thePricingofCorporateDebt(衍生证券的定价与保

2019/9/251Chapter12:OnthePricingofCorporateDebtXingluanHuangDepartmentofFinance,XMU2019/9/25212.1Introduction•ThevalueofcorporatedebtdependsonessentiallyonTherequiredrateofreturnofrisklessdebt;Thevariousprovisionsandrestrictionscontainedintheindenture;Theprobabilityofdefault.•Thepurposeofthischapteristopresentatheoryoftheriskstructureofinterestrates.2019/9/25312.2Onthepricingofcorporateliabilities•Someassumptions:Therearenotransactionscosts,taxes,orproblemswithindivisibilitiesofassets.Therearesufficientlymanyinvestorswithcomparablewealthlevels.Thereexistsanexchangemarketforborrowingandlendingatasamerateofinterest.Short-salesofallassets,withfulluseoftheproceedsisallowed.2019/9/254Assumptions(con.)Tradinginassetstakesplacecontinuouslyintime.TheMMtheoremobtains.Thetermstructureis“flat”andknownwithcertainty.Thedynamicsforthevalueofthefirm,V,canbedescriberas:VdzdtCVdV)(2019/9/255•Whereistheinstantaneousexpectedrateofreturnonthefirmperunittime;Cisthetotaldollarpayoutsbythefirmperunittimetoeitheritsshareholdersorliabilities-holdersifpositive,anditisthenetdollarsreceivedbythefirmfromnewfinancingifnegative;istheinstantaneousvarianceofthereturnonthefirmperunittime;anddzisastandardGauss-Wienerprocess.22019/9/256•SupposethereexistsasecuritywhosemarketvalueY=F(V,t).)2.12(])(21[)(21),,(:)1.12()(222dzVFdtFCVFVdtFdVFdVFdYtVFYthatgivenYdzdtCYdYvtvvtvvvyyyy2019/9/257Comparingtermsin(12.2)and(12.1))3.12()3.12()3.12()(2122cdzdzbVFFYaCFFCVFVFYyvyyytvvvyy2019/9/258Formingathree-securityportfolio•Let:W1bethenumberofdollarsoftheportfolioinvestedinthefirm;W2bethenumberofdollarsinvestedintheparticularsecurity;W3(=-W1+W2)bethenumberofdollarsinvestedinrisklessdebt.2019/9/259•Ifdxistheinstantaneousdollarreturntotheportfolio,then)4.12()()]()([)]()([21212121321dzWWdtrWrWdzWdzWdtrWrWrdtWYdtCdYWVCdtdVWdxyyyyyy2019/9/2510ChoosingW*)6.12()5.12(0))()5.12(0*2*1*2*1yyyyrrbrWrWaWW（2019/9/2511Substitutingfor)7.12()(210)6.12()(212222ytvvvvytvvvCFrFFCrVFVaVFrFCFFCVFVryy,2019/9/2512Remarkon(12.7)•Equation(12.7)mustbesatisfiedbyanysecuritywhosevaluecanbewrittenasafunctionofthevalueofthefirmandtime.•NotethatFdoesnotdependontheexpectedrateofreturnonthefirmnorontheriskpreferencesofinvestorsnoronthecharacteristicsofotherassetsavailabletoinvestorsbeyondthethreementioned.2019/9/251312.3Onthepricingof“risky”discountbond•Supposethecorporationhastwoclassesofclaims:(a)Asinglehomogeneousclassofdebtand(b)Theresidualclaim,equity.•Andsuppose:ThefirmpromisestopayatotalofBdollarsondateT.Intheeventthatthispaymentisnotmet,thebondholderstakeoverthecompany.Thefirmcannotissueanynewseniorclaimsonthefirmnorcanitpaycashdividendsordosharerepurchasepriortothematurityofthedebt.2019/9/2514IfFisthevalueofthedebtissueandfisthevalueoftheequity,wehavethat:)9.12(),min()0,()9.12(1),(),()9.12(0),0(),0(),(),(:0212cBVVFbVVFVVFafFVfVFVcontitionsboundaryFrFrVFFVvvv2019/9/2515Thevalueofequity)11.12(),0max()0,(:.)10.12(021),(),(22BVVftsfrfrVffVVFVVfvvvandboundaryconditions(12.9a)and(12.9b)2019/9/2516Comparing(12.10)withB-S-Mdifferentialequation•InspectionoftheB-SequationorMertonshowsthat(12.10)and(12.11)areidenticalwiththeequationforaEuropeancalloptiononanon-dividend-payingcommonstockwherefirmvaluein(12.10)-(12.11)correspondstostockpriceandBcorrespondstotheexerciseprice.2019/9/2517Black-Scholesformula2/1122/12122/121,])21()/[log()21exp()2(1)()12.12()()exp()(),(xxandrBVxdzzxwherexrBxVVfx2019/9/2518From(12.12)andF=V-f,wehavethat:2/12222/12212122)log(21),()log(21),()exp(:)13.12()]},([1)],([){exp(),(ddhddhVrBdwheredhddhrBVF2019/9/2519Itiscommonindiscussionsofbondpricingtotalkintermsofyieldsratherthanprice,wecanrewrite(12.13)as:BVFRwheredhddhrR),(])(exp[(:)14.12()]},([1)],([log{1)(2122istheyieldtomaturity.)(R2019/9/2520Derivationof(12.13)and(12.14)2121211122(,)(,)exp()()()exp()[()(1()]exp():(,)exp[()]exp[()]exp()[()(1()]exp():,FVVfVBrxVVxVBrxxBrwhereFVBRVRrxxBrlethxhxweget(12.13)andwillhavethat(12.14)2019/9/2521Reviewon(12.14)•Itseemsreasonabletocallariskpremiuminwhichcase(12.14)definesariskstructureofinterestrates.•Foragivenmaturity,theriskpremiumisafunctionofonlytwovariables:(a)thevarianceofthefirm’soperations;and(b)theratioofthepresentvalueofthepromisedpaymenttothecurrentvalueofthefirm.rR)(2019/9/252212.4Acomparativestaticsanalysisoftheriskstructure•Thevalueofdebtcanbewrittenas•wecanshowthatFisafirst-degreehomogeneousconcavefunctionofVandB.Furtherwehavethat:),,,,(2rBVF)15.12(00;00;0122rrBBvvfFfFfFfFfF2019/9/2523Theimplicationsof(12.15)•Thevalueofthedebtisanincreasingfunctionofthecurrentmarketvalueofthefirmandthepromisedpaymentatmaturity,andadecreasingfunctionofthetimetomaturity,thebusinessriskofthefirm,andtherisklessrateofinter