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TheMythoftheCreditSpreadPuzzlePeterFeldhutteryLondonBusinessSchoolStephenSchaeferzLondonBusinessSchoolMay12,2017AbstractWeaskwhetherastandardstructuralmodel(BlackandCox(1976))isabletoexplaincreditspreadsoncorporatebondsand,incontrasttomuchoftheliterature,we ndthatthemodelmatchesthelevelofinvestmentgradespreadswell.Modelspreadsforspeculativegradedebtaretoolowandwe ndthatbondilliquiditycontributestothisunderpricing.Ouranalysismakesuseofanewapproachforcalibratingthemodeltoaveragehistoricaldefaultratesandweshowviasimulationthatthisleadstomuchmorepreciseestimatesofinvestmentgradedefaultprobabilities.Keywords:Creditspreadpuzzle,Structuralmodels,Black-Coxmodel,Corporatebondspreads,Defaultprobabilities;JEL:C23;G12 WearegratefultotheQ-GroupfortheJackTreynorPrize.WearealsogratefulforveryvaluablecommentsfromtheeditorandananonymousrefereeandsuggestionsreceivedfromAntjeBerndt(discussant),HarjoatBhamra(discussant),HuiChen(discussant),MikeChernov,DarrellDue,StefanoGiglio(discussant),JeanHelwege(discussant),RalphKoijen,DavidLando,ErikLoualiche(discussant),GustavoManso(discussant),RobertMerton,Jens-DickNielsen,LasseHejePedersen,ScottRichardson,BatchimegSambalaibat(discussant),andseminarparticipantsattheQ-GroupSpring2016meeting,NBERAssetPricing2015conference,UtahWinterFinanceConfer-ence2015,EFA2015,HKUSTFinanceSymposium2014,WFA2014,SFSCavalcade2014,ESSFM2014inGerzensee,FifthRiskManagementConference2014,UNC'sRoundtable2013,UCLAAn-dersonSchoolofManagement,UCSanDiego,FinancialConductAuthority,AQR,LondonBusinessSchool,CassBusinessSchool,ViennaGraduateSchoolofFinance,StockholmSchoolofEconomics,TilburgUniversity,DuisenburgSchoolofFinanceAmsterdam,NHHBergen,RotterdamSchoolofManagement,UniversityofSouthernDenmark,CopenhagenBusinessSchool,BrunelUniversity,SurreyBusinessSchool,HenleyBusinessSchool,BankofEngland,andUniversityofCambridge.WeareparticularlygratefulfortheresearchassistanceprovidedbyIshitaSen.yLondonBusinessSchool,Regent'sPark,London,NW14SA,pfeldhutter@london.eduzLondonBusinessSchool,Regent'sPark,London,NW14SA,sschaefer@london.edu 1.IntroductionThestructuralapproachtocreditrisk,pioneeredbyMerton(1974)andothers,representstheleadingtheoreticalframeworkforstudyingcorporatedefaultriskandpricingcorporatedebt.Whilethemodelsareintuitiveandsimple,manystudies ndthat,oncecalibratedtomatchhistoricaldefaultandrecoveryratesandtheequitypremium,theyfailtoexplainthelevelofactualinvestmentgradecreditspreads,aresultreferredtoasthe\creditspreadpuzzle".Papersthat ndacreditspreadpuzzletypicallymakeuseofMoody'shistoricaldefaultrates,measuredoveraperiodofaround30yearsandstartingfrom1970.1Ourstartingpointistoshowthattheappearanceofacreditspreadpuzzledependsstronglyontheperiodoverwhichhistoricaldefaultratesaremeasured.Forexample,Chen,Collin-Dufresne,andGoldstein(2009)usedefaultratesfrom1970-2001and ndBBB-AAAmodelspreadsof57-79bps(dependingonmaturity),whicharesubstantiallylowerthanhistoricalspreadsof94-102bps.If,instead,weuseMoody'sdefaultratesfor1920-2001modelspreadsare91-112bpsandinlinewithhistoricalspreads.Usingsimulations,wedemonstratetwokeypointsabouthistoricaldefaultrates.The rstisthat,oversampleperiodsofaround30yearsthataretypicallyusedintheliterature,thereisalargesamplingerrorintheobservedaveragerate.Forexample,ifthetrue10-yearBBBcumulativedefaultprobabilitywere5.09%2,a95%con dencebandfortherealizeddefaultratemeasuredover31yearswouldbe[115%1278%].Intuitively,thelargesampleerrorarisesbecausedefaultsarecorrelatedand31yearsofdataonlygiverisetothreenon-overlapping10-yearintervals.Asaresultofthelargesamplingerror,whenhistoricaldefaultratesareusedasestimatesofex-antedefaultprobabilities,thedi erencebetweenactualspreadsandmodelspreadsneedstobeverylarge{muchlarger,forexample,thanthatfoundfortheBBB-AAAspreadmentionedabove{tobeinterpretedasstatisticallysigni cantevidenceagainstthemodel.Second,andequallycrucial,distributionsofaveragehistoricalinvestmentgradedefault 1SeeforexampleLeland(2006),Cremers,Driessen,andMaenhout(2008),Chen,Collin-Dufresne,andGoldstein(2009)andHuangandHuang(2012)2ThisisthenumberreportedbyMoody'sfor1970-2001.1 ratesarehighlyskewed.Mostofthetimeweseefewdefaultsbutoccasionallyweseemanyandthismeansthatthereisahighprobabilityofobservingaratethatisbelowtheactualmean.Skewnessislikelytoleadtotheconclusionthatastructuralmodelunderpredictsinvestmentgradespreadsevenifthemodeliscorrect.Thereasonforthepresenceofskewnessisthatdefaultsarecorrelatedacross rmsasaresultofthecommondependenceofindividual rmvaluesonsystematic(\market")shocks.Toseewhycorrelationleadstoskewness,wecanthinkofalargenumberof rmswithadefaultprobability(oversomeperiod)of5%andwheretheirdefaultsareperfectlycorrelated.Inthiscasewewillseenodefaults95%ofthetime(and100%defaults5%ofthetime)sotherealizeddefaultratewillunderestimatethedefaultprobability95%ofthetime.Iftheaveragedefaultrateiscalculatedoverthreeindependentperiods,therealizeddefaultratewillstillunderestimatethedefaultprobability0953=8574%ofthetime.Weproposeanewapproachtoestimatedefaultprobabilities.Insteadofusingthehis-toricaldefaultrateatasinglematurityandratingasanestimateofthedefaultprobabilityforthissamematurityandrating,weuseawidecrosssectionofdefaultratesatdi erentmaturitiesandratings.WeusetheBlackandCox(1976)modelandwhattiesdefaultprob-abilitiesfor rmswithdi erentratingstogetherinthemodelisthatweassumethattheywill,nonetheless,havethesamedefaultboundary.(Thedefaultboundaryisthevalueofthe rm,measuredasafractionofthefacevalueofdebt,belowwhichthe rmdefaults).Thisisreasonablesince,ifthe rmweretodefault,thereisnoobviousreasonwhythedefaultboundarywoulddependontheratingthe rmhadheldpreviously.Weshowinsimulationsthatourapproachresultsinmuchmorepreciseandlessskewedestimatesofinvestmentgradedefaultprobabilities.Fortheestimated10-yearBBBdefaultprobability,forexample,thestandarddeviationandskewnessusingthenewapproachareonly16%and4%respectivelyofthoseusingtheexistingapproach.Theimprovedprecisionispartlytheresultofthefactthatwecombineinformationacross20maturitiesand7ratings.But,toasigni cantextent,itistheresultofcombiningdefaultinformationoninvestmentgradeandhighyielddefaults.Becausedefaultsoccurmuchmorefrequentlyinhighyielddebt,these rmsprovidemoreinformationonthelocationofthedefaultboundary.Sincetheboundaryiscommontoinvestmentgradeandhighyielddebt,whenwecombineinvestment2 gradeandhighyielddefaultdata,we\import"theinformationonthelocationofthedefaultboundaryfromhighyieldtoinvestmentgradedebt.Thereductioninskewnessisalsotheresultofincludingdefaultratesthataresigni cantlyhigherthanthoseforBBBdebt.Whilelowinvestmentgradedefaultratesproduceapositiveskewinthedistributionofdefaults,adefaultrateof50%producesasymmetricdistributionand,forevenhigherdefaultrates,theskewisactuallynegative.WeuseourestimationapproachandtheBlack-Coxmodeltoinvestigatespreadsovertheperiod1987-2012.Ourdatasetconsistsof256,698corporatebondyieldspreadstotheswaprateofnoncallablebondsissuedbyindustrial rmsandismoreextensivethanthosepreviouslyusedintheliterature.OurimplementationoftheBlack-Coxmodelisnewtotheliteratureinthatitallowsforcross-sectionalandtime-seriesvariationin rmleverageandpayoutratewhilematchinghistoricaldefaultrates.Wethenapplyourproposedestimationapproachandestimatethedefaultboundarysuchthataveragemodel-implieddefaultprobabilitiesmatchaveragehistoricaldefaultratesfrom1920-2012.IncalibratingthedefaultboundaryweuseaconstantSharperatioandmatchtheequitypremium,butoncewehaveimpliedoutthesingle rm-widedefaultboundaryparameterwecompute rmandtime-speci cspreadsusingstandard\risk-neutral"pricingformulae.We rstexplorethedi erencebetweenaveragespreadsintheBlack-Coxmodelandactualspreads.Theaveragemodelspreadacrossallinvestmentgradebondswithamaturitybetween3and20yearsis111bpswhiletheaverageactualspreadis92bps.Acon dencebandforthemodelspreadthattakesintoaccountuncertaintyindefaultprobabilitiesis[88bps;128bps],thusthereisnostatisticaldi erencebetweenactualandmodelinvestmentspreads.Forspeculativegradebonds,theaveragemodelspreadis382bpswhiletheactualspreadis544bpsandherethedi erenceisstatisticallyhighlysigni cant.Wealsosortbondsaccordingtotheactualspreadand ndthatactualandmodel-impliedspreadsaresimilar,exceptforbondswithaspreadofmorethan1000bps.Forexample,forbondswithanactualspreadbetween100-150bpstheaverageactualspreadis136bpswhiletheaveragemodel-impliedspreadis121bps.Importantly,theresultsaresimilarifwecalibratethemodelusingdefaultratesfrom1970-2012ratherthan1920-2012,thusresolvingtheproblemdescribedabovethatearlierresultsintheliteraturedependsigni cantlyonthehistorical3 periodchosentobenchmarkthemodel.Tostudythetimeserieswecalculateaveragespreadsonamonthlybasisand ndthatforinvestmentgradebondsthereisahighcorrelationof93%betweenaverageactualspreadsandmodelspreads.Notethatthemodel-impliedspreadsare\out-of-sample"predictionsinthesensethatactualspreadsarenotusedinthecalibration.Furthermore,foragiven rmonlychangesinleverageandthepayoutrate{calculatedusingaccountingdataandequityvalues{leadtochangesinthe rm'screditspread.Forspeculativegradebondsthecorrelationisonly40%,showingthatthemodelhasamuchhardertimematchingspreadsforlow-quality rms.Althoughaverageinvestmentgradespreadsarecapturedwellonamonthlybasis,themodeldoeslesswellattheindividualbondlevel.RegressingindividualinvestmentgradespreadsonthoseimpliedbytheBlack-CoxmodelgivesanR2ofonly44%,soattheindividualbondlevellessthanhalfthevariationininvestmentgradespreadsisexplainedbythemodel.ForspeculativegradespreadsthecorrespondingR2isonly13%.Wealsoinvestigatethepotentialcontributionofbondilliquiditytocreditspreads.Weusebondageasliquiditymeasureanddouble-sortbondsonliquidityandcreditquality.Forinvestmentgradebondswe ndnorelationbetweenbondliquidityandspreads,consistentwiththeabilityofthemodeltomatchactualspreadsandthe ndinginDick-Nielsen,Feldhutter,andLando(2012)thatoutsidethe nancialcrisis2007-2008illiquiditypremiumsininvestmentgradebondswerenegligible.Forspeculativegradebondswe ndastrongrelationbetweenbondliquidityandyieldspreadsandtheresultssuggestthatbondliquiditymayexplainmuchoftheunderpricingofspeculativegradebonds.InthispaperweusetheBlackandCox(1976)modelasalensthroughwhichtostudythecreditspreadpuzzle.TheresultsinHuangandHuang(2012)showthatmanystructuralmodelswhichappearverydi erentinfactgeneratesimilarspreadsoncethemodelsarecalibratedtothesamedefaultprobabilities,recoveryrates,andtheequitypremium.ThemodelstestedinHuangandHuang(2012)includefeaturessuchasstochasticinterestrates,endogenousdefault,stationaryleverageratios,strategicdefault,time-varyingassetriskpre-mia,andjumpsinthe rmvalueprocess,yetallgenerateasimilarlevelofcreditspread.Thus,our ndingthattheBlack-Coxmodelmatchesaverageinvestmentgradespreadsis4 alsolikelytoholdforawiderangeofstructuralmodelsonceourestimationapproachisusedtoestimatedefaultprobabilities.Thereisanextensiveliteratureontestingstructuralmodels.Leland(2006),Cremers,Driessen,andMaenhout(2008),Chen,Collin-Dufresne,andGoldstein(2009),Chen(2010),HuangandHuang(2012),Chen,Cui,He,andMilbradt(2016),Bai(2016),Bhamra,Kuehn,andStrebulaev(2010),andZhang,Zhou,andZhu(2009))usethehistoricaldefaultrateatagivenratingandmaturitytoestimatethedefaultprobabilityatthatmaturityandrating.Weshowthatthistestisstatisticallyweak.Eom,Helwege,andHuang(2004),Ericsson,Reneby,andWang(2015),andBao(2009)allowforheterogeneityin rmsandvariationinleverageratios,butdonotcalibratetohistoricaldefaultrates.Bhamra,Kuehn,andStrebulaev(2010)observethatdefaultratesarenoisyestimatorsofdefaultprobabilities,butdonotproposeasolutiontothisproblemaswedo.2AmotivatingexampleThereisatraditioninthecreditriskliteratureofusingMoody'saveragerealizeddefaultrateforagivenratingandmaturityasaproxyforthecorrespondingex-antedefaultprobability.Thissectionprovidesanexampleshowingthattheapparentexistenceornon-existenceofacreditspreadpuzzledependsontheparticularperiodoverwhichthehistoricaldefaultrateismeasured.Laterinthepaperwedescribeanalternativeapproachforextractingdefaultprobabilityestimatesfromhistoricaldefaultratesthatnotonlyprovidesmuchgreaterprecisionbutisalsolesssensitivetothesampleperiodchosen.TounderstandhowMoody'scalculatesdefaultfrequencies,considerthe10-yearBBBcumulativedefaultfrequencyof5.09%usedinChen,Collin-Dufresne,andGoldstein(2009).3ThisnumberispublishedinMoody's(2002)andisbasedondefaultdatafortheperiod1970-2001.Fortheyear1970,Moody'sidenti esacohortofBBB-rated