《债券市场：分析与策略（第七版）》Ch22

Chapter22

CreditRiskModeling

22-1Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectivesAfterreadingthischapter,youwillunderstandthedifficultiesofmeasuringcreditriskexposurecomparedtointerestrateriskexposuretherearetwobasictypesofcreditriskmodels:structuralmodelsandreducedformmodelswhatastructuralmodelisandthelinktooptiontheorythebasicstructuralmodel(Black-Scholes-Mertonmodel)anditsextensions(Geskemodelandfirst-passagetimemodel)22-2Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectives

(continued)Afterreadingthischapter,youwillunderstandwhatismeantbydefaultcorrelationsandthereasonforusingcopulasinmeasuringportfoliocreditriskthebasicfeatureofreduced-formmodelstheroleofthePoissonprocessinreduced-formmodelsthedifferencesinthetwomajorreduced-formmodels:Jarrow-TurnbullmodelandDuffie-Singletonmodelwhatanincompleteinformationmodelis22-3Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallDifficultiesinCreditRiskModelingQuantifyinginterestriskexposureislesscomplicatedthanmodelingcreditriskexposure.Therearethreereasonswhythisisso.Creditdefaultriskisarareeventand,asaresult,thehistoricaldataneededtocomputetheinputsintoacreditriskmodel(e.g.,defaultratesandrecoveryrates)areconsiderablylessincomparisontothedataavailableforthemodelingofinterestraterisk.Itismuchmoredifficulttodrawanymeaningfulandpossiblypredictiveconclusionsabouttheprobabilityofdefaultbecauseofthediversityofthecorporationsinvolvedandthelackofcompleteinformationregardingcorporatepractices.Therearevariouscausesofdefaultbyacorporateborrowerthatmakedefaulthardtopredict.22-4Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallDifficultiesinCreditRiskModeling(continued)WhileourfocusinthischapterwillbeoncreditriskmodelingforU.S.corporations,applyingthesemodelstonon-U.S.entitiesiscomplicatedbythefactthatdefaultisnotauniversalconcept.Everycountryhasitsownbankruptcycodetodealwithdefaults.Thereisnoassurancethattheadministratorsofthebankruptcylawwillapplythelawinamannerthatisconsistentwiththebankruptcycode.Thereisapreponderanceofevidencethatshowsthatstrictabsolutepriorityhasnotbeenupheldbythecourtsincorporatereorganizations.22-5Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallDifficultiesinCreditRiskModeling(continued)Eventhoughitmaybeunlikelythatcreditriskmodelingwillyieldsignificantinformation,creditriskmodelshavelongbeenemployedinthefinanceandinsuranceindustries.Thefocusoftheearlymodelswasongeneratingforecastsofdefaultrates,creditratings,andcreditspreads.Sincethemid-1990s,moresophisticatedapproachestocreditriskmodelinghavebeenproposedandmadecommerciallyavailabletoportfoliomanagers.Foroneoftheseapproachesthatiscommerciallyavailable,thetheoreticalfoundationofthemodeldatesbacktotheearly1970s.22-6Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallOverviewofCreditRiskModelingCreditriskmodelingisusedtoestimatethedefaultprobability,priceindividualcorporatebonds,andmeasureaportfolio’screditrisk.Thedefaultprobabilityisthelikelihoodthataborrowerwilldefaultsometimeoverthelifeofthedebtobligation.Bydefaultitismeantthattheborrowfailstohonorthetermsoftheagreement,suchasthefailuretomakeaprincipalorcouponpaymentrequiredundertheagreement,ortheviolationofacovenant.22-7Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallOverviewofCreditRiskModeling(continued)Toestimatethedefaultprobabilityforoneyear,acreditriskmodelrequiresthefollowing:adefinitionofwhatconstitutesadefaulteventamodelofinvestoruncertaintyhowthatinformationwillevolveovertime22-8Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallOverviewofCreditRiskModeling(continued)Givenacreditriskmodelandobservedmarketpricesforcorporatebondsand