TheJournalofFinanceinfluenceofsuchobservablefactorsForexampleratingagenciestypicallyestimatetheprobabilityoflossestoseniortranchesofCDOswhichareintendedtooccuronlywhentheunderlyingportfoliolossese ID: 331801
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THEJOURNALOFFINANCEVOL.LXIV,NO.5OCTOBER2009FrailtyCorrelatedDefaultDARRELLDUFFIE,ANDREASECKNER,GUILLAUMEHOREL,andLEANDROSAITATheprobabilityofextremedefaultlossesonportfoliosofU.S.corporatedebtismuchgreaterthanwouldbeestimatedunderthestandardassumptionthatdefaultcorrela-tionarisesonlyfromexposuretoobservableriskfactors.Atthehighconfidencelevelsatwhichbankloanportfolioandcollateralizeddebtobligation(CDO)defaultlossesaretypicallymeasuredforeconomiccapitalandratingpurposes,conventionallybasedlossestimatesaredownwardbiasedbyafullorderofmagnitudeontestportfolios.OurestimatesarebasedonU.S.publicnonfinancialfirmsbetween1979and2004.Wefindstrongevidenceforthepresenceofcommonlatentfactors,evenwhencontrollingforobservablefactorsthatprovidethemostaccurateavailablemodeloffirm-by-firmdefaultprobabilities.HISPAPERPROVIDESamorerealisticassessmentoftheriskoflargedefaultlossesonportfoliosofU.S.corporatedebtthanhadbeenavailablewithpriormethodologies.Atthehighconfidencelevelsatwhichportfoliodefaultlossesaretypicallyestimatedformeetingbankcapitalrequirementsandratingcol-lateralizeddebtobligations(CDOs),ourempiricalresultsindicatethatconven-tionalestimatorsaredownwardbiasedbyafullorderofmagnitudeontypicaltestportfolios.OurestimatesarebasedonportfoliosofU.S.corporatedebtex-istingbetween1979and2004.Forestimatinghigh-quantileportfoliolosses,conventionalmethodologiessufferfromtheirfailuretocorrectforasignificantdownwardomittedvariablebias.Wefindstrongevidencethatfirmsareex-posedtoacommondynamiclatentfactordrivingdefault,evenaftercontrollingforobservablefactorsthatontheirownprovidethemostaccurateavailablemodeloffirm-by-firmdefaultprobabilities.Bothuncertaintyaboutthecurrentlevelofthisvariable,aswellasjointexposuretofuturemovementsofthisvariable,causeasubstantialincreaseintheconditionalprobabilityoflargeportfoliodefaultlosses.Aconventionalportfoliolossriskmodelassumesthatborrower-levelcondi-tionaldefaultprobabilitiesdependonmeasuredfirm-specificormarketwidefactors.PortfoliolossdistributionsaretypicallybasedonthecorrelatingDuffieisattheGraduateSchoolofBusiness,StanfordUniversity.EcknerandHorelareatBankofAmerica.SaitaisatBarclaysCapital.WearegratefulforfinancialsupportfromMoodyÕsCorporationandMorganStanley;fordatafromMoodyÕsandEdAltman;forresearchassistancefromSabriOncuandVineetBhagwat;forremarksfromTorbenAndersen,AndreLucas,RichardCantor,StavGaon,TylerShumway,JunS.Liu,Xiao-LiMeng,andespeciallyMichaelJohannes;andforsuggestionsbyareferee,anassociateeditor,andtheeditor,CampbellHarvey. TheJournalofFinanceinfluenceofsuchobservablefactors.Forexample,ratingagenciestypicallyesti-matetheprobabilityoflossestoseniortranchesofCDOs,whichareintendedtooccuronlywhentheunderlyingportfoliolossesexceedahighconfidencelevel,byrelyingontheobservablecreditratingsoftheunderlyingcollateraldebtin-struments.Modeledco-movementoftheratingsoftheborrowersrepresentedinthecollateralpoolisintendedtocapturedefaultcorrelationandthetailsofthetotallossdistribution.However,iftheunderlyingborrowersarecommonlyex-posedtoimportantriskfactorswhoseeffectisnotcapturedbyco-movementsofborrowerratings,thentheportfoliolossdistributionwillbepoorlyestimated.Thisisnotmerelyanissueofestimationnoise;afailuretoincluderiskfac-torsthatcommonlyincreaseanddecreaseborrowersdefaultprobabilitieswillresultinadownwardbiasedestimateoftaillosses.Forinstance,inordertoreceiveatriple-Arating,aCDOistypicallyrequiredtosustainlittleornolossesataconfidencelevelsuchas99%.Althoughanymodelofcorporatedebtportfoliolossescannotaccuratelymeasuresuchextremequantileswiththelim-itedavailablehistoricaldata,ourmodeloftaillossesavoidsalarge,downwardomittedvariablebias,andsurvivesgoodness-of-fittestsassociatedwithlargeportfoliolosses.Wheneveritispossibletoidentifyandmeasurenewsignificantriskfac-tors,theyshouldbeincludedinthemodel.Wedonotclaimtohaveidentifiedandincludedallrelevantobservableriskfactors.Althoughourobservableriskfactorsincludefirm-levelandmacroeconomicvariablesleadingtohigheraccu-racyratiosforout-of-sampledefaultpredictionthanthoseofferedbyanyotherpublishedmodel,furtherresearchwillundoubtedlyuncovernewsignificantobservableriskfactorsthatshouldbeincluded.Wediscusssomeproposedin-clusionslaterinthispaper.Itisinevitable,however,thatnotallrelevantriskfactorsthatarepotentiallyobservablebytheeconometricianwillendupbeingincluded.Thereisalsoapotentialforimportantriskfactorsthataresimplynotobservable.Adownwardbiasintail-lossestimatesisthusinevitablewithoutsomeformofbiascorrection.Ourapproachistodirectlyallowforunobservedriskfactorswhosetime-seriesbehaviorandwhoseposteriorconditionaldistri-butioncanbothbeestimatedfromtheavailabledatabymaximumlikelihoodForexample,subprimemortgagedebtportfoliosrecentlysufferedlossesinexcessofthehighconfidencelevelsthatwereestimatedbyratingagencies.Thelossesassociatedwiththisdebaclethathavebeenreportedbyfinancialinstitutionstotalapproximately$800billionasofthiswriting,andarestillac-cumulating.Anexampleofanimportantfactorthatwasnotincludedinmostmortgageportfoliodefaultlossmodelsisthedegreetowhichborrowersandmortgagebrokersprovidedproperdocumentationofborrowerscreditqualities.Withhindsight,moreteamsresponsiblefordesigning,rating,intermediating,andinvestinginsubprimeCDOsmighthavedonebetterbyallowingforthepossibilitythatthedifferencebetweenactualanddocumentedcreditqualitieswouldturnouttobemuchhigherthanexpected,ormuchlowerthanexpected,NewworkbyLandoandNielsen(2008)suggestsadditionalhelpfulcovariates. FrailtyCorrelatedDefaultinamannerthatiscorrelatedacrossthepoolofborrowers.IncorporatingthisadditionalsourceofuncertaintywouldhaveresultedinhigherpricesforCDOfirst-lossequitytranches(aconvexityeffect).SeniorCDOswouldhavebeendesignedwithmoreconservativeover-collateralization,oralternativelywouldhavehadlowerratingsandlowerprices(aconcavityeffect),ontopofanyre-latedriskpremiaeffects.Accordingly,moremodelerscouldhaveimprovedtheirmodelsbyaddingproxiesforthismoralhazardeffect.Itseemsoptimistictobe-lievethattheywerepreparedtodoso,however,fordespitetheclearincentives,manyapparentlydidnot.Thissuggeststhatitisnoteasy,exante,toincludeallimportantdefaultcovariates,andfurther,thatthenexteventofextremeportfoliolosscouldbebasedonadifferentomittedvariable.Itthereforeseemsprudent,goingforward,toallowformissingdefaultcovariateswhenestimatingtaillossesondebtportfolios.Asamotivatinginstanceofmissingriskfactorsinthecorporatedebtarenaonwhichwefocus,thedefaultsofEnronandWorldCommayhaverevealedfaultyaccountingpracticesthatcouldhavebeeninuseatotherfirms,andthusmayhavehadanimpactontheconditionaldefaultprobabilitiesofotherfirms,andthereforeonportfoliolosses.ThebasicideaofourmethodologyisanapplicationofBayessRuletoupdatetheposteriordistributionofunobservedriskfactorswheneverdefaultsarrivewithatimingthatismoreorlessclus-teredthanwouldbeexpectedbasedontheobservableriskfactorsalone.Inthestatisticsliteratureoneventforecasting,theeffectofsuchanunobservedcovariateiscalledfrailty.Inthepriorstatisticsliterature,frailtycovariatesareassumedtobestatic.Itwouldbeunreasonabletoassumethatlatentriskfactorsinfluencingcorporatedefaultarestaticoverour25-yeardataperiod,soweextendthepriorstatisticalmethodologysoastoallowafrailtycovari-atetovaryovertimeaccordingtoanautoregressivetime-seriesspecification.WeuseMarkovChainMonteCarlo(MCMC)methodstoperformmaximumlikelihoodestimationandtofilterfortheconditionaldistributionofthefrailtyprocess.Whileourempiricalresultsaddressthearrivalofdefaultevents,ourmethod-ologycanbeappliedinothersettings.Recently,forinstance,Chernobai,Jorion,andYu(2008)adoptedourmethodologytoestimateamodelofoperationalriskevents.Ourmodelcouldalsobeusedtotreattheimplicationsofmissingcovari-atesformortgagepre-payments,employmentevents,mergersandacquisitions,andotherevent-basedsettingsinwhichtherearetime-varyinglatentvariables.Theremainderofthepaperisorganizedasfollows.SectionIgivesanoverviewofourmodelingapproachandresults.SectionIIplacesourworkinthecontextoftherelatedliteratureandclarifiesourincrementalcontribution.SectionIIIspecifiesthepreciseprobabilisticmodelforthejointdistributionofdefaulttimes.SectionIVdescribesourdatasources,providesthefittedmodel,andsummarizessomeoftheimplicationsofthefittedmodelforthedistributionoflossesonportfoliosofU.S.corporatedebt.SectionVexaminesthefitofthemodelandaddressessomepotentialsourcesofmisspecification,providingrobustnesschecks.SectionVIconcludes.Appendicesprovidesomekeytechnicalinforma-tion,includingourestimationmethodology,whichisbasedonacombination