to use symbols such as , , etc.However, the absence of such symbols should not be taken to indicate absence of trademarkprotection; anyone wishing to use product names in the public domain should first clear such usewith the product owner.

vAngelo Arvanitis is head of the risk department at Egnatia Bank in Greece.He is on the board of directors of the banks portfolio management sub-sidiary, as well as being a member of the assets and liabilities managementcommittee and the audit committee. Prior to that, Angelo headed the quan-titative credit, insurance and risk research at Paribas worldwide, which hefounded in 1997. His responsibilities included the development of pricing,hedging and risk management models for credit derivatives, convertiblebonds and insurance derivatives. He was also responsible for analysing theeconomic capital of the fixed income division, covering market, credit andoperational risks. Previously, he was a vice president in the derivatives andthe emerging markets departments at Lehman Brothers in New York. Hehas also spent time in structured products at BZW in London, specialisingin tax arbitrage. He has had various papers published in both professionaland academic journals, his main subject being the management of creditand insurance risks. Angelo gained a BA in physics at the University ofAthens. He continued his studies at the University of California, theLondon School of Economics and the doctoral programme in finance at theUniversity of Chicago.

Jon Gregory is global head of credit derivatives research at BNP Paribas,based in London. He joined Paribas in 1997 and was responsible for thedevelopment of the internal model for analysing the economic capital ofthe fixed income division. In addition to his work on capital management,he has worked on pricing models for the interest-rate derivatives, convert-ible bond and credit derivatives desks. He has also been involved indeveloping models for insurance derivatives. His main interest lies inreconciling theoretical and practical approaches for pricing, hedging andmanaging risk. Jon gained a BSc in chemistry from the University of Bristoland then continued his studies at Cambridge University, where he studiedthe quantum mechanical behaviour of small clusters of water molecules.This work has answered some important, fundamental questions (such aswhen water actually becomes wet) and has been published widely inboth academic and popular science journals. Jon was awarded his PhDfrom Cambridge in 1996 and was offered a research fellowship to enablehim to continue research in Cambridge. Instead, he decided upon a changeof field and joined Salomon Brothers where he worked in the fixed incomedivision before joining Paribas in 1997.

About the Authors

Introduction xi

PART I CREDIT RISK MANAGEMENT

1 Overview of Credit Risk 31.1 Components of Credit Risk 41.2 Factors Determining the Credit Risk of a Portfolio 51.3 Traditional Approaches to Managing Credit Risk 81.4 Market Risk versus Credit Risk 101.5 Historical Data 121.6 An Example of Default Loss Distribution 151.7 Credit Risk Models 211.8 Conclusion 29

2 Exposure Measurement 312.1 Introduction 312.2 Exposure Simulation 332.3 Typical Exposures 362.4 Conclusion 44

3 A Framework for Credit Risk Management 453.1 Credit Loss Distribution and Unexpected Loss 453.2 Generating the Loss Distribution 483.3 Example One Period Model 623.4 Multiple Period Model 663.5 Loan Equivalents 693.6 Conclusion 71Appendix A Derivation of the Formulas for Loan

Equivalent Exposures 71Appendix B Example Simulation Algorithm 73

4 Advanced Techniques for Credit Risk Management 754.1 Analytical Approximations to the Loss Distribution 754.2 Monte Carlo Acceleration Techniques 794.3 Extreme Value Theory 854.4 Marginal Risk 904.5 Portfolio Optimisation 994.6 Credit Spread Model 1064.7 Conclusion 115

Contents

PART II PRICING AND HEDGING OF CREDIT RISK

5 Credit Derivatives 1215.1 Introduction 1215.2 Fundamental Credit Products 1225.3 Fundamental Ideas on Pricing Credit 1265.4 Pricing Fundamental Credit Products 1275.5 Credit Spread Options 1465.6 Multiple Underlying Products 1565.7 Conclusion 177

6 Pricing Counterparty Risk in Interest Rate Derivatives 1836.1 Introduction 1856.2 Overview 1866.3 Expected Loss versus Economic Capital 1906.4 Portfolio Effect 1916.5 The Model 1926.6 Interest Rate Swaps 1936.7 Cross-Currency Swaps 1996.8 Caps and Floors 2026.9 Swaptions 2026.10 Portfolio Pricing 2046.11 Extensions of the Model 2096.12 Hedging 2116.13 Conclusion 220Appendix A Derivation of the Formula for the Expected

Loss on an Interest Rate Swap 221Appendix B The Formula for the Expected Loss on an

Interest Rate Cap or Floor 226Appendix C Derivation of the Formula for the Expected

Loss on an Interest Rate Swaption (Hull and WhiteInterest Rate Model) 227

Appendix D Derivation of the Formula for the ExpectedLoss on a Cancellable Interest Rate Swap 236

Appendix E Market Parameters used for the Computations 246

7 Credit Risk in Convertible Bonds 2457.1 Introduction 2457.2 Basic Features of Convertibles 2467.3 General Pricing Conditions 2477.4 Interest Rate Model 2487.5 Firm Value Model 2497.6 Credit Spread Model 2607.7 Comparison of the two Models 2697.8 Hedging of Credit Risk 2727.9 Conclusion 274Appendix A Firm Value Model Analytic Pricing Formulae 274

Appendix B Derivation of Formulae for Trinomial Treewith Default Branch 275

Appendix C Effect of Sub-optimal Call Policy 276Appendix D Incorporation of Smile in the Firm

Value Model 278

8 Market Imperfections 2818.1 Liquidity Risk 2848.2 Discrete Hedging 2958.3 Asymmetric Information 3008.4 Conclusion 311

Appendix 3131. Credit Swap Valuation Darrel Duffie 3152. Practical use of Credit Risk Models in Loan Portfolio and

Counterparty Exposure Management Robert A. Jarrowand Donald R. van Deventer 338

3. An Empirical Analysis of Corporate Rating Migration,Default and Recovery Sean C. Keenan, Lea V. Cartyand David T. Hamilton 350

4. Modelling Credit Migration Bill Demchak 3765. Haircuts for Hedge Funds Ray Meadows 3896. Generalising with HJM Dmitry Pugachevsky 400

Glossary 409

Index 421

In this book, we aim to provide a comprehensive treatment of credit, with-out artificial barriers, while developing the links between pricing, hedgingand risk management. Study of this book should provide readers with thetools necessary for addressing the credit problems they face in their ownareas of specialisation. We first develop the fundamental framework, andthen apply this to a broad range of products. We hope that this commondenominator will be apparent and prevent this book being approached as aseries of independent studies. Our focus is mainly capital markets ratherthan corporate banking products, because this is our area of concentration,and because only limited material is available in the financial literature,with the exception of academic papers.

We believe that the timing of this book is right. On the one hand, creditderivative products are being used more, while, on the other, the need tomeasure and guard against credit risk is heightened. There is also increasedregulatory pressure for the banks to control and manage their aggregatecredit exposure. The banks have started to view their credit risk globally,and this is the approach that we follow here.

This book has been written with the practitioner in mind. Our mainaudience comprises chief risk officers, credit risk managers, researchers,quantitative credit analysts, portfolio managers, senior credit derivativeand interest rate traders and marketers, as well as anyone else interested inthe subject. We do not intend this to be a book on mathematical financewith an emphasis on credit. We have avoided unnecessary use of complexmathematics, while keeping the book self-sufficient by providing technicalappendices where needed. The material should be accessible to the non-technical reader, but analytical aptitude and knowledge of standardderivatives, calculus and statistics are helpful. Selectively, it can also beused as a textbook for MBA and advanced undergraduate students major-ing in economics or business.

In Chapter 1, after reviewing some of the fundamental concepts in creditrisk management, we demonstrate how the normal approximation of thecredit loss distribution can lead to gross errors in the computation ofvarious statistics, even for simple loan portfolios. The normal distributiondoes not capture the skewness in typical credit portfolios, and this leadsto underestimation of the tail probabilities. We discuss the scarcity of

xi

Introduction

historical time series, and briefly review some of the available data fordefault and credit migration probabilities, default correlation and recoveryrates. At the end of this chapter, we critically review and compare the com-mercially available models: JP Morgans CreditMetrics, Credit SuisseFinancial Products CreditRisk+, KMVs Portfolio Manager and McKinseyCompanys Credit Portfolio View.

Chapter 2 introduces the concept of stochastic exposures for fixed-income instruments, which, unlike loan portfolios, are marked-to-market.In the following chapters, we demonstrate how stochastic exposurescomplicate the computation of the credit loss distribution, and show howapproximating by a constant exposure, quite often referred to as loanequivalent, can provide erroneous results. The credit exposure is definedas the percentile of the distribution of the future present value of a parti-cular transaction at a certain confidence level. This idea is closely relatedto the credit lines, which are commonly used by financial institutions incontrolling their exposure to each counterparty they transact with.