/orcreditderivatives,afairvalueforthecreditspreadforanilliquidorunpricedcorporatebondwithagivencreditratingorothercredit-basedcharacteristiccanbeestimated.Thiscreditspreadisreferredtoasthefairmarketcreditspread.Toestimatethefairmarketcreditspread,acreditriskmodelrequiresamodelthatestimatesrecoveryifadefaultoccursamodelthatshowsthecreditspreadthatinvestorswantinordertoacceptsystematiccreditriskandidiosyncraticriskamodeloftherisk-freerate22-9Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallCreditRatingsVersusCreditRiskModelsAlong-termcreditratingisapredictionofthelikelihoodthatanissuerorissuewilldefaultandtheseverityoftheloss.Therearethreereasonswhyonecannotsimplyrelyoncreditratingsasaforecasterofdefault:Unlikedefaultprobabilities,creditratingsarediscretewithalimitednumberofratinggrades.Whileratingsareupdatedveryinfrequently,defaultprobabilitiescanbeestimatedonareal-timebasis.Thereisnoclearmaturityforacreditrating.22-10Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModelsTheBlack-Scholes-Merton(BSM)optionpricingmodelanditsextensionsarereferredtoasstructuralmodels.Thefundamentalfeaturethatiscommontoallstructuralmodelsisthatdefaultcanbeviewedassometypeofoptionbytheequityownersontheassetsofthefirm,andthattheoptionistriggered(i.e.,thecorporationdefaults)whenthevalueofthecorporation’sassetsdeclinesbelowacertaindefaultpoint.Theoutputsofstructuralmodelsshowhowthecreditriskofacorporatebondisafunctionoftheissuer’sleverageandthevolatilityoftheissuer’sassets.22-11Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)Structuralmodelshavebeenusedbybanksinmakingcreditdecisionsandbybondportfoliomanagers.Structuralmodelsmayperformwellinoneareaofapplicationinbondportfoliomanagementbutturnouttobeuselessforotherapplications.Whenconsideringthepotentialuseofstructuralmodels,itisimportanttobeawareoftheunderlyingassumptionsofthemodelbecauseitistheseassumptionsthatmaylimittheusefulnessofamodel.22-12Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)FundamentalsoftheBlack-Scholes-MertonModelIntheBSMmodel,thefollowingassumptionsaremade:Assumption1:Acorporationhasonlyonetypeofbondoutstandinginitsdebtstructure.Assumption2:Thebondoutstandingisazero-couponbondthatmaturesinTyears.Assumption3:Thereisaconstantrisk-freeinterestrateoverthelifeofthebond.Assumption4:Thepaymenttobondholdersinthecaseofdefaultofthecorporationismadeinaccordancewiththeprincipleofabsolutepriority.Assumption5:Volatilityisassumedtobeconstant.22-13Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)ExtensionsoftheBSMmodelattempttorelaxthemoreunrealisticassumptions.Assumption1isobviouslyunrealistic,andextensionsoftheBSMmodelallowformultipletypesofbondissues.Assumption2isimportantbecauseitprecludesacorporationfromdefaultingpriortothematuritydateofthezero-couponbond.Thereasonisthattherearenopaymentsthatmustbemadeand,therefore,nodefaultbasedonmissedpaymentsiftheoutstandingdebtisazero-couponbond.22-14Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)TheBSMmodelincludesthefollowingvariables.E(t)isthevalueofthecorporation’sequityattimet.A(t)isthevalueofthecorporation’sassetattimet.Kisthematurityvalueofthezero-couponbondissuedbythecorporation.Atthematuritydateofthezero-couponbond,T,thevalueofthecorporation’sequityisE(t)andthevalueofthecorporation’sassetsisA(t).22-15Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)Letuslookatwhatcanhappenatthematuritydateofthezero-couponbond.Thereareonlythreepossiblescenariosatthematuritydateofthezero-couponbond(T):Scenario1:

Totalassetsexceedthematurityvalueofthezero-couponbond.Thatis,A(T)>K.Scenario2:

Totalassetsarelessthanthematurityvalueofthezero-couponbond.Thatis,A(T)<K.Scenario3:

Totalassetsareequaltothematurityvalueofthezero-couponbond.Thatis,A(T)=K.ThevalueoftheequityattimeT,E(T),isthedifferencebetweenthevalueoftheassets,A(T),andthematurityvalueofthezero-couponbond,K.Thatis,E(T)=A(T)

–

K.22-16Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)ForScenario1,thebondholdersarepaidinfullandstockholdersgettherest.Wehave:E(T)=A(T)–K

>0.ForScenario2,bondholderswouldreceivelessthanthematurityvalueofthebondandtakeoverthecompany.Wehave:E(T)=A(T)–K

<0.ForScenario3,thestockholderswouldpayoffthebondholdersinfullbutownacorporationwithzerovalue.Wehave:E(T)=A(T)–K

=0orA(T)=K.IfweletB(T)denotethevalueofthecorporation’szero-couponbond,thenitsvalueatthematuritydatecanbeexpressedas:B(T)=A(T)–max[A(T)–K,0].Thenotationmax[A(T)–K,0]meansthemaximumofA(T)–K

andzero.22-17Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)ForScenario1,A(T)–KispositivesothemaximumvalueisA(T)–KandthevalueofthebondisKis:B(T)=A(T)–[A(T)–K]=K.ForScenario2,A(T)–Kisnegativeandthemaximumvalueiszeroandthevalueofthebondis:B(T)=A(T)–0=A(T).ForScenario3,thevalueofthebondissimplyK.22-18Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)Thetermmax[A(T)–K,0]isthepayoffofacalloptionwithastrikepriceofKthatexpiresatT.Sincethetermentersintotheequationwithanegativesign,thismeansashortpositioninacalloption(i.e.,thesaleofacalloption).Thus,thebondholderhasalongpositioninthecorporation’sassetsandhassoldacalloptiontothecommonstockholdersonthecorporation’sassets.Thevalueofacorporatebondisthevalueofthetotalassetsreducedbythevalueofthecalloption.ThecalloptioncanbevaluedbyusinganoptionpricingmodelsuchastheBlack-Scholesmodel.22-19Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)IfwerewriteB(T)=A(T)–max[A(T)–K,0],wehaveanotherinterpretationthatisuseful.Theequationcanberewrittenas

B(T)=K–max[K–A(T),0]Theresultsforeachofthethreescenariosforthisequationarethesameasbefore:B(T)=A(T)–max[A(T)–K,0]22-20Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)Theterm[K–A(T)]isthepayoffofaputoptionattimeTwrittenonthecorporation’sassetswithastrikepriceK.SincethistermentersintoB(T)=K–max[K–A(T),0]withanegativesign,itisthepayoffofashortputposition.OnecaninterpretthepositiongivenbyB(T)=K–max[K–A(T),0]asapositioninarisk-freebondreducedbythevalueoftheputpositionthatthestockholderssoldtothebondholdersonthecorporation’sassets.22-21Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels

(continued)Tovaluetheoptionusingthisapproachtocorporatebondvaluationusinganoptionpricingmodel,thefollowinginputsarerequired:thecorporation’scapitalstructurethecorporation’smarketvaluethevolatilityofthemarketvalueofthecorporation22-22Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)ExtensionsoftheBSMModelConsiderAssumption1(thecorporationhasonlyonetypeofbondoutstanding):Ifthecompanyhasaseriesofzero-couponbondsoutstandingwithdifferentmaturities,thenitisquiteeasyfortheBSMmodeltocharacterizedefaultatdifferenttimes.AnotherseriesofmodelshavebeenproposedtoextendtheBSMmodeltothecasewheredefaultcanoccurnotonlyatmaturitybutatanytimepriortomaturity.Theunderlyinglegalprinciplehereisthattherearetypicallycovenantsinatypicalbondindenturegrantingthebondholderstherighttorestructurethecorporationshouldthevalueofthecorporateassetsfallbelowagivenamount,referredtoasadefaultbarrier.Thesemodelsarereferredtoasfirst-passagetimemodelswiththefirstsuchmodelbeingproposedbyBlackandCox.22-23Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)Moody’sKMVModelAnumberofsoftware/consultingcompanieshavedevelopedcreditriskmodelsbasedonstructuralmodels.ThetwomostpopularmodelsuseBSMtomodeldefaultsusinglargedatabasesofhistoricaldata.IntheMoody’sKMVmethodology,informationcontainedinequitypricesandthebalancesheetofcorporatebondissuersisusedtoextracttheprobabilityofdefault,whichitreferstoastheexpecteddefaultfrequency(EDF)andistheprobabilityofdefaultingwithinaspecifiedtimeperiod.So,acorporationwithanEDFforaone-yeartimeperiodof3%hasa3%probabilityofdefaultingwithinthenext12months.22-24Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)Moody’sKMVModelMoreover,eachEDFcanbeassociatedwithacreditspreadcurveandacreditrating.Thecreditratingassignedbythemodelbasedonmarketpricesiscalledamarketimpliedrating.Insteadofbeingaggregatedintoratingclasses,corporationsarecategorizedintheMoody’sKMVmethodologyusinga“distance-to-defaultindex”measure.22-25Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)AdvantagesandDisadvantagesofStructuralModelsFromatheoreticalperspective,structuralmodelsanalyzedefaultbasedonareasonableassumptionthatitisaresultofthevalueofthecorporateissuer’sassetsfallingbelowthevalueofitsdebt.Inadditiontoprovidingdefaultprobabilities,thesemodelsallowabondportfoliomanagertoseehowthecreditriskofcorporatedebtisafunctionoftheleverageandtheassetvolatilityoftheissuer.Accordingly,theimpactofanewstockorbondofferingthatwillchangethecapitalstructureofacorporationcanbeassessed.22-26Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)AdvantagesandDisadvantagesofStructuralModelsWhilesuperiortowhatwaspreviouslyavailable,therearetwoconcernsthathavebeenexpressedaboutstructuralmodels:difficulttocalibrateandcomputationallyburdensome.Tocalibrateastructuralmodeltopriceacorporatebondrequirescalibrationtoassetvolatility,assetvalue,facevalueofthecorporateissuer’sdebt,thedefaultbarrier(inthecaseoffirst-passagetimemodels),andtherisk-freerate.Forfirst-passagetimemodels,asuitabledefaultbarriermustbeestimated.Becauseofthisdifficulty,itisarguedthatstructuralmodelsarenotsuitableforthefrequentmarkingtomarketofcreditcontingentsecurities.22-27Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStructuralModels(continued)AdvantagesandDisadvantagesofStructuralModelsFromacomputationalperspective,thepricingofacorporatezero-couponbondisjustlikepricinganoptiononabond.However,forcoupon-bearingcorporatebondstheproblembecomesoneofpricingacompoundoption,amoredifficultproblem.Topriceasubordinatedbond,itisnecessarytosimultaneouslyvalueallofthemoreseniordebt(bondsandloans).Consequently,thereisreluctancebysomemarketparticipantstousestructuralmodelswherethereisaneedforrapidandaccuratepricingofcorporatebonds.Themainapplicationofstructuralmodelsinpracticeappearstobeintheareaofcreditriskanalysis.22-28Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallEstimatingPortfolioCreditRisk:DefaultCorrelationandCopulasForaportfolioofcorporatebonds,thereistheriskthatsomeeventthattriggersthedefaultofoneofthecorporatebondsintheportfoliowilladverselyimpactanothercorporatebondintheportfolio,therebyincreasingtheprobabilityofthedefaultofthatsecondcorporation.Acommonlyusedstatisticalconcepttogaugethedependencebetweentwovariablesiscorrelation.Increditriskmanagement,thistypeofriskisreferredtoasdefaultcorrelation.Onewouldexpectthatforcorporateissuersinthesameindustrysector,defaultcorrelationishigh.22-29Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallEstimatingPortfolioCreditRisk:DefaultCorrelationandCopulas