rmsandthenrecordshowmanyofthesedefaultoverthenext10years.The10-yearBBBdefaultfrequencyfor1970isthenumberofdefaulted rmsdividedbythenumberinthe1970cohort.Theaverage 3Moody'sreporta10-yearBBBdefaultrateof5.09%(Exhibit32)whileChen,Collin-Dufresne,andGoldstein(2009)use4.89%.WeuseMoody'sreportednumber.5 defaultrateof5.09%iscalculatedastheaverageofthetwenty-two10-yeardefaultratesforthecohortsformedatyearlyintervalsovertheperiod1970-1991AlargepartoftheliteraturehasfocussedontheBBB-AAAspreadat4-yearand10-yearmaturities.Inourmainempiricalanalysis(Section4)westudyamuchwiderrangeofratingsandmaturitiesbutfornow,tokeepourexamplesimple,wealsofocusontheBBB-AAAspread.ForagivensampleperiodweusetheBBBandAAAaveragedefaultratesforthe4-yearand10-yearhorizonsreportedbyMoody's.Followingtheliterature(e.g.,Chen,Collin-Dufresne,andGoldstein(2009),HuangandHuang(2012)andothers)we rstbenchmarkamodeltomatchthesedefaultrates,oneatatime.Usingthebenchmarkedparameterswethencomputerisk-neutraldefaultprobabilitiesand,fromthese,creditspreads.FollowingEom,Helwege,andHuang(2004),Bao(2009),HuangandHuang(2012),andothersweassumethatifdefaultoccurs,investorsreceive(atmaturity)afractionoftheoriginallypromisedfacevalue,butnowwithcertainty.Thecreditspread,s,isthencalculatedas:s=y r= 1 Tlog[1 (1 R)Q(T)](1)whereRistherecoveryrate,TisthebondmaturityandQ(T)istherisk-neutralde-faultprobability.ThroughoutouranalysisweemploytheBlack-Coxmodel(BlackandCox(1976));AppendixAprovidesthemodeldetails.Weuseouraverageparametervaluesfortheperiod1987-2012estimatedinSection4andChen,Collin-Dufresne,andGoldstein(2009)'sestimatesoftheSharperatioandrecoveryrate.Weestimatethedefaultboundarybymatchinganobserveddefaultfrequency.Thedefaultboundaryisthevalueofthe rm,measuredasafractionofthefacevalueofdebt,belowwhichthe rmdefaults.FollowingChen,Collin-Dufresne,andGoldstein(2009)andotherswecarrythisoutseparatelyforeachmaturityandratingsuchthat,conditionalontheotherparameters,themodeldefaultprobabilitymatchesthereportedMoody'sdefaultfrequency.Foreachmaturityandratingwethenusethebenchmarkeddefaultboundaryandcalculatethecreditspreadusingequation(1)above.ThesolidbarsinFigure1showestimatesoftheactualBBB-AAAcorporatebondcreditspreadfromanumberofpapers.Forboththe4-yearand10-yearmaturities,theestimated6 BBB-AAAspreadisintherangeof98-109basispointswiththenotableexceptionofHuangandHuang(2012)'sestimateofthe10-yearBBB-AAAof131basispoints.(HuangandHuangusebothcallableandnon-callablebondsintheirestimateofthespreadandthismayexplainwhyitishigher).UsingMoody'saveragedefaultratesfromtheperiod1970-2001,the4-yearand10-yearBBB-AAAspreadsintheBlack-Coxmodelare52and72basispointsrespectively.Thesemodelestimatesaresubstantiallybelowactualspreadsandthis ndingiswhathasbeencoinedthecreditspreadpuzzle.Figure1alsoshowsthemodel-impliedspreadsusingMoody'saveragehistoricaldefaultratesfrom1920-2002(defaultratesfrom1920-2001arenotavailable).Themodel-impliedspreadsusingdefaultratesfromthislongerperiodaresubstantiallyhigher:the4-yearand10-yearBBB-AAAspreadsare87and104basispointsrespectively.Thus,whenweusedefaultratesfromalongertimeperiodthepuzzlelargelydisappearsToemphasisethatthisconclusionisnotspeci ctotheBlack-Coxmodel,Figure1alsoshowsthefourspreadscomputedundertheMertonmodel(andusingtheparametersandmethodgiveninChen,Collin-Dufresne,andGoldstein(2009)).Thesespreadsareverysimilarto,andjustalittlehigherthan,theBlack-Coxspreads.Whatremainsunchangedisthe ndingthattheappearanceofacreditspreadpuzzledependsonthesampleperiod.IntheexamplewearecomparingcorporatebondyieldsrelativetoAAAyieldstobeconsistentwithCDGandothers.Inourlateranalysisweusebondyieldsrelativetoswaprates.Theaveragedi erencebetweenswapratesandAAAyieldsissmall:overoursampleperiod1987-2012theaverage5-yearand10-yearAAA-swapspreadis4bpsrespectively6bps.Weuseswapratesinourlateranalysis,becausethetermstructureofswapratesisreadilyavailableonadailybasiswhilethereareveryfewAAA-ratedbondinthelaterpartofoursampleperiodandwewouldnotbeabletocalculateaAAAyieldatdi erentmaturities.Insummary,realisedaveragedefaultratesvarysubstantiallyovertimeand,asaresult,whenthesearetakenasex-antedefaultprobabilitiesthehistoricalperiodoverwhichtheyaremeasuredhasastrongin\ruenceonwhetherornottherewillappeartobeacreditspreadpuzzle.Inthenextsectionweexplorethestatisticaluncertaintyofhistoricaldefaultratesinmoredetailandproposeadi erentapproachtoestimatingdefaultprobabilitiesthatexploits7 theinformationcontainedinhistoricaldefaultratesmoreecientlythanhasbeenthecaseintheliteraturesofar.3Estimatingex-antedefaultprobabilitiesTheexistingliteratureonthecreditspreadpuzzleandmorebroadlytheliteratureoncreditrisktypicallyusestheaverageex-posthistoricaldefaultrateforasinglematurityandratingasanestimateoftheex-antedefaultprobabilityforthissamematurityandrating.We rstexplorethestatisticaluncertaintyassociatedwiththeseestimatesandconcludethatitislarge.Wethenproposeanewapproachthatuseshistoricaldefaultratesforallmaturitiesandratingssimultaneouslytoextracttheex-antedefaultprobabilityforagivenmaturityandrating.Simulationsshowthatourapproachgreatlyreducesstatisticaluncertainty.3.1Existingapproach:extractingtheex-antedefaultprobabilityfromasingleex-postdefaultfrequencyIntheliteratureonthepricingofcorporatedebtthereisalongtraditionofusingMoody'saveragehistoricaldefaultrateforasinglematurityandratingasaproxyfortheex-antedefaultprobabilityforthatspeci cmaturityandrating.4Therearetwosigni cantreasonswhyanex-postrealizeddefaultfrequencymaybeanunreliableestimateoftheex-antedefaultprobability.The rstisthatthelowlevelofdefaultfrequency,particularlyforinvestmentgrade rms,leadstoasamplesizeproblemwithdefaulthistoriesasshortasthosetypicallyusedintheliteraturewhentestingstandardmodels.Thesecondisthat,eventhoughtheproblemofsamplesizeispotentiallymitigatedbythepresenceofalargenumberof rmsinthecross-section,defaultsarecorrelatedacross rmsandsothebene tofalargecross-sectioninimprovingprecisionisgreatlyreduced. 4ExamplesincludeChen,Cui,He,andMilbradt(2016),Gomes,Jermann,andSchmid(2016),Christof-fersenandElkamhi(2016),Bai(2016),Zhang,Zhou,andZhu(2009),Chen(2010),Leland(2006),Cremers,Driessen,andMaenhout(2008),Chen,Collin-Dufresne,andGoldstein(2009),Chen(2010),HuangandHuang(2012),Campello,Chen,andZhang(2008),andMcQuade(2013).8 Howseverearethesestatisticalissues?Weaddressthisquestioninasimulationstudyandbaseoursimulationparametersontheaverage10-yearBBBdefaultrateof5.09%over1970-2001usedinChen,Collin-Dufresne,andGoldstein(2009).Inaneconomywheretheexante10-yeardefaultprobabilityis5.09%forall rms,wesimulatetheexpostrealized10-yeardefaultfrequencyover31years.Weassumethatinyear1wehave446identical rms,equaltotheaveragenumberof rmsinMoody'sBBBcohortsovertheperiod1970-2001.IntheBlack-Cox(andMerton)model rmi'sassetvalueunderthenaturalmeasurefollowsaGBMdVit Vit=( )dt+dWPit(2)whereisthedriftof rmvaluebeforepayoutofthedividendyieldandisthevolatilityof rmvalue.AsinSection2aboveweuseouraverageparametervaluesfortheperiod1987-2012estimatedinSection4:=1005%,=472%and=246%.WeintroducesystematicriskbyassumingthatWPit=p Wst+p 1 Wit(3)whereWiisaWienerprocessspeci cto rmiWsisaWienerprocesscommontoall rms,andisthepairwisecorrelationbetweenpercentagechangesin rmvalue.AlltheWienerprocessesareindependent.The rmdefaultsthe rsttimeassetvaluehitsaboundaryequaltoafractiondofthefacevaluedebtF,i.e.the rsttimeV dF.Therealized10-yeardefaultfrequencyintheyear-1cohortisfoundbysimulatingonesystematicand446idiosyncraticprocessesinequation(3).Inyear2weformacohortof446new rms.The rmsinyear2havecharacteristicsthatareidenticaltothoseoftheyear1cohortatthetimeofformation.Wecalculatetherealized10-yeardefaultfrequencyoftheyear-2cohortaswedidfortheyear-1cohort.Crucially,thecommonshockforyears1-9fortheyear-2cohortisthesameasthecommonshockforyears2-10for rmsintheyear-1cohort.Werepeatthesameprocessfor21yearsandcalculatetheoverallaveragerealizedcumulative10-yeardefaultfrequencyintheeconomybytakinganaverageofthedefaultfrequenciesacrossthe21cohorts.Finally,werepeatthisentiresimulation25,000times.9 Toestimatethecorrelationparameterwecalculatepairwiseequitycorrelationsforratedindustrial rmsintheperiod1987-2012.Speci cally,foreachyearwecalculatetheaveragepairwisecorrelationofdailyequityreturnsforallindustrial rmsforwhichStandard&Poor'sprovidearating,andthencalculatetheaverageofthe26yearlyestimates1987-2012.Weestimatetobe20.02%.Tosetthedefaultboundaryweproceedasfollows.First,withoutlossofgeneralityweassumethattheinitialassetvalueofeach rmisequaltoone.Thismeansthatthe rm'sleverage,LF V=F,andwesetthedefaultboundary,dF(=dL)suchthatthemodel-implieddefaultprobabilitygiveninequation(12)intheAppendixmatchesthe10-yeardefaultrateof5.09%.5Panel(a)ofFigure2showsthedistributionoftherealizeddefaultrateinthesimulationstudyandtheblackverticallineshowstheexantedefaultprobabilityof5.09%.The95%con denceintervalfortherealisedaveragedefaultrateiswideat[1.15%;12.78%].Giventhatwesimulate9366 rmsoveraperiodof31years,itmightbesurprisingthattherealizeddefaultratecanbefarfromtheexantedefaultprobability.Thereasonissimplythepresenceofsystematicriskintheeconomywhichinducescorrelationindefaultsacross rms.Wealsoseethatthedefaultfrequencyissigni cantlyskewedtotheright,i.e.,themodalvalueofaround3%issigni cantlybelowthemeanof5.09%.Thismeansthatthedefaultfrequencymostoftenobserved{e.g.,theestimatefromtheratingagencies{isbelowthemean.Speci cally,thetrue10-yeardefaultprobabilityis5.09%,buttheprobabilitythattheobservedaverage10-yeardefaultrateover31yearsishalfthatlevelorlessis19.9%.ThisskewnessmeansthatthenumberreportedbyMoody's(5.09%)ismorelikelytobebelowthetruemeanthanaboveitand,inthiscase,ifspreadsre\rectthetrueexpecteddefaultprobability,theywillappeartoohighrelativetotheobservedhistoricallossrate.Moody'snowpublishdefaultratesstartingfrom1920and,otherthingsequal,alongertimeperiodoverwhichwemeasureaveragedefaultrateswillleadtoimprovedstatistical 5Wesimulate rmvaluesonaweeklybasis.Thereisasmalldownward-biasindefaultratesbecauseadefaultonlyoccursonaweeklybasisandnotcontinuouslyandweadjustforthisbiasbymultiplyingaveragedefaultratesineachofthe25,000simulationswith5.09%dividedbytheaverageofthe25,000averagedefaultrates.10 precision.However,evenifthedefaultrateismeasuredoveratimeperiodof92yearsthereremainssigni cantstatisticaluncertaintywhenestimatesofthedefaultprobabilityarebasedonasingleratingandmaturity.Keepingthedefaultprobability xedat5.09%andincreasingthesimulatedtimeperiodfrom31yearsto92years,leadstoa95%con denceintervalof[2.47%;8.95%].Thusevenwith92yearsofdefaultdatathereisstillsigni cantuncertaintyregardingthetrueexantedefaultprobability.3.2Anewapproach:extractingtheex-antedefaultprobabilityfromacross-sectionofex-postdefaultfrequenciesWenowdescribeamethodtoestimatedefaultprobabilitiesthatusesrealisedcumulativedefaultratesfromawiderangeofratingsandmaturities.Thecentralidea,asdescribedearlier,isthatbecauselowcreditqualitybondsdefaultmorefrequently,theyprovidemuchmoreinformationonthelocationofthedefaultboundarythanhighcreditqualitybondsandwecanthereforeobtainbetterestimatesofthedefaultprobabilityonthelatterwhenwealsoincludedefaultrateinformationontheformer.Chen,Collin-Dufresne,andGoldstein(2009) ndthedefaultboundaryparameterdsuchthatthemodel-implieddefaultprobabilitymatchesthehistoricaldefaultrateatasingleratingandmaturity.Whencalculatingthe10-yearBBBspread,andgivenestimatesofP,thisapproachamountsto ndingdLsuchthatthemodel-implieddefaultprobabilityP(dL;P;T)inequation(12)isequaltothehistorical10-yearBBBdefaultrate.Chen,Collin-Dufresne,andGoldstein(2009)thenusethisdefaultboundarytocalculatethe10-yearBBBspreadaccordingtoequation(1).Ourapproachissimilarbut,crucially,we ttothehistoricaldefaultratesonallavailableratingsandmaturities.Weestimatethedefaultboundaryparameterdbyminimisingthesumofabsolutedeviationsbetweenannualizedmodel-impliedandhistoricaldefaultrates:minfdgCXa=AAA20XT=11 T P(dLaP;T) ^PaT (4)where^PaTisthehistoricalcumulativedefaultrateforratingaandmaturityT6 6Theremaybealternativeweightingpatternsthatareevenmoreecientbutinmakingthischoiceour11 Howmuchofanimprovementisthisapproachrelativetothestandardapproachdescribedintheprevioussection?Weanswerthisquestionbysimulatingcumulativedefaultratesandapplyingequation(4)inordertoobtainthedistributionoftheestimateddefaultboundary.Followingtheproceduredescribedintheprevioussection,wesimulateoveraperiodof31yearsbutnowforeachofthesevenmajorratingcategories.Weassumethatall