Chapter 3 builds on Chapter 2 and couples the exposure distributionwith the default process, introducing the concept of credit portfolio man-agement. It provides standard material for the computation of credit lossdistributions and the simulation of binary default events. The first part ofthe chapter is based on work originally presented in the CreditMetricsframework. In the second part, we discuss a few other related topics.We expand the standard model so that the recovery rate is driven by theseverity of default. We show how we can use default correlation as aninput to the simulation, rather than asset return correlation. We investigatethe sensitivity of the economic capital to default correlation, stochasticrecovery rates, stochastic exposures and multiple periods.

Chapter 4, which expands on Chapter 3, is probably the most technicalchapter in the book. The reader who is not interested in computationaltechniques can skip most of the material without loss of continuity. Wepresent analytical techniques for computing capital under rather restrictiveassumptions, Monte Carlo speeding-up techniques and extreme valuetheory for fitting the tails of the credit loss distribution. An importantsubject that we cover is the computation of risk contributions and portfoliooptimisation, while the normality assumption is being relaxed. Lastly,we present a model for the evolution of the term structure of the creditspreads. This model can be used for marking-to-market the portfolioscredit spread when computing the credit loss distribution.

Chapters 5, 6 and 7 constitute the main credit pricing (as opposed to riskmanagement) chapters of this book. In Chapter 5, we discuss the pricingand hedging of credit derivatives. First, we explore standard interpolationtechniques for deriving a credit-spread curve from the market quotes ofliquid default swaps. We analyse various products on single underlyings,such as credit spread options, credit contingent contracts, cancellable swaps

INTRODUCTION

xii

and quanto default swaps. Next, we expand the analysis to cover multipleunderlying credit derivative products, with particular emphasis on baskets ie, worst of and first m or last k to default in a basket of n assets.We decided to focus on this product class because of both its complexityand its importance in structured credit derivative transactions. We showhow these products relate to standard collateralised debt obligations.We explore different hedging strategies and show how traditional deltahedging can result in poorly managed positions. A multitude of examplesfor pricing and hedging various types of portfolios have been provided toclarify the theoretical ideas.

Chapters 3 and 4 covered the computation of the credit loss distri-bution of a derivative portfolio, assuming that counterparty risk is nothedgeable. We pointed out the importance of economic capital, in orderfor the financial institution to sustain potential future defaults withoutdisruption of its operations. In Chapter 6, we use standard default swapsas the fundamental instruments for hedging counterparty default andcredit spread risk. This approach provides a link between two areasthat have developed independently within the financial community:credit risk management and credit derivatives trading. If credit risk isassumed to be non-hedgeable, the credit spread of a new transactionshould compensate for its marginal contribution to the financial institu-tions capital, while if it is hedgeable, it should cover its expected loss. Weconsider pricing risky transactions on a stand-alone and on a port-folio basis. In the latter case, existing deals within a master agreementare taken into account, as they are netted in the event of default. Finally,we extend the hedging techniques developed in the credit derivativechapters with the aim of immunising a portfolio against defaults andcredit spread volatility.

In Chapter 7, we incorporate a credit element to the pricing of convert-ible bonds. The importance of the US convertible high yield market alongwith the expected growth of a similar market in Europe with the adventof the euro have been the motivation behind this work. We present twomodels: the Credit Spread and the Firm Value model. In the former, thecredit spread is mainly driven by the value of the firms stock, while in thelatter we model the process followed by the value of the firm explicitly.The pros and cons of each approach are explored in detail.

In the last chapter we discuss three topics that are related to credit, eventhough for simplicity the first two applications are presented in an equitycontext: impact of market illiquidity, non-continuous rebalancing of hedg-ing portfolios and asymmetric information among market participants. Thediscussion is rather brief and our purpose has been only to introduce thesetopics and the underlying fundamental economic ideas. Even though wehave not yet applied these ideas to credit, we believe that they can providemore realistic models that emulate more closely the functioning of the real

INTRODUCTION

xiii

market. We realise that the models complexity and difficult calibrationissues could limit their applicability.

In the Appendix we have reprinted articles from either Risk magazine orRisk Books that we consider important, and they are complementary to thematerial covered in the book.

We would like to thank our colleagues, academics and conferenceparticipants, where we have presented much of this work, for stimulatingdiscussions. Special thanks are due to Jean-Michel Lasry, Jean-Paul Laurentand Olivier Scaillet. Angelo Arvanitis would like to thank Darrell Duffie fornumerous discussions on the fundamentals of pricing credit. We thankDouglas Long for refereeing Chapter 4. We express our appreciation toConrad Gardner, Head of Risk Books Publishing, for his enthusiasm inundertaking this project. We are also grateful to Nick Dunbar for initialdiscussions regarding the subject matter and focus of this book. Specialthanks go to Sarah Jenkins for her tireless work in coordinating the wholeprocess.

While writing this book we have used C++ to implement all the modelsthat are presented and we have used the software to produce the examplesin the book. In the near future, we intend to provide companion software.This will give readers the opportunity to experiment with and challengethe models we describe.

We have independently implemented full versions of all of the modelsdescribed here. These models have been used exclusively for all of theexamples presented. We have been careful to implement numericallyefficient algorithms which have been tested thoroughly. Needless to say,we are fully responsible for any errors. We hope that this book will triggerdiscussions and further work on the subjects we have addressed. Needlessto say, we are fully responsible for any errors. We hope that this effort willtrigger further work on the subjects that we have addressed.1

1 If readers would like to contribute any comments or ideas, the authors can be contacted at:[email protected]; [email protected].

INTRODUCTION

xiv

Part I

Credit Risk Management

Chapter 1 9/4/01 2:58 pm Page 1

Measuring and managing credit risk, whether for loans, bonds or deriva-tive securities, has become a key issue for financial institutions. The riskanalysis can be performed either for stand-alone trades or for portfolios ofdeals. The latter approach takes into account risk diversification acrosstrades and counterparties. As will be shown later, ignoring risk diversifi-cation can lead to erroneous risk management decisions, increasing ratherthan reducing the risk exposure of the financial institution. Diversificationoccurs both at the counterparty (obligor) level and at a more global level.At the counterparty level, diversification is the result of the offsetting of theexposures of different trades with the same transacting counterparty.Portfolio diversification also arises across counterparties, since differentfirms do not default at the same time. As a result, the equity that must beheld against defaults for a well-diversified portfolio is only a fraction of thetotal exposure of the portfolio.

Traditionally, credit risk encompasses two distinct risk types: counter-party risk and issuer risk. Counterparty risk is relevant for loans andderivative transactions, whereas issuer risk refers mostly to bonds. Thisbook deals mainly with the former. Credit derivative contracts are subjectto both types of risk. Counterparty risk is analysed over a long time hori-zon, as the positions are illiquid and it is difficult (if at all possible) to tradeout of them. Issuer risk for bonds is computed over a time horizon of a fewdays similarly to traditional market risk. Bonds are considered liquid ie,they can be traded easily if the need arises; in practice, however, this is notalways the case.

Consider a portfolio with market value V. In order to assess the marketrisk of this portfolio over a certain time horizon, we need to estimate theworst value of V over this period. This value could be negative if cashflowsare exchanged between the two transacting counterparties, as is the case inan interest rate swap. Market risk is generally driven by the variability ofcontinuous market variables, such as interest rates and stock prices. Theperiod over which market risk should be assessed is often only a few days,since it is usually possible to trade out of positions over this time horizon.For example, a liquid stock that has plummeted in value can usually be

3

1

Overview of Credit Risk


sold fairly easily (albeit at a loss), whereas credit-risky assets, such as loans,are not often traded on the secondary market. Derivative contracts, such asswaps, are even more illiquid, since it is not usually possible to trade oreven unwind them.

If V is negative, there is no loss or gain to the holder of this position if hiscounterparty defaults, since the same amount of money is still to be paid. Itis important to stress that credit risk is still present, even though mostlikely reduced, since it is still possible that V becomes positive over the lifeof the deal. Credit risk is more significant when V is positive, as moniesowed at default will not be paid in full. Only the recovery value will bereceived. Therefore credit risk is driven by two components: the overallvalue of the positions and the actual credit state of the counterparty. Thefirst is similar to market risk (although in the opposite direction) and hasthe same underlying variables driving the exposure. The second is particu-lar to credit risk and can be simply thought of as default or no-default. Thisintroduces a binary element into credit risk, which is not an element ofmarket risk. The credit state can be enriched by assigning credit ratings,which are characterised by the default probabilities and the credit migra-tion matrix.

