credit risk modeling

©2003 CRC Press LLC Preface In banking, especially in risk management, portfolio management, and structured finance, solid quantitative know-how becomes more and more important. We had a two-fold intention when writing this book: First, this book is designed to help mathematicians and physicists leaving the academic world and starting a profession as risk or portfolio managers to get quick access to the world of credit risk management. Second, our book is aimed at being helpful to risk managers looking for a more quantitative approach to credit risk. Following this intention on one side, our book is written in a Lecture Notes style very much reflecting the keyword “introduction” already used in the title of the book. We consequently avoid elaborating on technical details not really necessary for understanding the underlying idea. On the other side we kept the pres entation mathematically pre- cise and included some proofs as well as many references for readers interested in diving deeper into the mathematical theory of credit risk management. The main focus of the text is on portfolio rather than single obligor risk. Consequently correlations and factors play a major role. More- over, most of the theory in many aspects is based on probability theory. We therefore recommend that the reader consult some standard text on this topic before going through the material presented in this book. Nevertheless we tried to keep it as self-contained as possible. Summarizing our motivation for writing an introductory text on credit risk management one could say that we tried to write the book we would have liked to read before starting a profession in risk management some years ago. Munich and Frankfurt, August 2002 Christian Bluhm, Ludger Overb eck, Christoph Wagner ©2003 CRC Press LLC Acknowledgements Christian Bluhm would like to thank his wife Tabea and his children Sarah and Noa for their patience during the writing of the manuscript. Without the support of his great family this project would not had come to an end. Ludger Overbeck is grateful to his wife Bettina and his children Leonard, Daniel and Clara for their ongoing support. We very much appreciated feedback, support, and comments on the manuscript by our colleagues. Questions and remarks of the audiences of several conferences, sem- inars and lectures, where parts of the material contained in this book have been presented, in many ways improved the manuscript. We al- ways enjoyed the good discussions on credit risk modeling issues with colleagues from other financial institutions. To the many people dis- cussing and sharing with us their insights, views, and opinions, we are most grateful. Disclaimer This book reflects the personal view of the authors and not the opin- ion of Hyp oVe reinsbank, Deutsche Bank, or Allianz. The contents of the book has been written for educational purposes and is neither an of- fering for business nor an instruction for implementing a bank-internal credit risk model. The authors are not liable for any damage arising from any application of the theory presented in this book. ©2003 CRC Press LLC About the Authors Christian Bluhm works for HypoVereinsbank's group portfolio management in Munich, with a focus on portfolio modeling and risk management instruments. His main responsibilities include the analytic evaluation of ABS transactions by means of portfolio models, as introduced in this book. His first professional position in risk management was with Deutsche Bank, Frankfurt. In 1996, he earned a Ph.D. in mathematics from the University of Erlangen-Nuernberg and, in 1997, he was a post-doctoral member of the mathematics department of Cornell University, Ithaca, New York. He has authored several papers and research articles on harmonic and fractal analysis of random measures and stochastic processes. Since