rms,regardlessofrating,havethesameparametersand,thesamecorrelationstructure(giveninequation(3))andthesamecorrelationparameter,.Thedefaultboundaryforeachratingcategoryisset,asintheprevioussection,suchthatthehistorical10-yearcumulativedefaultrateismatched.Sincediscommonacrossall rms,itmeansthat rmsindi erentratingcategoriesdi eronlyintheirinitialleverage.Wethensimulate1,...,20yeardefaultrates.Wesetthenumberof rmsineachratingcohortequaltotheaveragenumberof rmsintheMoody'scohortsforthatratingcategoryintheperiod1970-2001.Wecarryoutthissimulation25,000timesandforeachsimulationwe rstestimatedaccordingtoequation(4)andthencalculatethe10-yearBBBcumulativedefaultprobabilityinequation(12).Panel(b)ofFigure2showsthedistributionoftheestimateddefaultprobability.Weseethatthedistributionofthe10-yearBBBdefaultprobabilityisbothmuchtighterandlessskewedthanwhentheestimateisbasedonsolelythe10-yearBBBdefaultrate.Thedistributionismuchtighterbecauseweusedefaultratesfromallmaturitiesandratings,inparticularfromlowratings,insteadofjustone.Theskewnessisreducedbecauseweincludedefaultratesintheestimationthataresigni cantlyhigherthanBBBdefaultrates.Toseewhythisisthecase,considertheexamplediscussedearlierofalargenumberof rmswithadefaultprobabilityof5%andwheretheirdefaultsareperfectlycorrelated.Inthiscasewewillseenodefaults95%ofthetimeandtherealizeddefaultratewillunderestimatethedefaultprobability95%ofthetimeandthedistributionwillexhibitpositiveskewness.Bythesamelogic,ahigherdefaultprobabilityreducestheskewnessandfordefaultprobabilitiesgreaterthan50%theskewnessisinfactnegative.Table1comparesthestandarddeviationoftheestimateddefaultprobabilitiesintheexistingapproachandourapproachforalltheratingsandmaturitieswestudyinthenextsection.Ourapproachresultsinasubstantialreductioninthestandarddeviation,except objectivehasbeentokeepourmethodsassimpleandtransparentaspossible.12 fortheshortestmaturitiesofthelowestratings.Forexample,thestandarddeviationfortheestimated10-yearBBBdefaultprobabilityis0.48%comparedto3.05%usingtheexistingapproach.Table2showsthatforinvestmentgraderatings,ourapproachalsoresultsinasubstantialreductionintheskewness.Overall,ourproposedmethodgreatlyreducesboththestandarddeviationandskewnessofestimatedinvestmentgradedefaultprobabilities.74.AnewperspectiveonthecreditspreadpuzzleIntheprevioussectionweproposedanewmethodtoestimatedefaultprobabilitiesbycombininginformationonhistoricaldefaultratesfromthecrosssectionofratingcategoriesandmaturitiesandshowedthat,comparedtotheexistingapproachofusingasingledefaultrate,itgreatlyimprovesstatisticalprecision.Inthissectionweapplyourmethodtoalargedatasetofbondquotesover1987-2012toshednewlightonthecreditspreadpuzzle.4.1DataFortheperiodJanuary1,1997toJuly1,2012,weusedailyquotesprovidedbyMerrillLynch(ML)onallcorporatebondsincludedintheMLinvestmentgradeandhigh-yieldindices.Thesedataareusedby,amongothers,SchaeferandStrebulaev(2008)andAcharya,Amihud,andBharath(2013).WeobtainbondinformationfromtheMergentFixedIncomeSecuritiesDatabase(FISD)andlimitthesampletoseniorunsecured xedrateorzerocouponbonds.Weexcludebondsthatarecallable,convertible,putable,perpetual,foreigndenominated,Yankee,havesinkingfundprovisions,orhavecovenants.8FortheperiodApril1987toDecember1996weusemonthlydatafromtheLehmanBrothersFixedIncomeDatabase. 7InoursimulationswematchMoody'shistorical10-yeardefaultratesover1970-2001forratingsAAA,...,C.IntheInternetAppendixweshowthatthedistributionofthedefaultboundaryparameterd{andthereforealsothedistributionofthedefaultprobability{issimilarifwematchMoody'sdefaultratesatothermaturitiesthan10year.8Forbondrating,weusethelowerofMoody'sratingandS&P'srating.Ifonlyoneofthetworatingagencieshaveratedthebond,weusethatrating.Wetrackratingchangesonabond,sothesamebondcanappearinseveralratingcategoriesovertime.13 ThisdataisusedbyamongothersDu ee(1998),HuangandHuang(2012),andAcharya,Amihud,andBharath(2013).WeincludeonlydatafromtheLehmandatabasethatareactualquotes(incontrasttodatabasedonmatrix-pricing).TheLehmandatabasestartsin1973,buttherearetworeasonswhywestartfromApril1987.First,therearefewnoncallablebondsbeforethemid-80s(seeDu ee(1998))andsecond,wecalculatecreditspreadsrelativetotheswaprateandwedonothavedataonswapratespriortoApril1987.Weuseonlybondsissuedbyindustrial rmsandrestrictoursampletobondswithamaturityoflessthan20yearstobeconsistentwiththematuritiesofthedefaultratesweuseaspartoftheestimation.9Intotalwehave256,698observations.WeshowintheInternetAppendixthatdealerquotesareunreliableforshort-maturitybondsduetoquotesbeingbidquotesandwhenwereportbondspreadswethereforeexcludebondswithamaturitylessthanthreeyears.Table3showssummarystatisticsforthecorporatebondsample.ThetableshowsthatthenumberofbondswithalowratingofBorCissmall;forexamplethereisonlyoneC-ratedbondinthematuritygroup13-20y.Thereasonforthisisthatspeculativegradebondsfrequentlycontaincalloptionswhichleadstotheirexclusionfromoursample(seealsoBooth,Gounopoulos,andSkinner(2014)).InthefollowingwereportresultsforratingsBandC,butwiththecaveatthattheseresults{particularlyforlongmaturities{arebasedonfewobservationsandthereforenoisy.TopriceabondintheBlack-Coxmodelweneedtheissuing rm'sassetvolatility,leverageratio,andpayoutratioalongwiththebond'srecoveryrate.Leverageratioiscalculatedasthebookvalueofdebtdividedby rmvalue(where rmvalueiscalculatedasbookvalueofdebtplusmarketvalueofequity).Payoutratioiscalculatedasthesumofinterestpaymentstodebt,dividendpaymentstoequity,andnetstockrepurchasesdividedby rmvalue.AnimportantparameteristheassetvolatilityandherewefollowtheapproachofSchaeferandStrebulaev(2008)incalculatingassetvolatility.Since rmvalueisthesumofthedebtandequityvalues,assetvolatilityisgivenby:2t=(1 Lt)22E;t+L2t2D;t+2Lt(1 Lt)ED;t(5) 9IntheInternetAppendixweshowthatourresultsaresimilarifweuseTRACEtransactionsdatafortheshorterperiod2002-2012.14 wheretisthevolatilityofassets,D;tvolatilityofdebt,ED;tthecovariancebetweenthereturnsondebtandequity,andLtisleverageratio.Ifweassumethatdebtvolatilityiszero,assetvolatilityreducestot=(1 Lt)E;t.Thisisalowerboundonassetvolatility.SchaeferandStrebulaev(2008)(SS)computethislowerboundalongwithanestimateofassetvolatilitythatimplementsequation(5)infull.They ndthatforinvestmentgradecompaniesthetwoestimatesofassetvolatilityaresimilarwhileforjunkbondsthereisasigni cantdi erence.Wecomputethelowerboundofassetvolatility,(1 Lt)E;t,andmultiplythislowerboundwithSS'sestimateoftheratioofassetvolatilitycomputedfromequation(5)tothelowerbound.Speci cally,weestimate(1 Lt)E;tandmultiplythisby1ifLt025,1.05if025Lt035,1.10if035Lt045,1.20if045Lt055,1.40if055Lt075,and1.80ifLt&#x-333;&#x.094;075.10Thismethodhastheadvantageofbeingtransparentandeasytoreplicate.Foragiven rmwethencomputetheaverageassetvolatilityoverthesampleperiodandusethisconstantassetvolatilityforeverydayinthesampleperiod.All rmvariablesareobtainedfromCRSPandCompustatanddetailsaregiveninAppendixB.Summarystatisticsforthe rmsinoursampleareshowninTable4.Theaverageleverageratiosof0.13forAAA,0.14forAA,0.27forA,and0.37forBBBaresimilartothosefoundinotherpapers:HuangandHuang(2012)usealeverageratioof0.13forAAA,0.21forAA,0.32forA,and0.43forBBBwhileSchaeferandStrebulaev(2008) ndanaverageleverageof0.10forAAA,0.21forAA,0.32forA,and0.37forBBB.Averageequityvolatilityismonotonicallyincreasingwithrating,consistentwithaleveragee ect.TheestimatesaresimilartothoseinSSforA-AAAratings,whiletheaverageequityvolatilityforBBB rmsof0.38ishigherthanthevalueof0.33giveninSS.AssetvolatilitiesareslightlyincreasinginratingandbroadlyconsistentwiththeestimatesinSS.Wesettherecoveryrateto378%whichisMoody's(2013)'saveragerecoveryrate,asmeasuredbypost-defaulttradingprices,forseniorunsecuredbondsfortheperiod1982-2012.Finally,theriskfreerate,r,istheswaprateforthesamematurityasthebond.Traditionally, 10ThesefractionsarebasedonTable7inSSapartfrom1.80whichwedeemtobereasonable.ResultsareinsensitivetootherreasonablechoicesofvaluesforL�075.SeealsoCorreia,Kang,andRichardson(2014)foranassessmentofdi erentapproachestocalculatingassetvolatility.15 Treasuryyieldshavebeenusedasriskfreerates,butrecentevidenceshowsthatswapratesareabetterproxythanTreasuryyields.AmajorreasonforthisisthatTreasurybondsenjoyaconvenienceyieldthatpushestheiryieldsbelowriskfreerates(FeldhutterandLando(2008),KrishnamurthyandVissing-Jorgensen(2012),andNagel(2014)).TheconvenienceyieldisforexampleduetotheabilitytopostTreasuriesascollateralwithasigni cantlylowerhaircutthanother nancialsecurities,ane ectoutsidethescopeofthemodel.4.2EstimationofthedefaultboundaryInthissectionweestimatethedefaultboundary,asingleparameterthatwethenusetopricebondsacrossratingsandmaturities.Althoughestimatingthedefaultboundaryparameter,d,by ttingtohistorical(natural)defaultratesrequiresanestimateoftheSharperatio,whenwecomputeyieldspreadsweusestandard\risk-neutral"pricing.WefollowthemethodoutlinedinSection3.2.Speci cally,ifweobserveaspreadonbondiwithatime-to-maturityTissuedby rmjondatet,wecalculatethe rm'sT-yeardefaultprobabilityP(dLjtPjt;T)wherePjt=(jt;j;jt)usingequation(12).Here,jisthe rm'sconstantassetvolatility,Ljtjtandjtarethetime-testimatesofthe rm'sleverageratio,assetvaluedriftandpayoutrate.TocalculatejtweassumeaconstantSharperatiosuchthatjt=j+rTt jt,whererTtistheT-yearriskfreerate.WeuseChen,Collin-Dufresne,andGoldstein(2009)'sestimateoftheSharperatioof0.22.ForagivenratingaandmaturityT-roundeduptothenearestintegeryear-we ndallbondobservationsinthesamplewiththecorrespondingratingandmaturity.Foragivencalendaryearywecalculatetheaveragedefaultprobability Py;aT(d)andwethencalculatetheoverallaveragedefaultprobabilityforratingaandmaturityT PaT(d),bycomputingthemeanacrosstheNyears, PaT(d)=1 NPNy=1 Py;aT(d).Wedenoteby^PaTthecorrespondinghistoricaldefaultfrequencygivenbyMoody'fortheperiod1920-2012.Forallmajorratings(AAA,AA,A,BBB,BB,B,C)andhorizonsof1-20years(Moody'sonlyreportsdefaultratesforuptoahorizonof20years)we ndthevalueofdthatminimizesthesumofabsolutedi erencesbetweentheannualizedhistoricalandmodel-implieddefaultratesbysolvingminfdgCXa=AAA20XT=11 T PaT(d) ^PaT (6)16 Usingthisapproachourestimateis^d=08944.4.3ThetermstructureofdefaultprobabilitieswiththeestimateddefaultboundaryOurestimateofdmatchestheaveragedefaultprobabilityof rmsissuingstraightcouponbulletbondstoaveragehistoricaldefaultratesandtheremaybeatleastthreeconcernswiththisapproach.First, rmsissuingstraightcouponbulletbondsmaybedi erentfromtheaverage rminMoody'ssample.Second,althoughtheaveragehistoricaldefaultrateacrossmaturitiesismatched,thetermstructureofdefaultratesmightnotbematchedaccurately.Third,theremaybesystematicdi erencesintheabilityofthemodeltomatchdefaultratesacrossratings.Toaddresstheseconcernsweusetheestimateddefaultboundarytocomputetheaveragedefaultprobabilitiesforallratedindustrial rmsinCompustat{amuchbroadersample{andcomparethesetoMoody'shistoricaldefaultrates.Speci cally,weextractfromCapitalIGtheissuerseniordebtratingassignedbyStandard&Poor's.Therearealmostnoratingobservationsbefore1985,sooursampleperiodis1985-2012.Table5givessummarystatisticsonthemainparameters{asinTable4{forthisnewsample.Comparedtothesampleof rmsusedtoestimatedthedefaultboundary(inTable4)thesampleis4.6timesaslargeandhasareasonablylargenumberofspeculativegrade rms.Wecomputeaveragemodeldefaultprobabilitiesforagivenratingainamanneranal-ogoustothewayMoody'scalculateshistoricaldefaultrates.Speci cally,onthe naldayofeachmonthwe ndall rmsforwhichwehavedataandthathaveratinga,andforthose rmsanddateswecalculatethetermstructureofdefaultprobabilities,i.e.defaultprobabilitiesforhorizonsof1,2,...,20-years.Wethencalculatetheaveragetermstructureofdefaultprobabilitiesforeachyear1985,...,2012and, nally,wecomputetheaveragetermstructureacrossyears.Weusethe10-yearTreasuryCMTrateastheriskfreeratebecausewedonothaveswapratesinthe rstyears.Figure3andTable6showtheaveragemodel-implieddefaultprobabilitiesandMoody'shistoricaldefaultratesfor1920-2012.Inboththe gureandtableweshow95%con dence17 bandsforthehistoricaldefaultrate.Thecon dencebandisobtainedbyfollowingthesim-ulationprocedureinSection3.1wherewesimulateover92yearsanduseMoody'shistoricaldefaultratesfortheperiod1920-2012asinput.InTable6,caseswherethemodel-implieddefaultprobabilityisoutsidethe95%and99%con dencebandareindicatedby`*'and`**'respectively.WeseethatintheBlack-Coxmodelthereisastatisticallysigni cantunderestimationof1-2yearAAandAdefaultprobabilitiesandoverestimationofshort-termspeculativegradedefaultprobabilities.Thetermstructureofmodel-implieddefaultprobabilitiesisclosetohistoricaldefaultratesforAbeyondthreeyearsandforBBBrated