As will be shown in the following chapters, the characterisation of acredit portfolios risk allows the identification of concentration in creditexposures, which leads naturally to the active management of credit riskthrough trading in the secondary markets, if they exist, use of creditderivatives and securitisation.

1.1 COMPONENTS OF CREDIT RISKCredit risk arises from potential changes in the credit quality of a counter-party in a transaction. It has two components: default risk and creditspread risk.

Default riskDefault risk is driven by the potential failure of a counterparty to makepromised payments, either partly or wholly. In the event of default, afraction of the obligations will normally be paid. This is known as therecovery value.

Credit spread riskIf a counterparty does not default, there is still risk due to the possiblewidening of the credit spread or worsening in credit quality. We make thedistinction between two components of credit spread risk:

Jumps in the credit spread. These may arise from a rating change (ie, anupgrade or a downgrade). It will usually be firm-specific and mayresult from some adverse or positive information becoming availablein the market.

CREDIT: THE COMPLETE GUIDE TO PRICING, HEDGING AND RISK MANAGEMENT

4


Credit spread volatility (continuous changes in the credit spread). This ismore likely to be driven by the markets appetite for certain levels ofrisk. For example, the spreads on high-grade bonds may widen ortighten, although this need not necessarily be taken as an indicationthat they are more or less likely to default.

If the credit spread of a portfolio is not marked-to-market, only defaultrisk is important, since it represents the sole immediate loss. However, ifa portfolios credit spread is marked-to-market, then an increase in thecredit spread will lead to an immediate realised loss, due to the expectedfuture losses being increased. In Chapter 5 we discuss in detail thediscounting of future risky cashflows. Loan portfolios are not usuallymarked-to-market. Consequently, the only important factor is whether ornot the loan is in default today (since this is the only credit event that canlead to an immediate loss). Capital market portfolios are marked-to-market. This means that a transaction with a counterparty whose creditquality has worsened (though not defaulted) will be marked at a loss.The reason for this is that the future cashflows with that counterparty arenow more risky and should therefore be discounted at a higher dis-count rate.

1.2 FACTORS DETERMINING THE CREDIT RISK OF A PORTFOLIOThree groups of factors are important in determining the credit risk of aportfolio, as illustrated in Figure 1.1:

deal-level factors corresponding to a single transaction; counterparty-level factors corresponding to a portfolio of transactions

with a single counterparty; and portfolio-level factors corresponding to a portfolio of transactions with

more than one counterparty.

These are depicted in Figure 1.1

1.2.1 Deal levelExposureIt is important to model the variability of the future exposure in a deriva-tive transaction, since the related market risk of a dynamic exposure ismuch harder to manage than that of a static one. It is therefore necessaryto compute the distribution of the future exposures (present value (PV)discounted at the risk-free rate) at the deal level at all possible defaulttimes. This exposure can change dramatically over time, so it is vital tocapture the future variability of such a deal. In fixed-income instrumentsthis variability is driven by the evolution of the underlying variables, ie,interest rates and foreign exchange rates. Crucial parameters are thevolatilities and the correlation of the above variables.

OVERVIEW OF CREDIT RISK

5


Default probabilitiesThe default probability is the likelihood that a counterparty will be bank-rupt or will not honour its obligations at the times when they become dueover a given period. Since exposures can vary significantly with time, it isuseful to have a series of marginal default probabilities over some reason-ably fine intervals, such as three months. The marginal default probabilitygives the probability of default in a certain period (eg, six to nine months).

Credit migration probabilitiesLosses may also occur as a result of less drastic deterioration in the creditquality of a counterparty, such as a downgrade to a lower rating. Althougha downgrade does not mean that the counterparty will fail to make a pay-ment (as in the event of default), it does mean that all future cashflowsshould be discounted at a higher risky rate, or, equivalently, that theexpected future losses will have increased. For this reason, if a position ismarked-to-market, a downgrade will result in a decrease in the PV and aloss. In some circumstances, it may not be appropriate or even possible tovalue a deal or a portfolio (eg, an illiquid loan), and hence credit migrationevents may not be considered. Furthermore, the severity of losses due tocredit migration is generally lower than that of losses due to default(although the corresponding probabilities are almost always greater). Onthe other hand, there are reasonable situations where losses due to credit


6

Exposure

Defaultprobabilities

Joint defaultprobabilities

Creditmigration

probabilities

Joint creditmigration

probabilities

Recoveryrates

Recoveryrates

Aggregatedexposure

Aggregatedexposure

DEALLEVEL

PORTFOLIOLEVEL

COUNTERPARTYLEVEL

Netting

Collateral

Figure 1.1 Factors determining the credit risk of a portfolio


deterioration can be of substantial magnitude. (This is addressed inChapter 3.)

Recovery ratesEven in the event of default, there will normally be some recovery on theposition. This is defined by the recovery rate. As we will present later, his-torical recovery rates have had a very large variability, spanning thewhole interval between zero and one.

Correlation of exposure and default parametersA final point to consider is whether a high exposure to a counterpartyimplies an increase in its default probability (or vice versa). Such an effectwould significantly increase the expected loss. There are clearly extremecircumstances where this may be the case, such as when a company sells aput on its own equity. In this case, there is clearly a non-zero correlationbetween the probability of default and the exposure. There has been littleempirical research relating market variables to default parameters, and it isgenerally ignored in modelling. In Chapter 5, we will present a simple wayto model this effect.

1.2.2 Counterparty levelAggregated exposureMoving from the deal level to the counterparty level, we must aggregatethe possible future exposure of all deals with the counterparty in question.This aggregation must take into account netting agreements and collateral,which we discuss briefly in the next section.

1.2.3 Portfolio levelJoint default probabilitiesEven if we know the values of the default probabilities of each counter-party in a portfolio, the distribution of defaults and, in particular, the tail ofthis distribution will still depend on the joint default probabilities. If wehave individual default probabilities P1 and P2, the joint default probability(probability that both default) is only P1 P2 if the events are independent.Historically, default rates have varied substantially from year to year.There are two potential interpretations of this observation, as outlinedbelow.

1. The default probabilities themselves are random quantities that dependon a set of underlying variables. The underlying variables driving thedefault probabilities depend to some extent on the nature of the counter-party in question (eg, geographical location, industry), but also onmore general factors (eg, the state of the economy). The default proba-bilities decrease (increase) under good (bad) economic conditions.


7


This is the approach taken by KMV, CreditRisk+ and McKinseys, asdescribed later in this chapter.

2. Although the probabilities do not change, the underlying events arethemselves correlated. A counterparty specific part (idiosyncratic orfirm-specific risk) and country/industry contributions determine thiscorrelation. A high correlation will occur among counterparties in simi-lar countries or industries while unrelated entities will have only a smallcorrelation. CreditMetrics advocates this approach.

The second assumption is the one we will follow in most of our analysis hence we will now describe default events as being correlated rather thandefault probabilities being stochastic. However, the estimation of defaultcorrelation is a rather difficult issue. In equity markets, correlation can beeasily measured by observing liquid stock prices. However, for relativelyinfrequent default events, this is much more difficult, and the confidenceintervals on the estimated parameters will be high.

Joint credit migration probabilitiesFollowing on from the above discussion of joint default probabilities, itseems reasonable to assume that if default events are correlated, then so arechanges in credit quality. For example, in the case of an investment gradeportfolio, even if there are no defaults, there may be a significant chancethat a rather high proportion of the counterparties will be downgraded.This would lead to a large mark-to-market loss in the portfolio. In anotherexample, suppose that two counterparties, believed to be very highly cor-related, have different default probabilities, so they cannot always defaultat the same time. However, given that they are so highly correlated, thedefault of one counterparty should surely imply that there is some effect onthe other one. The answer to this may be that the second counterparty maybe downgraded if the first one defaults. For estimation and implementa-tion, it would therefore seem natural that the same model should be usedfor generating both joint defaults and joint credit migrations.

1.3 TRADITIONAL APPROACHES TO MANAGING CREDIT RISKThe advent of the secondary markets for loans and other debt securities,together with the growth of the credit derivatives market and the develop-ment of securitisation techniques, has opened up new opportunities foractive credit risk management. For completeness, we review some of thetraditional approaches to credit risk management, which are still com-monly used, such as credit limits, netting agreements and collateralarrangements.

1.3.1 Credit limitsTraditionally, credit limits have been used to control the amount of


8


exposure a financial institution can have to each transacting counterparty.In particular, there has been a cap on the total exposure, the value of whichdepends on the credit rating of the counterparty. However, this is at oddswith the Bankers Paradox, since it implies that a financial institutionshould stop transacting with a counterparty with whom it has a very goodrelationship, but high exposure. This is the case because the risk of a posi-tion will increase with increasing exposure. Although it is likely that a bankwill have a number of large exposures with certain clients, these must becontrolled where possible, and the returns made must truly reflect the riskstaken. Looking at credit risk from a portfolio perspective allows the risksand returns of each position to be properly quantified. This approach issafer and more efficient than the somewhat arbitrary approach of creditlines, as the latter considers counterparties on an individual basis and nottheir contributions to the portfolio as a whole.