TheJournalofFinanceoftheMonteCarloexpectationsmaximization(EM)algorithmandtheGibbssampler.I.ModelingApproachandResultsInordertofurthermotivateourapproachandsummarizeourmainempiri-calresults,webrieflyoutlineourspecificationhere,andlaterprovidedetails.Ourobjectiveistoestimatetheprobabilitydistributionofthenumberofde-faultsamonggivenfirmsoveranypredictionhorizon.Foragivenfirmourmodelincludesavectorofobservabledefaultpredictioncovariatesthatarespecifictofirm.Thesevariablesincludethefirmdistancetodefault,awidelyfollowedvolatility-correctedleveragemeasurewhoseconstructionisreviewedlaterinthispaper,aswellasthefirmstrailingstockreturn,animportantauxiliarycovariatesuggestedbyShumway(2001).Allowingforun-observedheterogeneity,weincludeanunobservablefirm-specificcovariateWealsoincludeavectorofobservablemacroeconomiccovariates,includinginterestratesandmarketwidestockreturns.Inrobustnesschecks,weexplorealternativeandadditionalchoicesforobservablemacro-covariates.Finally,weincludeanunobservablemacroeconomiccovariateinfluenceonportfoliodefaultlossesisourmainfocus.Ifallofthesecovariateswereobservable,ourmodelspecificationwouldimplythattheconditionalmeanarrivalrateofdefaultoffirmattimeforcoefficients,andtobeestimated.Ifallcovariateswereobservable,thiswouldbeastandardproportionalhazardsspecification.Theconditionalmeanarrivalrateisalsoknownasadefaultintensity.Forexample,aconstantan-nualintensityof0.01meansPoissondefaultarrivalwithanannualprobabilityofdefaultof1arenotobservable,theirposteriorprobabilitydistributionsareestimatedfromtheavailableinformationset,whichincludesthepriorhistoryoftheobservablecovariates,whereandalsoincludespreviousobservationsoftheperiodsofsurvivalandtimesofdefaultsofallfirms.Becausepublicfirmdefaultsarerelativelyrare,werelyon25yearsofdata.Weincludeall2,793U.S.publicnonfinancialfirmsforwhichwewereabletoobtainmatchingdatafromtheseveraldatasetsonwhichwerely.Ourdata,describedinSectionIV.A,coverover400,000firm-months.Wespecifyanau-toregressiveGaussiantime-seriesmodelfor()thatwillbedetailedlater.Becauseisunobservable,wefindthatitisrelativelydifficulttotiedownitsmeanreversionratewiththeavailabledata,butthedatadoindicatehassubstantialtime-seriesvolatility,increasingtheproportionalvolatil-ityofbyabout40%aboveandbeyondthatinducedbytime-seriesvariationOurmainfocusistheconditionalprobabilitydistributionofportfoliodefaultlossesgiventheinformationactuallyavailableatagiventime.Forexample, FrailtyCorrelatedDefaultconsidertheportfolioofthe1,813firmsfromourdatasetthatwereactiveatthebeginningof1998.Forthisportfolio,weestimatetheprobabilitydistribu-tionofthetotalnumberofdefaultingfirmsoverthesubsequent5years.Thisdistributioncanbecalculatedfromourestimatesofthedefaultintensityco-efficients,and;ourestimatesofthetime-seriesparametersgoverningthejointdynamicsof();andtheestimatedposteriordistributionofgiventheinformationavailableatthebeginningofthis5-yearperiod.Thedetailedestimationmethodologyisprovidedlaterinthepaper.The95thand99thpercentilesoftheestimateddistributionare216and265de-faults,respectively.Theactualnumberofdefaultsduringthisperiodturnedouttobe195,slightlybelowthe91%confidenceleveloftheestimateddistribution.Withhindsight,weknowthat2001to2002wasaperiodofparticularlyseverecorporatedefaults.InSectionIV,weshowthatafailuretoallowforafrailtyeffectwouldhaveresultedinaseveredownwardbiasofthetailquantilesoftheportfoliolossdistribution,tothepointthatonewouldhaveincorrectlyassignednegligibleprobabilitytotheeventthatthenumberofdefaultsactuallyrealizedwouldhavebeenreachedorexceeded.Asarobustnesscheck,weprovideaBayesiananalysisoftheeffectofajointpriordistributionforthemeanreversionrateandvolatilityofontheposteriordistributionoftheseparametersandontheposteriordistributionofportfoliodefaultlosses.Wefindthatthisparameteruncertaintycausesaddi-ofthetailoftheportfoliolossdistribution,notablyatextremequantiles.Moregenerally,weprovidetestsofthefitoffrailty-basedtailquantilesthatsupportourmodelspecificationagainstthealternativeofano-frailtymodel.Weshowthattherearetwoimportantpotentialchannelsfortheeffectofthefrailtyvariableonportfoliolossdistributions.First,aswithanobservablemacrovari-able,thefrailtycovariatecausescommonupwardanddownwardadjustmentsoffirm-levelconditionaldefaultintensitiesovertime.Thiscauseslargeport-foliolossestobemorelikelythanwouldbethecasewithamodelthatdoesnotincludethisadditionalsourceofdefaultintensitycovariation.Second,be-causethefrailtycovariateisnotobservable,uncertaintyaboutthecurrentlevelofatthebeginningoftheforecastperiodisanadditionalsourceofcorrelationacrossfirmsoftheeventsoffuturedefaults.Thissecondeffectontheportfoliolossdistributionwouldbeimportanteveniftherewerecertaintobenofuturechangesinthisfrailtycovariate.Inanillustrativeexample,weshowthatthesetwochannelsofinfluenceofthefrailtyprocesshavecomparablylargeimpactsontheestimatedtailquantilesoftheportfoliolossAftercontrollingforobservablecovariates,wefindthatdefaultswerepersis-tentlyhigherthanexpectedduringlengthyperiodsoftime,forexample,1986to1991,andpersistentlylowerinothers,forexample,duringthemid-1990s.Fromtroughtopeak,theestimatedimpactofthefrailtycovariateontheaveragedefaultrateofU.S.corporationsduring19802004isroughlyafactoroftwoormore.Asarobustnesscheck,andasanexampleoftheimpactonthemagnitudeofthefrailtyeffectofaddinganobservablefactor,wereestimatethemodelincludingasanadditionalobservablemacro-covariatethetrailing TheJournalofFinanceaveragerealizedrateofdefault,whichcouldproxyforanimportantfactorthathadbeenomittedfromthebase-casemodel.Weshowthatthistrailingdefaultratecovariateisstatisticallysignificant,butthatthereremainsanimportantroleforfrailtyincapturingthetailsofportfoliolossdistributions.II.RelatedLiteratureAstandardstructuralmodelofdefaulttimingassumesthatacorporationdefaultswhenitsassetsdroptoasufficientlylowlevelrelativetoitsliabilities.Forexample,themodelsofBlackandScholes(1973),Merton(1974),Fisher,Heinkel,andZechner(1989),andLeland(1994)taketheassetprocesstobeageometricBrownianmotion.Inthesemodels,afirmsconditionaldefaultprobabilityiscompletelydeterminedbyitsdistancetodefault,whichisthenumberofstandarddeviationsofannualassetgrowthbywhichtheassetlevel(orexpectedassetlevelatagiventimehorizon)exceedsthefirmsliabilities.Anestimateofthisdefaultcovariate,usingmarketequitydataandaccountingdataforliabilities,hasbeenadoptedinindustrypracticebyMoodysAnalytics,aleadingproviderofestimatesofdefaultprobabilitiesforessentiallyallpub-liclytradedfirms.Basedonthistheoreticalfoundation,weincludedistancetodefaultasacovariate.Inthecontextofastandardstructuraldefaultmodelofthistype,DuffieandLando(2001)showthatifdistancetodefaultcannotbeaccuratelymeasured,thenafilteringproblemarises,andtheresultingdefaultintensitydependsonthemeasureddistancetodefaultandonothercovariates,bothfirm-specificandmacroeconomic,thatmayrevealadditionalinformationaboutthefirmscondi-tion.If,acrossfirms,thereiscorrelationintheobservationnoisesofthevariousfirmsdistancestodefault,thenthereisfrailty.Forreasonsoftractability,wehavechosenareduced-formspecificationoffrailty.Altman(1968)andBeaver(1968)areamongthefirsttoestimatereduced-formstatisticalmodelsofthelikelihoodofdefaultofafirmwithinoneaccount-ingperiod,usingaccountingdata.Althoughthevoluminoussubsequentem-piricalliteratureaddressingthestatisticalmodelingofdefaultprobabilitieshastypicallynotallowedforunobservedcovariatesaffectingdefaultprobabilities,thetopicofhiddensourcesofdefaultcorrelationhasrecentlyreceivedsomeat-tention.Collin-Dufresne,Goldstein,andHelwege(2003)andZhangandJorionWearegratefultoarefereeforsuggestingthis.SeeCrosbieandBohn(2002)andKealhofer(2003).EarlyintheempiricalliteratureondefaulttimedistributionsistheworkofLane,Looney,andWansley(1986)onbankdefaultprediction,usingtime-independentcovariates.LeeandUrrutia(1996)usedadurationmodelbasedonaWeibulldistributionofdefaulttimes.Durationmodelsbasedontime-varyingcovariatesincludethoseofMcDonaldandVandeGucht(1999),inamodelofthetimingofhigh-yieldbonddefaultsandcallexercises.RelateddurationanalysisbyShumway(2001),Kavvathas(2001),ChavaandJarrow(2004),andHillegeistetal.