Chp11A partial-equilibrium one-period model(连续时间金融

[e-mail],XiamenUniversityMainline:1Apartial-equilibriumone-periodmodel2Ageneralintertemporalequilibriummodeloftheassetmarket,includesthreemodels(model1isbasedonaconstantinterestrateassumption,model2isano-riskless-assetcase,model3isthegeneralmodel).ⅠApartial-equilibriumone-periodmodelWefollowsthewarrantpricingapproachusedinChapter7,thatis,investorschooseamongthreeassets:thewarrant,thestockofthefirmandarisklessassettoformoptimalportfolioswhichmaximizetheirexpectedutility.Consideraneconomymadeupofonlyonefirmwithcurrentvalue,andthereexistsa“representativeman”actssoastomaximizetheexpectedutilityofwealthattheendofaperiodoflength,thatis,……①DefinearandomvariableZbyandassumeitsprobabilitydistributionisknownatpresent,moreimportantly,max()tEUVt()()VtZVt(,)PZVtisindependentoftheparticularstructureofthefirm,thisisconsistentwiththeMM(Modigliani-Miller)theorem.Defineasthecurrentvalueoftheithtypeofsecurityissuedbythefirm.Thedifferenttypesofsecuritiesaredistinguishablebytheirterminalvalue.Foradebtissue(i=1),…②(,)PZ,iFV,0iFVZ1,0min(,)FVZBVZBecauseeachofthesecuritiesappearsseparatelyinthemarket,so:andDefine,sowecanrewrite①asamaximizationunderconstraint:…③1,niVFV1,niVZFVZ,iiwFVV11,0max1,innitiiwiFVZEUVwwFVThecorrespondingfirst-orderconditionsare:Thiscanberewrittenintermsofutil-probdistributionsQas:…④1,0,0,,niitiiiFVZFVZEUVwFVFV00,0,0exp(),,jiijFVZFVZdQdQFVFVWhereandisanewmultiplierrelatedto.dQisindependentofthefunctionsbytheassumptionthatthevalueofthefirmisindependentofitscapitalstructure,so④isasetofintegralequationslinearinthe,andwecanrewrite④as0,,UZVdPZdQUZVdPZexpiFiF…*Supposethefirmissuesjustonetypeofsecurity--equity,thenSubstitutingin④,wehave11,,,,0FVVandFVZVZ0exp,ZdQZ0,exp,0,iiFVFVZdQZFrom④,wecanseethattheexpectedreturnonallsecuritiesinutil-probspacemustbeequated.IfUwaslinear,thendQ=dPand④wouldimplytheresultfortherisk-neutralcase.Hence,theutil-probdistributionisthedistributionofreturnsadjustedforrisk.SomeexamplesExample1:Firmissuestwotypesofsecurities,debtandequitywithcurrentvalueandrespectively.From②and④,wehave⑤:1,FV2,FV/10//0,exp,,expexp,BVBVBVFVZVdQZBdQZBBZVdQZSupposeorforthenas.Inthelimit,thedebtbecomesriskless,sowillbereplacedbyr.Anotherusefulformof⑤isSinceinequilibrium0BV,0dQZ0ZBV1,expFVB0BV10//,exp,,exp,BVBVFVrZVdQZZVBdQZVrZVBdQZ12,,VFVFVSo,.ThisisidenticaltothewarrantpricingequationderivedinChapter7.Thisequationcanalsobederiveddirectlyfromtheterminalvalueofequityinthesamewayasdebt.2/,exp,BVFVrZVBdQZ2,0max0,FVZVZBExample2:Firm’scapitalstructuremadeupfromthreetypesofsecurities:debt,equity(NsharesoutstandingwithcurrentpricepershareofS,i.e.),warrants(exercisepriceis).AssumetherearenwarrantsoutstandingwithcurrentmarketvalueperwarrantofW,2,FVNSSi.e..Becausethewarrantisajuniorsecuritytothedebt,thecurrentvalueofthedebtwillbethesameasinthefirstexample.Thecurrentvalueoftheequitywillbe…⑥3,FVnW/2//,exp[,,]VBVVFVrZVBdQZNZVnSBdQZnNWhere.Rewrite⑥as…⑦NSB2//1/,exp[,1,]exp,,BVVVFVrZVBdQZNnNnSZVBdQZnNNNrnVFVnSZVBdQZnNNInequilibrium,.Sofrom⑦wehave⑧Ifwedefinenormalizedpriceofthefirmas312,,,FVVFVFV3/,exp,VnFVrZVdQZnNVnNVyNSBnNAnddefinethenormalizedpriceofawarrantas,then⑧canberewrittenaswhichisofthesameformasequation(7.24).3,FVwnnN/,exp1,VwyrZydQZExample3:Firm’scapitalstructurecontainstwosecurities:convertiblebondswithatotalterminalclaimonthefirmofeitherBdollarsoralternativelythebondscanbeexchangedforatotalofnsharesofequity;andNsharesofequitywithcurrentpricepershareofSdollars.So,,andWhereisdeterminedtobe.2,0max0,min,FVZVZBNVZnN/2//,exp,,VBVVFVrVZBdQZNVZdQZnNnNBn/120//,,exp[,,,]BVBVVFVVFVrVZdQZnBdQZVZdQZnNByinspectionofthisequation,wehavethewell-knownresultthatthevalueofaconvertiblebondisequaltoitsvalueasastraightbondplusawarrantwithexerciseprice.SBnExample4:A“dual”fund:itissuestwotypesofsecuritiestofinanceitsassets:namely,capitalshares(equity)whichareentitledtoalltheaccumulatedcapitalgains(inexcessofthefixedterminalpayment);andincome-shares(atypeofbond)whichareentitledtoalltheordinaryincomeinadditiontoafixedterminalpayment.Letbetheinstantaneousfixedproportionoftotalassetvalueearnedasordinaryincome,VbethecurrentassetvalueofthefundandZthetotalreturnonthefund.LetbethecurrentvalueoftheincomeshareswithterminalclaimonthefundofBdollarsplusallinterestanddividendsearned,bethecurrentvalueofthecapitalshares.So,fromdefinition,wehave1,FV2,FV2,0max0,expFVZVZBAndWhere.Thecurrentvalueof