2008-Forecasting-default-with-the-Merton-distance-

ForecastingDefaultwiththeMertonDistancetoDefaultModelSreedharT.BharathRossSchoolofBusiness,UniversityofMichiganTylerShumwayRossSchoolofBusiness,UniversityofMichiganWeexaminetheaccuracyandcontributionoftheMertondistancetodefault(DD)model,whichisbasedonMerton’s(1974)bondpricingmodel.Wecomparethemodeltoa“na¨ıve”alternative,whichusesthefunctionalformsuggestedbytheMertonmodelbutdoesnotsolvethemodelforanimpliedprobabilityofdefault.Wefindthatthena¨ıvepredictorper-formsslightlybetterinhazardmodelsandinout-of-sampleforecaststhanboththeMertonDDmodelandareduced-formmodelthatusesthesameinputs.Severalotherforecastingvariablesarealsoimportantpredictors,andfittedvaluesfromanexpandedhazardmodeloutperformMertonDDdefaultprobabilitiesoutofsample.ImplieddefaultprobabilitiesfromcreditdefaultswapsandcorporatebondyieldspreadsareonlyweaklycorrelatedwithMertonDDprobabilitiesafteradjustingforagencyratingsandbondcharacteristics.WeconcludethatwhiletheMertonDDmodeldoesnotproduceasufficientstatisticfortheprob-abilityofdefault,itsfunctionalformisusefulforforecastingdefaults.(JELG12,G13,G33)1.IntroductionDuetotheadventofinnovativecorporatedebtproductsandcreditderivatives,academicsandpractitionershaverecentlyshownrenewedinterestinmodelsthatforecastcorporatedefaults.Oneinnovativeforecastingmodel,whichhasbeenwidelyappliedinbothacademicresearch1andpractice,isaparticularapplicationofMerton(1974)thatwasdevelopedbytheproprietorsoftheKMVcorporation.2WerefertothismodelastheMertondistancetodefaultmodel,Previouslycirculatedwiththetitle“ForecastingDefaultwiththeKMV-MertonModel.”Thisresearchwassup-portedbytheNTTfellowshipoftheMitsuiLifeCenterattheRossSchoolofBusiness,UniversityofMichigan.WethankseminarparticipantsatMichigan,BostonCollege,IndianSchoolofBusiness,Moody’s2006CreditRiskConference,andStanford.WealsothankBillBeaver,JeffBohn,DarrellDuffie,EricFalkenstein,WayneFerson,RaviJagannathan,KyleLundstedt,KenSingleton,andJorgeSobehartfortheircomments.Allerrorsareours.AddresscorrespondencetoeitherSreedharT.BharathandTylerShumway,RossSchoolofBusi-ness,UniversityofMichigan,701TappanStreet,E7606,AnnArbor,MI48109;telephone:[phone];e-mail:[e-mail],RossSchoolofBusiness,UniversityofMichigan,Michi-ganBusinessSchool,701TappanStreet,AnnArbor,MI48109-1234;telephone:[phone];e-mail:[e-mail].1ThemodelisdiscussedinDuffieandSingleton(2003)andSaundersandAllen(2002).ItisappliedbyVassalouandXing(2004);Duffie,Saita,andWang(2007);andCampbell,Hilscher,andSzilagyi(2007),amongothers.2WedonotintendtoimplythatweareusingexactlythesamealgorithmthatMoody’sKMV(whichacquiredKMVin2002)usestocalculatedistancetodefault.DifferencesbetweenourmethodandthatofMoody’sKMVarediscussedinSection2.2andinTable2.C
TheAuthor2008.PublishedbyOxfordUniversityPressonbehalfofTheSocietyforFinancialStudies.Allrightsreserved.ForPermissions,pleasee-mail:[e-mail]:10.1093/rfs/hhn044AdvanceAccesspublicationMay19,2008TheReviewofFinancialStudies/v21n32008ortheMertonDDmodel.ThispaperassessestheaccuracyandthecontributionoftheMertonDDmodel.TheMertonDDmodelappliestheframeworkofMerton(1974),inwhichtheequityofthefirmisacalloptionontheunderlyingvalueofthefirmwithastrikepriceequaltothefacevalueofthefirm’sdebt.Themodelrecognizesthatneithertheunderlyingvalueofthefirmnoritsvolatilityisdirectlyobservable.Underthemodel’sassumptionsbothcanbeinferredfromthevalueofequity,thevolatilityofequity,andseveralotherobservablevariablesbyusinganiterativeproceduretosolveasystemofnonlinearequations.Afterinferringthesevalues,themodelspecifiesthattheprobabilityofdefaultisthenormalcumulativedensityfunctionofaz-scoredependingonthefirm’sunderlyingvalue,thefirm’svolatility,andthefacevalueofthefirm’sdebt.TheMertonDDmodelisacleverapplicationofclassicfinancetheory,buthowwellitperformsinforecastingdependsonhowrealisticitsassumptionsare.Themodelisasomewhatstylizedstructuralmodelthatrequiresanumberofassumptions.Amongotherthings,themodelassumesthattheunderlyingvalueofeachfirmfollowsgeometricBrownianmotionandthateachfirmhasissuedjustonezero-couponbond.Ifthemodel’sstrongassumptionsareviolated,itshouldbepossibletoconstructareduced-formmodelwithmoreaccuracy.Weexaminethreehypothesesinthispaper.First,weaskwhethertheprob-abilityofdefaultgivenbytheMertonDDmodelisasufficientstatisticforforecastingbankruptcy.IftheMertonmodelisliterallytrue,itshouldbeim-possibletoimproveonthemodel’simpliedprobabilityforforecasting.Ifitispossibletoconstructareduced-formmodelwithbetterpredictiveproperties,wecanconcludethattheprobabilityimpliedbytheMertonDDmodel(πMerton)isnotasufficientstatisticforforecastingdefault.Thesecondhypothesiswetestisthatthez-scorefunctionalformusedbytheMertonDDmodelisanimportantconstructforpredictingdefault.Wehypothesizethattheprobabilitycalculatedwiththez-scorefunctionalformcannotbecompletelyreplacedinaforecastingmodelbyalinearcombinationofsimplevariables,includingthevariablesusedtocalculatetheprobability.Inotherwords,wetestwhetherasufficientstatisticfordefaultprobabilitycanbecalculatedwithoutconsi