1.3.2 Netting agreementsNetting is relevant only when there is more than one transaction with acounterparty. A netting agreement is a contract that is legally bindingthrough the existence of a signed master agreement. It allows exposuresunder the master agreement to be aggregated in the event of default. Theamount owed by a defaulted counterparty can be offset against anyamount that is to be received and only the net outstanding amount needsto be paid. This will reduce the overall loss. Without a master agreement,all money owed to a counterparty must be paid, while a fraction of allmoney owed will be received.

1.3.3 Collateral agreementsA way to reduce a credit exposure is to take collateral from the counter-party, as this can be liquidated in the event of default and set aside againstany losses incurred. However, there is still the risk that the exposure at thetime of default will be more than the market value of the collateral. Theremay, therefore, also be postings of collateral to account for the change inthe exposure of the position. Even then, there remains the risk that theexposure will rise and default will occur before a required margin call. Tomodel the effect of collateral appropriately, we should first consider theinitial threshold (or haircut) which determines the cushion with respectto the initial exposure. The probability that more collateral will be neededthen depends on the volatility of the exposure. The minimum call amount(and the time between a margin call and the due payment) will influencethe amount that can be lost between margin calls. Finally, we should alsoconsider the correlation between the exposure and the collateral value. Ifthese values are positively correlated, the credit risk on the position will befurther decreased, as the value of the collateral is likely to increase with theexposure. For example, if the collateral for a cross-currency swap position


9


is denominated in the currency we receive, we are better protected againstthe strengthening or weakening of that currency. In Chapter 8 we discusscollateral liquidation in an illiquid market, which is a situation that is quiteoften encountered in practice.

1.4 MARKET RISK VERSUS CREDIT RISKA major difference between credit risk and market risk lies in their respec-tive return distributions. For simplicity, we demonstrate this effect throughan illiquid corporate bond. Very illiquid bonds could in principle be (andquite often are) included in the computation of the global credit exposure.Consider a corporate bond of face value 100 and maturity 10 years, payinga semi-annual coupon of 7.5% issued by a Baa-rated corporate. In


10

Actual

Normal approximation

0

5

10

15

20

25

30

35

Prob

abili

ty (%

)

40 50 60 70 80 90 100 110 120 130

Bond price

Figure 1.2 Actual and simulated distribution of prices of a 10-year Baa-ratedcorporate bond. (UK sterling interest-rate distribution over the period fromFebruary 22, 1999 to March 21, 2000)

Table 1.1 Data used to calculate credit returns

Rating Transition probability Spread Bond price

Aaa 0.02% 5bps 110.0Aa 0.33% 10bps 109.6A 5.95% 20bps 108.9Baa 86.93% 50bps 106.7Ba 5.30% 200bps 96.8B 1.17% 400bps 85.4Caa 0.12% 1000bps 60.2Default 0.18% 44.1


Figure 1.2, we show a histogram of the distribution of market prices for thisbond, based on the actual UK sterling interest rate distribution fromFebruary 22, 1999 to March 21, 2000. We see that the actual distribution isfairly well approximated by a normal distribution.

Next, we investigate the impact of credit risk on the price of this corpo-rate bond, using historical data on ratings changes and some potentialvalues of the credit spread in each rating class, as shown in Table 1.1. Thisdataset is discussed in the next section. This table shows that, for example,there is a 5.95% probability that a Baa- rated bond will be upgraded to anA, resulting in the spread tightening from 50bps to 20bps and leading to anincrease in its price. We assume that in the event of default, the recoveryvalue is 40% of the bond price just before the default event. In Figure 1.3,we show the distribution of returns due to credit migration and default forthis bond. It is clear that a normal distribution is not a good approximation,since it does not capture the large losses that can occur. The normal distri-bution is symmetric, while the actual one has a fat tail to tail to left.

An obvious question to ask is whether the normal approximation,although not valid for a single bond, will be a reasonable approximationfor a large portfolio. We will see that, usually, the answer to this question isthat it will not. Unlike market returns, credit returns are not well approxi-mated by a normal distribution. In Figure 1.4, we show typical market andcredit returns for a portfolio consisting of a large number of assets. Thecredit-return distribution is characterised by a large negative skew, whicharises from the significant (albeit small) probability of incurring a large loss.


11

Actual

Caa

Baa

B Ba A Aa AaaDefault

Normal approximation

0

5

10

15

20

25

30

35

Prob

abili

ty (%

)

40 50 60 70 80 90 100 110 120 130

Bond price

Figure 1.3 Distribution of prices for a 10-year Baa-rated corporate bond drivenby credit events (Moodys historical one-year transition probabilities andpotential spreads shown in Table 1.2)


This is also true for a relatively homogeneous, well-diversified portfoliowith small individual exposures.

1.4.1 Time horizonAnother important difference between credit risk and market risk is therelevant time horizon. For market risk, regulatory guidelines state that thisneed only be 10 days. Positions can usually be liquidated within such aperiod, assuming that there is enough liquidity in the market. For creditrisk, the period to use is less clear, but must be considerably longer, ascredit risk is not readily tradable.

It is becoming a convention that credit risk measurement is done on anannual time horizon, in part because budgets are annual. However, theactual time horizon used should reflect the security type, liquidity, matu-rity and credit worthiness of the underlying exposures. An interest rateswap may have a peak exposure several years in the future, which wouldgive rise to the biggest risk, so it should therefore be risk-managed withrespect to a longer horizon. A high-grade institution may have only a slightchance of default over a short period, but, because of the possibility ofdeterioration of its credit quality, it may be more likely to default over alonger horizon. In practice, more than one time horizon should be used,and they should be chosen following a detailed analysis of the portfolio ofthe financial institution.

1.5 HISTORICAL DATA1.5.1 Default and credit migration probabilitiesHistorical transition matrices, such as those provided by Moodys (1997a),


12

P ro b

a bili

ty

Return

Credit

Market

Small but significantchance of a large loss

Nodefaults

Figure 1.4 Typical credit and market return distributions


provide average rating transition probabilities sampled over many thou-sands of firm years of data. The transition probabilities are obtained byanalysing time-series data of credit-rating transitions for many differentfirms. A Moodys one-year transition matrix is shown in Table 1.2. The firstcolumn represents the initial rating and the first row the final one. Forexample, the probability of a Baa-rated company being downgraded to Bain one year is 4.76%. The last column represents the default probabilities.The probability of a Baa defaulting in one year is estimated to be 0.15%.

There are clearly some inconsistencies in the above transition matrix. Forexample, the probability of default of an A-rated counterparty is zero; yetthere is a non-zero probability that a supposedly more credit-worthyAa-rated counterparty will go bankrupt. Such inconsistencies are clearly aresult of the scarcity of data for these extremely unlikely events. We alsonote that, regardless of category, the most likely outcome is that the creditrating remains unchanged. Moodys (1997a) also make available transitionmatrices for longer periods. It is also possible to construct, by modificationof a one-year matrix under certain assumptions, transition matrices for anyperiod required. In this case, there are certain properties that such a com-puted transition matrix should be required to have. For example, it isreasonable to require that low-rated firms are always more risky. Note alsothat there is seemingly a mean-reversion effect in credit ratings, sincegood ratings (Aaa, Aa, A) are more likely to get worse rather than better,whereas poor ratings (Ba, B, Caa) have a greater probability of improving(assuming they do not default first!).

1.5.2 Recovery ratesStudies of recovery data have shown that the recovery value, in case ofdefault, can vary substantially, depending on the seniority of the under-lying debt. For example, Table 1.3 shows the recovery rates as reported byMoodys (1997b). There is clearly a large variance around the mean value,even within the same debt class. It is important to incorporate this uncer-tainty in the recovery rate to account for the fact that a worst case scenario


13

Table 1.2 Moodys one-year transition matrix

Aaa Aa A Baa Ba B Caa Default

Aaa 93.4% 5.94% 0.64% 0.00% 0.02% 0.00% 0.00% 0.00%Aa 1.61% 90.55% 7.46% 0.26% 0.09% 0.01% 0.00% 0.02%A 0.07% 2.28% 92.44% 4.63% 0.45% 0.12% 0.01% 0.00%Baa 0.05% 0.26% 5.51% 88.48% 4.76% 0.71% 0.08% 0.15%Ba 0.02% 0.05% 0.42% 5.16% 86.91% 5.91% 0.24% 1.29%B 0.00% 0.04% 0.13% 0.54% 6.35% 84.22% 1.91% 6.81%Caa 0.00% 0.00% 0.00% 0.62% 2.05% 4.08% 69.20% 24.06%


may correspond to an especially low recovery rate, even if the expectedrecovery rate is high.