(2004)predictbankruptcy.Shumway(2001)usesadiscretedurationmodelwithtime-dependentcovariates.Duffie,Saita,andWang(2007)providemaximumlikelihoodestimatesoftermstructuresofdefaultprobabilitiesbyusingajointmodelfordefaultintensitiesandthedynamicsoftheunderlyingtime-varyingcovariates. FrailtyCorrelatedDefault(2007)findthatamajorcrediteventatonefirmisassociatedwithsignificantincreasesinthecreditspreadsofotherfirms,consistentwiththeexistenceofafrailtyeffectforactualorrisk-neutraldefaultprobabilities.Collin-Dufresne,Goldstein,andHuggonier(2004),Giesecke(2004),andSchonbucher(2003)ex-plorelearningfromdefaultinterpretations,basedontheexpectedeffectofun-observablecovariates.Yu(2005)findsempiricalevidencethat,otherthingsequal,areductioninthemeasuredprecisionofaccountingvariablesisasso-ciatedwithawideningofcreditspreads.Dasetal.(2007),usingroughlythesamedatastudiedhere,provideevidencethatdefaultsaresignificantlymorecorrelatedthanwouldbesuggestedbytheassumptionthatdefaultriskiscap-turedbytheobservablecovariates.Theydonot,however,estimateamodelwithunobservedcovariates.Here,wedepartfromtraditionalduration-basedspecificationsofdefaultpre-diction,suchasthoseofCoudercandRenault(2004),Shumway(2001),andDuffie,Saita,andWang(2007),byallowingfordynamicunobservedcovari-ates.Independentofourwork,andwithasimilarthrust,Delloy,Fermanian,andSbai(2005)andKoopman,Lucas,andMonteiro(2008)estimatedynamicfrailtymodelsofratingtransitions.Theysupposethattheonlyobservablefirm-specificdefaultcovariateisanagencycreditrating,andthatallintensitiesofdowngradesfromoneratingtothenextdependonacommonunobservablefactor.Becausecreditratingsareincompleteandlaggingindicatorsofcreditquality,asshown,forexample,byLandoandSkdeberg(2002),onewouldex-pecttofindsubstantialfrailtyinratings-basedmodelssuchasthese.AsshownbyDuffie,Saita,andWang(2007),whoestimateamodelwithoutfrailty,theobservablecovariatesthatweproposeoffersubstantiallybetterout-of-sampledefaultpredictionthandoespredictionbasedoncreditratings.Evenwiththebenefitoftheseobservablecovariates,however,inthispaperweexplicitlyin-corporatetheeffectofadditionalunincludedsourcesofdefaultcorrelationandshowthattheyhavestatisticallyandeconomicallysignificantimplicationsforthetailsofportfoliodefaultlossdistributions.III.ADynamicFrailtyModelTheintroductionhasgivenabasicoutlineofourmodel.Thissectionpro-videsaprecisespecificationofthejointprobabilitydistributionofcovariatesanddefaulttimes.Wefixaprobabilityspace()andaninformationfil-.Foragivenborrowerwhosedefaulttimeis,wesaythatanonnegativeprogressivelymeasurableprocessisthedefaultintensityoftheborrowerif,asoftime,theborrowerhasnotyetdefaulted;istheconditionalmeanarrivalrateofdefault,measuredineventsperunitoftime.WesupposethatallfirmsdefaultintensitiesattimedependonaMarkovstatevectoroffirm-specificandmacroeconomiccovariates.Wesuppose,how-ever,thatisonlypartiallyobservabletotheeconometrician.Withcompleteobservationof,thedefaultintensityoffirmattimewouldbeoftheformPrecisely,amartingaleisdefinedby1 TheJournalofFinance),whereisaparametervectortobeestimatedand)isthecomponentofthestatevectorthatisrelevanttothedefaultintensityoffirmWeassumethat,conditionalonthepathoftheunderlyingstateprocessdeterminingdefaultandotherexitintensities,theexittimesoffirmsarethefirsteventtimesofindependentPoissonprocesseswithtime-varyingin-tensitiesdeterminedbythepathof.Thisdoublystochasticmeansthat,giventhepathofthestatevectorprocess,themergerandfailuretimesofdifferentfirmsareconditionallyindependent.Whilethisconditional-independenceassumptionistraditionalfordurationmodels,wedepartinanimportantwayfromthetraditionalsettingbyassumingthatisnotfullyobservabletotheeconometrician.Thus,fromtheviewpointoftheeconome-sinformation,defaultsarenotdoublystochastic,andwecannotusestandardestimationmethods.Onemayentertainvariousalternativeapproaches.Forexample,thereisthepossibilityofbywhichthedefaultofonefirmcouldhaveadirectinfluenceontherevenues(orexpensesorcapital-raisingopportunities)ofanotherfirm.Inthispaper,weexamineinsteadtheimplicationsoffrailty,bywhichmanyfirmscouldbejointlyexposedtooneormoreunobservableriskfactors.Werestrictattentionforsimplicitytoasinglecommonfrailtyfactorandtofirm-by-firmidiosyncraticfrailtyfactors,althougharichermodelandsufficientdatacouldallowfortheestimationofadditionalfrailtyfactors,forexample,atthesectorallevel.Weletbeafirm-specificvectorofcovariatesthatareobservableforfirmfromwhenitfirstappearsinthedataatsometimeuntilitsexittimeWeletdenoteavectorofmacroeconomicvariablesthatareobservableatalltimes,andletbeavectorofunobservablefrailtyvariables.Thecompletestatevectoristhen,...,),whereisthetotalnumberoffirmsinthedataset.Atime-seriesmodelof,tobedescribed,isdeterminedbyavectorofparameterstobeestimated.Welet)bethevectorofobservedcovariatesforcompany(includingaconstant).Thelastobservationtimeofcompanycouldbethetimeofadefaultoranotherformofexit,suchasamergeroracquisition.Whilewetakethefirstappearancetimetobedeterministic,ourresultsarenotaffectedbyallowingtobeastoppingtimeunderadditionaltechnicalconditions.Theeconometriciansinformationfiltration(isthatgeneratedbytheobservedvariablesisthedefaultindicatorprocessofcompany(whichiszerobeforedefault,oneafterwards).Thecomplete-informationfiltration(isgen-eratedbythevariablesinaswellasthefrailtyprocessBecauseweobservethesecovariatesonamonthlybasisbutmeasuredefaulttimescontinu-ously,wetake,where)isthetimeofthemostrecentmonthend. FrailtyCorrelatedDefaultWeassumethat),where).Wetakethepro-portionalhazardsformforaparametervector)commontoallfirms,whereisaparameterwhoserolewillbedefinedlater.Beforeconsideringtheeffectofotherexitssuchasmergersandacquisitions,themaximumlikelihoodestimators(MLE)of-conditionalsurvivalprobabil-ities,portfoliolossdistributions,andrelatedquantitiesareobtainedundertheusualsmoothnessconditionsbytreatingtheMLEoftheparametersasthoughtheyarethetrueparameters(WewillalsoexaminetheimplicationsofBayesianuncertaintyregardingcertainkeyparameters.Tofurthersimplifynotation,let)denotethevectorofob-servedcovariateprocessesforallcompanies,andlet)denotethevectorofdefaultindicatorsofallcompanies.Iftheeconometricianweretobegivencompleteinformation,Proposition2ofDuffie,Saita,andWang(2007)wouldimplyalikelihoodofthedataattheparameters()oftheformformDititt(1Dit)].