rms.TheseratingcategoriesaccountformorethanhalfoftheUScorporatebondmarketvolume(measuredbythenumberoftransactions).11ItmayseemsurprisingthattheBlack-CoxmodelcapturesthetermstructureofdefaultratesforBBB-ratedbondswellbecausethereareanumberofpapersshowingthat,forhorizonsbelow3-4years,structuralmodelsimplyessentiallyzerodefaultprobabilitiesforinvestmentgrade rms(Zhou(2001),Leland(2004),Leland(2006),Cremers,Driessen,andMaenhout(2008),Zhang,Zhou,andZhu(2009)andothers).WeshowinAppendixCthatresultsintheexistingliteraturedocumentingafailureofstructuralmodelstocaptureshort-termdefaultratesarestronglybiasedduetoa\convexitye ect"arisingfromJensen'sinequality.Thebiasariseswhenusingarepresentative rmwithaverageleverage(andaverageassetvolatilityandpayoutrate)tocalculateshort-termdefaultprobabilitiesbecausethedefaultprobabilityusingaverageleverageissubstantiallylowerthantheaveragedefaultprobabilitycalculatedusingthedistributionofleverage.Ourresultsshowthatoncewedealwiththeconvexitybiasbyusingdataonindividual rmstheBlack-Coxmodelcapturesshort-termdefaultratesmuchbetterthanpreviouslyreported4.4AveragecorporatebondcreditspreadsWecalculateaveragespreadsbyfollowingthecalculationsinDu ee(1998).Speci cally,wecalculateamonthlyaverageactualspreadforagivenratinga,maturityrangeM1toM2 11AccordingtoTRACEFactBook2012,53%ofallU.S.corporatebondtransactionvolumein2012wasinAorBBBratedbonds(TablesC24andC25).18 andmontht.Toeasenotation,weindexthecombinationofrating,maturityrangeandmonthbyh=(a;;M1M2;t).Foragivenh,we ndallNhbondobservationswithratingaandindividualbondmaturitiesTh1;Th2;:::;ThNh,whereM1ThiM2,observedondays h1; h2;:::; hNhinmontht.Denotingthecorrespondingyieldobservationsasyh1;yh2;:::;yhNhandtheswapratesassw( h1;Th1);sw( h2;Th2);:::;sw( hNh;ThNh),theaverageyieldspreadforratinga,maturityrangeM1toM2,andmonthtissh=1 NhNhXi=1 yhi sw( hi;Thi)(7)Theaverageyieldspreadforagivenratingandagivenmaturityintervalisthentheaverageofthemonthlyvalues.Similarly,wecalculatemodel-impliedspreadsbyreplacingtheactualspreadyhi sw( hi;Thi)withthespreads(^d;Qjhi;Ljhi; hi;Thi)impliedbytheBlack-Coxmodelgiveninequation(1),whereQjhi=(sw( hi;Thi);jhi;jhi; hi;R),jhiistheassetvolatilityof rmjthatissuedbondiLjhi; hiandjhi; hiaretheleverageratioandpayoutraterespectivelyof rmjonday hiR=378%istherecoveryrate,andthedefaultboundary^d=08944,asestimatedinSection4.2.Wecomputecon denceintervalsforthemodel-impliedspreadsinthefollowingway.InSection3.2wecalculatethedistributionofthedefaultboundary.WeredothiscalculationusingMoody'sdefaultratesfrom1920-2012.Thatis,wesimulateover92yearsandsettheleverageratiosforeachratingsuchthatthehistorical10-yearcumulativedefaultratesforAAA,AA,...,Cfortheperiod1920-2012arematchedanduseforeachratingtheaveragecohortsizefortheperiod1970-2012inoursimulation.Fortheestimateddefaultboundarydistributionwecalculatethe2.5%and97.5%quantiles,q2:5%andq97:5%.Sincethemodel-impliedspreadismonotoneinthedefaultboundary,wecalculatealowerandupperboundonthemodel-impliedspreadbycomputingspreadsusingq2:5%^dandq97:5%^drespectively.4.4.1SortingaccordingtoratingTable7showsactualandmodel-impliedbondspreadsinoursample,calculatedusingtheapproachjustdescribed.Weseethattheaverageactualinvestmentgradespreadacrossmaturityis92bpswhiletheaveragemodel-impliedspreadis111bpsandtakingintoaccount19 theuncertaintyofdefaultprobabilitiesthedi erenceisstatisticallyinsigni cant.Wealsoseethatthereisagoodcorrespondencebetweenmodel-impliedandactualinvestmentgradespreadswhenwelookattheindividualmaturities3-7,7-13and13-20years.Thus,theBlack-Coxmodelcapturesaverageinvestmentgradespreadswell.Turningtospeculativegradespreadsweseeanunderpredictionofspreadsacrossmaturitywithaverageactualspreadsat544bpsandaveragemodelspreadsat382bpswiththedi erencebeingstatisticallysigni cant.WhenwelookatindividualinvestmentgraderatingsinTable7theBlack-CoxmodelslightlyunderpredictsspreadsonAAA-andAA-ratedbonds(by13-15bps)andoverpredictsspreadsonA-ratedbondsby24bps.ForA-ratedbondstheaverageactualspreadacrossmaturityis61bpswhileitis85bpsinthemodel.TheaverageactualBBBspreadacrossmaturityis146bpswhiletheaveragemodel-impliedspreadis169bps.Thus,theaveragespreadofbondswheremosttradingtakesplaceintheUScorporatebonds{bondswitharatingofAorBBB{iscapturedwellbytheBlack-Coxmodel.Forspeculativegradebonds,underpredictionincreasesaswemovedowntheratingscale.ForBB-ratedbonds,thereisnosigni cantunderpredictionformaturitiesbelow13years,whilemodelspreadsaretoolowforlongermaturities.Overall,theaveragelevelofactualandmodel-impliedinvestmentgradespreadsarestatisticallynotdi erent.Incontrastwe ndthatthemodelunderpredictsspeculativegradespreads;here,theaveragespreadacrossmaturityis544bpsinthedataand382bpsinthemodelandthedi erenceisstatisticallysigni cant.4.4.2SortingaccordingtoactualyieldspreadTheliteraturehastraditionallycomparedmodel-impliedandactualcreditspreadswithinratingcategories.Thereareseveralreasonsforthis.First,Moody'sprovidesyielddataanddefaultratesfrom1920andthereisthereforealonghistoryofdefaultandyieldorganisedaccordingtorating.Infact,sofarweareaware,theonlypubliclyavailabledataonaggregatedefaultratesareorganisedbyrating.Second,spreadsorganisedaccordingtoratingshowalargevariationinthemean,withlowerrated rmshavinghigheraveragespreads;matchingaveragebondspreadsorganisedbyratinghasprovidedahardtestforstructuralmodels.Althoughaveragedefaultratesareavailableonlybyrating,wecanneverthelesssort20 bondsinotherwaysinordertocomparemodel-impliedandactualspreads.Ifthereisasubstantialdi erence,themodelismisspeci edinsomedimension.Sinceanyusefulsortshouldresultinsigni cantvariationinspreads,themostobviouschoiceistosortaccordingbyactualspreads.Table8showsmodelspreadssortedaccordingtothesizeoftheactualspread.Weseethatforspreadsbelow1,000bpsthereisnostatisticaldi erencebetweenmodelspreadsandactualspreadswhenaveragedacrossmaturity.Forexample,theactualspreadforbondswithspreadsbetween100-150bpsis121bpswhileitis136bpsinthemodel.However,forbondspreadsabove150bpswestarttoseeamodestunder-predictionatlongmaturitieswhichbecomesstrongonlywhenspreadsareabove300bps.However,above1000bpsthemodelunderestimatesspreadssubstantiallyandheretheaveragemodelspreadisonlyaroundhalfoftheaverageactualspread.Overall,theresultswhensortingbyactualspreadaresimilartothosesortedbyrating,namelythatspreadsforlowcreditrisk rmsarematchedwellwhilespreadsforthehighestcreditrisk rms,particularforbondswithlongmaturity,areunder-predicted.4.5TimeseriesvariationinyieldspreadsHavingestablishedthattheBlack-Coxmodelcanmatchtheaveragesizeofinvestmentgradecreditspreads,wenextexaminewhetherthemodelcanalsocapturetheirtimeseriesvariation.Ineachmonth,wecalculatetheaverageactualyieldspreadforagivenratingaccordingtoequation(7)(wherethespreadisrelativetotheswaprate)alongwiththecorrespondingmodel-impliedaveragespreadandinvestigatethemonthlytimeseries.Toprovideanoverallassessmentofthemodel'sabilitytocaptureinvestmentgradespreads,wegrouptogetherallinvestmentgradespreadsandallmaturitiesbetween3-20yearsandplottheactualandmodel-impliedspreadsinFigure4.Weseethatthemodel-impliedspreadtrackstheactualspreadwellwithacorrelationis93%TotestmoreformallytheabilityoftheBlack-Coxmodeltocapturethetimeseriesvariationinspreads,weregressthemonthlytimeseriesoftheactualspread,st,onthe21 model-impliedspread,^stst= + ^st+t(8)andreportthe andtheR2oftheregressioninTable9PanelA.Thetableshowsthat{forallbondswithmaturitiesbetween3-20years{theregressionofactualinvestmentgradespreadsonmodel-impliedinvestmentgradespreadsgivesaslopecoecientof0.88andanR2equalto87%showingthatonceinvestmentgradespreadsareaggregatedmodel-impliedspreadstrackactualspreadsverywell.TheR2'sfortheseparateaggregateregressionsforAandBBBspreadsarehighat70%and88%respectively.ForspeculativegradeandAAA/AAratingstheBlack-Coxmodel'sabilitytocapturethetimeseriesvariationismuchlower.Morenoiseduetofewerobservationsisonefactorcontributingtothedeteriorating t.PanelBshowstheregressioninchanges,st+1 st= + (^st+1 ^st)+t+1(9)andweseethattheR2'saresubstantiallylower.Thisimpliesthatthereissigni cantvariationinmonthlychangesincreditspreadsthatisnotexplainedbytheBlack-CoxmodelandCollin-Dufresne,Goldstein,andMartin(2001)andFeldhutter(2012)linkthisvariationtosupply/demandshocks.4.6SpreadpredictionsonindividualbondsOurmainresultisthat,whencalibratedtomatchhistoricaldefaultrates,theBlack-Coxmodelwithaconstant1)Sharperatio,2)recoveryrate,and3)defaultboundary,andnopricedrisksbeyonddi usionriskcanmatchtheaveragespreadofinvestmentgradebonds.Thisresultdoesnotnecessarilyimplythatthemodelcanmatchspreadsonindividualbondswithahighdegreeofprecision,becauseaveragespreadsmaywellmasksigni cantindividualpricingerrors.Whileourinterestliesmainlyinaskingwhetherthemodelcancaptureaveragespreads,weneverthelesscarryoutanexploratoryanalysisontheabilityofthemodeltocapturethecross-sectionofspreads(foramoreextensiveanalysisseeBao(2009)).22 The rstcolumninTable10showstheR2'sfromregressingactualspreadsonmodel-impliedspreads(andaconstant)attheindividualbondlevel.ForinvestmentgradebondstheR2is44%,substantiallybelowtheR2of87%obtainedusingmonthlyaveragespreadsandreportedinTable9.Forspeculativegradebondstheexplanatorypoweroftheregressionattheindividualbondlevelisonly13%,showingthatthemodelhasonlylimitedabilitytopricespeculativegradebonds.Togiveanindicationonhowthemodelorparameterestimatesmightbeimproved,wecorrelatethepricingerror{thedi erencebetweentheactualandmodel-impliedspread{withvariablesusedintheestimation.Table10showstheresults.Thepricingerrorsforinvestmentgradebondshaveacorrelationof-0.45withleverageand-0.34withthepayoutrate.Thissuggeststhatestimatesforindividualbondscouldbeimprovedeitherbyestimatingleverageandpayoutrateinadi erentwayorindeedbychangestothemodel.Correlationsbetween(equityandasset)volatilitiesandpricingerrorsaremodestandrangefrom-0.14to0.12.Thismayindicatethatabettermeasurementofvolatilityand/orincorporationofstochasticassetvolatilityintothemodelmaybelessimportantinimprovingcrosssectionalaccuracy.4.7TheroleofbondilliquidityAnumberofpapersexaminetheimpactofilliquidityoncorporatebondspreads;theseincludeDick-Nielsen,Feldhutter,andLando(2012)(DFL),Bao,Pan,andWang(2011),Friewald,Jankowitsch,andSubrahmanyam(2012)andLin,Wang,andWu(2011).Thesepapersexaminetransactionsdatafromtherelativelyrecentpast,typicallyfrom2004,butsinceweusespreaddatastartingbackin1987,wecanprovideevidenceontheimpactofilliquidityoncreditspreadsoveralongerhistoricaltimeperiod.Thedrawbackofourlongertimeperiodisthatwecannotcalculatetransactions-basedmeasuresofilliquidity.Insteadweusebondageasameasureofbondilliquiditysinceageisknowntoberelatedtoilliquidity(seeBao,Pan,andWang(2011)andHouweling,Mentink,andVorst(2005)andthereferencestherein).InTable11we rstsortaveragecreditspreadresidualsintoquintilesaccordingtobondageandthenaccordingtowhetherthebondisinvestmentgradeorspeculativegrade.Thetablereportstheaveragecreditspreadresidual,de nedasthe23 di erencebetweenthemodel-impliedandactualcreditspread,andtestswhetherthisaverageisdi erentfromzero.Forinvestmentgradebondsthereisessentiallynorelationbetweentheaveragepricingerrorandbondilliquidity.Thisisconsistentwiththe ndingsinDFL,Friewald,Jankowitsch,andSubrahmanyam(2012)andLin,Wang,andWu(2011)thatthepotentialimpactofilliquidityonpricesofinvestmentgradebondsismuchsmallerthanforspeculativegrade.Incontrast,Table11showsastrongrelationbetweenpricingerrorsandbondilliquidityforspeculativegradebonds.Forliquidspeculativegradebondsthepricingerrorismodestlypositive,butaswemovetomoreilliquidbondsastrongmodelunderpredictionemergesandtheaveragepricingerrordi erencebetweenbondsintheleastandmostliquidquintilesis470bps.Themagnitudeofthepricingerrordi erenceacrossbondilliquiditysuggeststhatmuch,ifnotall,oftheunderpredictionofspeculativegradecreditspreadscanbeexplainedbybondilliquidity.Thisinturnsuggeststhattopricespeculativegradebondsinstructuralmodels,itisimportanttoincorporateilliquidityinthesemodels.Asaninterestingclassofsuchmodels,HeandMilbradt(2014)andChen,Cui,He,andMilbradt(2016)incorporatesearchfrictionsinastructuralmodelofcreditriskinsuchawaythatilliquidityismoreimportantforspeculativegradebonds.Duringthe2008-2009 nancialcrisisDFL ndanilliquiditypremiuminAAA,AA,A,BBB,andspeculativegradebondsof5bps,42bps,51bps,93bps,and197bpsrespectively.IfwerestricttheanalysisinSection4.4tothecrisisperiodidenti