he started to work in risk management, he has continued to publish in this area and regularly speaks at risk management conferences and workshops. Christoph Wagner works on the risk methodology team of Allianz Group Center. His main responsibilities are credit risk and operational risk modeling, securitization and alternative risk transfer. Prior to Allianz he worked for Deutsche Bank's risk methodology department. He holds a Ph.D. in statistical physics from the Technical University of Munich. Before joining Deutsche Bank he spent several years in postdoctoral positions, both at the Center of Nonlinear Dynamics and Complex Systems, Brussels and at Siemens Research Department in Munich. He has published several articles on nonlinear dynamics and stochastic processes, as well as on risk modeling. Ludger Overbeck heads the Research and Development team in the Risk Analytics and Instrument department of Deutsche Bank's credit risk management function. His main responsibilities are the credit portfolio model for the group-wide RAROC process, the risk assesement of credit derivatives, ABS, and other securitization products, and operational risk modeling. Before joining Deutsche Bank in 1997, he worked with the Deutsche Bundesbank in the supervision department, examining internal market risk models. He earned a Ph.D. in Probability Theory from the University of Bonn. After two post-doctoral years in Paris and Berkeley, from 1995 to 1996, he finished his Habilitation in Applied Mathematics during his affiliation with the Bundesbank. He still gives regular lectures in the mathematics department of the University in Bonn and in the Business and Economics Department at the University in Frankfurt. In Frankfurt he received a Habilitation in Business and Economics in 2001. He has published papers in several forums, from mathematical and statistical journals, journals in finance and economics, including RISK Magazine and practioners handbooks. He is a frequent speaker at academic and practioner conferences. ©2003 CRC Press LLC ©2003 CRC Press LLC Contents 1TheBasicsofCreditRiskManagement 1.1.1TheDefaultProbability 1.1.1.1Ratings 1.1.1.2CalibrationofDefaultProbabilitiesto Ratings 1.1.2TheExposureatDefault 1.1.3TheLossGivenDefault 1.1ExpectedLoss 1.2.1EconomicCapital 1.2.2TheLossDistribution 1.2.2.1MonteCarloSimulationofLosses 1.2.2.2AnalyticalApproximation 1.2.3ModelingCorrelationsbyMeansofFactorModels 1.2UnexpectedLoss 1.3RegulatoryCapitalandtheBaselInitiative 2ModelingCorrelatedDefaults 2.1.1AGeneralBernoulliMixtureModel 2.1.2UniformDefaultProbabilityandUniformCorre- lation 2.1TheBernoulliModel 2.2.1AGeneralPoissonMixtureModel 2.2.2UniformDefaultIntensityandUniformCorrelation 2.2ThePoissonModel 2.3BernoulliVersusPoissonMixture 2.4.1CreditMetrics TM andtheKMV-Model 2.4.2CreditRisk + 2.4.3CreditPortfolioView 2.4.3.1CPVMacro 2.4.3.2CPVDirect 2.4.4DynamicIntensityModels 2.4AnOverviewofToday’sIndustryModels ©2003 CRC Press LLC 2.5.1TheCreditMetrics TM /KMVOne-FactorModel 2.5.2TheCreditRisk + One-SectorModel 2.5.3ComparisonofOne-FactorandOne-SectorModels 2.5One-Factor/SectorModels 2.6.1Copulas:VariationsofaScheme 2.6LossDistributionsbyMeansofCopulaFunctions 2.7WorkingExample:EstimationofAssetCorrelations 3AssetValueModels 3.1IntroductionandaSmallGuidetotheLiterature 3.2.1GeometricBrownianMotion 3.2.2PutandCallOptions 3.2AFewWordsaboutCallsandPuts 3.3.1CapitalStructure:Option-TheoreticApproach 3.3.2AssetfromEquityValues 3.3Merton’sAssetValueModel 3.4.1Itˆo’sFormula“Light” 3.4.2Black-ScholesPartialDifferentialEquation 3.4TransformingEquityintoAssetValues:AWorkingAp- proach 4TheCreditRisk+Model 4.1TheModelingFrameworkofCreditRisk + 4.2ConstructionStep1:IndependentObligors 4.3.1SectorDefaultDistribution 4.3.2SectorCompoundDistribution 4.3.3SectorConvolution 4.3ConstructionStep2:SectorModel 5AlternativeRiskMeasuresandCapitalAllocation 