(2)Wesimplifybysupposingthatthefrailtyprocessisindependentoftheobservablecovariateprocess.WithrespecttotheeconometricianslimitedInthesenseofProposition4.8.4ofJacobsen(2006),theeconometriciansdefaultintensityforfirmItisgenerallytruethattheconditionalprobabilityofsurvivaltoafuturetimetheeffectofmergersandotherexits)isgivenbytheusualformula).Rather,forafirmthathassurvivedtotime,theprobabilityofsurvivaltotime(againneglectingotherexits)).Thisisjustifiedbythelawofiteratedexpectationsandthedoublystochasticpropertyonthecomplete-informationfiltration(),whichimpliesthatthe-conditionalsurvivalprobabilityis).SeeCollin-Dufresne,Goldstein,andHuggonier(2004)foranotherapproachtothiscalculation.Ifotherexits,forexample,duetomergersandacquisitions,arejointlydoublystochasticwithdefaultexits,andotherexitshavetheintensityprocess,thentheconditionalprobabilityattimethatfirmwillnotexitbeforetime).Forexample,itisimpossibleforafirmtodefaultbeginningin2yearsifithasalreadybeenacquiredbyanotherfirmwithin2years. TheJournalofFinancefiltration(),thelikelihoodisthenthenDititt(1Dit)]W,D,(3)wherepY()istheunconditionalprobabilitydensityofthepathoftheunob-servedfrailtyprocess.Thefinalexpectationof(3)iswithrespecttothatdensity.MostofourempiricalresultsarepropertiesoftheMLE()for(whenconsideringotherexitssuchasthoseduetoacquisitions,()isthefullMLEfor()becausewehaveassumedthatallformsofexitarejointlydoublystochasticontheartificiallyenlargedinformationfiltration(Inordertoevaluatetheexpectationin(3),onecouldsimulatesamplepathsofthefrailtyprocess.Sinceourcovariatedataaremonthlyobservationsfrom1979to2004,evaluating(3)bydirectsimulationwouldthenmeanMonteCarlointegrationinahigh-dimensionalspace.Thisisextremelynumericallyintensivebybrute-forceMonteCarlo,giventheoverlyingsearchforparameters.Wenowturntoaspecialcaseofthemodelthatcanbefeasiblyestimated.WesupposethatisanOrnsteinUhlenbeck(OU)process,inthat0,(4)isastandardBrownianmotionwithrespectto()),andisanonnegativeconstant,themean-reversionrateof.Withoutlossofgenerality,wehavefixedthevolatilityparameteroftheBrownianmotiontobeunitybecausescalingtheparameter,whichdeterminesin(1)thedependenceofthedefaultintensitieson,playspreciselythesameroleinthemodelasscalingthefrailtyprocessTheOUmodelforthefrailtyvariablecouldcapturetheaccumulativeef-fectovertimeofvariousdifferentunobservedfundamentalcommonshockstodefaultintensities.Forexample,assuggestedintheintroduction,aborrowermeasuredcreditqualitiescouldbesubjecttoacommonsourceofreportingnoise.Whilesuchanaccountingfailurecouldbemitigatedovertimewithim-provedcorporategovernanceandaccountingstandards,somenewformofcom-monunobservedshiftindefaultintensitiescouldarise,suchastheincentiveeffectsofachangeinbankruptcylawthattheeconometricianfailedtoconsider,Fornotationalsimplicity,expression(3)ignoresthepreciseintramonthtimingofdefault,al-thoughitwasaccountedforintheparameterestimationbyreplacinginthecasethatcompanydefaultsinthetimeinterval( FrailtyCorrelatedDefaultoracorrelatedshiftintheliquidityofbalancesheetsthatwentunobserved,andsoon.Themean-reversionparameterisintendedtocapturetheexpectedrateofdecayofthecumulativeeffectofpastunobservedshockstodefaultin-tensities.AlthoughanOUprocessisareasonablestartingmodelforthefrailtypro-cess,onecouldallowmuchricherfrailtymodels.FromtheBayesiananalysisreportedinSectionIV,however,wehavefoundthatevenourrelativelylargedatasetistoolimitedtoidentifymuchofthetime-seriespropertiesoffrailty.Thisisnotsosurprising,giventhatthesamplepathsofthefrailtyprocessarenotobserved,andgiventherelativelysparsedefaultdata.Forthesamereason,wehavenotattemptedtoidentifysector-specificfrailtyeffects.Thestartingvalueandlong-runmeanoftheOUprocessaretakentobezero,sinceanychange(ofthesamemagnitude)ofthesetwoparameterscanbeabsorbedintothedefaultintensityinterceptcoefficient.However,wedolosesomegeneralitybytakingtheinitialconditionfortobedeterministicandtobeequaltothelong-runmean.AnalternativewouldbetoaddoneormoreadditionalparametersspecifyingtheinitialprobabilitydistributionofWehavefoundthattheposterioroftendstoberobusttotheassumedinitialdistributionof,forpointsintimethatareayearortwoaftertheinitialdateofoursample.WeestimatethemodelparametersusingacombinationoftheEMalgorithmandtheGibbssamplerthatisdescribedinAppendixA.IV.MajorEmpiricalResultsThissectiondescribesourdata,presentstheestimatedmodel,andprovidesitsimplicationsforthedistributionofportfoliodefaultlossesrelativetoamodelwithoutfrailty.A.DataOurdataset,drawingelementsfromBloomberg,Compustat,CRSP,ands,isalmostthesameasthatusedtoestimatetheno-frailtymodelsofDuffie,Saita,andWang(2007)andDasetal.(2007).WehaveslightlyimprovedthedatabyusingTheDirectoryofObsoleteSecuritiesandtheSDCdatabasetoidentifyadditionalmergers,defaults,andfailures.WehavecheckedthatthefewadditionaldefaultsandmergersidentifiedthroughthesesourcesdonotchangesignificantlytheresultsofDuffie,Saita,andWang(2007).Ourdatasetcontains402,434firm-monthsofdatabetweenJanuary1979andMarch2004.Becauseofthemannerinwhichwedefinedefaults,itisappropriatetousedataonlyuptoDecember2003.Forthetotalof2,793companiesinthisimproveddataset,TableIshowsthenumberoffirmsineachexitcategory.Oftheto-talof496defaults,176firstoccurredasbankruptcies,althoughmanyoftheotherdefaultseventuallyledtobankruptcy.WerefertheinterestedreadertoSection3.1ofDuffie,Saita,andWang(2007)foranin-depthdescriptionoftheconstructionofthedatasetandanexactdefinitionoftheseeventtypes. TheJournalofFinanceTableINumberofFirmExitsofEachTypebetween1979and2004 ExitType OtherdefaultMerger-acquisitionOtherexits 19801985199019952000YearNumberofdefaults Figure1.Yearlynumberofdefaults.Thenumberofdefaultsinourdatasetforeachyearbetween1980and2003.Figure1showsthetotalnumberofdefaults(bankruptciesandotherdefaults)ineachyear.Moodys13thannualcorporatebonddefaultstudyprovidesadetailedexpositionofhistoricaldefaultratesforvariouscategoriesoffirmssince1920.Themodelofdefaultintensitiesestimatedinthispaperadoptsaparsimo-nioussetofobservablefirm-specificandmacroeconomiccovariates:1.Distancetodefault,avolatility-adjustedmeasureofleverage.Ourmethodofconstruction,basedonmarketequitydataandCompustatbookliabilitysInvestorService,HistoricalDefaultRatesofCorporateBondIssuers,1920 FrailtyCorrelatedDefaultdata,isthatusedbyVassalouandXing(2004),CrosbieandBohn(2002),andHillegeistetal.(2004).Althoughtheconventionalapproachtomea-suringdistancetodefaultinvolvessomeroughapproximations,BharathandShumway(2008)provideevidencethatdefaultpredictionisrelativelyrobusttovaryingtheproposedmeasurewithsomerelativelysimpleal-ternatives.2.Thefirmstrailing1-yearstockreturn,acovariatesuggestedbyShumway(2001).Althoughwedonothaveinmindaparticularstructuralinterpre-tationforthiscovariate,likeShumway,wefindthatitofferssignificantincrementalexplanatorypower,perhapsasaproxyforsomeunobservedfactorthathasaninfluenceondefaultriskbeyondthatofthefirmsmea-sureddistanceofdefault.3.The3-monthTreasurybillrate,whichplaysaroleintheestimatedmodelconsistentwiththeeffectofamonetarypolicythatlowersshort-terminterestrateswhentheeconomyislikelytobeperformingpoorly.4.Thetrailing1-yearreturnontheS&P500index.Theinfluenceofthiscovariate,whichisstatisticallysignificantbut,inthepresenceofdistancetodefault,ofonlymoderateeconomicimportance,willbediscussedlater.Duffie,Saita,andWang(2007)giveadetaileddescriptionofthesecovariatesanddiscusstheirrelativeimportanceinmodelingcorporatedefaultintensities.Asrobustnesschecks,weexaminetheinfluenceofGDPgrowthrates,indus-trialproductiongrowthrates,averageBBBAAAcorporatebondyieldspreads,industryaveragedistancetodefault,andfirmsize,measuredasthelogarithmofthemodel-impliedassets.Eachoftheseisfoundtobeatbestmarginallysignificantaftercontrollingforourbasiccovariates,distancetodefault,trail-ingreturnsofthefirmandtheS&P500,andthe3-monthTreasurybillrate.Laterinthispaper,wealsoconsidertheimplicationsofaugmentingourlistofmacro-covariateswiththetrailingaveragedefaultrate,whichcouldproxyforimportantmissingcommoncovariates.Thisvariablemightalsocaptureadi-rectsourceofdefaultcontagion,inthatwhenagivenfirmdefaults,otherfirmsthathaddependedonitasasourceofsalesorinputsmayalsobeharmed.Thiswasthecase,forexample,intheeventssurroundingthecollapseofPennCen-tralin1970to1971.AnotherexampleofsuchacontagioneffectistheinfluenceofthebankruptcyofautopartsmanufacturerDelphiinNovember2005onthesurvivalprospectsofGeneralMotors.Wedonotexploretheroleofthisformofcontagion,whichcannotbetreatedwithinourmodelingframework.B.TheFittedModelTableIIshowstheestimatedcovariateparametervectorandfrailtypa-,togetherwithestimatesofasymptoticstandarderrors.Sizemaybeassociatedwithmarketpower,managementstrategies,orborrowingability,allofwhichmayaffecttheriskoffailure.Forexample,itmightbeeasierforabigfirmtorenegotiatewithitscreditorstopostponethepaymentofdebt,ortoraisenewfundstopayolddebt.Inasense,firmsizemayalsonegativelyinfluencefailureintensity.ThestatisticalsignificanceofsizeasadeterminantoffailureriskhasbeendocumentedbyShumway(2001).Forourdataandourmeasureoffirmsize,however,thiscovariatedoesnotplayastatisticallysignificantrole. TheJournalofFinanceTableIIMaximumLikelihoodEstimatesofIntensityModelParametersThefrailtyvolatilityisthecoefficientofdependenceofthedefaultintensityontheOUfrailty.EstimatedasymptoticstandarderrorsarecomputedusingtheHessianmatrixoftheexpectedcompletedataloglikelihoodat.Themeanreversionandvolatilityparametersarebasedonmonthlytimeintervals. CoefficientStd.Error 