edinDFL(2007:Q2-2009:Q2),formaturitiesbetween3and20yearswe ndthattheaveragedi erencebetweenactualandmodel-impliedspreadsforAAA,AA,A,BBB,andspeculativegradebondsis18bps,63bps,-43bps,-33bps,and262bpsrespectively(whencalculatedasinTable7).ForAAAandAAratedbondsthemodelunderpredictioniscomparabletotheliquiditypremiumfoundinDFLandforspeculativegradebondstheunderpredictionisaround100basispointslarger.Onehastobecarefulininterpretingpointestimatesofaveragespreaddi erencesinbasispointsoverarelativelyshortperiodwherespreadswereatahistoricalhighandveryvolatile,butifwetakethemodeloverpredictionof43bpsand33bpsforAandBBBratedbondsatfacevalue,itsuggeststhepresenceofsomemodelmisspeci cationduringthe nancialcrisis.24 4.8Usingdefaultdatafrom1970-2012toestimatethedefaultboundaryWesawinSection2thatwhenthehistoricalBBBandAAAdefaultratesareusedone-at-a-timeasestimatesoftheBBBandAAAdefaultprobabilities,theappearanceofacreditspreadpuzzledependsstronglytimeperiodoverwhichhistoricaldefaultratesarecalculated.Toseewhetherourproposedmethodsu ersfromthesamedrawback,weestimatethedefaultboundaryasdescribedinSection3.2but ttingtoMoody'sdefaultratesfrom1970-2012ratherthanthan1920-2012asinthemainanalysis.Inthiscasethedefaultboundaryisestimatedtobe^d=09302andTable12showsaveragespreadsusingthisvalue.Thetableshowsthatthereareonlymodestchangesinthemodel-impliedspreads.Forexample,theaverageinvestmentgradeandspeculativegradespreadsare122bpsand420bpsrespectivelywhenusingdefaultratesfrom1970-2012comparedto111bpsand382bpsrespectivelywhenusingdefaultratesfrom1920-2012.Figure5showstheresults,usingourproposedapproach,inthesameformatasFigure1calibratedbothto1970-2012and1920-2012.Unliketheearlierresultsweseethatresultsareverysimilar.Indeed,giventhedi erencesweobserveinFigure1,thestabilityofthemodel-impliedspreadsisstrikingandsuggeststhatbyusingacross-sectionofdefaultratestocalibratethemodel,wecanprovideboth rmerandmorestableconclusions.5.ConclusionMuchoftheexistingliteratureontestingstructuralmodelsreliesheavilyonestimatesofdefaultprobabilitiesobtainedfromhistoricaldefaultfrequencies.One,muchusedapproachusesthehistoricaldefaultrateforasingleratingandmaturityasanestimateofthedefaultprobabilitywhencalculatingthespreadatthatmaturityandrating.We ndthattheoutcomeofthisapproachdependsstronglyonthehistoricalperiodfromwhichthedefaultfrequenciesareobtainedandweshowinsimulationsthatasinglehistoricaldefaultrateisaverynoisyestimatorofthedefaultprobability.Furthermore,thedistributionofthehistoricaldefaultrateforanyinvestmentgraderatingisskewed,implyingthattheobserved25 defaultratemostlikelyisbelowtheex-antedefaultprobability.Thisinturnimpliesthatwhentestinginvestmentgradespreadpredictionsofastructuralmodelcalibratedtothehistoricaldefaultrate,mostlikelythemodel-impliedspreadwillappeartoolowrelativetotheactualspreadevenifthestructuralmodelisthetruemodel.Weproposeanewmethodtocalibratestructuralmodelstohistoricaldefaultrates.Inthisapproachweextractthedefaultboundary{thevalueofthe rm,measuredasafractionofthefacevalueofdebt,belowwhichthe rmdefaults{byminimizingthedi erencebetweenactualandmodel-implieddefaultratesacrossawiderangeofmaturitiesandratings.Weshowthatthisapproachdramticallyimprovesthestatisticalpropertiesofestimatedinvestmentgradedefaultprobabilities,bothintermsofstandarddeviationandskewness.UsingourproposedapproachwetesttheBlack-CoxmodelusingUSdataonspreadsfromindividual rmsovertheperiod1987-2012.We ndthatmodelspreadsmatchaverageactualinvestmentgradecreditspreadswell.Inotherwordswedonot ndevidenceofa\creditspreadpuzzle".Goingbeyondtestingthepuzzle,we ndthatthetimeseriesofmodel-impliedinvestmentgradespreadstracksaverageactualinvestmentgradespreadswellwithacorrelationof93%.Incontrast,we ndthatthemodelunderpredictsspeculativegradespreadssigni cantly.Weexplorethepotentiale ectofbondilliquiditybysortingpricingerrors{thedi erencebetweenmodel-impliedandactualspreads{onbondage,aproxyforbondilliquidity.We ndnorelationbetweenpricingerrorsandbondilliquidityininvestmentgradebonds.How-ever,thereisastronglymonotonerelationbetweenaveragepricingerrorsandbondilliquidityinspeculativegradebonds,suggestingthattoalargeextentthemodelunderpredictionforspeculativegradebondsisduetoanilliquiditypremium.Ourresultsshowthatthecreditspreadpuzzle-theperceivedfailureofstructuralmodelstoexplainlevelsofcreditspreadsforinvestmentgradebonds-haslesstodowithde cienciesofthemodelsthanwiththewayinwhichtheyhavebeenimplemented.WefocusourattentionontheBlack-Coxmodel,butthemodelisonlyaconvenienttoolandourresultshaveimportantimplicationsforstructuralmodelsingeneral.TheresultsinHuangandHuang(2012)showthatmanystructuralmodelswhichappearverydi erentinfactgenerate26 similarspreadsoncethemodelsarecalibratedtothesamehistoricaldefaultrates,recoveryrates,andtheequitypremium.ThemodelstestedinHuangandHuang(2012)includefeaturessuchasstochasticinterestrates,endogenousdefault,stationaryleverageratios,strategicdefault,time-varyingassetriskpremia,andjumpsinthe rmvalueprocess,yetallgenerateasimilarlevelofcreditspread.Chen,Collin-Dufresne,andGoldstein(2009)pointoutthesigni canceofthis ndinganddiscusstherelationbetweenthepricingkernel,thedefaulttimeandLGDthatwouldresultinhigherspreads.Fromtheno-arbitragecondition,thepriceofacorporatebondcanbewrittenasP=1 rEEXs cov(ms;Xs)(10)whereXsisthebondcash\rowinstatesmsisthepricingkernelandristherisklessrate.Calibratingdi erentmodelstothesamehistoricaldefaultrateandrecoveryratewillresultinthe rstterminequation(10)beingsimilaracrossmodels.FixingtheSharperatio,recoveryrate,anddefaultratewillresultinthesecondtermbeingsimilarifdi erentmodelsresultinsimilarvaluesoftheelasticityofthebondpricetothe rm'sassetvalue.TheimplicationofthestrikingresultsinHHisthatforthewiderangeofmodelstheyconsidered,theelasticitiesareindeedsimilar.Therefore,allthestructuralmodelsconsideredbyHuangandHuang(2012){and,possiblystillothersthatgeneratesimilarelasticities{thatarecalibratedtothesamehistoricalrecoveryrate,thehistoricalSharperatio,anddefaultprobabilitieswillgeneratespreadsthataresimilar,bothtoeachotherandtotheBlack-Coxmodel,andtherefore,aswehavefound,consistentwithhistoricalinvestmentgradespreads.27 ATheBlack-CoxmodelWeassumethata rm'sassetvaluefollowsaGeometricBrownianMotionunderthenaturalmeasure,dVt Vt=( )dt+dWPt(11)whereisthepayoutratetodebtandequityholders,istheexpectedreturnonthe rm'sassetsandisthevolatilityofreturnsontheasset.The rmis nancedbyequityandasinglezero-couponbondwithfacevalueFandmaturityT.The rmdefaultsthe rsttimetheassetvalueisbelowsomefractiondofthefacevalueofdebt.Oneinterpretationofthedefaultboundaryisthatthebondhascovenantsinplacethatallowbondholderstotakeoverthe rmif rmvaluefallsbelowthethreshold.ThecumulativedefaultprobabilityintheBlack-CoxmodelattimeTis(seeBao(2009))P(dL;P;T)=Nh ( log(dL)+ 2 2T p Ti+exp2log(dL)( 2 2) 2Nhlog(dL)+ 2 2T p Ti(12)whereL=F V0istheleverageandP=(;;).Therisk-neutraldefaultprobability,Q,isobtainedbyreplacingwithrinequation(12).BFirmdataTocomputebondpricesintheMertonmodelweneedtheissuing rm'sleverageratio,payoutratio,andassetvolatility.ThisAppendixgivesdetailsonhowwecalculatethesequantitiesusingCRSP/Compustat.FirmvariablesarecollectedinCRSPandCompustat.Todosowematchabond'sCUSIPwithCRSP'sCUSIP.Intheorythe rst6digitsofthebondcusipplusthedigits`10'correspondstoCRSP'sCUSIP,butinpracticeonlyasmallfractionof rmsismatchedthisway.Evenifthereisamatchwecheckiftheissuing rmhasexperiencedM&Aactivityduringthelifeofthebond.Ifthereisnomatch,wehand-matchabondcusipwith rmvariablesinCRSP/Compustat.28 Leverageratio :Equityvalueiscalculatedonadailybasisbymultiplyingthenumberofsharesoutstandingwiththepriceofshares.DebtvalueiscalculatedinCompustatasthelatestquarterobservationoflong-termdebt(DLTTQ)plusdebtincurrentliabilities(DLCQ).LeverageratioiscalculatedasDebtvalue Debtvalue+EquityvaluePayoutratio :Thetotalout\rowtostakeholdersinthe rmisinterestpaymentstodebtholders,dividendpaymentstoequityholders,andnetstockrepurchases.Interestpaymentstodebtholdersiscalculatedasthepreviousyear'stotalinterestpayments(previousfourthquarter'sINTPNY).Dividendpaymentstoequityholdersistheindicatedannualdividend(DVI)multipliedbythenumberofshares.Theindicatedannualdividendisupdatedonadailybasisandisadjustedforstocksplitsetc.Netstockrepurchaseisthepreviousyear'stotalrepurchaseofcommonandpreferredstock(previousfourthquarter'sPRSTKCY).Thepayoutratioisthetotalout\rowtostakeholdersdividedby rmvalue,where rmvalueisequityvalueplusdebtvalue.Ifthepayoutratioislargerthan0.13,threetimesthemedianpayoutinthesample,wesetitto0.13.Equityvolatility :Wecalculatethestandarddeviationofdailyreturns(RETinCRSP)inthepastthreeyearstoestimatedailyvolatility.Wemultiplythedailystandarddeviationwithp 255tocalculateannualizedequityvolatility.Iftherearenoreturnobservationsonmorethanhalfthedaysinthethreeyearhistoricalwindow,wedonotcalculateequityvolatilityanddiscardanybondtransactionsonthatday.29 CConvexitybiaswhenusinga\representative rm"tocalculatedefaultprobabilitiesOur ndingintheSection4.3thattheBlack-CoxmodelmatchesdefaultprobabilitiesforBBB-rated rmsevenforhorizonsasshortasoneyearissurprising,sinceitisanestablishedstylizedfactintheliteraturethatshort-rundefaultprobabilitiesinstructuralmodelswithonlydi usion-riskaremuchtoolow.Papersshowingthatdefaultprobabilitiesatshorthori-zonsaretoolowincludeamongothersZhou(2001),Leland(2004),Leland(2006),Cremers,Driessen,andMaenhout(2008),Zhang,Zhou,andZhu(2009),andMcQuade(2013).Thereasonthatwearriveatadi erentconclusionisthatweallowforcross-sectionalvariationinassetvolatilityandbothcross-sectionalandtimeseriesvariationinleverageandpayoutrates.Incontrast,theexistingliteratureusesa\representative rm"withaverageassetvolatility,leverage,andpayoutratewithinagivenratingcategory.Usingarepresentative rmleadstoabiasduetoJensen'sinequalitybecausethedefaultprobabilityistypicallyconvexinassetvolatilityandleverage(whileitisclosetolinearinthepayoutrate).ThisconvexitybiasinthecaseofleverageisillustratedinFigureA1.Theconvexitybiaswhenusingarepresentative rmtocalculatespreadsisknowntotheliterature,butimportantlytheimpactoftheconvexitybiasontheshort-rundefaultprobabilitieshasnotbeenrecognizedintheliterature.12Todocumenttheimpactoftheconvexitybiasonshort-rundefaultprobabilitieswefocusonheterogeneityinleverageandcarryoutasimulationof100,000 rms.Foreach rmweuseanassetvolatilityof23%,apayoutrateof3.3%,andaSharperatioof0.22.The rmsdi eronlyintheirleverageratiosandwedraw100,000valuesfromanormaldistributionwithmean0.29andastandarddeviationof0.18.13ThechosenvaluesaremedianvaluesforBBB rmsandthestandarddeviationofleverageinthesimulationisequaltotheempiricalstandarddeviationofBBB rmsinthesample.Finally,theriskfreerateis5%.Foreach rmwecalculatethecumulativedefaultprobabilityfordi erentmaturities.PanelAinTable 12Bhamra,Kuehn,andStrebulaev(2010)presentastructural-equilibriummodelwithmacro-economicriskandsimulatedefaultratesover5and10yearsand ndthereisasubstantiale ectofallowingfor rmheterogeneity.Theydonotlookatdefaultprobabilitiesbelow veyears,ourmainfocus.13Ifasimulatedleverageratioisnegativewesetittozero.Thisimpliesthattheaverageleverageratioisslightlyhigherthan0.29,namely0.2937inoursimulation.30 A1showstheaveragedefaultprobabilityandthecorrectassetvolatilityof23%thatisusedforall rmsandatallmaturities.Zhou(2001),Leland(2004),Leland(2006),andMcQuade(2013)usevaluesofthelever-ageratio,payoutrate,andassetvolatilityaveragedovertimeand rmstocalculatemodel-implieddefaultprobabilitiesforarepresentative rmandthencomparethesewithhistoricalaverages.ToseetheextentoftheconvexitybiasintheBlack-Coxmodelwhenusingtheirapproach,wecalculatethetermstructureofdefaultprobabilitiesinPanelBofTableA1forarepresentative rmwithaleverageratioequaltothemeaninoursimulationof0.29.ThereisadownwardbiasindefaultprobabilitiesrelativetothecorrectvaluesgiveninPanelAandthebiasbecomesmorepronouncedatshortermaturities.Forexample,theone-yeardefaultprobabilityoftherepresentative rminPanelBis0.00%whilethetrueaveragedefaultprobabilityinPanelAis0.42%.Theaforementionedpaperscomparethedefaultprobabilityoftherepresentative rmwiththeaveragehistoricaldefaultrateandsincethehistoricaldefaultratere\rectstheaveragedefaultprobabilitiesinPanelA,theirresultsforparticularlyshort-maturitydefaultprobabilitiesarestronglybiased.Cremers,Driessen,andMaenhout(2008),Zhang,Zhou,andZhu(2009),andHuangandHuang(2012)letarepresentative