FRM-The Pricing of Portfolio Credit Risk

BISWorkingPapersNo214ThepricingofportfoliocreditriskbyNikolaTarashevandHaibinZhuMonetaryandEconomicDepartmentSeptember2006BISWorkingPapersarewrittenbymembersoftheMonetaryandEconomicDepartmentoftheBankforInternationalSettlements,andfromtimetotimebyothereconomists,andarepublishedbytheBank.TheviewsexpressedinthemarethoseoftheirauthorsandnotnecessarilytheviewsoftheBIS.Copiesofpublicationsareavailablefrom:BankforInternationalSettlementsPress&CommunicationsCH-4002Basel,SwitzerlandE-mail:[e-mail]:[phone]and+41612808100ThispublicationisavailableontheBISwebsite().©BankforInternationalSettlements2006.Allrightsreserved.Limitedextractsmaybereproducedortranslatedprovidedthesourceisstated.ISSN1020-0959(print)ISSN1682-7678(onlineThePricingofPortfolioCreditRisk∗NikolaTarashev†HaibinZhu‡Thisdraft:September2006AbstractEquityandcredit-default-swap(CDS)marketsareindisagreementastotheextenttowhichassetreturnsco-moveacrossfirms.Thissuggestsmarketsegmentationandcastsambiguityabouttheasset-returncorrelationsunderpinningobservedpricesofportfoliocreditrisk.Theambiguitycouldbeeliminatedby–currentlyunavailable–datathatrevealthemarketvaluationoflow-probability/large-impactevents.Atpresent,judiciousassumptionsaboutthisvaluationcanbeusedtoreconcileobservedpriceswithasset-returncorrelationsimpliedbyeitherequityorCDSmarkets.TheseconclusionsarebasedonananalysisoftranchespreadsofapopularCDSindex,whichincorporatearathersmallpremiumforcorrelationrisk.JELClassificationNumbers:G13,C15Keywords:CDSindextranche,Jointdistributionofassetreturns,Correlationriskpremium,Copula∗TheviewspresentedherearesolelythoseoftheauthorsanddonotnecessarilyrepresentthoseoftheBankforInternationalSettlements.WearegratefultoClaudioBorio,AndyFilardo,PeterHoerdahl,SrichanderRamaswamy,KostasTsatsaronis,HaoZhouandseminarparticipantsattheBankforInternationalSettle-mentsandthe2006FDICDerivativesConferencefordiscussionsandcommentsonearlierversionsofthepaper.WealsothankFrankZhangforsharingMatlabcodesandMarcusJellinghausfordatasupport.†NikolaTarashev:ResearchandPolicyAnalysis,BankforInternationalSettlements,Basel,Switzerland.Tel.:[phone].Fax:[phone].E-mail:[e-mail].‡HaibinZhu:ResearchandPolicyAnalysis,BankforInternationalSettlements,Basel,Switzerland.Tel.:[phone].Fax:[phone].E-mail:[e-mail].1IntroductionPortfoliocreditriskhasthreekeycomponents:probabilityofdefault(PD),lossgivendefault(LGD)andtheprobabilitydistributionofjointdefaults.1Thelastcomponent,whichhasreceivedleastattentionowingtothetraditionalfocusofacademicresearchersandmarketpractitionersonsingle-namecreditevents,hasrecentlygainedinimportanceasaresultoftherapiddevelopmentofinnovativeproductsinstructuredfinance.Suchproducts,whichallowinvestorstotradeportfoliocreditrisk,includecollateralizeddebtobligations(CDOs),CDOsofCDOs(orCDO2),nth-to-defaultcreditdefaultswap(CDS)andCDSindices.2Thepricesofthesefinancialinstrumentsrelyheavilyonestimatedprobabilitiesofdefaultclustering(seeHullandWhite,2004;Gibson,2004).Thereisnoconsensus,however,onhowmarketparticipantsconstructsuchestimates.Theliteraturehasproposedtwomainapproachestoestimatingthelikelihoodthataparticularnumberofdefaultswilloccurwithinaportfolio.Thefirst,direct,approachreliesexclusivelyondefaultdata(Danielsetal.,2005;Demeyetal.,2004;JarrowandvanDeventer,2005).Sincedefaultsarerareevents,however,thisapproachleadstolargeestimationerrors,especiallyforportfoliosconsistingofinvestment-gradeentities.Thesecond,indirect,approachisbasedontheMerton(1974)frameworkandexploitsthenotionthatadefaultoccurswhentheassetsofaborrowerfallbelowathresholdvalue.Thisnotionallowsonetocombinesingle-namePDswiththecorrespondingcorrelations,thirdandhighermomentsofassetreturnsinordertoconstructaprobabilitydistributionofdefaults.Twoofthebuildingblocksofthisdistribution,PDsandasset-returncorrelations,aretypicallyestimated.3Bycontrast,theliteraturemakesassumptionsaboutthethirdandhighermomentsofassetreturns,whichrelatetotheskewnessofdensityfunctionsandthefatnessoftheirtails.Such,largelyarbitrary,assumptionscaninfluencesubstantially,andpotentiallyunduly,theestimateddistributionofdefaultsforanyestimateofPDsandasset-returncorrelations.Thisisthefirstpapertoinvestigatewhetherpricesofsecuritiesassociatedwithindividualcompanies–iecreditspreadsorequityprices–couldshedlightonmarketvaluationofcreditriskassociatedwithaportfolioofcompanies.Toanalyzethisvaluation,wecompareobservedtranchespreadsofapopularCDSindex–DowJonesCDXNorthAmerica5-1ThemainstreamofthecreditriskliteraturefocusesonPD:seeDuffieandSingleton(2003)foranoverview.ThegrowingliteratureonLGDincludesAltmanandKishore(1996),Jarrow(2001)andCovitzandHan(2004).2Forageneraldiscussionofproductsusedfortradingportfoliocreditrisk,seeBCBS(2004).3InthespecialcaseofGaussianassetreturns,thecorrelationofthesereturnsdescribesfullytheirco-movement.Asaresult,thedistributionofjointdefaultscanbederivedsolelyonthebasisofPDsandasset-returncorrelations.Zhou(2001)goesbeyondthisspecialcaseandstudiesanalytically