1.5.3 Default correlationOne of the most challenging tasks in credit risk management is the model-ling of the correlation among default events. High default correlation candrastically increase the severity of the losses in a portfolio of many assets.Historical observations over many years can also be used to show thatdefault events and credit movements among different firms are, in general,positively correlated. These correlations are mostly influenced by macro-economic factors and depend on the general state of the economy. Forexample, see CreditMetrics (1997) for a convincing argument for the exis-tence of positive default correlation and estimates of the correlationbetween counterparties according to ratings.

The main implication of positive default correlation is the increasedprobability of suffering abnormally large losses due to multiple badcredit events. Indeed, we will later show that this effect can be very sub-stantial. There are two issues with modelling default correlations. First,there is the introduction of correlation among binary variables and thesimulation from this distribution; unlike for a multivariate normal distri-bution, this is not a trivial issue. Second, there is the estimation of therequired parameters.

The estimation of default correlations can be made directly from histori-cal data or indirectly using an approach such as the KMV Asset ValueModel (KMV, 1996). The advantage of the latter treatment is that com-pany-specific information, such as industry and region, is naturallycaptured in the correlation estimation. Alternatively, we can enrich thehistorical data with some sort of macroeconomic analysis, such as in themodel used by McKinsey & Company (Wilson, 1997) to incorporate thecurrent state of the economy. At the end of this chapter, we will review thedifferent approaches that have become standard in the industry.

Given the importance of default correlation, we will discuss it in detailin the next chapter. We will show how correlation between credit migra-tion events and recovery rates can also be incorporated in one unifiedframework.


14

Table 1.3 Moodys (1997b) recovery rates based on debt seniority

Seniority Mean Standard deviation

Senior secured 53.80% 26.86%Senior unsecured 51.13% 25.45%Senior subordinated 38.52% 23.81%Subordinated 32.74% 20.18%Junior subordinated 17.09% 10.90%


1.6 AN EXAMPLE OF DEFAULT LOSS DISTRIBUTION1.6.1 A simple loan portfolioIn this section, we present an example of the quantification of credit risk fora simple hypothetical portfolio, making some important observations. Bygreatly simplifying matters, we aim to highlight the main features ofcredit risk that are important from a risk management perspective. We willkeep the mathematics simple. For a further study of the relevant mathe-matics, the reader should consult any standard textbook on statistics.

Suppose we have a portfolio of n loans of the same maturity andnotional values denoted by Xi . Assume that there is a probability pi that theith loan will not be repaid at maturity. We use a binary variable di todenote whether or not the ith loan is to be repaid. For this analysis, weignore credit migration. We assume that there is no recovery in case ofdefault.

The distribution of di is such that, if we draw a large number of realisations,their average value will be pi . In other words, the expected value of di is pior E[di] = pi.

1.6.2 Expected lossWe can now define the loss, Lp, on the portfolio of loans as:

(1)

depending on the realised values of di. Since the Xis are constant we canrewrite the above expression as:

(2)

We now simplify further, and assume that the notional amounts andprobabilities of default are all equal, ie,

pi = p i and Xi = X i

EL E L E X d X E d X pp p i ii

n

ii

n

i ii

n

i= [ ] =

= [ ] == = =

1 1 1

L X dp i ii

n

=

=

1

di

1 Default Loss = Xi

0 No default Loss = 0


15


In this case, the expected loss is simply given by:

ELp = p

1.6.3 Distribution of lossesWe further assume that the default events of the various loans are inde-pendent. Therefore the probability of a given number of defaults k,resulting in a portfolio loss of kX, is given by the binomial distribution:

(3)

The number of combinations represents the number of ways of achieving kdefaults. It is given by:

Suppose that there are a total of 100 loans (n = 100), each of which has anotional value of 100 (X = 100), and also that the probability of default ofeach loan is 1% (p = 0.01). In Table 1.4, we show the probability of sufferingvarious losses as given by the binomial distribution.

n

k

n

k n k

= ( )

!

! !

Prob

Number ofcombinations

Probability that kloans default

Probability that other nkloans do not default

L kXn

kp pp

k n k=( ) = ( ) 1


16

Table 1.4 Loss probabilities corresponding to n = 100, X = 100and p = 0.01

k Lp pk(1p)n k Pr (Lp = kX)

0 0 1 0.366 0.3661 100 100 0.004 0.3702 200 4,950 ~105 0.1853 300 161,700 ~107 0.0614 400 3,921,225 ~109 0.0155 500 ~107 ~1011 0.0036 600 ~109 ~1013 ~104

7 700 ~1010 ~1015 ~105

8 800 ~1011 ~1017 ~106

9 900 ~1012 ~1019 ~107

10 1000 ~1013 ~1021 ~108

20 2000 ~1020 ~1041 ~1020

30 3000 ~1025 ~1061 ~1035

40 4000 ~1028 ~1081 ~1053

50 5000 ~1029 ~10101 ~1072

n

k


As the loss increases, the probability of achieving the required number ofdefaults ( pk(1 p)n k) decreases, but the number of possible ways toachieve them, , increases. However, the former effect is dominant, so theprobability of more than 10% of the portfolio defaulting is very small. Infact, it is so small that it can be neglected. We can represent the losses in asimpler manner as in the frequency distribution shown in Figure 1.5. Thisis a direct consequence of the assumption that the default correlation iszero. If the default correlation was positive, then the occurrence of largelosses would be a lot more likely. This would result in a longer tail for thedistribution shown in Figure 1.5.

The expected loss for this portfolio is 100 and the probability of this lossoccurring is 37%. The probability of losing more than the expected loss is26%. Indeed, the probability of losing four times the expected loss is stillsignificant at 1.5%. The expected loss alone is not sufficient for us to charac-terise the risk of the portfolio. We need, in addition, to have some measureof the deviation of the potential losses around the mean.

The variability of a distribution of losses around the mean value can becharacterised by the standard deviation that is the square root of the vari-ance of the distribution. This is given by:1

(4)

We can compute the standard deviation using the above expression:

L L L

L L

nn

jj

n

1 2

2

1, ,...,( ) = ( )=

n

k


17

0

5

10

15

20

25

30

35

40

Prob

abili

ty (%

)

0 100 200 300 400 500 600 700 800 900 1000

Loss

Figure 1.5 Loss distribution for n = 100, X = 100 and p = 0.01


For demonstration purposes, we also compute the standard deviationusing properties of the binary distribution which should give the sameresult. As 2(di ) = pi(1 pi ) and as the total variance is simply the sum ofeach individual one, we write:

(5)

It is a straightforward task to calculate the standard deviation of a port-folio, even when the notionals and the default probabilities are not thesame for each asset.

1.6.4 The normal approximationThe Central Limit theorem states that the sum of a large number of inde-pendent random variables converges to a normal distribution. A normaldistribution is characterised entirely by its mean and variance 2 (or stan-dard deviation).

The probability density function of a normal distribution is given by:

2 2

1

2

1

2

1

1 1 99 5

L X d

X d X p p X np p

p i ii

n

i ii

n

i i ii

n

( ) =

= ( ) = ( ) = ( ) ==

= =

.

L p( ) = ( ) + ( ) + ( ) +

=

0 366 0 100 0 185 200 100 0 061 300 100

99 5

2 2 2. . . ...

.


18

Table 1.5 Accuracy of the normal approximation of the credit lossdistribution for n = 100, X = 100 and p = 0.01

Lp Actual Normal

0 0.366 0.242100 0.370 0.401200 0.185 0.242300 0.061 0.053400 0.015 0.004500 0.003 ~104

600 ~104 ~106

700 ~105 ~109

800 ~106 ~109

900 ~107 ~1015

1000 ~108 ~1019


The probability of a value lying within one standard deviation of the meanis 66%. This means that the probability of experiencing a loss of more thanone standard deviation from the mean is 16%. We can calculate the prob-ability of exceeding any loss by computing the necessary number ofstandard deviations. For example, a loss that is exceeded no more than 1%of the time corresponds to 2.33 standard deviations.

The normal distribution seems to give a reasonable approximationaround the mean of the credit loss distribution. However, when we con-sider the values shown in Table 1.5 for the tail (far right-hand side) of thedistribution, this is clearly not the case. For example, we would under-estimate the probability of a loss of 500 by a factor of 23 and underestimatethe probability of a loss greater than 500 by over 100 times. This is clearly acrucial point and the normal approximation, although a very powerful andconvenient tool, must be used with extreme caution when analysing creditportfolios. The approximation will perform even more poorly if the defaultcorrelation is positive, which will be the case for most portfolios.