rmmatchhistoricaldefaultratesbybackingoutassetvolatility.Toexaminehowtheconvexitybiasin\ruencestheimpliedassetvolatilityweproceedasfollows.Foragivenmaturity,wecomputetheassetvolatilitythatallowstherepresentative rmtomatchtheaveragedefaultprobabilityintheeconomyatthatgivenmaturity.PanelCshowstheimpliedassetvolatilitiesandweseethattherearetwoproblemswiththisapproach.The rstproblemisthatassetvolatilityisbiased:all rmsintheeconomyhaveanassetvolatilityof23%andyettheimpliedassetvolatilityrangesfrom27.0%atthe10-yearhorizonto43.3%attheone-yearhorizon.The ndingthatimpliedassetvolatilityinthedi usion-typestructuralmodelsistoohigh,particularatshorterhorizonshasbeenseenasafailureofthemodels,butthisexampleshowsthatthehighimpliedassetvolatilityarisesmechanicallyfromtheuseofarepresentative rm.Thesecondproblemisthatitisnotpossibletomatchthetermstructureofdefaultprobabilitieswithoutcounterfactuallychangingtheassetvolatilitymaturity-by-maturity.Cremers,Driessen,andMaenhout(2008)andZhang,Zhou,andZhu(2009)usearep-31 resentative rmtoimplyoutassetvolatilitybymatchinglong-termdefaultratesandthenusethisassetvolatilitytocalculatethetermstructureofdefaultprobabilities.Wereplicatethisapproachbyimplyingouttheassetvolatilitythatmakestherepresentative rm'sde-faultprobabilitymatchtheaveragedefaultprobabilityforthe10-yearbondintheeconomyandthencalculatethetermstructureofdefaultprobabilitiesforthisrepresentative rm.Theimpliedassetvolatilityis27.0%andthetermstructuresareinPanelD.Thedi erencebetweentheimpliedassetvolatilityof27.0%andthetruevalueof23%re\rectsamoderateconvexitybiasatthe10-yearhorizon,butsincethebiasbecomesmoresevereatshorterhorizons,thestrongdownwardbiasindefaultprobabilitiesreappearsasmaturitydecreases.Thus,thebiasinshort-termdefaultprobabilitiespersistswhenusingarepresentative rmandimputingassetvolatilitybymatchingalong-termdefaultrate14Insummary,weshowthatthetermstructureofdefaultprobabilitiesintheBlack-Coxmodelisdownwardbiased,andmoresoatshortmaturities,whenusingarepresentative rm.Thisislikelytobetrueforanystandardstructuralmodel:defaultprobabilitiesarestronglybiasdownwardsatshortmaturities.Existingevidence(showingthatdefaultprobabilitiesatshorthorizonsaremuchtoolow)inZhou(2001),Leland(2004),Leland(2006),Cremers,Driessen,andMaenhout(2008),Zhang,Zhou,andZhu(2009),andMcQuade(2013)issubjecttothisstrongbiasandthereforenotreliable. 14OurresultsclarifythoseinBhamra,Kuehn,andStrebulaev(2010).Withintheframeworkoftheirstructural-equilibriummodel,theycomparearepresentative rmwithacrosssectionof rmsand ndthattheslopeofthetermstructureofdefaultprobabilitiesis\ratterforthecrosssectionof rms.Intheirexperiment,thecrosssectionof rmshaveanaveragedefaultprobabilitythatismorethanthreetimesaslargeasthedefaultprobabilityoftherepresentative rm(theirTable3,PanelsBandC).Sincethetermstructureofdefaultprobabilitiesbecomes\ratterforarepresentative rmatthesametimeasdefaultriskincreases,itisnotclearifitiscrosssectionalvariationortheriseindefaultprobabilitythatdrivesthe\ratteningofthetermstructure.Sinceweholdthe10-yeardefaultprobability xedinPanelsAandD,itisclearinouranalysisthatthe\rattertermstructureisdrivenbycross-sectionalvariationinleveragealone.32 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 0.5 1 1.5 leverage ratiodefault probabilitiy in percent1-year default probability of a typical investment firm in the Black-Cox model low leverage high leverage average leverageodef prob of average leverage oaverage def prob Fig.A1ConvexitybiaswhencalculatingthedefaultprobabilityintheBlack-Coxmodelusingaverageleverageandcomparingittotheaveragedefaultprobability.Itiscommonintheliteraturetocomparehistoricaldefaultratestomodel-implieddefaultprobabilities,wherethelatterarecalculatedusingaverage rmvariables.Thisintroducesabiasbecausethedefaultprobabilityinstructuralmodelsisanon-linearfunctionof rmvariables.The gureillustratesthebiasincaseoftwo rmobservationswiththesamerating,onewithalowleverageratioandonewithahighleverageratio.Thetwoobservationscanbetwodi erent rmsatthesamepointintimeorthesame rmattwodi erentpointsintime.Assetvolatilityis5%,dividendyield3.7%,Sharperatio0.22,andriskfreerate5%.33 maturity12345678910 PanelA:Trueeconomy(thereisvariationinleverageratios) averagedefaultprobability0.130.591.241.982.753.504.234.925.586.20assetvolatility25.025.025.025.025.025.025.025.025.025.0PanelB:Representative rm(averageleverageratioused) defaultprobability0.000.000.050.200.490.891.371.902.463.03assetvolatility25.025.025.025.025.025.025.025.025.025.0PanelC:Representative rm,averagedef.prob.atbondmaturityismatched defaultprobability0.130.591.241.982.753.504.234.925.586.20impliedassetvolatility44.337.334.332.631.430.529.929.328.928.6PanelD:Representative rm,averagedef.prob.at10-yearbondmaturityismatched defaultprobability0.000.030.240.741.472.343.304.285.256.20impliedassetvolatility28.628.628.628.628.628.628.628.628.628.6TableA1ConvexitybiaswhencalculatingdefaultprobabilitiesintheBlack-Coxmodelusingtherepresentative rmapproach.Itiscommonintheliteraturetocompareaverageactualdefaultratestomodel-implieddefaultprobabilities,wheremodel-implieddefaultprobabilitiesarecalculatedusingaverage rmvariables.ThisintroducesabiasbecausethedefaultprobabilityandspreadintheMertonmodelisanon-linearfunctionof rmvariables.Thistableshowsthemagnitudeofthisbias.PanelAshows,formaturitiesbetweenoneand10years,theaveragedefaultprobabilityfor100,000 rmsthathavedi erentleverageratiosbutareotherwiseidentical.Theircommonassetvolatilityis25%andpayoutrate3.7%.Theirleverageratiosaresimulatedfromanormaldistributionwithmean0.28andstandarddeviation0.18(truncatedatzero).Theriskfreerateis5%.PanelBshowsthedefaultprobabilityofarepresentative rmwheretheaverageleverageratioisused.InPanelC,foreachmaturity{oneatatime{anassetvolatilityiscomputedsuchthat,forarepresentative rmwithaleverageratioequaltotheaverageleverageratio,thedefaultprobabilityisequaltotheaveragedefaultprobabilityintheeconomy(giveninthe rstrowinPanelAandagaininPanelC).Thisisdoneseparatelyforeachmaturity.Thepanelshowstheresultingimpliedassetvolatility.PanelDshowstheresultsofacalculationsimilartothatinPanelCexceptheretheassetvolatilityusedtocomputethedefaultprobabilityforeachmaturityisthevaluethatmatchestheaverage10-yeardefaultprobabilityintheeconomy.34 ReferencesAcharya,V.,Y.Amihud,andS.Bharath(2013).Liquidityriskofcorporatebondreturns:conditionalapproach.JournalofFinancialEconomics110(2),358{386.Bai,H.(2016).Unemploymentandcreditrisk.WorkingpaperBao,J.(2009).StructuralModelsofDefaultandtheCross-SectionofCorporateBondYieldSpreads.WorkingpaperBao,J.,J.Pan,andJ.Wang(2011).TheilliquidityofCorporateBonds.JournalofFinance66,911{946.Bhamra,H.S.,L.-A.Kuehn,andI.A.Strebulaev(2010).TheLeveredEquityRiskPre-miumandCreditSpreads:AUni edFramework.ReviewofFinancialStudies23(2),645{703.Black,F.andJ.Cox(1976).ValuingCorporateSecurities:SomeE ectsofBondIndentureProvisions.JournalofFinance31,351{367.Booth,L.,D.Gounopoulos,andF.Skinner(2014).Thechoicebetweencallableandnoncallablebonds.TheJournalofFinancialResearch37(4),435{459.Campello,M.,L.Chen,andL.Zhang(2008).Expectedreturns,yieldspreads,andassetpricingtests.ReviewofFinancialStudies21(3),1297{1338.Chen,H.(2010).Macroeconomicconditionsandthepuzzlesofcreditspreadsandcapitalstructure.JournalofFinance65(6),2171{2212.Chen,H.,R.Cui,Z.He,andK.Milbradt(2016).QuantifyingLiquidityandDefaultRisksofCorporateBondsovertheBusinessCycle.WorkingPaperChen,L.,P.Collin-Dufresne,andR.S.Goldstein(2009).Ontherelationbetweenthecreditspreadpuzzleandtheequitypremiumpuzzle.ReviewofFinancialStudies223367{3409.Christo ersen,P.andD.D.R.Elkamhi(2016).RareDisasters,Credit,andOptionMarketPuzzles.forthcoming,ManagementScience35 Collin-Dufresne,P.,R.Goldstein,andS.Martin(2001).TheDeterminantsofCreditSpreadChanges.JournalofFinance56,2177{2207.Correia,M.,J.Kang,andS.Richardson(2014).Assetvolatility.WorkingpaperCremers,M.,J.Driessen,andP.Maenhout(2008).Explainingthelevelofcreditspreads:Option-impliedjumpriskpremiaina rmvaluemodel.ReviewofFinancialStudies212209{2242.Dick-Nielsen,J.,P.Feldhutter,andD.Lando(2012).Corporatebondliquiditybeforeandaftertheonsetofthesubprimecrisis.JournalofFinancialEconomics103,471{492.Du ee,G.R.(1998).TheRelationBetweenTreasuryYieldsandCorporateBondYieldSpreads.JournalofFinance53(6),2225{2241.Eom,Y.H.,J.Helwege,andJ.-Z.Huang(2004).Structuralmodelsofcorporatebondpricing:Anempiricalanalysis.ReviewofFinancialStudies17(2),499{544.Ericsson,J.,J.Reneby,andH.Wang(2015).Canstructuralmodelspricedefaultrisk?Evidencefrombondandcreditderivativemarkets.QuarterlyJournalofFinance5(3).Feldhutter,P.(2012).Thesamebondatdi erentprices:identifyingsearchfrictionsandsellingpressures.ReviewofFinancialStudies25,1155{1206.Feldhutter,P.andD.Lando(2008).DecomposingSwapSpreadsJournalofFinancialEconomics88,375{405.Friewald,N.,R.Jankowitsch,andM.G.Subrahmanyam(2012).Illiquidityorcreditdete-rioration:astudyofliquidityintheUScorporatebondmarketduring nancialcrises.105,18{36.JournalofFinancialEconomics.Gomes,J.,U.Jermann,andL.Schmid(2016).StickyLeverage.forthcoming,AmericanEconomicReviewHe,Z.andK.Milbradt(2014).EndogenousLiquidityandDefaultableDebt.Economet-rica82(4),1443{1508.Houweling,P.,A.Mentink,andT.Vorst(2005).Comparingpossibleproxiesofcorporatebondliquidity.JournalofBankingandFinance29,1331{1358.36 Huang,J.andM.Huang(2012).HowMuchoftheCorporate-TreasuryYieldSpreadisDuetoCreditRisk?ReviewofAssetPricingStudies2(2),153{202.Krishnamurthy,A.andA.Vissing-Jorgensen(2012).TheAggregateDemandforTreasuryDebt.JournalofPoliticalEconomy120(2),233{267.Leland,H.(2004).PredictionsofDefaultProbabilitiesinStructuralModelsofDebt.JournalofInvestmentManagement2Leland,H.(2006).StructuralModelsofCorporateFinancialChoice.PrincetonLecturesinFinance,Lecture1Lin,H.,J.Wang,andC.Wu(2011).Liquidityriskandexpectedcorporatebondreturns.JournalofFinancialEconomics99,628{650.McQuade,T.J.(2013).StochasticVolatilityandAssetPricingPuzzles.WorkingPaper,HarvardUniversityMerton,R.(1974).OnthePricingofCorporateDebt:TheRiskStructureofInterestRates.JournalofFinance29,449{470.Moody's(2002).DefaultandrecoveryRatesofCorporateBondissuers-AStatisticalReviewofMoody'sRatingsperformance1970-2001.Moody'sinvestorsService,1{52.Moody's(2013).AnnualDefaultStudy:CorporateDefaultandRecoveryRates,1920-2012.Moody'sInvestorsService,1{64.Nagel,S.(2014).TheLiquidityPremiumofNear-MoneyAssets.WorkingpaperSchaefer,S.andI.Strebulaev(2008).Structuralmodelsofcreditriskareuseful:Evidencefromhedgeratiosoncorporatebonds.JournalofFinancialEconomics90,1{19.Zhang,B.Y.,H.Zhou,andH.Zhu(2009).ExplainingCreditDefaultSwapSpreadswiththeEquityVolatilityandJumpRisksofIndividualFirms.ReviewofFinancialStudies22(12),5101{5131.Zhou,C.(2001).Thetermstructureofcreditspreadswithjumprisk.JournalofBankingandFinance25,2015{2040.37 horizon(years)4567891011121314151617181920 AAA Newmethod0.000.010.020.030.050.070.090.110.140.160.180.200.220.240.260.280.29Existingmethod0.020.050.110.210.340.520.720.951.201.461.732.002.272.542.793.033.27Ratio10%14%15%15%14%13%13%12%11%11%10%10%10%10%9%9%9% AA Newmethod0.000.010.020.040.060.080.100.130.150.170.200.220.240.260.280.290.31Existingmethod0.010.050.110.220.370.560.781.031.291.561.852.132.412.682.943.193.44Ratio17%18%17%16%15%14%13%12%12%11%11%10%10%10%9%9%9% A Newmethod0.010.020.040.070.100.140.170.200.230.260.290.320.340.360.380.400.42Existingmethod0.030.100.220.400.640.911.231.571.912.262.622.973.303.633.944.234.51Ratio24%22%20%18%16%15%14%13%12%12%11%11%10%10%10%9%9% BBB Newmethod0.080.150.220.290.360.420.480.520.570.600.640.670.690.710.730.750.77Existingmethod0.240.530.931.401.922.473.053.634.194.725.255.766.226.667.067.447.78Ratio32%27%24%21%19%17%16%14%14%13%12%12%11%11%10%10%10% BB Newmethod0.981.171.311.401.471.521.551.581.591.601.611.611.621.621.611.611.61Existingmethod2.273.284.235.156.026.847.648.389.099.7610.4010.9911.5512.0612.5312.9413.32Ratio43%36%31%27%24%22%20%19%18%16%15%15%14%13%13%12%12% B Newmethod3.113.093.052.992.932.872.822.772.722.682.642.602.562.532.502.472.44Existingmethod5.356.277.067.778.419.019.5510.0610.5511.0211.4711.8912.2812.6512.9813.3013.59Ratio58%49%43%38%35%32%30%28%26%24%23%22%21%20%19%19%18% C Newmethod5.214.844.564.334.154.003.873.753.653.573.493.423.353.303.243.193.15Existingmethod5.155.455.725.986.236.456.676.877.097.297.497.687.868.038.198.368.50Ratio101%89%80%72%67%62%58%55%52%49%47%44%43%41%40%38%37%Table1Standarddeviationoftheestimateddefaultprobability.Thistableshowsthestandarddeviationoftheestimateddefaultprobability(inpercent)whentheestimateisbasedonaveragehistoricaldefaultratesover31years.Thetableisbasedon25,000simulationsofaveragedefaultratesover31years.