thePricingofCorporateDebt衍生证券的定价与保

Chapter12:OnthePricingofCorporateDebtXingluanHuangDepartmentofFinance,XMU1/10/2024112.1IntroductionThevalueofcorporatedebtdependsonessentiallyonTherequiredrateofreturnofrisklessdebt;Thevariousprovisionsandrestrictionscontainedintheindenture;Theprobabilityofdefault.Thepurposeofthischapteristopresentatheoryoftheriskstructureofinterestrates.1/10/2024212.2OnthepricingofcorporateliabilitiesSomeassumptions:Therearenotransactionscosts,taxes,orproblemswithindivisibilitiesofassets.Therearesufficientlymanyinvestorswithcomparablewealthlevels.Thereexistsanexchangemarketforborrowingandlendingatasamerateofinterest.Short-salesofallassets,withfulluseoftheproceedsisallowed.1/10/20243Assumptions(con.)Tradinginassetstakesplacecontinuouslyintime.TheMMtheoremobtains.Thetermstructureis“flat〞andknownwithcertainty.Thedynamicsforthevalueofthefirm,V,canbedescriberas:1/10/20244Whereistheinstantaneousexpectedrateofreturnonthefirmperunittime;Cisthetotaldollarpayoutsbythefirmperunittimetoeitheritsshareholdersorliabilities-holdersifpositive,anditisthenetdollarsreceivedbythefirmfromnewfinancingifnegative;istheinstantaneousvarianceofthereturnonthefirmperunittime;anddzisastandardGauss-Wienerprocess.1/10/20245SupposethereexistsasecuritywhosemarketvalueY=F(V,t).1/10/20246Comparingtermsin(12.2)and(12.1)1/10/20247Formingathree-securityportfolioLet:W1bethenumberofdollarsoftheportfolioinvestedinthefirm;W2bethenumberofdollarsinvestedintheparticularsecurity;W3(=-W1+W2)bethenumberofdollarsinvestedinrisklessdebt.1/10/20248Ifdxistheinstantaneousdollarreturntotheportfolio,then1/10/20249ChoosingW*1/10/202410Substitutingfor1/10/202411Remarkon(12.7)Equation(12.7)mustbesatisfiedbyanysecuritywhosevaluecanbewrittenasafunctionofthevalueofthefirmandtime.NotethatFdoesnotdependontheexpectedrateofreturnonthefirmnorontheriskpreferencesofinvestorsnoronthecharacteristicsofotherassetsavailabletoinvestorsbeyondthethreementioned.1/10/20241212.3Onthepricingof“risky〞discountbondSupposethecorporationhastwoclassesofclaims:(a)Asinglehomogeneousclassofdebtand(b)Theresidualclaim,equity.Andsuppose:ThefirmpromisestopayatotalofBdollarsondateT.Intheeventthatthispaymentisnotmet,thebondholderstakeoverthecompany.Thefirmcannotissueanynewseniorclaimsonthefirmnorcanitpaycashdividendsordosharerepurchasepriortothematurityofthedebt.1/10/202413IfFisthevalueofthedebtissueandfisthevalueoftheequity,wehavethat:1/10/202414Thevalueofequityandboundaryconditions(12.9a)and(12.9b)1/10/202415Comparing(12.10)withB-S-MdifferentialequationInspectionoftheB-SequationorMertonshowsthat(12.10)and(12.11)areidenticalwiththeequationforaEuropeancalloptiononanon-dividend-payingcommonstockwherefirmvaluein(12.10)-(12.11)correspondstostockpriceandBcorrespondstotheexerciseprice.1/10/202416Black-Scholesformula1/10/202417From(12.12)andF=V-f,wehavethat:1/10/202418Itiscommonindiscussionsofbondpricingtotalkintermsofyieldsratherthanprice,wecanrewrite(12.13)as:istheyieldtomaturity.1/10/202419Derivationof(12.13)and(12.14)weget(12.13)andwillhavethat(12.14)1/10/202420Reviewon(12.14)Itseemsreasonabletocallariskpremiuminwhichcase(12.14)definesariskstructureofinterestrates.Foragivenmaturity,theriskpremiumisafunctionofonlytwovariables:(a)thevarianceofthefirm’soperations;and(b)theratioofthepresentvalueofthepromisedpaymenttothecurrentvalueofthefirm.1/10/20242112.4AcomparativestaticsanalysisoftheriskstructureThevalueofdebtcanbewrittenaswecanshowthatFisafirst-degreehomogeneousconcavefunctionofVandB.Furtherwehavethat:1/10/202422Theimplicationsof(12.15)Thevalueofthedebtisanincreasingfunctionofthecurrentmarketvalueofthefirmandthepromisedpaymentatmaturity,andadecreasingfunctionofthetimetomaturity,thebusinessriskofthefirm,andtherisklessrateofinterest.1/10/202423From(12.13)wehavethat:1/10/202424Derivation1/10/202425AnotherratioFromtheno-arbitragecondition,wehavethat1/10/202426Derivationof(12.19)1/10/202427StudyP(d,T)Wecanrewrite(12.17)and(12.18)inelasticityformintermsofgas1/10/202428StaticsanalysisIfwedefine,thenfrom(12.14)wehavethat1/10/2024290“Quasi〞Debt-to-FirmValueRatiodTermPremiumR-rFigure12.1:TermPremiumvs“Quasi〞Debt-to-FirmValueRatio1/10/202430Figure12.2:TermPremiumvsVarianceoftheFirm0VarianceoftheFirmTermPremiumR-r1/10/202431Figure12.3:TermPremiumvsTimetoMaturityTermPremiumR-r0TermuntilMaturity1/10/202432HandrTocompletetheanalysisoftheriskstructureasmeasuredbythetermpremium,weshowthatthepremiumisadecreasingfunctionoftherisklessrateofinterest,i.e.1/10/202433AnimportantquestionCanoneassertthatifR-rislargerforonebondthanforanother,thentheformerisriskierthanthelatter?Toanswerthisquestion,onemustfirstestablishanappropriatedefinitionof“riskier〞.Thenaturalchoiceasameasureofriskisthestandarddeviationofthereturnonthebond,1/10/202434