1.6.5 PercentilesThe normal approximation is inadequate, as it can give completely inaccu-rate estimates of the probabilities of experiencing certain losses, especiallyin the tail of the distribution, where the biggest losses occur. The normaldistribution does not capture the skews in typical credit portfolios, and thisleads to an underestimate of the tail probabilities.

However, instead of the standard deviation, we can use a percentile as ameasure of the variability of losses. The kth percentile of the distributioncorresponds to a loss Lk , such that Pr (Lp > Lk) = or equivalentlyPr (Lp < Lk) = 1 . This means that the probability of exceeding a loss of Lkis 1 . The percentile therefore gives either the true probability that agiven loss will be exceeded or the maximum loss for a given confidencelevel, without making any assumptions about the underlying distribution.

1

2

12

2

exp

x


19

Table 1.6 Accuracy of the normal approximation in percentileestimation

Percentile Actual loss Normal approximation

95th 400 263.799th 400 331.599.9th 500 407.599.97th 600 441.5


In Table 1.6 we illustrate some percentiles for the distribution of losses inthe above example, and compare them to the normal approximation.2

From Table 1.6 we see that the normal approximation leads to an under-estimate as expected. A legitimate question to ask is when the normalapproximation is valid. This is illustrated in Figure 1.6, where we comparethe ratio of the 99.9th percentile, as estimated using the portfolio variance(3.09 standard deviations), to the true 99.9th percentile for various homo-geneous portfolios (ie, one in which all asset returns follow the samedistribution and the loans have the same notional). We can see that as thenumber of assets in the portfolio and the default probability increase, theratio slowly approaches one. However, given that the average defaultprobability for a typical portfolio is low, it is clear that this measure willsubstantially underestimate the true unexpected loss.

Finally, we also notice that for this portfolio, the 95th and 99th per-centiles are equal; this is due to the fact that the distribution we are lookingat is highly discrete. In practice, for large portfolios this is rarely an issue.However, there are certain circumstances where we need to be aware ofthis drawback. We will need to address this point in Chapter 4 when wediscuss a method for credit portfolio optimisation.


20

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

200

600

1,00

0

1,40

0

1,80

0

1

3

5

7

9

pi (%)

n

Figure 1.6 Ratio of the unexpected loss at the 99.9% confidence level of ahomogeneous portfolio as estimated by the portfolio variance (3.09 standarddeviations) and the actual percentile


1.7 CREDIT RISK MODELSDuring recent years, specialist teams in major financial institutions havedeveloped several models that have either remained proprietary or becomecommercially available. In this section we review the latter models thathave attracted a lot of attention from the financial community: JP MorgansCreditMetrics, Credit Suisse Financial Products (1997), CreditRisk+, KMVsPortfolio Manager and McKinsey & Companys Credit Portfolio View(Wilson, 1997). In this section we review these models and point out boththeir commonalities and differences. (See also Koyluoglu and Hickman,1998.)

1.7.1 KMV/CreditMetricsThese models use Mertons firm value model (1974) to calculate defaultprobabilities based on the firms capital structure and the asset returnvolatility. The Merton model states that a firm defaults when the value ofits assets is lower than that of its liabilities when its debt matures.

In this model, sampling from a multivariate normal distribution, whichrepresents the asset returns of the firm, can generate default events.

The Merton model uses an option valuation framework based on Blackand Scholes (1973). The firms assets are assumed to follow a lognormalrandom walk. Under the assumption that the firm will default when itsliabilities exceed its assets, the default probability is given by3

(6)

where A0 is the asset value of the firm today, X is the strike or face valueof the firms debt, is the asset volatility, T is the maturity of its debt and ris the risk-free rate. The above model is known to underestimate defaultprobabilities, since default can often occur before a firms assets exceeds itsliabilities. KMV therefore use a modified approach to create an empiricalestimate of default probabilities based on the distance to default (DD),given by:

(7)

Firms with a higher volatility or gearing will therefore have a shorter DD,and a higher default probability. The default probability is obtained bycomparing the DD to KMVs proprietary database of empirically observedDDs for companies that have actually defaulted. KMV claim that this rela-tionship is stable across time and in different environments.

DDDefault point Asset value

Asset volatility=

( )1

p NA X r T

T=

( ) + ( )

ln 012

2

2


21


CreditMetrics adopts a similar approach to KMV, but rather than esti-mate default probabilities directly, it uses historical data based on ratings(see Chapter 3). By adding additional thresholds, credit migrations (tran-sition between ratings, ie upgrades and downgrades) may also beincorporated.

1.7.2 Correlation structure in KMV/CreditMetricsThe KMV/CreditMetrics framework is very powerful, since the jointdefault probability for a pair of assets:

(8)

where denotes the cumulative bivariate normal density function that iscomputationally tractable (see Chapter 3). More generally, the probabilityof the assets one to k defaulting (note that this is not the probability ofk defaults) is an integral over the multivariate normal distribution:

(9)

The above formulae require an estimate of the asset return correlation, ,among all pairs of assets. The assets returns are calculated using easilyobservable equity returns. Although this method has the drawback of over-looking the differences between equity and asset correlations, it is moreaccurate than using a fixed correlation, and is based on data that are morereadily available than, say, credit spreads or actual rating changes.It is inefficient to produce correlations for every pair of obligors that a usermight need, this is because of the scarcity of data for some obligors, and theresulting size of the correlation matrix. CreditMetrics therefore use cor-relations within a set of indices and a mapping scheme to build theobligor-by-obligor correlations from the index correlations. There are twosteps, as follows:

1. Use industry indices in particular countries to construct a matrix of cor-relations between these industries (so one particular element might bethe correlation between UK finance and US steel).

2. Map individual obligors by industry participation. For example, a com-pany might be mapped as 80% UK and 20% US, 70% steel and 30%finance. By multiplying out, we have: 56% UK steel, 24% UK finance,14% US steel, 6% US finance.

CreditMetrics uses the following:

(10)r w r w r w rk k= + + +0 0 1 1Company

idiosyncraticSectorial

123L

1 2444 3444

p N p N p N pk k121

11

21

K K= ( ) ( ) ( )( ) , , , ;

p N p N p121

11

2= ( ) ( )( ) , ;


22


in which r and the (ri ) are Gaussian of mean 0 and variance 1; r is the nor-malised asset return of the company, r0 is the specific risk and theremaining rs are the normalised asset returns of the country/industryindices that affect the obligor in question. The random variable r0 is uncor-related with all the others, while the remaining rs we have Corr(ri , rj ) = i j.The obligor specific risk (OSR) is related to w0 by

and is a given parameter. Let i denote the participation ratios of theobligor in the various sectors (in the example above, these were 56%, 24%,14%, 6%), and i denote the sector volatilities. The remaining weights (wi )are given by the equation

(11)

Note that

which must be so if r is to have unit variance. KMV adopts a similar butmore complex strategy.

The full details of the estimation procedure are proprietary parts of the

Firm return

Firm-specific effect

Country-specific effects

Industry-specific effects

Industry sector effects

Regional economic effects

Global economic effects

[ ] =

[ ]+ [ ]+ [ ]+ [ ]+ [ ]+ [ ]

w w wi j iji jn

02

11+ =

= ,

wii i

i j i j iji j

n=

=

,

( )

1

1 OSR

OSR = 1 1 02w


23

Table 1.7 Number of defaults at the 99.97% confidence level for ahomogeneous portfolio of 50 assets with default probabilities of 0.1%using KMV/CreditMetrics model for different asset return correlations

0% 30% 50% 70%

Default correlation, 0.0% 1.39% 5.33% 15.87%Number of defaults 2 5 10 17



24

Figure 1.7 Loss distributions for a homogeneous portfolio of 50 assets withdefault probabilities of 0.1% using KMV/CreditMetrics model for different assetreturn correlations

0.06000

0.05000

0.04000

0.03000

0.02000

0.01000

0.000001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Prob

abili

ty

Number of defaults

0.03500

0.03000

0.02500

0.02000

0.01500

0.01000

0.00500

0.000001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Prob

abili

ty

Number of defaults

(a) = 0% ( = 0%)

(b) = 30% ( = 1.39%)

KMV software. Table 1.7 and Figure 1.7 illustrate the loss distributions fordifferent values of the asset return correlation. Below, we summarise theadvantages and disadvantages of the KMV/CreditMetrics approaches.

The advantages are that:

a Monte Carlo simulation is easy to perform; credit migration can be incorporated; and it is relatively easy to estimate the underlying multivariate Gaussian

structure.