`Existingmethod'referstothestandardapproachintheliteratureofusingtheaveragehistoricaldefaultrateforasingleratingandmaturityasanestimateoftheex-antedefaultprobability.`Newmethod'referstothemethodproposedinSection3.2wherethedefaultprobabilityestimateforasingleratingandmaturityisextractedusingtheBlack-Coxmodeland1,2,...,20yeardefaultratesforratingsAAA,AA,A,BBB,BB,BandC.`Ratio'isthethestandarddeviationofdefaultprobabilityestimatesusingthenewmethoddividedbythestandarddeviationofdefaultprobabilityestimatesusingtheexistingmethod.38 horizon(years)4567891011121314151617181920 AAA Newmethod0.510.360.250.180.120.070.040.01-0.02-0.04-0.06-0.07-0.09-0.10-0.11-0.12-0.13Existingmethod3.572.872.562.352.232.051.901.761.631.511.401.301.211.131.051.000.94Ratio14%12%10%8%5%4%2%0%-1%-3%-4%-6%-7%-9%-10%-12%-14% AA Newmethod0.500.350.240.170.110.070.030.00-0.02-0.04-0.06-0.08-0.09-0.10-0.11-0.12-0.13Existingmethod3.353.042.692.442.232.061.911.771.631.511.391.291.211.121.051.000.94Ratio15%11%9%7%5%3%2%0%-1%-3%-4%-6%-7%-9%-11%-12%-14% A Newmethod0.440.300.200.130.080.040.01-0.02-0.04-0.06-0.08-0.09-0.11-0.12-0.13-0.14-0.14Existingmethod2.892.562.362.172.001.871.731.601.491.381.281.191.111.040.980.920.87Ratio15%12%9%6%4%2%0%-1%-3%-5%-6%-8%-10%-11%-13%-15%-17% BBB Newmethod0.290.180.100.050.01-0.03-0.05-0.08-0.09-0.11-0.12-0.13-0.14-0.15-0.16-0.17-0.17Existingmethod1.751.631.541.481.401.341.281.211.151.091.020.950.900.840.790.740.69Ratio17%11%7%3%0%-2%-4%-6%-8%-10%-12%-14%-16%-18%-20%-23%-25% BB Newmethod0.080.00-0.05-0.08-0.11-0.13-0.14-0.16-0.17-0.18-0.19-0.19-0.20-0.20-0.21-0.21-0.22Existingmethod0.730.690.660.640.630.630.620.610.600.580.570.550.530.500.470.440.41Ratio10%0%-7%-13%-17%-20%-23%-26%-28%-30%-33%-35%-38%-41%-44%-48%-53% B Newmethod-0.08-0.12-0.15-0.17-0.19-0.20-0.21-0.21-0.22-0.22-0.23-0.23-0.24-0.24-0.24-0.24-0.24Existingmethod0.240.210.190.170.160.150.140.130.120.100.090.080.070.060.040.02-0.00Ratio-35%-60%-79%-99%-115%-132%-149%-170%-189%-215%-241%-278%-330%-404%-605%-1301%28766% C Newmethod-0.20-0.21-0.23-0.23-0.24-0.24-0.25-0.25-0.25-0.25-0.26-0.26-0.26-0.26-0.26-0.26-0.26Existingmethod-0.10-0.13-0.16-0.18-0.21-0.22-0.24-0.26-0.28-0.30-0.32-0.34-0.36-0.38-0.41-0.44-0.47Ratio195%159%142%129%116%109%102%95%89%85%80%76%72%68%64%59%56%Table2Skewnessoftheestimateddefaultprobability.Thistableshowstheskewnessoftheestimateddefaultprobability(inpercent)whentheestimateisbasedonaveragehistoricaldefaultratesover31years.Thetableisbasedon25,000simulationsofaveragedefaultratesover31years.`Existingmethod'referstothestandardapproachintheliteratureofusingtheaveragehistoricaldefaultrateforasingleratingandmaturityasanestimateoftheex-antedefaultprobability.`Newmethod'referstothemethodproposedinSection3.2wherethedefaultprobabilityestimateforasingleratingandmaturityisextractedusingtheBlack-Coxmodeland1,2,...,20yeardefaultratesforratingsAAA,AA,A,BBB,BB,BandC.`Ratio'isthetheskewnessofdefaultprobabilityestimatesusingthenewmethoddividedbytheskewnessofdefaultprobabilityestimatesusingtheexistingmethod.39 3-7-yearbondmaturity AAAAAABBBBBBCall Numberofbonds211093272891163614753Meannumberofbondsprmonth1.697.5422.3186.941.710.3758.4Meannumberofquotesprmonth1.68745.68105.7109.857.9913.314.497338.7Age2.054.735.195.894.373.072.505.08Coupon7.196.307.007.577.629.6111.727.36Amountoutstanding($mm)265329277321414300231322Time-to-maturity4.744.734.874.774.925.255.234.857-13-yearbondmaturity AAAAAABBBBBBCall Numberofbonds16942882761002611680Meannumberofbondsprmonth1.287.9820.918.15.040.930.354.4Meannumberofquotesprmonth1.91328.1967.370.5227.125.613.35204Age7.196.955.674.882.865.219.465.26Coupon7.217.297.307.847.769.239.957.64Amountoutstanding($mm)400308265294535222270317Time-to-maturity10.378.969.269.148.748.9410.149.1213-20-yearbondmaturity AAAAAABBBBBBCall Numberofbonds32181753091173Meannumberofbondsprmonth0.3072.068.286.262.090.4730.023319.5Meannumberofquotesprmonth3.732.0626.0417.945.4472.2430.4157.87Age14.632.407.745.649.6411.019.317.66Coupon8.298.048.178.258.417.289.228.19Amountoutstanding($mm)687208332218227157183297Time-to-maturity15.1916.7716.6115.8416.9517.2714.0716.33 Table3Bondsummarystatistics.Thesampleconsistsofnoncallablebondswith xedcouponsissuedbyindustrial rms.Thistableshowssummarystatisticsforthedataset.Bondyieldquotescovertheperiod1987Q2-2012Q2.`Numberofbonds'isthenumberofbondsthatappear(inaparticularratingandmaturityrange)atsomepointinthesampleperiod.`Meannumberofbondsprmonth'istheaveragenumberofbondsthatappearinamonth.`Meannumberofquotesprmonth'isthetotalnumberofquotesinthesampleperioddividedbythenumberofmonths.Foreachquotewecalculatethebond'stimesinceissuanceand`Age'istheaveragetimesinceissuanceacrossallquotes.`Coupon'istheaveragebondcouponacrossallquotes.`Amountoutstanding'istheaverageoutstandingamountofabondissueacrossallquotes.`Time-to-maturity'istheaveragetimeuntilthebondmaturesacrossallquotes.40 # rmsMean10th25thMedian75th90th Leverageratio AAA100.130.070.080.090.140.23AA530.140.070.100.130.170.23A1700.270.130.170.240.340.46BBB1970.370.170.250.360.480.58BB1000.460.180.290.470.610.73B400.520.210.320.480.730.83C60.760.600.700.800.920.96all3930.330.110.180.290.450.61Equityvolatility AAA100.190.150.170.180.200.24AA530.270.180.220.250.330.37A1700.300.200.240.300.360.41BBB1970.370.240.270.340.420.55BB1000.460.250.310.420.530.74B400.510.310.360.480.650.78C60.730.340.640.730.791.02all3930.350.210.250.320.400.54Assetvolatility AAA100.180.150.150.190.190.19AA530.230.190.220.230.250.28A1700.240.170.220.240.280.29BBB1970.250.170.190.240.280.36BB1000.270.170.230.250.280.40B400.280.160.220.250.390.42C60.260.170.180.220.260.42all3930.250.170.220.240.280.33Payoutratio AAA100.0360.0120.0170.0300.0490.071AA530.0410.0150.0270.0400.0530.066A1700.0470.0190.0300.0430.0580.079BBB1970.0500.0170.0270.0450.0650.098BB1000.0450.0200.0270.0400.0570.078B400.0460.0180.0290.0410.0610.077C60.0680.0370.0490.0560.0960.103all3930.0470.0180.0280.0430.0590.083Table4Firmsummarystatistics,industrial rmsinCompustatwithstraightbulletbondsoutstandingForeachbondyieldobservation,theleverageratio,equityvolatility,assetvolatility,andpayoutratioarecalculatedfortheissuing rmonthedayoftheobservation.Leverageratioistheratioofthebookvalueofdebttothemarketvalueofequityplusthebookvalueofdebt.Equityvolatilityistheannualizedvolatilityofdailyequityreturnsfromthelastthreeyears.Assetvolatilityistheunleveredequityvolatility,calculatedasexplainedinthetext.Payoutratioisyearlyinterestpaymentsplusdividendsplussharerepurchasesdividedby rmvalue.FirmvariablesarecomputedusingdatafromCRSPandCompustat.41 # rmsMean10th25thMedian75th90th Leverageratio AAA190.120.020.030.070.150.34AA950.150.040.070.110.180.29A3850.200.060.100.170.270.39BBB6500.280.080.150.250.370.51BB9980.390.130.230.370.540.70B10140.530.210.350.530.720.86C1620.720.380.580.770.890.95all20870.350.080.160.300.500.71Equityvolatility AAA190.260.170.210.260.300.35AA950.280.190.220.270.320.37A3850.310.210.250.300.360.42BBB6500.370.240.280.340.420.52BB9980.480.310.370.450.560.68B10140.650.380.480.610.770.94C1620.880.530.620.801.051.24all20870.450.240.300.400.550.72Assetvolatility AAA190.230.180.220.230.260.26AA950.240.200.210.240.260.29A3850.260.200.220.240.280.35BBB6500.280.200.220.270.320.38BB9980.320.210.250.300.380.44B10140.340.200.250.320.410.51C1620.330.180.250.310.400.50all20870.300.200.230.280.350.43Payoutratio AAA190.0270.0080.0140.0230.0350.050AA950.0260.0050.0110.0200.0340.053A3850.0320.0070.0140.0260.0430.067BBB6500.0370.0090.0160.0300.0490.078BB9980.0390.0090.0180.0330.0540.078B10140.0490.0100.0240.0440.0680.092C1620.0670.0220.0450.0680.0890.108all20870.0390.0080.0170.0320.0540.081Table5Firmsummarystatistics,industrial rmsinCompustatwitharating.Foreach rminCompustatforwhichthereisanS&PratinginCapitalIQ,theleverageratio,equityvolatility,assetvolatility,andpayoutratioarecalculatedonDecember31ineachyear1985-2012.Leverageratioistheratioofthebookvalueofdebttothemarketvalueofequityplusthebookvalueofdebt.Equityvolatilityistheannualizedvolatilityofdailyequityreturnsfromthelastthreeyears.Assetvolatilityistheunleveredequityvolatility,calculatedasexplainedinthetext.Payoutratioisyearlyinterestpaymentsplusdividendsplussharerepurchasesdividedby rmvalue.FirmvariablesarecomputedusingdatafromCRSPandCompustat.42 horizon(years)123456810121520 AAA Model0.010.040.100.160.220.280.410.540.670.871.18Actual0.000.010.030.090.170.250.520.871.161.381.7195%c.b.(NaN;NaN)(0.00;0.04)(0.00;0.09)(0.01;0.21)(0.04;0.38)(0.06;0.58)(0.16;1.17)(0.26;1.96)(0.34;2.69)(0.36;3.37)(0.37;4.52) AA Model0.010.080.190.310.450.590.871.161.451.892.58Actual0.070.220.350.540.831.171.832.503.344.525.8595%c.b.(0.03;0.13)(0.11;0.37)(0.17;0.62)(0.25;0.99)(0.39;1.54)(0.53;2.17)(0.79;3.48)(1.03;4.88)(1.34;6.59)(1.70;9.09)(2.0;12.3) A Model0.020.140.340.600.911.262.002.773.554.676.37Actual0.100.310.640.991.381.782.663.624.615.997.9395%c.b.(0.05;0.17)(0.17;0.51)(0.34;1.06)(0.51;1.69)(0.70;2.39)(0.88;3.16)(1.24;4.82)(1.62;6.69)(1.99;8.64)(2.4;11.5)(2.9;15.8) BBB Model0.300.951.782.703.644.576.378.049.5411.5314.24Actual0.290.861.542.293.103.905.467.118.7210.8713.7695%c.b.(0.18;0.44)(0.52;1.29)(0.92;2.38)(1.33;3.60)(1.75;4.95)(2.14;6.31)(2.87;9.01)(3.6;11.9)(4.3;14.8)(5.2;18.9)(6.2;24.5) BB Model2.235.618.7611.5413.9916.1519.7822.7225.1528.0931.74Actual1.353.275.467.759.9211.9715.7119.2722.4726.6532.0695%c.b.(0.95;1.83)(2.27;4.48)(3.73;7.58)(5.2;10.8)(6.6;14.0)(7.8;17.0)(10.1;22.6)(12.2;27.8)(14.2;32.6)(16.6;38.8)(19.4;47.0) B Model8.4316.4022.3927.0430.7833.8638.6542.2445.0548.3152.17Actual3.808.7113.7218.1622.0625.5431.4135.8939.5844.2249.1495%c.b.(2.93;4.78)(6.6;11.1)(10.4;17.6)(13.7;23.3)(16.5;28.4)(19.0;33.0)(23.3;40.6)(26.5;46.4)(29.0;51.3)(32.2;57.3)(35.3;63.9) C Model23.3236.1143.7949.0752.9956.0560.5763.8066.2469.0072.17Actual14.0223.8131.2136.8641.4044.7849.6353.8858.0263.7671.3495%c.b.(11.8;16.4)(19.9;28.1)(26.0;37.0)(30.6;43.8)(34.2;49.1)(36.9;53.2)(40.6;59.0)(44.0;64.0)(47.6;68.8)(52.8;74.8)(60.0;82.2)Table6AveragedefaultprobabilitiesintheBlack-Coxmodelandhistoricaldefaultrates.Wemerge rmdatafromCRSP/CompustatwithratingsfromStandard&Poorsandforevery rmandeveryyear1985-2012wecalculatea1-,2-,...,19-,20-yeardefaultprobabilityintheBlack-Coxmodel.`Model'showstheaveragedefaultprobabilities.`Actual'showsMoody'saveragehistoricaldefaultrates1920-2012.`95%c.b.'shows95%con dencebandsforthehistoricaldefaultratescalculatedfollowingtheapproachinSection3.1.`*'and`**'showwhenthemodel-implieddefaultprobabilityisoutsidethe95%and99%con dencebandrespectively.43 3-20y 3-7y7-13y13-20y InvActualspread 92 898787Modelspread 111(88;128) 107(82;127)107(87;121)88(76;96)Di erence 19( 4;36) 18( 8;38)20(0;34)1( 11;9)Observations 294 294293244 SpecActualspread 544 560417461Modelspread 382(305;440) 376(289;443)392(336;429)314(279;337)Di erence 162( 239; 104) 184( 271; 117) 25( 81;12) 147( 182; 124)Observations 289 276229141 AAAActualspread 16 4622Modelspread 3(2;4) 3(1;6)1(0;1)2(1;2)Di erence 13( 14; 12) 0( 3;2) 6( 6; 5) 20( 20; 20)Observations 132 707091 AAActualspread 23 173426Modelspread 9(6;10) 2(1;3)14(11;17)19(15;22)Di erence 15( 17; 13) 15( 16; 14) 20( 23; 17) 7( 11; 4)Observations 289 27926493 AActualspread 61 506563Modelspread 85(67;99) 67(48;82)102(82;117)83(71;90)Di erence 24(6;38) 17( 2;32)37(17;51)19(8;27)Observations 294 294293223 BBBActualspread 146 141141144Modelspread 169(134;195) 165(126;195)166(137;186)131(112;143)Di erence 23( 12;49) 24( 15;54)25( 4;46) 14( 32; 1)Observations 291 291257198 BBActualspread 377 370290398Modelspread 349(282;397) 320(247;374)337(285;372)255(223;277)Di erence 27( 94;21) 51( 124;4)46( 5;82) 142( 175; 121)Observations 259 240216114 BActualspread 675 723427445Modelspread 445(360;509) 480(376;560)441(376;485)323(294;342)Di erence 229( 314; 166) 243( 347; 163)15( 51;59) 122( 150; 103)Observations 243 20313482 CActualspread 1442 12111948661Modelspread 958(828;1041) 1097(920;1209)783(709;829)525(449;575)Di erence 484( 615; 401) 114( 291; 2) 1165( 1239; 1119) 136( 212; 86)Observations 96 65427 Table7Actualandmodelyieldspreads.Thistableshowsactualandmodel-impliedcorporatebondyieldspreads.Spreadsaregroupedaccordingtoremainingbondmaturityatthequotationdate.