5.1CoherentRiskMeasuresandConditionalShortfall 5.2.1Variance/CovarianceApproach 5.2.2CapitalAllocationw.r.t.Value-at-Risk 5.2.3CapitalAllocationsw.r.t.ExpectedShortfall 5.2.4ASimulationStudy 5.2ContributoryCapital 6TermStructureofDefaultProbability 6.1SurvivalFunctionandHazardRate 6.2Risk-neutralvs.ActualDefaultProbabilities ©2003 CRC Press LLC 6.3.1ExponentialTermStructure 6.3.2DirectCalibrationofMulti-YearDefaultProba- bilities 6.3.3MigrationTechniqueandQ-Matrices 6.3TermStructureBasedonHistoricalDefaultInformation 6.4TermStructureBasedonMarketSpreads 7CreditDerivatives 7.1TotalReturnSwaps 7.2CreditDefaultProducts 7.3BasketCreditDerivatives 7.4CreditSpreadProducts 7.5Credit-linkedNotes 8CollateralizedDebtObligations 8.1.1TypicalCashFlowCDOStructure 8.1.1.1OvercollateralizationTests 8.1.1.2InterestCoverageTests 8.1.1.3OtherTests 8.1.2TypicalSyntheticCLOStructure 8.1IntroductiontoCollateralizedDebtObligations 8.2.1TheOriginator’sPointofView 8.2.1.1RegulatoryArbitrageandCapitalRelief 8.2.1.2EconomicRiskTransfer 8.2.1.3FundingatBetterConditions 8.2.1.4ArbitrageSpreadOpportunities 8.2.2TheInvestor’sPointofView 8.2DifferentRolesofBanksintheCDOMarket 8.3.1Multi-StepModels 8.3.2CorrelatedDefaultTimeModels 8.3.3StochasticDefaultIntensityModels 8.3CDOsfromtheModelingPointofView 8.4RatingAgencyModels:Moody’sBET 8.5Conclusion 8.6SomeRemarksontheLiterature ©2003 CRC Press LLC References List of Figures 1.1 Calibration of Moody's Ratings to Default Probabilities 1.2 The Portfolio Loss Distribution 1.3 An empirical portfolio loss distribution 1.4 Analytical approximation by some beta distribution 1.5 Correlation induced by an underlying factor 1.6 Correlated processes of obligor's asset value log-returns 1.7 Three-level factor structure in KMV's Factor Model 2.1 Today's Best-Practice Industry Models 2.2 Shape of Ganima Distributions for some parameter sets 2.3 CreditMetrics/KMV One-Factor Model: Conditional default probability as a function of the factor realizations 2.4 CreditMetrics/KMV'One-Factor Model: Conditional default probability as a function of the average one-year default probability 2.5 The probability density f ρς , 2.6 Economic capital EC α in dependence on α 2.7 Negative binomial distribution Nvith parameters ( α , β ) = (1,30) 2.8 t (3)-deilsity versus N(0,1)-density 2.9 Normal versus t -dependency with same linear correlation 2.10 Estimated economic cycle compared to Moody's average historic default frequencies 3.1 Hedging default risk by a long put 3.2 Asset-Equity relation 5.1 Expected Shortfall 5.2 Shortfall contribution versus var/covar-contribution 5.3 Shortfall contribution versus Var/Covar-contribution for business units 6.1 Curnulative default rate for A-rated issuer 6.2 Hazard rate functions 7.1 Total return swap 7.2 Credit default swap 7.3 Generating correlated default times via the copula approach 7.4 The averages of the standard deviation of tire default times, first-to- default- and last-to-default-time 7.5 kth-to-default spread versus correlation for a basket with three underlyings 7.6 Default spread versus correlation between reference asset and swap counterparty 7.7 Credit spread swap 7.8 Example of a Credit-linked Note 8.1 Classification of CDOs 8.2 Example of a cash flow CDO 8.3 Example of waterfalls in a cash flow CDO 8.4 Example of a synthetic CDO 8.5 Equity return distribution of a CDO 8.6 CFO modeling scheme 8.7 CDO modeling workflow based on default times 8.8Diversification Score as a function of m 8.9 Fitting loss distributions by the BET 8.10 Tranching a Loss Distribution ©2003 CRC Press LLC ©2003 CRC Press LLC Chapter1 TheBasicsofCreditRisk Management Whyiscreditriskmanagementanimportantissueinbanking?To answerthisquestionletusconstructanexamplewhichis,although simplified,neverthelessnottoounrealistic:Assumeamajorbuilding companyisaskingitshousebankforaloaninthesizeoftenbillion