The disadvantages are that::

the model is analytically intractable we must use a simulation to com-pute loss distribution;

there is uncertainty as to whether probability of multiple defaults is real-istic, or not; and

for large portfolios, the computational load scales as O(N2).

1.7.3 Credit Portfolio ViewThe McKinsey model is an econometric model that takes the current


25

Figure 1.7 (continued)

0.02500

0.02000

0.01500

0.01000

0.00500

0.000001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Prob

abili

ty

Number of defaults

0.01000

0.00800

0.00600

0.00400

0.00200

0.000001 2 3 4 5 6 7 8 9 10 11 12 13 1415 16 17 18 19 20

Prob

abili

ty

Number of defaults

(c) = 50% ( = 5.33%)

(d) = 70% ( = 15.87%)


macroeconomic environment into account. Rather than using historicaldefault rate averages calculated from decades of data, the McKinsey modeluses default probabilities that are conditional on the current state of theeconomy.

The default probability of a particular obligor is obtained from the (nor-mally distributed) macroeconomic explanatory variable by the logistictransformation:

(12)

The independent macroeconomic factors used as explanatory variables inthe model are then simulated to create numerous possible states of theeconomy over the horizon. These outcomes are then used to determine theexpected default rates.

The model is fitted using the historical average default rates andeconomic data for different rating, industry and country pairings. Onepractical difficulty is the lack of data with which to calibrate the model. Asthe data are further refined to make them more granular (eg, by increasingthe number of rating classes, industries or geographical regions), theobservations become more sparse and the standard error of the estimatesincreases. It becomes difficult to verify the accuracy of such an economicmodel other than by observing default probabilities increasing duringeconomic downturns. Another problem is model risk itself: does the modelcorrectly simulate the default process based on the selected economicfactors? Are these factors correctly modelled? Few institutions can predictinterest rates, let alone the resulting default rates based on these.

1.7.4 CreditRisk+This is an actuarial model, which means that it is based on probabilitiesalone and does not infer an underlying causality or default process. Defaultprobabilities are assigned to rating classes that are then mapped to indi-vidual exposures. CreditRisk+ also uses the volatility (standard deviation)of default rates as an input. A certain type of correlation structure isimposed; the parameters of this can be inferred from the default ratevolatility. This produces the tail of the portfolio. The CreditRisk+ docu-mentation recommends a volatility approximately equal to the default rateitself. Neither the theory nor the documentation provide an insight into theappropriate value.

The correlation structure is obtained as follows. The loss from the wholeportfolio is assumed to follow a Poisson distribution with parameter K (ie,if there are n unit exposures each with default probability p, then K = np)which is itself a random variable. In that case the number of defaults M hasthe distribution

pe y

=

+

1

1


26



27

Figure 1.8 Loss distributions for a homogeneous portfolio of 50 assets withdefault probabilities of 0.1% using CreditRisk+ model for different asset returncorrelations

0.03000

0.02500

0.02000

0.01500

0.01000

0.00500

0.000001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Prob

abili

ty

Number of defaults

0.01400

0.01200

0.01000

0.00800

0.00600

0.00400

0.00200

0.000001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Prob

abili

ty

Number of defaults

(a) = 1.39%

(b) = 5.33%

0.00600

0.00500

0.00400

0.00300

0.00200

0.00100

0.000001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Prob

abili

ty

Number of defaults(c) = 15.87%


(13)

where G is the distribution function of K. If K has a (K,K) distribution, asCreditRisk+ recommends, then

(14)

For a finite number of assets, it is more convenient to work with the distri-bution of p = K n, which will also have a Gamma distribution, but withparameters = K n and = K. The probability that one particular assetdefaults is /, and the probability that two particular assets default is 2

+ ( )2 (these being the first two moments of the Gamma distribution).The default correlation is therefore = 1 ( ).

One possible objection to this treatment is that the distribution of thedefault probability should be bounded above by 1. Therefore, the Beta dis-tribution would be a more appropriate choice. This has the same analyticaltractability as the Gamma case discussed above. For, if p is B (a, b)-distrib-uted, then the number of defaults follows this distribution:

(15)

where B (,) denotes the Beta function. Further, the probability that oneparticular asset defaults is -p = a (a + b) and the probability that two partic-

P M rp p

B a b

n

rp p dp

B a r b n r

B a b

n

r

a br n r( )

( )

( , )( )

( , )

( , )

= =

=

+ +

1 10

1 11

P M re k

r

e kdk

r

r

k r k K

K

K

Kr

K

K

K K

( )!

!

= = ( )=

+( )+( )( )

+

1

0

1

P M re k

rdG k

k r( )

!( )= =


28

Table 1.8 Number of defaults at the 99.97% confidence level for ahomogeneous portfolio of 50 assets with default probabilities of 0.1%using CreditRisk+ model for different default correlations

Default correlation, 0.0% 1.39% 5.33% 15.87%Number of defaults 2 4 9 17


ular assets default is a(a + 1) (a + b)(a + b + 1). The default correlation is = 1 (a + b + 1). Hence a = -p (1 1) and b = (1 -p)(1 1) .

To compare the results with those of KMV/CreditMetrics (see Table 1.7,Figure 1.7), we have used the same default correlations , and present thesame results as shown previously (see Table 1.8, Figure 1.8).

1.8 CONCLUSIONCredit risk encompasses counterparty risk for loan and derivative port-folios as well as issuer risk for bonds. In this book, we will cover theformer, with a few references to the latter where appropriate. The tradi-tional approach to managing credit risk is the use of credit limits, nettingagreements and collateral. These techniques, although useful, have provedinadequate for the plethora of capital market products that are traded.Internal models have been developed during recent years by major finan-cial institutions. New mathematical and statistical techniques have beenintroduced which were not required for developing market risk models.The two main differences between market and credit risk are as follows.

1. The analysis of credit risk is performed over a longer time horizon thanmarket risk. This is driven by the illiquidity of the correspondingpositions. It requires more refined simulation techniques for theevolution of the market exposure.

2. The credit loss distribution is highly asymmetric and the normalapproximation produces erroneous results. It is characterised by a largenegative skew, which arises due to the significant (albeit small) proba-bility of incurring a large loss.

Derivative portfolios are distinguished from loan portfolios by the fact thattheir exposures are random over time, since they are marked-to-market. Inorder to compute the credit loss distribution of a derivative portfolio, weneed to model:

1. The evolution of the underlying market variables, ie, interest rates, FXrates, and stock prices driven by the underlying volatilities and cor-relations; and

2. the credit parameters, ie, default and credit migration events, driven bydefault and credit migration probabilities, correlation and recoveryrates.

One of the most crucial parameters that drive the tails of the credit loss dis-tribution is the default correlation matrix. The financial industry hasplaced particular emphasis on estimating this matrix. Various packages,such as KMV/CreditMetrics, Credit Portfolio View and CreditRisk+, arecommercially available. KMV/CreditMetrics are option-based models thatcalculate the default probabilities based on the firms capital structure andthe asset return volatility. Credit Portfolio View is an econometric model


29


that takes into account the current macroeconomic environment.CreditRisk+ is an actuarial model. It is based on probabilities alone and itdoes not infer an underlying causality or default process.

1 Commonly, in order to derive an unbiased estimator, the denominator in this expression isn 1 instead of n. In this case, as the mean of the distribution is known, it is appropriate touse n.

2 The probabilities for the normal distribution can be determined by + c, ie, a certainnumber of standard deviations away from the mean. The values of c required in the aboveexample are 1.644, 2.326, 3.090 and 3.432 respectively.

3 This is actually under the risk-neutral measure. Under the historical measure, we shouldsubstitute (the actual return) for r in the above expression.


30

BIBLIOGRAPHY

Black, F., and M. Scholes, 1973, The Pricing of Options and Corporate Liabilities, Journal ofPolitical Economy 81, pp. 63759.

Carty, L., and D. Lieberman, 1996a, Corporate Bond Defaults and Default Dates19381995, Moodys Investor Services, Global Credit Research, January.

Carty, L., and D. Lieberman, 1996b, Defaulted Bank Loan Recoveries, Moodys InvestorServices, Global Credit Research, November.

Credit Suisse Financial Products, 1997, CreditRisk+ A Credit Risk Management Framework.

Gupton, G. M., C. C. Finger and M. Bhatia, 1997, CreditMetrics Technical Document,Morgan Guaranty Trust (New York: JP Morgan).

Jarrow, R., and D. R. van Deventer, 1999, Practical Use of Credit Risk Models in LoanPortfolio and Counterparty Exposure Risk Management, in D. Shimko (ed), Credit Risk:Models and Management (London: Risk Books), pp. 3019.