`Actualspread'istheaverageactualspreadtotheswaprate.`Modelspread'istheaverageBlack-Coxmodelspreadsofthebondsinagivenmaturity/ratingbucket.Theaveragespreadiscalculatedby rstcalculatingtheaveragespreadofbondsinagivenmonthandthencalculatingtheaverageofthesespreadsovermonths.`Di erence'isthedi erencebetweenthemodelspreadandtheactualspread.Inparenthesisare95%con dencebandscalculatedaccordingSection3.2;`*'impliessigni canceatthe5%leveland`**'atthe1%level.`Observations'isthenumberofmonthlyobservations.Thebondyieldspreadsarefromtheperiod1987-2012.44 3-20y 3-7y7-13y13-20y 20bpsActualspread 7 7810Modelspread 15(11;18) 14(9;18)18(14;22)19(16;22)Di erence 8(3;11) 7(2;11)10(6;13)9(5;12)Observations 279 279214138 20-40bpsActualspread 30 293031Modelspread 37(27;45) 30(20;39)54(41;63)36(30;41)Di erence 7( 3;15) 1( 9;10)23(11;32)5( 1;9)Observations 279 272233165 40-70bpsActualspread 55 545556Modelspread 72(55;85) 69(49;85)84(66;98)56(47;62)Di erence 17( 0;30) 15( 5;31)29(11;42)0( 9;7)Observations 284 277262191 70-100bpsActualspread 84 858484Modelspread 104(80;122) 115(81;142)118(94;135)87(74;96)Di erence 20( 4;38) 30( 4;57)34(10;50)3( 10;12)Observations 281 264246170 100-150bpsActualspread 121 121120121Modelspread 136(105;160) 145(102;179)140(112;160)121(104;133)Di erence 15( 16;39) 25( 19;59)19( 9;39) 0( 18;11)Observations 269 260254166 150-200bpsActualspread 172 172172171Modelspread 174(135;203) 167(120;204)188(153;213)144(123;158)Di erence 2( 37;31) 5( 52;32)15( 20;40) 27( 48; 13)Observations 252 222211120 200-300bpsActualspread 243 242243245Modelspread 257(202;297) 251(185;303)305(253;343)218(188;238)Di erence 14( 40;54) 9( 57;60)62(10;100) 28( 57; 8)Observations 267 22222099 300-1000bpsActualspread 499 523456517Modelspread 507(414;573) 558(442;642)499(430;546)368(332;391)Di erence 8( 85;74) 35( 81;119)43( 26;90) 149( 185; 127)Observations 268 244221150 �1000bpsActualspread 1744 174617891166Modelspread 909(747;1026) 1004(806;1151)735(646;792)513(455;550)Di erence 835( 997; 718) 742( 940; 595) 1055( 1143; 998) 653( 711; 616)Observations 132 895915 Table8Actualandmodelyieldspreadssortedaccordingtoactualspread.Thistableshowsactualandmodel-impliedcorporatebondyieldspreads.Spreadsaregroupedaccordingtothesizeoftheactualspreadandtheremainingbondmaturityatthequotationdate.`Actualspread'istheaverageactualspreadtotheswaprate.`Modelspread'istheaverageBlack-Coxmodelspreadsofthebondsinagivenmaturity/ratingbucket.Theaveragespreadiscalculatedby rstcalculatingtheaveragespreadofbondsinagivenmonthandthencalculatingtheaverageofthesespreadsovermonths.`Di erence'isthedi erencebetweenthemodelspreadandtheactualspread.Inparenthesisare95%con dencebandscalculatedaccordingSection3.2;`*'impliessigni canceatthe5%leveland`**'atthe1%level.`Observations'isthenumberofmonthlyobservations.Thebondyieldspreadsarefromtheperiod1987-2012.45 PanelA:Regressioninlevels 3-20y 3-7y7-13y13-20y Inv 0:88(0:06) 0:81(0:08)0:82(0:09)0:79(0:13)R2 0.87 0.840.690.59 Spec 0:85(0:38) 0:82(0:41)0:80(0:19)1:55(0:20)R2 0.16 0.150.250.75 AAA 1:17(0:44) 1:40(0:14)1:63(0:22)R2 0.15 0.260.22 AA 1:07(0:58) 0:81(0:34)1:57(0:22) 0:10(0:04)R2 0.22 0.030.620.08 A 0:65(0:08) 0:52(0:19)0:54(0:11)0:40(0:09)R2 0.70 0.470.570.41 BBB 0:82(0:07) 0:71(0:09)0:90(0:10)0:72(0:15)R2 0.88 0.810.740.62 BB 0:72(0:28) 0:70(0:31)0:64(0:14)1:66(0:28)R2 0.42 0.390.490.79 B 0:61(0:28) 0:64(0:30)0:31(0:19)1:14(0:23)R2 0.09 0.100.060.58 C 0:75(0:56) 0:34(0:56)R2 0.04 0.01PanelB:Regressioninchanges 3-20y 3-7y7-13y13-20y Inv 0:52(0:08) 0:47(0:08)0:37(0:09)0:62(0:10)R2 0.35 0.330.170.37 Spec 0:00(0:50) 0:42(0:49)0:76(0:22)1:30(0:24)R2 0.00 0.010.170.45 AAA 0:56(0:27) 0:54(0:49)0:57(0:41)R2 0.11 0.070.08 AA 0:25(0:11) 0:21(0:30)0:23(0:12) 0:08(0:05)R2 0.06 0.010.060.11 A 0:37(0:07) 0:50(0:07)0:18(0:06)0:47(0:08)R2 0.27 0.440.100.36 BBB 0:46(0:08) 0:33(0:08)0:31(0:11)0:67(0:09)R2 0.32 0.200.110.54 BB 0:66(0:11) 0:61(0:12)0:68(0:10)1:33(0:30)R2 0.34 0.310.470.41 B 1:02(0:32) 0:92(0:33) 0:18(0:26)1:39(0:37)R2 0.14 0.130.010.41 C 2:12(1:49) 2:37(2:28)R2 0.08 0.06 Table9Commonalityintimeseriesvariationofactualandmodel-impliedyieldspreads.Foragivenratingandmaturitygroupwecalculateamonthlyaveragespreadbycomputingtheaverageyieldspreadforbondswiththecorrespondingratingandmaturityobservedinthatmonth.Wedothisforbothmodel-impliedspreadsandactualspreads(totheswaprate)resultinginatimeseriesofmonthlyactualspreadss1;s2;:::;sTandimpliedspreadsfromtheBlack-Coxmodel^s1^s2;:::;^sTfortheperiod1987-2012.PanelAshowstheregressioncoecientintheregressionoftheactualspreadonthemodel-impliedspreadst= + ^st+t.Inparenthesisisthestandarderror,Newey-Westcorrectedwith12lagsand`*'impliesthat issigni cantlydi erentfromoneatthe5%leveland`**'atthe1%level.Insomemonthstheremaynotbeanyobservationsandiftherearelessthan100monthlyobservationswedonotreportregressioncoecients.PanelBshowsregressionresultsformonthlychanges,st+1 st= + (^st+1 ^st)+t+1.InparenthesisistheOLSstandarderrorand`*'impliesthat issigni cantlydi erentfromoneatthe5%leveland`**'atthe1%leve46 correlationofpricingerrorwith R2 Ltetatt Investmentgrade 0.44 -0.45-0.140.12-0.34Speculativegrade 0.13 -0.02-0.03-0.020.00Table10Explainingindividualpricingerrors.The rstcolumnshowstheR2fromrunningaregressionofactualspreadsonindividualbondsontheimpliedspreadsfromtheBlack-Coxwhereweusealltransactionsinthedatasample,separatedintoinvestmentgradeandspeculativegrade.Thenextcolumnsshowthecorrelationbetweenthepricingerror,de nedasthedi erencebetweentheactualspreadandmodel-impliedspread,andvariablesthatmaycontributetopricingerrors.Ltistheleverageratioonthedayofthetransaction,etistheestimatedequityvolatilityonthedayofthetransaction,atistheissuing rm'sassetvolatilitywhenestimatedday-by-day,tisthepayoutrateonthedayofthetransaction.Thebondyieldspreadsarefromtheperiod1987-2012. (1)(2)(3)(4)(5) Inv 18( 4;35)8( 11;23)19( 5;37) 5( 32;15)30(5;48) [28320][26855][26546][30998][31125]Spec 88(34;125)56( 9;101) 114( 203; 51) 356( 424; 306) 382( 430; 350) [7425][8940][9321][4865][4782]Table11Creditspreadresidualssortedonbondage.We rstsortallbondspreadobservationsforbondswithamaturitybetween3-20yearsintoquintilesbasedonthetimesincethebondwasissued.Forinvestmentgraderespectivelyspeculativegradebondswethencalculatetheaveragedi erencebetweenthemodel-impliedandactualspread.Thetableshowsthisaveragedi erenceinbasispoints.Inparenthesisare95%con dencebandscalculatedaccordingSection3.2;`*'impliessigni canceatthe5%leveland`**'atthe1%level.Thenumberofobservationsareinsquarebrackets.47 3-20y 3-7y7-13y13-20y InvActualspread 92 898787Modelspread 122(91;150) 120(85;154)116(90;140)93(78;106)Di erence 30( 1;59) 31( 4;65)30(3;53)7( 9;20)Observations 294 294293244 SpecActualspread 544 560417461Modelspread 420(315;517) 420(301;535)417(344;476)329(284;364)Di erence 124( 229; 27) 140( 259; 24) 0( 73;59) 132( 177; 97)Observations 289 276229141 AAAActualspread 16 4622Modelspread 4(2;6) 5(1;11)1(0;1)2(2;3)Di erence 12( 14; 10) 1( 2;7) 6( 6; 5) 20( 20; 19)Observations 132 707091 AAActualspread 23 173426Modelspread 10(7;13) 2(1;4)16(11;21)21(16;26)Di erence 14( 17; 11) 14( 15; 13) 18( 23; 13) 5( 10;0)Observations 289 27926493 AActualspread 61 506563Modelspread 94(69;117) 77(51;103)112(85;136)88(73;100)Di erence 33(8;56) 27(1;53)46(19;70)24(10;37)Observations 294 294293223 BBBActualspread 146 141141144Modelspread 186(139;230) 185(131;236)180(141;213)139(115;158)Di erence 40( 7;83) 43( 10;94)39(0;72) 6( 29;14)Observations 291 291257198 BBActualspread 377 370290398Modelspread 381(291;460) 356(256;448)360(292;416)270(228;303)Di erence 5( 85;84) 15( 114;78)70(2;126) 128( 170; 95)Observations 259 240216114 BActualspread 675 723427445Modelspread 487(371;594) 532(390;670)471(385;540)336(299;364)Di erence 187( 303; 80) 190( 333; 53)44( 42;113) 109( 146; 81)Observations 243 20313482 CActualspread 1442 12111948661Modelspread 1014(846;1136) 1173(946;1339)814(720;881)559(460;635)Di erence 429( 596; 306) 39( 265;128) 1134( 1228; 1066) 102( 201; 25)Observations 96 65427 Table12Actualandmodelyieldspreadswhenusingdefaultratesfrom1970-2012tocalibratethemodelInthemainanalysisthedefaultboundaryisestimatedusingMoody'sdefaultratesfrom1920-2012.ThistableshowsresultswhenthedefaultboundaryisestimatedusingMoody'sdefaultratesfrom1970-2012.Thetableshowsactualandmodel-impliedcorporatebondyieldspreads.Spreadsaregroupedaccordingtoremainingbondmaturityatthequotationdate.`Actualspread'istheaverageactualspreadtotheswaprate.`Modelspread'istheaverageBlack-Coxmodelspreadsofthebondsinagivenmaturity/ratingbucket.Theaveragespreadiscalculatedby rstcalculatingtheaveragespreadofbondsinagivenmonthandthencalculatingtheaverageofthesespreadsovermonths.`Di erence'isthedi erencebetweenthemodelspreadandtheactualspread.Inparenthesisare95%con dencebandscalculatedaccordingSection3.2;`*'impliessigni canceatthe5%leveland`**'atthe1%level.`Observations'isthenumberofmonthlyobservations.Thebondyieldspreadsarefromtheperiod1987-2012.48 Fig.1Actualandmodel-impliedBBB-AAAcorporatebondyieldspreadswhenusingexistingapproachintheliterature.This gureshowsactualandmodel-impliedBBB-AAAspreadsbasedondi erentestimatesofactualandmodel-impliedspreads.TheactualBBB-AAAyieldspreadsareestimatesfromDu ee(1998)[Duf],HuangandHuang(2012)[HH],Chen,Collin-Dufresne,andGoldstein(2009)[CDG]andCremers,Driessen,andMaenhout(2008)[CDM].TheblacksolidlinesshowspreadsintheBlack-CoxmodelbasedonMoody'sdefaultratesfromtheperiod1920-2002and1970-2001respectively.TheblackdashedlinesshowspreadsintheMertonmodelbasedonMoody'sdefaultratesfromtheperiod1920-2002and1970-2001respectively.49 Fig.2Distributionofestimated10-yearBBBdefaultprobabilitywhenusingdefaultratesmeasuredover31years.Theexistingapproachintheliteratureistouseanaveragehistoricaldefaultrateforaspeci cratingandmaturityasanestimateforthedefaultprobabilitywhentestingspreadpredictionsofstructuralmodels.OneexampleisChen,Collin-Dufresne,andGoldstein(2009)whousethe10-yearBBBdefaultrateof5.09%realizedovertheperiod1970-2001asanestimateforthe10-yearBBBdefaultprobability.Panel(a)showsthedistributionofthe10-yearBBBdefaultprobabilitywhenusinga31yearshistoryofthe10-yearBBBdefaultrateasanestimate.Besidesthisdistribution,Panel(b)alsoshowsthedistributionoftheestimated10-yearBBBdefaultprobabilitywhenextractedusingtheproposedapproachinSection3.2.Speci cally,thedefaultprobabilityisestimatedusingtheBlack-Coxmodeland1,2,...,20yeardefaultratesforratingsAAA,...,Caveragedover31years.50 05101520 0246default prob (percent) AAA Black-Cox model Historical 95pct confidence band 05101520 051015 AA 05101520 05101520 A 05101520 0102030 BBB 05101520 0204060 BB 05101520 020406080 B 05101520 050100 C Fig.3AveragedefaultprobabilitiesintheBlack-Coxmodelandhistoricaldefaultrates.Wemerge rmdatafromCRSP/CompustatwithratingsfromStandard&Poorsandforevery rmandeveryyear1985-2012wecalculatea1-,2-,...,19-,20-yeardefaultprobabilityintheBlack-Coxmodel.The gureshowstheaveragedefaultprobabilitiesalongwiththeaveragehistoricaldefaultrate1920-2012calculatedbyMoody's.95%con dencebandsforthehistoricaldefaultratesarecalculatedfollowingtheapproachinSection3.1.51 Jan90Jan92Jan94Jan96Jan98Jan00Jan02Jan04Jan06Jan08Jan10Jan12 0100200300400500600spread in basis points Actual spread Model spread 95% confidence band Fig.4Timeseriesvariationininvestmentgradespreads.Thisgraphshowsthetimeseriesofactualandmodel-impliedinvestmentgradecorporatebondspreads.Eachmonthalldailyyieldobservationsinbondswithaninvestmentgraderatingandwithamaturitybetween3-30yearsarecollectedandtheaverageactualspread(totheswaprate)andtheaveragemodel-impliedspreadintheBlack-Coxmodelarecomputed.Thegraphshowsthetimeseriesofmonthlyspreads.A95%con dencebandforthemodel-impliedspreadiscalculatedfollowingtheapproachinSection3.2.52 Fig.5Actualandmodel-impliedcorporatebondyieldspreadswhenusingdefaultratesfrom1970-2012and1920-2012.This gureshowsaverageactualandmodel-impliedcorporatebondyieldspreadsestimatedusingdefaultratesfromeither1970-2012or1920-2012.Model-impliedspreadsarecalculatedaccordingtoourproposedmethodwheremanydefaultratesacrossmaturityandratingareusedinthecalibrationofthemodelandthe gureshowsresultswhendefaultratesfromeither1970-2012or1920-2012areusedinthecalibration.Con dencebandstakeintoaccountuncertaintyaboutexantedefaultprobabilities.SpreadsarefromTables7and12.Actualbondyieldspreadsareaveragespreadstotheswapratefromnoncallablebondsissuedbyindustrial rmsandfromtheperiod1987-2012.53