(continued)Developersofcreditriskmodelsneedanestimateofthedefaultcorrelationsinordertoassessthecreditriskofaportfolioandcreditderivatives.Thetechniqueusedtoestimatethedefaultcorrelationvaries.Forexample,Moody’susesMonteCarlosimulationofhistoricaldataonratingtransitionsanddefaultsinitsanalysis.FitchRatings,usescorrelationsbasedonequitypricechanges.Therearereasonsthatacorrelationmeasureisnotasuitableoneinthecaseofcreditriskmodeling,forexample,thereisasymmetricaldependence.Manydevelopersofcreditriskmodelsusedifferentmeasuresofdependencetounderstandthemultivariaterelationshipbetweenallofthebondsinaportfolio.Thecombinationofindividualdefaultprobabilities(ordefaultdistributions)andtheirdependenceareknownmathematicallyasa“copula.”Whatisimportanttounderstandisthatbyusingcopulasratherthansimplecorrelationstogaugethenatureofthedependencybetweentwovariables,amodelercanbetterhandlethemodelingofextremeevents.22-30Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallReduced-FormModelsReduced-formmodelswereintroducedinthemid1990s.Themajordifferencebetweenreduced-formmodelsandstructuralmodelsishowdefaultistreated.Aswithalleconomicmodels,structuralandreduced-formmodelsaremerelyanabstractsimplifiedmathematicalrepresentationofrelationshipsbetweeneconomicvariables.Instructuralmodels,defaultisendogenous;inreduced-formmodelsitisexogenous.Asitturnsout,specifyingdefaultsexogenously,asisdoneinreduced-formmodels,greatlysimplifiescreditriskmodelingbecauseitignorestheconstraintofdefiningwhatcausesdefaultandsimplylooksatthedefaulteventitself.22-31Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallReduced-FormModels

(continued)Thekeyelementsinreduced-formmodelsare:(1)thedefault-time,(2)recoveryrateprocess,and(3)risk-freeinterestrate.Themodelingofwhenadefaultoccursandtherecoveryprocess,iftheissuerdefaults,ishowthereduced-formmodelsthathavebeenproposeddiffer.Accuratelymodelingthebankruptcyrecoveryprocessisnotsimple.Recognitionmustbegiventothetrade-offbetweenanalytictractabilityandpracticalapplicability.Basedonrestrictiveassumptionsaboutthedynamicsofthedefaultandrecoveryprocesses,aclosed-formsolutiontoreduced-formmodelshasbeenderivedbytheirproposers.Thetheoreticalframeworkforreduced-formmodelsisthePoissonprocess,whichisasimplestochasticprocess.22-32Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallReduced-FormModels

(continued)PoissonProcessAPoissonprocessisoneofthemostimportantclassesofstochasticprocesses.TounderstandthePoissonprocess,webeginwithasequence,whichcountsthenumberofsomedefinedeventoccurringfromaninitialpointintime.WedenotethevalueofthiscounterattimetasNt.Thatis:Nt=numberofoccurrencesintheinterval0tot.

Ntwillincreaseby1foreveryoccurrenceofanevent,andtheseincreasesarereferredtoas“increments.”Theprobabilityofaneventoccurringfromoneintegertothenextoverasmalltimeintervaldtisgivenby:Probability[Nt+dt+Nt–1=1]=λdt

wheretheparameterλiscalledtheintensityparameterofthePoissonprocess.Inreduced-formmodels,theeventinaPoissonprocessisdefinedasadefault.Theintensityparameterinreduced-formmodelsiscalledthedefaultintensityandisakeyparameterinthemodel.Inthecontextofareduced-formmodel,thedefaultintensityattimetcanbethoughtofintermsofaprobability.22-33Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallReduced-FormModels

(continued)TheJarrow-TurnbullModelTheJarrow-Turnbullmodelisasimplemodelofdefaultandrecovery.Itassumesthatnomatterwhendefaultoccurs,therecoverypaymentispaidatthematuritydate.Bymakingtheassumptionthattherecoverypaymentismadeatmaturity,JarrowandTurnbullassumeawayanydependen