1.0290.201Distancetodefault1.2010.037Trailingstockreturn0.6460.0763-monthT-billrate0.2550.033TrailingS&P500return1.5560.3005.2Latent-factorvolatility0.1250.0177.4Latent-factormeanreversion0.0180.0044.8 Ourresultsshowimportantrolesforbothfirm-specificandmacroeconomiccovariates.Distancetodefault,althoughahighlysignificantcovariate,doesnotonitsowndeterminethedefaultintensity,butdoesexplainalargepartofthevariationofdefaultriskacrosscompaniesandovertime.Forexample,anegativeshocktodistancetodefaultbyonestandarddeviationincreasesthedefaultintensitybyroughly230%.The1-yeartrailingstockreturncovariateproposedbyShumway(2001)hasahighlysignificantimpactonde-faultintensities.Perhapsitisaproxyforfirm-specificinformationthatisnotcapturedbydistancetodefault.ThecoefficientlinkingthetrailingS&P500returntoafirmsdefaultintensityispositiveatconventionalsignificancelevels,andoftheunexpectedsignbyunivariatereasoning.Ofcourse,withmultiplecovariates,thesignneednotbeevidencethatagoodyearinthestockmarketisitselfbadnewsfordefaultrisk.Itcouldalsobethecasethat,afterboomyearsinthestockmarket,afirmsdistancetodefaultoverstatesitsfinancialTheestimate125ofthedependenceoftheunobservabledefaultin-tensitiesonthefrailtyvariablecorrespondstoamonthlyvolatilityofthisfrailtyeffectof12.5%,whichtranslatestoanannualvolatilityof43.3%,whichishighlyeconomicallyandstatisticallysignificant.TableIIIreportstheintensityparametersofthesamemodelafterremovingtheroleoffrailty.Thesigns,magnitudes,andstatisticalsignificanceofthecoefficientsoftheobservablecovariatesaresimilartothosewithfrailty,withtheexceptionofthecoefficientonthe3-monthTreasurybillrate,whichissmallerwithoutfrailtybutremainsstatisticallysignificant.Thereisalsothepotential,withthemomentumeffectsdocumentedbyJegadeeshandTitman(1993)andJegadeeshandTitman(2001),thattrailingreturnisaforecasteroffuturedistanceto FrailtyCorrelatedDefaultTableIIIMaximumLikelihoodEstimatesoftheIntensityParametersintheModelwithoutFrailtyEstimatedasymptoticstandarderrorswerecomputedusingtheHessianmatrixofthelikelihoodfunctionat CoefficientStd.Error 2.0930.121Distancetodefault1.2000.039Trailingstockreturn0.6810.0823-monthT-billrate0.1060.034TrailingS&P500return1.4810.9971.5 C.ThePosterioroftheFrailtyPathInordertointerpretthemodelandapplyittothecomputationofportfoliolossdistributions,wecalculatetheposteriordistributionofthefrailtyprocessgiventheeconometriciansinformation.First,wecomputethe-conditionalposteriordistributionofthefrailtypro-,whereisthefinaldateofoursample.Thisistheconditionaldistri-butionofthelatentfactorgivenallofthehistoricaldefaultandcovariatedatathroughtheendofthesampleperiod.Forthiscomputation,weusetheGibbssamplerdescribedinAppendixB.Figure2showstheconditionalmeanofthescaledlatentfactor,,estimatedastheaverageof5,000samplesofdrawnfromtheGibbssampler.One-standarddeviationbandsareshownaroundtheposteriormean.Weseesubstantialfluctuationsinthefrailtyeffectovertime.Forexample,themultiplicativeeffectofthefrailtyfactorondefaultintensitiesin2001isroughly,orapproximatelythreetimeslargerthanduring1995.WhileFigure2illustratestheposteriordistributionofthefrailtyeffectgivenallinformationavailableatthefinaltimeofthesampleperiod,mostapplicationsofadefaultriskmodelwouldcallfortheposteriordistribu-tionofgiventhecurrentinformation.Forexample,thisistherelevantinformationformeasurementbyabankoftheriskofaportfolioofcorporatedebt.AlthoughthecovariateprocessisGaussian,wealsoobservesurvivalsanddefaults,soweareinasettingoffilteringinnon-Gaussianstatespacemodels,towhichweapplytheforwardbackwardalgorithmofBaumetal.(1970),asexplainedinAppendixC.Figure3comparestheconditionaldensityofthefrailtyeffectattheendofJanuary2000,conditioningon(ineffect,theentiresampleofdefaulttimesandobservablecovariatesupto2004),withthedensityofwhencondi-tioningononly(thedataavailableuptoandincludingJanuary2000).GivenAcomparisonthatisbasedonreplacing)inne Y(t)t]withtheposteriormeanofworksreasonablywellbecausetheJenseneffectsassociatedwiththeexpectationsoffortimesin1995and2001areroughlycomparable. TheJournalofFinance 198019851990199520002005YearLatentFactor Figure2.Frailtyposteriordistribution.Conditionalposteriormean)ofthescaledlatentOrnstein-Uhlenbeckfrailtyvariable,withonestandarddeviationbandsbasedontheconditionalvarianceoftheadditionalinformationavailableattheendof2004,the-conditionaldis-tributionofismoreconcentratedthanthatobtainedbyconditioningononlytheconcurrentlyavailableinformation.TheposteriormeanofgiventheinformationavailableinJanuary2000islowerthanthatgivenallofthedatathrough2004,reflectingthesharpriseincorporatedefaultsin2001aboveandbeyondthatpredictedfromtheobservedcovariatesalone.Figure4showsthepathovertimeofthemean)ofthisposteriordensity.D.PortfolioLossRiskInordertoillustratetheroleofthecommonfrailtyeffectonthetailriskofportfoliolosses,weconsiderthedistributionofthetotalnumberofdefaultsfromahypotheticalportfolioconsistingofall1,813companiesinourdatasetthatwereactiveasofJanuary1998.Wecomputedtheposteriordistribution,conditionalontheinformationavailableforinJanuary1998,ofthetotalnumberofdefaultsduringthesubsequent5years,January1998toDecember2002.Figure5showstheprobabilitydensityofthetotalnumberofdefaultsinthisportfolioforthreedifferentmodels.Allthreemodelshavethesame FrailtyCorrelatedDefault 1 0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 latentfactor Figure3.Conditionalfrailtyposterior,January2000.Conditionalposteriordensityofthescaledfrailtyfactor,,forinJanuary2000,given,thatis,givenalldata(solidline),andgivenonlycontemporaneouslyavailabledatain(dashedline).ThesedensitiesarecalculatedusingtheforwardbackwardrecursionsdescribedinAppendixC.posteriormarginaldistributionforeachfirmsdefaultintensityprocessanddefaulttime,butthejointdistributionofdefaulttimesvariesacrossthethreemodels.Model(a)istheactualfittedmodelwithacommonfrailtyvariable.ForModels(b)and(c),however,weexaminethehypotheticaleffectsofreduc-ingtheeffectoffrailty.ForbothModels(b)and(c),thedefaultintensityischangedbyreplacingthedependenceofontheactualfrailtyprocesswithdependenceonafirm-specificprocessthathasthesamedistributionas.Formodel(b),theinitialconditioniscommontoallfirms,butthefutureevolutionofisdeterminednotbythecommonOU,butratherbyanOUprocessthatisindependentacrossfirms.Thus,Model(b)capturesthecommonsourceofuncertaintyassociatedwiththecurrentposteriordistributionof,buthasnocommonfuturefrailtyshocks.ForModel(c),thehypotheticalfrailtyprocessesofthefirms,,areindependent.Thatis,theinitialconditionisdrawnindependentlyacrossfirmsfromtheposteriordistributionof,andthefutureshockstoarethoseofanOUprocessthatisindependentacrossfirms.Onecanseethattheimpactofthefrailtyeffectontheportfoliolossdistribu-tionissubstantiallyaffectedbothbyuncertaintyregardingthecurrentlevel TheJournalofFinance 1975198019851990199520002005YearLatentFactor Figure4.Filteredfrailty.Conditionalmean)andconditionalone-standarddeviationbandsofthescaledfrailtyvariable,givenonlycontemporaneouslyavailabledata(ofcommonfrailtyinJanuary1998,andalsobycommonfuturefrailtyshockstodifferentfirms.Bothofthesesourcesofdefaultcorrelationareaboveandbeyondthoseassociatedwithexposureoffirmstoobservablemacroeconomicshocks,andexposureoffirmstocorrelatedobservablefirm-specificshocks(es-peciallycorrelatedchangesinleverage).Inparticular,weseeinFigure5thatthetwohypotheticalmodelsthatdonothaveacommonfrailtyvariableassignvirtuallynoprobabilitytotheeventofmorethan200defaultsbetweenJanuary1998andDecember2002.The95thand99thpercentilelossesofModel(c)withcompletelyindependentfrailtyvariablesare144and150defaults,respectively.Model(b),withindependentlyevolvingfrailtyvariableswiththesameinitialvalueinJanuary1998,hasa95thand99thpercentileof180and204defaults,respectively.Theactualnumberofdefaultsinourdatasetduringthistimeperiodwas195.The95thand99thpercentileofthelossdistributionoftheactualestimatedmodel(a),withacommonfrailtyvariable,are216and265defaults,respectively.Therealizednumberofdefaultsduringthiseventhorizon,195,isslightlybe-lowthe91stpercentileofthedistributionimpliedbythefittedfrailtymodel,thereforeconstitutingarelativelysevereevent. FrailtyCorrelatedDefault 