Euro.Somewhereinthebank’screditdepartmentasenioranalysthas thedifficultjobtodecideiftheloanwillbegiventothecustomeror ifthecreditrequestwillberejected.Letusfurtherassumethatthe analystknowsthatthebank’schiefcreditofficerhasknownthechief executiveofficerofthebuildingcompanyformanyyears,andtomake thingsevenworse,thecreditanalystknowsfromrecentdefaultstudies thatthebuildingindustryisunderhardpressureandthatthebank- internalrating 1 ofthisparticularbuildingcompanyisjustontheway downtoalowsubinvestmentgrade. Whatshouldtheanalystdo?Well,themostnaturalanswerwould bethattheanalystshouldrejectthedealbasedontheinformation sheorhehasaboutthecompanyandthecurrentmarketsituation.An alternativewouldbetogranttheloantothecustomerbuttoinsurethe losspotentiallyarisingfromtheengagementbymeansofsomecredit riskmanagementinstrument(e.g.,aso-calledcreditderivative). Admittedly,weintentionallyexaggeratedinourdescription,butsit- uationsliketheonejustconstructedhappenfromtimetotimeandit isnevereasyforacreditofficertomakeadecisionundersuchdifficult circumstances.Abrieflookatanytypicalbankingportfoliowillbesuf- ficienttoconvincepeoplethatdefaultingobligorsbelongtothedaily businessofbankingthesamewayascreditapplicationsorATMma- chines.Banksthereforestartedtothinkaboutwaysofloaninsurance manyyearsago,andtheinsuranceparadigmwillnowdirectlyleadus tothefirstcentralbuildingblockcreditriskmanagement. 1 Aratingisanindicationofcreditworthiness;seeSection1.1.1.1. ©2003 CRC Press LLC [...]... takes into account all the different risk characteristics of the loans in the portfolio Therefore it is clear that Monte Carlo simulation is the “state-of-the-art” in credit risk modeling, and whenever a portfolio contains quite different transactions from the credit risk point of view, one should not trust too much in the results of an analytical approximation 1.2.3 Modeling Correlations by Means of Factor... losses are charged to the customer, hereby taking the creditworthiness captured by the customer’s rating into account More risky customers have to pay a higher risk premium than customers showing high credit quality Third, the bank will also ask for some compensation for taking the risk of unexpected losses coming with the new loan into the bank’s credit portfolio The charge for unexpected losses is... KMV-Model in Section 1.2.3 and in Chapter 3 Another method for calibrating default probabilities from market data is based on credit spreads of traded products bearing credit risk, e.g., corporate bonds and credit derivatives (for example, credit default swaps; see the chapter on credit derivatives) • Calibration of default probabilites from ratings In this approach, default probabilities are associated... well developed as in the United States However, due to the strongly increasing market of credit derivatives and securitised credit products, one can expect that there will be a transparent and well-developed market for all types of loans in a few years 11 Synonymously called Capital at Risk (CaR) or credit Value-at -Risk (VaR) in the literature ©2003 CRC Press LLC in 9,998 out of 10,000 years, hereby assuming... be sufficient to protect the bank from insolvency This is the downside when calibrating risk capital by means of quantiles However, today most major banks use an EC framework for their internal credit risk model The reason for reducing the quantile qα by the EL is due to the “best practice” of decomposing the total risk capital (i.e., the quantile) into a first part covering expected losses and a second... additivity is lost Unfortunately this is the standard case, because correlations are “part of the game” and a main driver of credit risk In fact, large parts of this book will essentially ©2003 CRC Press LLC be dealing with correlation modeling The UL of a portfolio is the first risk quantity we meet where correlations respectively