JP Morgan & Co. Incorporated, 1997, CreditMetrics.

Keenan, S. C., L. V. Carty and D. T. Hamilton, 1999, An Empirical Analysis of CorporateRating, Migration, Default and Recovery, in Derivative Credit Risk: Further Advances inMeasurement and Management, Second Edition (London: Risk Books), pp. 3751.

KMV Corporation, 1996, Portfolio Manager Description, Usage and Specification, Version4.0, San Francisco.

Koyluoglu, H. U., and A. Hickman, 1998, Reconcilable Differences, Risk, October,pp. 5662.

Merton, R., 1974, On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,Journal of Finance, 29, pp. 44970.

Moodys Investor Services, 1997, Rating Migration and Credit Quality Correlation,19201996, Global Credit Research, July [http://www.moodysqra.com/research].

Standard and Poors, 1997, Ratings Performance 1996: Stability and Transition, SpecialReport, February [http://www.standardandpoors.com/ResourceCenter/index.html].

Wilson, T., 1997(a), Portfolio Credit Risk, Part I, Risk, September, pp. 11117.

Wilson, T., 1997(b), Portfolio Credit Risk, Part II, Risk, October, pp. 5661.


2.1 INTRODUCTIONA traditional and well-used credit risk management technique is control-ling the underlying market exposure to each counterparty. Banks requirethat there be a credit line for each counterparty they transact with.The credit line, set and controlled by a credit officer, will determine themaximum exposure that is allowed at a series of points in the future. Thecredit line will, therefore, limit the amount that could be lost at default.The size of the credit line must be set with a number of factors in mind.One of the most important factors is the probability of default. If this islow, a larger limit will be more tolerable than for a riskier counterparty.Other determining factors are collateralisation, netting agreements and thecharacteristics of the underlying transactions, ie, liquidity, complexity, etc.These factors may also determine the length of the credit line. Forexample, for transactions that would be hard to trade out of, were creditquality to deteriorate, the financial institution may decide to extendthe line only up to a certain point, thereby effectively limiting the maturityof the allowed trades. The main drawback of this approach is that itcan be overly conservative and difficult, if not impossible, to quantify for aportfolio consisting of deals with more than one counterparty. Also, it failsto take into account the diversification effect, ie the fact that defaultcorrelation is not equal to one.

To utilise credit lines for effective credit risk management, there needsto be a sensible method for consistently calculating the potential futureexposure on all deals. The potential exposure describes the projectedworst-case change in the present value of a deal, or several deals under amaster agreement. Since credit risk needs to be assessed over a relativelylong horizon, the variation of the potential exposure over time is impor-tant, since it shows the potential loss if default occurs at any point inthe future. The potential exposure is a useful measure that incorporatesboth the average value of a deal, or a portfolio of deals, and their futurevariability.

Two general factors are important in determining the credit exposureat some point in the future, and therefore the peak exposure, for a single

31

2

Exposure Measurement


transaction or a portfolio of transactions with the same counterparty.

1. Over time, there will be greater variability/uncertainty in the marketvariables, resulting in greater exposure as time progresses.

2. For a multitude of transactions, cashflows will be paid over time,causing the outstanding notional to decrease, eventually droppingto zero at maturity. Known as pull-to-par, this is the case for forexample interest rate swaps.

These two effects act in opposite directions, the former increasing the creditexposure over time and the latter decreasing it. It is, therefore, possible thatthe peak credit exposure will occur at some point between the inceptionand the maturity of the deal. Different instruments generate completelydifferent credit exposure profiles. An instrument with roughly equalpayments throughout its lifetime will have a smaller exposure than onewith a large payment made at maturity. An instrument whose valuedepends on a market variable that is mean-reverting, such as interest rates,will have a smaller future credit exposure due to the mean-reversion. Thefuture exposure profile of a contract is crucial for determining its creditrisk. In this chapter, we show typical exposure profiles over time for anumber of different products. The exposure is simply the percentile of thefuture PV of a particular transaction at a certain confidence level. Of course,this means that the precise definition of exposure is somewhat arbitrary.In the examples presented in this chapter, we use the 84th percentile,which, for a normal distribution, represents one standard deviation fromthe forward or the expected value. We do not use the standard deviationdirectly, since exposure distributions can deviate significantly fromnormality, particularly for derivative products.

In the computation of value-at-risk (VAR) for market risk, a commonassumption is to use the sensitivity of the portfolio to each of the under-lying market variables (the so-called delta approximation), instead ofperforming the full-blown simulation. This approximation can on occa-sions be dubious, since financial products, especially derivatives, tend tobe inherently non-linear. For market risk, where the measurements areonly over a period of a few days, this linear approximation is possiblyjustifiable, whereas for credit risk, where we consider much longerperiods, it is not.

A common way to generate potential exposures is to use Monte Carlosimulation based on some assumption regarding the joint distribution ofthe underlying variables. By simulating many realisations of interest rates,FX rates and equity paths, transactions can be valued, on a deal-by-dealbasis, at a number of time-points on these paths. The full correlation matrixof the underlying variables is introduced into the simulation, resultingin correlated simulated exposures. The exposure data is then aggregated.The aggregation is often on a counterparty basis, in compliance with


32


active netting agreements. Netting agreements, such as the InternationalSwaps and Derivatives (ISDA) Master Agreement, are legal contracts thatallow liabilities to a counterparty to be offset against exposures to thatsame counterparty in the case of default. A netting agreement decreases theexposure in case of default to the net difference between the liabilities inboth directions. The exposures need to be further adjusted to account forany collateral held against various positions. The treatment of collateral is acomplex problem that should take the following into account:

that the value of the collateral will not be perfectly correlated with thatof the underlying position; and

its amount will not change continuously to match the exposure, since itis called in blocks.

2.2 EXPOSURE SIMULATIONAs we will show in the next chapter, one of the most significant factorsdriving the capital of a typical credit portfolio is the process followed bythe underlying market variables. This process determines the evolution ofthe exposure of the portfolio at hand across time. We need financial mod-els, appropriately calibrated, that will then allow the financial institution tocapture the uncertainty of the future exposure, since greater uncertaintywill lead to greater risk. Note that generating exposures is inherentlydifferent from arbitrage pricing or risk-neutral valuation. For risk manage-ment, we are interested in the distribution of the actual exposure in thefuture. For the technically minded, we point out that the computations areperformed under the historical probability and not under the risk-neutral,which is the case for pricing.

The future exposure of a financial instrument will be driven by one ormore market variables. Next, we briefly discuss simple standard modelsfor the evolution of interest rates and FX rates. We also discuss the prosand cons of using rather simple models for the evolution of the underlyingcompared to using more sophisticated ones.

2.2.1 Interest ratesWe outline the important attributes that a model for generating a realisticevolution of the interest rates for our purposes should have.

It should match the current yield curve, in order to price correctly linearinterest rate products today, eg, spot and forward interest rate swaps.In other words, the model should not allow arbitrage.

It must account for different possible yield curve movements in thefuture (parallel shifts, steepenings and inversions).

The model should exhibit a realistic volatility structure. The volatilityparameters should be calibrated either on historical or market data,

EXPOSURE MEASUREMENT

33


depending on the application. In the second approach, the parametersshould be set such that the model correctly prices the fundamental instru-ments that will be used for pricing and hedging. For the purposes of thisanalysis, these instruments will be standard European caps and swaptions.

Probably the simplest possible model1 that meets the above criteria is theone-factor Hull and White (or extended Vasicek) model (Hull and White,1990). The short rate (ie, short-term interest rate) follows the stochasticprocess:

(1)dr = [(t) ar]dt + rdWt

In this model, r follows a normal random walk with mean-reversion. Theinstantaneous volatility depends on the level of the interest rate. Because ofmean-reversion, when the short rate is above a certain mean level,it reverts back to its mean. The speed of mean-reversion is determinedby a. The mean-reversion level (t) is time-dependent. This allows thismodel to be fitted to the initial yield curve. The parameters a and r need tobe estimated either from historical or market data. Although the yieldcurve is not modelled directly, it can be reconstructed at any point,knowing (t), a, r and the current short rate rt. The basic yield curve move-ments observed in practice (parallel shifts, steepenings and inversions)could be obtained, although more complex changes in the shape of theyield curve cannot be attained.

In the Hull and White model, the standard deviation at time t of a zero-coupon bond maturing at time T is given by:

(2)

If we can estimate the standard deviation (volatility) of zero-coupon bondprices of various maturities, we can imply es

Date post:	29-Oct-2015
Category:	Documents
Upload:	danghoangminhnhat
View:	148 times
Download:	1 times

Arvanitis a., Gregory J. Credit.. the Complete Guide to Pricing, Hedging and Risk Management (Risk...

Documents