150200250300NumberofdefaultsProbabilitydensity Figure5.Conditional5-yearportfoliolossdistributionin1998.Theconditionalprobabilitydensity,giveninJanuary1998,ofthetotalnumberofdefaultswithin5yearsfromtheportfolioofallactivefirmsatJanuary1998,in(a)thefittedmodelwithfrailty(solidline),(b)ahypotheticalmodelinwhichthecommonfrailtyprocessisreplacedwithfirm-by-firmfrailtyprocesseswithinitialconditionattimeequaltothatof,butwithcommonBrownianmotiondrivingfrailtyforallfirmsreplacedwithfirm-by-firmindependentBrownianmotions(dashedline),and(c)ahypotheticalmodelinwhichthecommonfrailtyprocessisreplacedwithfirm-by-firmindependentfrailtyprocesseshavingthesameposteriorprobabilitydistributionasline).ThedensityestimatesareobtainedwithaGaussiankernelsmoother(bandwidthequaltofive)appliedtoaMonteCarlo-generatedempiricaldistribution.V.AnalysisofModelFitandSpecificationThissectionexaminestheabilityofourmodeltosurvivetestsofitsfit.Wealsoexamineitsout-of-sampleaccuracy,anditsrobustnesstosomealternativespecifications.A.FrailtyversusNoFrailtyInordertojudgetherelativefitofthemodelswithandwithoutfrailty,wedonotusestandardtests,suchasthechi-squaretest.Instead,wecomparethemarginallikelihoodsofthemodels.Thisapproachdoesnotrelyonlarge-sampledistributiontheoryandhastheintuitiveinterpretationofattachingpriorprobabilitiestothecompetingmodels. TheJournalofFinanceSpecifically,weconsideraBayesianapproachtocomparingthequalityoffitofcompetingmodelsandassumepositivepriorprobabilitiesforthetwomodels(themodelwithoutfrailty)and(themodelwithacommonfrailtyvariable).Theposterioroddsratiois (noFW,D)F(öF,ö FW,D) noFnoFnoF ,(5)denotetheMLEandthelikelihoodfunctionforacertain,respectively.Plugging(3)into(5)gives (noFW,D)(öFW)F(ö FW,D) noFnoFnoF (noF)F(ö FW,D) noFnoF usingthefactthatthetime-seriesmodelforthecovariateprocessisthesameinbothmodels.Thefirstfactorontheright-handsideof(6)issometimesknownastheBayesfactor.FollowingKassandRaftery(1995)andEraker,Johannes,andPolson(2003),wefocusonthesizeofthestatisticgivenbytwicethenaturallogarithmoftheBayesfactor,whichisonthesamescaleasthelikelihoodratioteststatistic.Avalueforbetween2and6providespositiveevidence,avaluebetween6and10strongevidence,andavaluelargerthan10providesverystrongevidenceforthealternativemodel.Thiscriteriondoesnotnecessarilyfavormorecomplexmodelsduetothemarginalnatureofthelikelihoodfunctionsin(6).SmithandSpiegelhalter(1980)discussthepenalizingnatureoftheBayesfactor,sometimesreferredtoasthefullyautomaticOccamsrazor.Inourcase,theoutcomeoftheteststatisticis22.6.Inthesenseofthisapproachtomodelcomparison,weseestrongevidenceinfavorofincludingafrailtyvariable.B.MisspeciÞcationofProportionalHazardsAcomparisonofFigures1and2showsthatthefrailtyeffectisgenerallyhigherwhendefaultsaremoreprevalent.Inlightofthis,onemightsuspectmisspecificationoftheproportionalhazardsintensitymodel(1),whichwouldautomaticallyinduceameasuredfrailtyeffectifthetrueintensitymodelhasahigher-than-proportionaldependenceondistancetodefault,whichisbyfarthemosteconomicallyandstatisticallysignificantcovariate.Iftheresponseofthetruelogintensitytovariationindistancetodefaultisfasterthanlinear,thentheestimatedlatentvariableinourcurrentformulationwouldbehigherwhendistancestodefaultarewellbelownormal,asin1991and2003.InanUnfortunately,theBayesfactorcannotbeusedforcomparingthemodelwithfrailtytothemodelwithfrailtyandunobservedheterogeneity,becauseforthelattermodelevaluatingthelike-lihoodfunctioniscomputationallyprohibitivelyexpensive. FrailtyCorrelatedDefaultInternetAppendix,weprovideanextensionofthemodelthatincorporatesnon-parametricdependenceofdefaultintensitiesondistancetodefault.Theresultsindicatethattheproportionalhazardsspecificationisunlikelytobeasignificantsourceofmisspecificationinthisregard.Theresponseoftheesti-matedlogintensitiesisroughlylinearindistancetodefault,andtheestimatedposteriorofthefrailtypathhasroughlytheappearanceshowninFigure2.C.UnobservedHeterogeneityItmaybethatasubstantialportionofthedifferencesacrossfirmsrisksisduetoheterogeneityinthedegreetowhichdifferentfirmsaresensitivetothecovariates,perhapsthroughadditionalfirm-specificomittedvariables.Failuretoallowforthiscouldresultinbiasedandinefficientestimation.Weconsideranextensionofthemodelbyintroducingafirm-specificheterogeneityforfirm,sothatthecomplete-information()defaultintensityoffirmisoftheformareindependentlyandidenticallygamma-distributeddomvariablesthatarejointlyindependentoftheobservablecovariatesthecommonfrailtyprocessFixingthemeanoftheheterogeneityfactortobeonewithoutlossofgen-erality,wefindthatmaximumlikelihoodestimationdoesnotpindownthevarianceoftoanyreasonableprecisionwithourlimitedsetofdata.Wean-ticipatethatfarlargerdatasetswouldbeneeded,giventhealreadylargedegreeofobservableheterogeneityandthefactthatdefaultis,onaverage,relativelyunlikely.Intheend,weexaminethepotentialroleofunobservedheterogene-ityfordefaultriskbyfixingthestandarddeviationofat0.5.Itiseasytocheckthatthelikelihoodfunctionisagaingivenby(3),whereinthiscasethefinalexpectationiswithrespecttotheproductofthedistributionsofInanInternetAppendix,weshowthatourgeneralconclusionsregardingtheeconomicsignificanceofthecovariatesandtheimportanceofincludingatime-varyingfrailtyvariableremaininthepresenceofunobservedhetero-geneity.Moreover,theposteriormeanpathofthetime-varyinglatentfactorisessentiallyunchanged.ThisInternetAppendixisavailableonlineintheSupplementsandDataSetssectionathttp://www.afajof.org/supplements.asp.PicklesandCrouchery(1995)showinsimulationstudiesthatitisrelativelysafetomakeconcreteparametricassumptionsaboutthedistributionofstaticfrailtyvariables.Inferenceisexpectedtobesimilarwhetherthefrailtydistributionismodeledasgamma,lognormal,orsomeotherparametricfamily,butforanalyticaltractabilitywechoosethegammadistribution. TheJournalofFinanceD.ParameterUncertaintyUntilthispoint,ouranalysisisbasedonmaximumlikelihoodestimationofthefrailtymeanreversionandvolatilityparameters,.Uncertaintyregardingtheseparameters,inaBayesiansense,couldleadtoanincreaseinthetailriskofportfoliolosses,whichweinvestigatenext.Wearealsointerestedinexaminingourabilitytolearntheseparameters,inaBayesiansense.AmongotherimplicationsofourBayesiananalysis,wewillseethatthemeanreversionisparticularlyhardtotiedown.Thestationaryvarianceofthefrailtyvariable Motivatedbythehistoricalbehavioroftheposteriormeanofthefrailty,wetakethepriordensityofthestationarystandarddeviation,,tobeGammadistributedwithameanof0.5andastandarddeviationof0.25.ThepriordistributionforthemeanreversionrateisalsoassumedtobeGamma,withameanoflog236(whichcorrespondstoahalf-lifeof3yearsforshockstothefrailtyvariable)andastandarddeviationoflog272.Thejointpriordensityofisthereforeoftheform 2 3exp8 2 3exp 144 log2Figure6showsthemarginalposteriordensitiesofthevolatilityandmeanre-versionparametersofthefrailtyvariable.Figure7showstheirjointposteriordensity.Thesefiguresindicateconsiderableposterioruncertaintyregardingtheseparameters.Fromtheviewpointofsubjectiveprobability,estimatesofthetailriskoftheportfoliolossdistributionthatareobtainedbyfixingthesecommonfrailtyparametersattheirmaximumlikelihoodestimatesmightsig-nificantlyunderestimatetheprobabilityofcertainextremeevents.Althoughparameteruncertaintyhasaminorinfluenceontheportfoliolossdistributionatintermediatequantiles,Figure8revealsamoderateimpactofparameteruncertaintyontheextremetailsofthedistribution.Forexample,whenfixingthefrailtyparametersattheirmaximumlikelihoodesti-mates,the99thpercentileoftheportfoliodefaultdistributionis265defaults.Takingposteriorparameteruncertaintyintoaccount,thisquantilerisesto275defaults.E.DoTrailingDefaultsProxyforUnobservedCovariates?TableIVreportsthefittedmodelcoefficientsforamodelwithoutfrailty,butwithtrailing1-yearaverageyearlydefaultrateasacovariates.Weemphasizethatthismodelviolatestheassumptionsthatjustifyourlikelihoodfunction,fortheobviousreasonthatdefaultscannotbeindependentacrossdifferentfirmsconditionalonthepathofthecovariateprocessifweincludeaveragerealized