covariances play a fundamental role: ˜ ULP F = V[LP F ] (1 7) m m EADi... F Therefore it does not come much as a surprise that ˜ the distribution of LP F , the so-called loss distribution of the portfolio, plays a central role in credit risk management In Figure 1.2 it is illustrated that all risk quantities of the credit portfolio can be identified by means of the loss distribution of the portfolio This is an important observation, because it shows that in cases where the... random variables driven by a common underlying factor Now, so far we have always looked at the credit risk of a single facility, although banks have to manage large portfolios consisting of many different products with different risk characteristics We therefore will now indicate how one can model the total loss of a credit portfolio For this purpose we consider a portfolio consisting of m loans ˜ Li = EADi... institute is especially critical for obligors with former good credit quality, because banks tend to focus more on critical than on good customers regarding credit lines (bad customers get much more attention, because the bank is already “alarmed” and will be more sensitive in case of early warnings of financial instability) Any stochastic modeling of EAD should take these aspects into account 6 Collateral... induce some concentration risk Summarizing one can say the EL charges are portfolio independent, but EC charges are portfolio dependent This makes the calculation of the contributory EC in pricing tools more complicated, because one always has to take the complete reference portfolio into account Risk contributions will be discussed in Chapter 5 An alternative to EC is a risk capital based on Expected . main responsibilities are credit risk and operational risk modeling, securitization and alternative risk transfer. Prior to Allianz he worked for Deutsche Bank's risk methodology department LLC 1.1.1TheDefaultProbability Thetaskofassigningadefaultprobabilitytoeverycustomerinthe bank’screditportfolioisfarfrombeingeasy.Thereareessentiallytwo approachestodefaultprobabilities: •Calibrationofdefaultprobabilitiesfrommarketdata. Themostfamousrepresentativeofthistypeofdefaultprobabil- itiesistheconceptofExpectedDefaultFrequencies(EDF)from KMV 2 Corporation.WewilldescribetheKMV-ModelinSection 1.2.3andinChapter3. Anothermethodforcalibratingdefaultprobabilitiesfrommarket dataisbasedoncreditspreadsoftradedproductsbearingcredit risk, e.g.,corporatebondsandcreditderivatives(forexample, creditdefaultswaps;seethechapteroncreditderivatives). •Calibrationofdefaultprobabilitesfromratings. Inthisapproach,defaultprobabilitiesareassociatedwithratings, andratingsareassignedtocustomerseitherbyexternalrating agencieslikeMoody’sInvestorsServices,Standard&Poor’s (S&P),orFitch,orbybank-internalratingmethodologies.Be- causeratingsarenotsubjecttobediscussedinthisbook,we willonlybrieflyexplainsomebasicsaboutratings.Anexcellent treatmentofthistopiccanbefoundinasurveypaperbyCrouhy etal.[22]. Theremainingpartofthissectionisintendedtogivesomebasic indicationaboutthecalibrationofdefaultprobabilitiestoratings. 1.1.1.1Ratings Basicallyratingsdescribethecreditworthinessofcustomers.Hereby quantitativeaswellasqualitativeinformationisusedtoevaluatea client.Inpractice,theratingprocedureisoftenmorebasedonthe judgementandexperienceoftheratinganalystthanonpuremathe- maticalprocedureswithstrictlydefinedoutcomes.Itturnsoutthat intheUSandCanada,mostissuersofpublicdebtareratedatleast bytwoofthethreemainratingagenciesMoody’s,S&P,andFitch. 2 KMVCorp.,founded13yearsago,headquarteredinSanFrancisco,developsanddis- tributescreditriskmanagementproducts;seewww.kmv.com. ©2003. responsibilities are the credit portfolio model for the group-wide RAROC process, the risk assesement of credit derivatives, ABS, and other securitization products, and operational risk modeling. Before joining