FromthedefinitionofGand(12.19),wehavethat:1/10/20243512.5OntheMMtheoremwithbankruptcyThefundamentalequationforpricingofcorporatedebts(12,7)isderivedontheassumptionthatMMtheoremheld.Becauseofbankruptcycostsorcorporatetaxes,theMMtheoremdoesnotobtainandthevaluedoesdependonthedebt-equityratio,thentheformalanalysisofthechapterisstillvalid.However,thelinearpropertyof(12.7)wouldbelost,andanonlinearsimultaneoussolutionwouldberequired.1/10/202436ExaminingadebtissueforsinglefirmAsdiscussedinsection12.4,thecorrectmeasureofthisrelativeriskinessisFrom(12,16)and(12.19),wehavethat1/10/202437ImplicationsThedebtofthefirmcanneverbemoreriskythanthefirmasawhole,andasacorollary,theequityofaleveredfirmmustalwaysbeatleastasriskyasthefirm.Inparticular,from(12.13)and(12.31),whenAs1/10/202438StaticsanalysisofgNotingthat,wehavefrom(12.19)and(12.27)thatThatis,therelativeriksinessofthedebtisanincreasingfunctionofdand1/10/202439FurtherstudyofgFurther,wehavethatThus,ford=1,independentofthebusinessriskofthefirmorthelengthoftimeuntilmaturity,thestandarddeviationofthereturnonthedebtequalshalfthestandarddeviationofthereturnonthewholefirm.1/10/202440AclassicproblemincorporatefinanceGivenafixedinvestmentdecision,howdoestherequiredreturnondebtandequitychange,asalternativedebt-equitymixesarechosen?1/10/202441DefinitionDefinethemarketdebt-to-equitytobeXwhichequaltoF/f=F/(V-F).From(12.20),therequiredexpectedrateofreturnonthedebt:Thus,forafixedinvestmentpolicy,1/10/202442Derivationof(12.38)FormthedefinitionofXand(12.13),wehavethatSincewehavefrom(12.32),(12.36),and(12.37)that1/10/202443Furtheranalysisof(12.38)startsoutasaconvexfunctionofX,passesthroughaninflectionpointwhereitbecomesconcaveandapproachesasymptoticallyasXtendstoinfinity.1/10/202444DeterminethepathoftherequiredreturnonequityHasaslopeofatX=0andisaconcavefunctionboundedfromabovebytheline1/10/2024450MarketdebttoequityratioXExpectedreturn1/10/20244612.6OnthepricingofriskycouponbondsIntheusualanalysisofdefault-freebondsintermstructurestudies,thederivationofapricingrelationforpurediscountbondsforeverymaturitywouldbesufficient.However,nosuchsimpleformulaexistsforriskycouponbonds.Theapparatusdevelopedintheprevioussectionsissufficienttosolvethecouponbondproblem.1/10/202447AssumptionAssumethesamesimplecapitalstructureandindentureconditionsasinsection12.3exceptmodifytheindentureconditiontorequirepaymentsatacouponrateperunittime,.Fromtheindenturerestriction(c),wehavethat,inequation(12.7),1/10/202448NewPDEHencethecouponbondvaluewillsatisfythepartialdifferentialequation

subjecttothesameboundaryconditions(12.9).