FrailtyCorrelatedDefault 0.020.040.060.080.10.150.20.250.30.35FrailtyvolatilityFrailtymeanreversionDensityofgivenDensityofgiven Figure6.Marginalfrailtyparameterposteriordistribution.Marginalposteriordensities,,ofthefrailtyvolatilityparameterandthefrailtymeanreversionrateintheBayesianapproachofSectionV.D. 0.080.10.120.140.160.180.20.220.24FrailtyvolatilityFrailtymeanreversion Figure7.Jointfrailtyparameterposteriordistribution.Isocurvesofthejointposteriordensity,given,ofthefrailtyvolatilityparameterandmeanreversionrate TheJournalofFinance 150200250300350400450NumberofdefaultsProbabilitydensity(logscale) Figure8.Portfoliolossdensitycomparison.Density,onalogarithmicscale,ofthenumberofdefaultsintheportfoliowhenfixingthevolatilityandmeanreversionparameterattheirMLEestimates(dashedline),andintheBayesianestimationframework(solidline).Thedensityes-timateswereobtainedbyapplyingaGaussiankernelsmoother(withabandwidthof10)totheMonteCarlo-generatedempiricaldistribution.TableIVMaximumLikelihoodEstimatesoftheIntensityParametersintheModelwithoutFrailtybutwithTrailing1-YearAverageYearlyDefaultRateasaCovariateEstimatedasymptoticstandarderrorswerecomputedusingtheHessianmatrixofthelikelihoodfunctionat CoefficientStd.Error 2.3640.955Distancetodefault1.1890.052Trailingstockreturn0.6780.3013-monthT-billrate0.0860.135TrailingS&P500return1.7661.0011.8Trailing1-yeardefaultrate7.1541.0007.2 FrailtyCorrelatedDefaultdefaultratesasacovariate.Itmaybe,however,thattrailingdefaultrateswillproxyforanimportantsourceofdefaultriskcovariationthatisotherwiseunobserved,andreducetherelativeimportanceoffrailty.Thesigns,magnitudes,andstatisticalsignificanceofthecoefficientsontheobservablecovariatesaresimilartothoseofthemodelthatdoesnotincludethetrailingdefaultrateasacovariate.Thetrailingdefaultrateplaysamoderatelyimportantauxiliaryrole.Forexample,fixingothercovariates,ifthetrailingaveragedefaultrateweretoincreaseby1%peryear,alargebutplausibleshiftgivenourdataset,themodelestimatesimplyaproportionalincreaseintheconditionalmeanarrivalratesofallfirmsofabout7.1%.Thiswouldcauseashiftinthedefaultintensityofaparticularfirmfrom,say,2%toabout2.14%.Forthereasondescribedabove(thedistributionoftrailingdefaultisanen-dogenouspropertyofthedefaultintensitymodel),wecannotexaminethein-fluenceoftrailingdefaultontheposteriorofthefrailtyprocess.Weareable,though,toseewhetherincludingtrailingdefaultratesisaneffectivealterna-tivetofrailtyincapturingthedistributionofportfoliotaillosses.InthesenseofthetestsdescribedinSectionV.F,itisnot.F.PortfolioDefaultQuantileTestsWeturntotherealismwithwhichthefrailty-basedmodelestimatesthequantilesofportfoliodefaults.Wewillfocusonthequantilesoftheconditionaldistributionsofthetotalnumberofdefaultsover1-yearhorizons,fromtheportfoliosofallactivefirmsatthebeginningoftherespectiveyears.Intermsoffirm-by-firmdefaultprediction,Duffie,Saita,andWang(2007)showthattheobservablecovariatesofourbasicmodelalreadyprovidethehighestout-of-sampleaccuracyratiosdocumentedinthedefaultpredictionlit-erature.Allowingforfrailtydoesnotaddsignificantlytofirm-by-firmdefaultprediction.InanInternetAppendix,weshowthataccuracyratioswithfrailtyareessentiallythesameasthosewithout.Likewise,accuracyratiosareroughlyunaffectedbyaddingthetrailingaveragedefaultrateasacovariate.Atthelevelofindividualfirms,mostofourabilitytosortfirmsaccordingtodefaultproba-bilityiscomingfromthefirm-levelcovariates,particularlydistancetodefault.Thecoefficientsonthesevariablesarerelativelyinsensitivetothealternativespecificationsthatwehaveexamined.Ourmainfocusisthedistributionofportfoliolosses.Inordertogaugetheabilityofourmodeltocapturethisdistribution,weproceedasfollows.Atthebeginningofeachyearbetween1980and2003,wecalculateforthecompaniesinourdatasetthemodel-implieddistributionofthenumberofdefaultsduringthesubsequent12months.Wethendeterminethequantileoftherealizednumberofdefaultswithrespecttothisdistribution.Figure9showsthesequantilesfor(i)ourbenchmarkmodelwithfrailty,(ii)ourbenchmarkmodeladjustedbyremovingfrailty,and(iii)themodelwithoutfrailtybutincludingthetrailing1-yearaveragedefaultrateasanadditionalco-variate.Thequantilesofthetwomodelswithoutfrailtyseemtoclusteraroundzeroandone,whichsuggeststhatthesemodelsunderestimatetheprobabilities TheJournalofFinance 19851990YearQuantile Figure9.Realizedportfoliolossquantiles.Quantileoftherealizednumberofdefaultswithrespecttothepredicted1-yearportfoliolossdistributionasimpliedbythemodelwiththefrailtyvariable(crosses),withoutthefrailtyvariable(circles),andwithoutthefrailtyvariablebutwiththetrailing1-yearaveragedefaultrateasthecovariate(triangles).ofunusuallylowportfoliolossesandofunusuallyhighportfoliolosses.Forex-ample,in1994therealizednumberofdefaultsliesbelowtheestimated1stpercentileoftheportfoliodefaultdistributionforthemodelwithoutfrailty,whilein1990and2001therealizednumberofdefaultsliesabovethe99.9thpercentileoftheestimateddistribution.Forthemodelthatalsoincludesthetrailing1-yearaveragedefaultrateasacovariate,thesequantilesareonlyslightlylessextreme.Ontheotherhand,thequantilesforthemodelwithfrailtyaredistributedrelativelyevenlyintheunitinterval,indicatingamoreaccurateassessmentofcreditriskontheportfoliolevel.Moreover,theforecastingerrorsforthetwomodelswithoutfrailtytendtobeseriallycorrelatedovertime,whichismostevidentfortheperiods1994to1997aswellas2000to2003.Thenullhypothesisofnoserialcorrelationinthequantilesisindeedrejectedatthe1%significancelevelforthemodelwithoutfrailty(-valueof0.004).Forthemodelwithoutthefrailtyvariablebutwiththetrailing1-yearaveragedefaultrateasacovariate,thenullhypothesisofnoserialcorrelationinthequantilescanstillberejectedatthe5%significancelevel(-valueof0.019).Ontheotherhand,witha-valueof0.62,thenullhypothesisofnoserialcorrelationinthequantilescannotberejectedforthemodelwithfrailty. FrailtyCorrelatedDefaultVI.ConcludingRemarksWefindsubstantialevidenceamongU.S.publiccorporatesofacommonunob-servedsourceofdefaultrisk,relativetotheinformationprovidedbyapowerfulsetofobservablefactorsforpredictingindividualfirmdefaults.Accordingtoourestimates,failuretoallowforunobservedfactorsinthissettingleadstodramaticdownwardbiasesinvalue-at-riskestimatesforlargecorporatedebtportfolios.Ourresultshaveimportantimplicationsfortheriskmanagementofportfo-liosofcorporatedebt.Forexample,asbackingfortheperformanceoftheirloanportfolios,banksretaincapitalatlevelsdesignedtowithstanddefaultcluster-ingatextremelyhighconfidencelevels,suchas99.9%.Somebanksdosoonthebasisofmodelsinwhichdefaultcorrelationisassumedtobecapturedbycommonriskfactorsdeterminingconditionaldefaultprobabilities,asinVa-sicek(1987)andGordy(2003).If,however,defaultsaremoreheavilyclusteredintimethancurrentlycapturedinthesedefaultriskmodels,thensignificantlygreatercapitalmightberequiredinordertosurvivedefaultlosseswithhighconfidencelevels.Anunderstandingofthesourcesanddegreeofdefaultcluster-ingisalsocrucialfortheratingandriskanalysisofstructuredcreditproductsthatareexposedtocorrelateddefaults,suchasCDOsandoptionsonportfoliosofdefaultswaps.Whilewedonotaddressthepricingofcreditriskinthispa-per,frailtycouldplayausefulroleinthemarketvaluationofrelativelyseniortranchesofCDOs,whichsufferalossofprincipalonlywhenthetotaldefaultlossesoftheunderlyingportfolioofbondsisextreme.WeestimateourmodelondataforU.S.firmsbetweenJanuary1979andMarch2004.Wefindthatrealizedcorporatedefaultratesvaryovertimewellbeyondlevelsthatcanbeexplainedbyamodelthatincludesonlyourobserv-ablecovariates.Ingoodness-of-fitandquantiletests,themodelswithoutfrailtythatweexaminearerejectedandsignificantlyunderestimatetheprobabilityofextremepositiveaswellasnegativeeventsinportfoliosofcorporatecredits.Forourdataandmodel,weestimatethatunobservedfrailtyhasanimpactondefaultintensitiesthataddsaproportionalannualvolatilityofroughly40%.Theestimatedrateofmeanreversionofthefrailtyfactorisapproximately1.8%permonth,althoughthismeanreversionrateisdifficulttopindownwiththeavailabledata.Ourmethodologycanbeappliedtoothersituationsinwhichacommonun-observablefactorissuspectedtoplayanimportantroleinthetime-variationofarrivalsforagivenclassofevents,forexample,operationalriskevents,mergersandacquisitions,ormortgageprepaymentsanddefaults.AppendixA:ParameterEstimationThisappendixprovidesourestimationmethodology.Theparametervectordeterminingthetime-seriesmodelfortheobservablecovariateprocessspecifiedandestimatedinDuffie,Saita,andWang(2007).Thismodel,sum-marizedinanInternetAppendix,isvector-autoregressiveGaussian,withanumberofstructuralrestrictionschosenforparsimonyandtractability.We