Asset-Pricing——John-H.-Cochrane(教材及答案

SolutionstoproblemsinAssetPricingJohnH.Cochrane∗GraduateSchoolofBus[e-mail]26,2001Thisisaverypreliminarydraft;it’sincompleteandI’msurefulloftypos.Still,IwelcomecommentsonanyproblemsyouÞndwiththesenotes.1ProblemsforChapter11.aandbaretrivial.Forc,c2/c1d(c1/c2)dR/R=−dc1c1−dc2c2dRR.TheÞrstorderconditionsareu0(c1)=λβu0(c2)=λR.DifferentiatingtheÞrstorderconditions,γdc1c1=c1u00(c1)u0(c1)dc1c1=dλλγdc2c2=c2u00(c2)u0(c2)dc2c2=dλλ−dRR2.Theexpectedreturnoftheassetisthesameasthatofitsmimickingportfolio,proj(R|m)3.(a)Weknowtherearea,b,suchthatm=a+bRmv.Determinea,b,bypricingRmvandtheriskfreerateRf1=E(mRmv)=E[(a+bRmv)(Rmv)]1=E(mRf)=hE(a+bRmv)Rfi∗Copyrightc°JohnH.Cochrane200111=aE(Rmv)+bE³Rmv2´1=aRf+bE(Rmv)Rfa=E(Rmv)Rf−E(Rmv2)E(Rmv)2Rf−E(Rmv2)Rf=E(Rmv2)−E(Rmv)RfRfvar(Rmv)=var(Rmv)+³E(Rmv)−Rf´E(Rmv)Rfvar(Rmv)=1Rf1+³E(Rmv)−Rf´E(Rmv)var(Rmv)b=E(Rmv)−RfE(Rmv)2Rf−E(Rmv2)Rf=−1RfE(Rmv)−Rfvar(Rmv)b=−1RfE(Rmv)−Rfvar(Rmv)a=1Rf−bE(Rmv).Aneasierwaytodothisistoparameterizethelinearfunctionbyameanandshock:|ρ|=1:m=E(m)+a(Rmv−E(Rmv))E(m)=1/Rf:m=1/Rf+a(Rmv−E(Rmv))1=E(mRmv):1=E(Rmv)Rf+aσ2(Rmv)a=−E(Rmv)−RfRfσ2(Rmv)m=1Rf−E(Rmv)−RfRfσ2(Rmv)(Rmv−E(Rmv))(b)WehadE(Ri)=Rf+βi,mλmWehavecov(Ri,a+bRmv)=bcov(Ri,Rmv).4.No.TheSharperatioboundappliestoanyexcessreturnE(Ri)−E(Rj)σ(Ri−Rj)≤σ(m)E(m)=E(Rmv)−Rfσ(Rmv)5.σ£(ct+1/ct)−γ¤=qE(e−2γ∆lnct+1)−E(e−γ∆lnct+1)2=qe−2γE(∆lnct+1)+2γ2σ2(∆lnct+1)−e−2γE(∆lnct+1)+γ2σ2(∆lnct+1)=e−γE(∆lnct+1)+12γ2σ2(∆lnct+1)qeγ2σ2(∆lnct+1)−1Eh(ct+1/ct)−γi=E³e−γln∆ct+1´=e−γE(∆lnct+1)+12γ2σ2(∆lnct+1).Dividing,wegettheÞrstresult.Forthesecondresult,usetheapproximationforsmallxthatex≈1+x.26.Youwouldn’tputallyourmoneyinsuchanasset,butyoumightwellputsomeofyourmoneyinsuchanassetifitprovidesinsurance—ifitsbetaislow.(Graph!)7.(a)Ratherobviously,usetheequationattandt+1,i.e.startwithpt+1=Et+1µβu0(ct+2)u0(ct+1)dt+2+β2u0(ct+3)u0(ct+1)dt+3+...¶(b)Substituterecursively,pt=Et·βu0(ct+1)u0(ct)pt+1¸+Et·βu0(ct+1)u0(ct)dt+1¸=Et·β2u0(ct+2)u0(ct)pt+2¸+Et·β2u0(ct+2)u0(ct)dt+2¸+Et·βu0(ct+1)u0(ct)dt+1¸...=Et∞Xj=1βju0(ct+j)u0(ct)dt+j+limT→∞Et·βTu0(ct+T)u0(ct)pt+T¸Thelasttermisnotautomaticallyzero.Forexample,ifu0(c)isaconstant,thenpt=βtorgreatergrowthwillleadtosuchaterm.Italsohasaninterestingeconomicinterpretation.Eveniftherearenodividends,ifthelasttermispresent,itmeansthepricetodayisdrivenentirelybytheexpectationthatsomeoneelsewillpayahigherpricetomorrow.Peoplethinktheyseethisbehaviorin“speculativebubbles”andsomemodelsofmoneyworkthisway.TheabsenceofthelasttermisaÞrstorderconditionforoptimizationofaninÞnitely-livedconsumer.Ifpt()EtP∞j=1βju0(ct+j)u0(ct)dt+j,hecanbuy(sell)moreoftheasset,eatthedividendsastheycome,andincreaseutility.Thislowersct,increasesct+j,untiltheconditionisÞlled.Ifmarketsarecomplete—ifhecanalsobuyandsellclaimstotheindividualdividends—thenhecandoevenmore.Forexample,ifpt,thenhecanselltheasset,buyclaimstoeachdividend,paythedividendstreamoftheassetwiththeclaims,andmakeasure,instantproÞt.Hedoesnothavetowaitforever.(Advocatesofbubblespointoutthatyouhavetowaitalongtimetoeatthedividendstream,buttheyoftenforgettheopportunitiesforimmediatearbitragethatabubblecaninduce.Theplausibilityofbubblesreliesonincompletemarkets.)Bubbletypesolutionsshowupofteninmodelswithoverlappinggenerations,nobequestmotive,andincompletemarkets.TheOGgetsridoftheindividualÞrstorderconditionthatremovesbubbles,andtheincompletemarketsgetsridofthearbitrageopportunity.ThepossibilityofbubblesÞguresintheevaluationofvolatilitytests.8.Λ=e−δtuc(c,l)dΛ=−δΛdt+e−δt·uccdc+ucldl+12ucccdc2+12uclldl2+uccldcdl¸dΛΛ=−δdt+·uccucdc+ucluccdl+12ucccucdc2+12ucllucdl2+ucclucdcdl¸3AftermultiplicationbydP/Ponlythedcanddltermswillhaveanythingleft,soEtµdpp¶+Dpdt−rftdt=EtµdppdΛΛ¶=uccucEtµdppdc¶+uclucEtµdppdl¶or,Et(Ri)−Rf≈uccuccovt(Ri,c)+ucluccovt(Ri,l)thisisyourÞrstviewofamultifactormodel,onewithmultiplebetasorfactorsontherighthandside.Ofcourse,thereisnothingdeepaboutmultiplefactors—thesamemodelisexpressedwiththesingleΛontherighthandside.ButtheremaybemoreeconomicintuitioninhavingthecandlseparatelyratherthancombiningthetwointoΛ.9.1=E(elnm+lnR)eE(lnm)+E(lnR)0E(lnm)+E(lnR)−E(lnm)E(lnR)IfyouincreaseleverageαinR=(1−α)Rf+αRmyouincreasemeanandvolatility.IfRcangetanywherenearzero,lnRgoesoffto-∞.Thus,increasingαeventuallyleadstoadecreaseinElnR.Forexample,ifreturnsarenormal,thenE(R)=eE(lnR)+12σ2(R)lnE(R)=E(lnR)+12σ2(R)E(lnR)=lnE(R)−12σ2(R)E(lnR)=lnhαE(Rm)+(1−α)Rfi−12α2σ2(Rm).Asαincreases,thesecondtermeventuallydominates.2ProblemsforChapter21.(a)pt=EtXβjµct+jct¶−γct+jptct=EtXβjµct+jct¶1−γ.Ifγ=1,pc=β/(1−β)=1δwhereβ=1/(1+δ).4(b)Ifγ1,thenariseinct+jraisespt.Ifγ1,however,ariseinct+jlowerspt.Anypieceofnewshastwopossibleeffects:cashßowsanddiscountrates.Inthiscasethediscountraterisesfasterthanthepayoffs,sothepriceactuallydeclines.2.(a)TheÞrstorderconditionsarect−c∗=Et[Rβ(ct+1−c∗)]withR=1+r,andhencect=Et(ct+1).Iteratethetechnologyforward,kt+2=R(Rkt+it)+it+1=R2kt+Rit+it+1kt+3=R3kt+R2it+Rit+1+it+21R3kt+3=kt+1R·it+1Rit+1+1R2it+2¸β3kt+3=kt+βhit+βit+1+β2