TheJournalofFinancefocushereontheestimationoftheparametervectorofthedefaultintensityWeuseavariantoftheEMalgorithm(seeDemptser,Laird,andRubin(1977)),aniterativemethodforthecomputationoftheMLEofparametersofmodelsinvolvingmissingorincompletedata.SeealsoCappe,Moulines,anden(2005),whodiscussEMinthecontextofhiddenMarkovmodels.Inmanypotentialapplications,explicitlycalculatingtheconditionalexpectationrequiredintheofthealgorithmmaynotbepossible.Nevertheless,theexpectationcanbeapproximatedbyMonteCarlointegration,whichgivesrisetothestochasticEMalgorithm,asexplained,forexample,byCeleuxandDiebolt(1986)andNielsen(2000),ortotheMonteCarloEMalgorithm(WeiandTanner(1990)).Maximumlikelihoodestimationoftheintensityparametervectorthefollowingsteps:1.Initializeanestimateof)at05,0),whereisthemaximumlikelihoodestimatorofinthemodelwithoutfrailty,whichcanbeobtainedbymaximizingthelikelihoodfunction(2)bystandardmethodssuchastheNewtonRaphsonalgorithm.2.(E-step)Giventhecurrentparameterestimateandtheobservedco-variateanddefaultdata,respectively,drawindependentsam-plepathsfromtheconditionaldensity)ofthelatentOUfrailtyprocess.WedothiswiththeGibbssamplerdescribedinAppendixB.Welet))(A1),(A2)denotesexpectationwithrespecttotheprobabilitymeasureassociatedwithaparticularparametervector.Thisexpectedcomplete-dataloglikelihoodintermediatequantity,asitiscommonlycalledintheEMliterature,canbeapproximatedwiththesamplepathsgeneratedbytheGibbssampleras 3.(M-step)Maximize)withrespecttotheparametervector,forexample,byNewtonRaphson.Themaximizingchoiceofisthenewparameterestimate4.Replace1,andreturntoStep2,repeatingtheE-stepandtheM-stepuntilreasonablenumericalconvergenceisachieved.Onecanshow(Demptser,Laird,andRubin(1977)orGelmanetal.(2004))).Thatis,theobserveddatalikelihood FrailtyCorrelatedDefault(3)isnon-decreasingineachstepoftheEMalgorithm.Underregularitycon-ditions,theparametersequencethereforeconvergestoatleastalocalmaximum(seeWu(1983)forapreciseformulationintermsofstationar-itypointsofthelikelihoodfunction).Nielsen(2000)givessufficientconditionsforglobalconvergenceandasymptoticnormalityoftheparameterestimates,althoughtheseconditionsareusuallyhardtoverify.Aswithmanymaximiza-tionalgorithms,asimplewaytomitigatetheriskthatonemissestheglobalmaximumistostarttheiterationsatmanypointsthroughouttheparameterspace.Underregularityconditions,theFisherandLouisidentitiesimplythatTheHessianmatrixoftheexpectedcomplete-datalikelihood(A2)canthere-forebeusedtoestimateasymptoticstandarderrorsfortheMLEparameterestimators.Wealsoestimateageneralizationofthemodelthatincorporatesunobservedheterogeneity,usinganextensionofthisalgorithmthatisprovidedintheInternetAppendix.AppendixB:ApplyingtheGibbsSamplerwithFrailtyAcentralquantityofinterestfordescribingandestimatingthehistoricaldefaultdynamicsistheposteriordensity)ofthelatentfrailty.Thisisacomplicatedhigh-dimensionaldensity.Itisprohibitivelycomputationallyintensivetodirectlygeneratesamplesfromthisdistribution.Nevertheless,MCMCmethodscanbeusedforexploringthisposteriordistribu-tionbygeneratingaMarkovchainover,denoted,whoseequilibriumdensityis).Samplesfromthejointposteriordistributioncanthenbeusedforparameterinferenceandforanalyzingthepropertiesofthefrailty.Forafunction)satisfyingregularityconditions,theMonteCarloestimateofoff(Y)W,D, ] f(y)pY(yW,D, )dy(B1)isgivenby Underconditions,theergodictheoremforMarkovchainsguaranteesthecon-vergenceofthisaveragetoitsexpectationas.OnesuchfunctionofSee,forexample,Proposition10.1.6ofCappe,Moulines,andRyden(2005). TheJournalofFinanceinterestistheidentity,sothatthatf(Y)W,D, ]E[YW,D, ] E(YtT):0 t T ,theposteriormeanofthelatentOUfrailtyprocess.ThelinchpintoMCMCisthatthejointdistributionofthefrailtypathcanbebrokendownintoasetofconditionaldistributions.AgeneralversionoftheCliffordHammersley(CH)Theorem(HammersleyandClifford(1970)andBesag(1974))providesconditionsunderwhichasetofconditionaldistributionscharacterizesauniquejointdistribution.Forexam-ple,inoursetting,theCHTheoremindicatesthatthedensity)isuniquelydeterminedbytheconditionaldistributions:usingthenaturallysuggestedinterpretationofthisinformalnotation.WerefertheinterestedreadertoRobertandCasella(2005)foranextensivetreatmentofMonteCarlomethods,aswellasJohannesandPolson(2003)foranoverviewofMCMCmethodsappliedtoproblemsinfinancialeconomics.Inourcase,theconditionaldistributionofatasinglepointintimegiventheobservablevariables()andgiven,issomewhattractable,asshownfurther.ThisallowsustousetheGibbssampler(GemanandGeman(1984)orGelmanetal.(2004))todrawwholesamplepathsfromtheposteriordistributionofbythefollowingalgorithm:1.Initialize0for2.For1,2,,drawanewvalueoffromitsconditionaldistribution.Foramethod,seelater.3.StorethesamplepathandreturntoStep2untilthedesirednumberofpathshasbeensimulated.WeusuallydiscardthefirstseveralhundredpathsasabecauseinitiallytheGibbssamplerhasnotapproximatelyconvergedtotheposteriordistributionofItremainstoshowhowtosamplefromitsconditionaldistributiongiven.Recallthat)denotesthecomplete-informationlikelihoodWeusevariousconvergencediagnostics,suchastraceplotsofagivenparameterasafunctionofthenumberofsamplesdrawn,toassurethattheiterationshaveproceededlongenoughforapproximateconvergenceandtoassurethatourresultsdonotdependmarkedlyonthestartingvaluesoftheGibbssampler.SeeGelmanetal.(2004),Chapter11.6,foradiscussionofvariousmethodsforassessingconvergenceofMCMCmethods. FrailtyCorrelatedDefaultoftheobservedcovariatesanddefaults,where).For0,wehave wherewerepeatedlymakeuseofthefactthattermsnotinvolvingarecon-FromtheMarkovpropertyitfollowsthattheconditionaldistributionofisthesameastheconditionaldistributionof.Therefore, p(Yt1,Yt1 )p(Yt1,Yt,Yt1 )p(Yt1,Yt )p(Yt1Yt1,Yt, )p(Yt1,Yt ) )istheone-steptransitiondensityoftheOUprocess(4).Hence,Equation(B3)determinesthedesiredconditionaldensityofinanimplicitform.Althoughitisnotpossibletodirectlydrawsamplesfromthisdistribution,wecanemploytheRandomWalkMetropolisalgorithm(MetropolisandUlam(1949),andHastings(1970)).Weusethepro-posaldensity,4),thatis,wetakethemeantobethevalueoffromthepreviousiterationoftheGibbssampler,andthevariancetobetwicethevarianceofthestandardBrownianmotionAlternatively,wecoulddiscretizethesamplespaceandapproximatetheconditionaldistri-butionbyadiscretedistribution,anapproachcommonlyreferredtoastheGriddyGibbsmethod(Tanner(1998)).However,theMetropolisHastingsalgorithmisusuallyacoupleoftimesfasterincasesinwhichtheconditionaldensityisnotknownexplicitly. TheJournalofFinance.TheMetropolisHastingssteptosampleinthethiterationoftheGibbssamplerthereforeworksasfollows:1.Drawacandidate,4).2.Compute 3.Drawwiththeuniformdistributionon(0,1),andletotherwise.Thechoiceoftheacceptanceprobability(B4)ensuresthattheMarkovchainsatisfiesthedetailedbalanceequationdenotesthedensityofanormaldistributionwithmean.Moreover,)asitsstationarydistribution(see,forexample,Theorem7.2inRobertandCasella(2005)).AppendixC:ForwardÐBackwardFilteringforFrailtydenotethesetoffirmsthatarealiveat,and1):bethesetoffirmsthatdefaultedattime.Adiscrete-timeapproximationofthecomplete-informationlikelihoodoftheobservedsurvivalsanddefaultsattimeThenormalizedversionoftheforwardbackwardalgorithmallowsustocalcu-latethefiltereddensityofthelatentOrnsteinUhlenbeckfrailtyvariablebytherecursion Wecalculatetheconditionaldensityforvariouspointsintimenumericallytoassurethatitdoesnothaveanyfattails.ThiswasindeedthecasesothatusinganormalproposaldensitydoesnotjeopardizetheconvergenceoftheMetropolisHastingsalgorithm.SeeMengersenandTweedie(1996)fortechnicalconditions. 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