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AP_solutions

臺北不是我的家
2014-03-10 0人阅读 举报 0 0 暂无简介

简介:本文档为《AP_solutionspdf》,可适用于经济金融领域

SolutionstoproblemsinAssetPricingJohnHCochrane∗GraduateSchoolofBusinessUniversityofChicagoEthStChicagoILjohncochranegsbuchicagoeduMarch,Thisisaverypreliminarydraftit’sincompleteandI’msurefulloftyposStill,IwelcomecommentsonanyproblemsyouÞndwiththesenotesProblemsforChapteraandbaretrivialForc,ccd(cc)dRR=−dcc−dccdRRTheÞrstorderconditionsareu(c)=λβu(c)=λRDifferentiatingtheÞrstorderconditions,γdcc=cu(c)u(c)dcc=dλλγdcc=cu(c)u(c)dcc=dλλ−dRRTheexpectedreturnoftheassetisthesameasthatofitsmimickingportfolio,proj(R|m)(a)Weknowtherearea,b,suchthatm=abRmvDeterminea,b,bypricingRmvandtheriskfreerateRf=E(mRmv)=E(abRmv)(Rmv)=E(mRf)=hE(abRmv)Rfi∗Copyrightc°JohnHCochrane=aE(Rmv)bE³Rmv´=aRfbE(Rmv)Rfa=E(Rmv)Rf−E(Rmv)E(Rmv)Rf−E(Rmv)Rf=E(Rmv)−E(Rmv)RfRfvar(Rmv)=var(Rmv)³E(Rmv)−Rf´E(Rmv)Rfvar(Rmv)=Rf³E(Rmv)−Rf´E(Rmv)var(Rmv)b=E(Rmv)−RfE(Rmv)Rf−E(Rmv)Rf=−RfE(Rmv)−Rfvar(Rmv)b=−RfE(Rmv)−Rfvar(Rmv)a=Rf−bE(Rmv)Aneasierwaytodothisistoparameterizethelinearfunctionbyameanandshock:|ρ|=:m=E(m)a(Rmv−E(Rmv))E(m)=Rf:m=Rfa(Rmv−E(Rmv))=E(mRmv):=E(Rmv)Rfaσ(Rmv)a=−E(Rmv)−RfRfσ(Rmv)m=Rf−E(Rmv)−RfRfσ(Rmv)(Rmv−E(Rmv))(b)WehadE(Ri)=Rfβi,mλmWehavecov(Ri,abRmv)=bcov(Ri,Rmv)NoTheSharperatioboundappliestoanyexcessreturnE(Ri)−E(Rj)σ(Ri−Rj)≤σ(m)E(m)=E(Rmv)−Rfσ(Rmv)σ£(ctct)−γ¤=qE(e−γ∆lnct)−E(e−γ∆lnct)=qe−γE(∆lnct)γσ(∆lnct)−e−γE(∆lnct)γσ(∆lnct)=e−γE(∆lnct)γσ(∆lnct)qeγσ(∆lnct)−Eh(ctct)−γi=E³e−γln∆ct´=e−γE(∆lnct)γσ(∆lnct)Dividing,wegettheÞrstresultForthesecondresult,usetheapproximationforsmallxthatex≈xYouwouldn’tputallyourmoneyinsuchanasset,butyoumightwellputsomeofyourmoneyinsuchanassetifitprovidesinsurance—ifitsbetaislow(Graph!)(a)Ratherobviously,usetheequationattandt,iestartwithpt=Etµβu(ct)u(ct)dtβu(ct)u(ct)dt¶(b)Substituterecursively,pt=Et·βu(ct)u(ct)pt¸Et·βu(ct)u(ct)dt¸=Et·βu(ct)u(ct)pt¸Et·βu(ct)u(ct)dt¸Et·βu(ct)u(ct)dt¸=Et∞Xj=βju(ctj)u(ct)dtjlimT→∞Et·βTu(ctT)u(ct)ptT¸ThelasttermisnotautomaticallyzeroForexample,ifu(c)isaconstant,thenpt=βtorgreatergrowthwillleadtosuchatermItalsohasaninterestingeconomicinterpretationEveniftherearenodividends,ifthelasttermispresent,itmeansthepricetodayisdrivenentirelybytheexpectationthatsomeoneelsewillpayahigherpricetomorrowPeoplethinktheyseethisbehaviorin“speculativebubbles”andsomemodelsofmoneyworkthiswayTheabsenceofthelasttermisaÞrstorderconditionforoptimizationofaninÞnitelylivedconsumerIfpt<(>)EtP∞j=βju(ctj)u(ct)dtj,hecanbuy(sell)moreoftheasset,eatthedividendsastheycome,andincreaseutilityThislowersct,increasesctj,untiltheconditionisÞlledIfmarketsarecomplete—ifhecanalsobuyandsellclaimstotheindividualdividends—thenhecandoevenmoreForexample,ifpt>,thenhecanselltheasset,buyclaimstoeachdividend,paythedividendstreamoftheassetwiththeclaims,andmakeasure,instantproÞtHedoesnothavetowaitforever(Advocatesofbubblespointoutthatyouhavetowaitalongtimetoeatthedividendstream,buttheyoftenforgettheopportunitiesforimmediatearbitragethatabubblecaninduceTheplausibilityofbubblesreliesonincompletemarkets)Bubbletypesolutionsshowupofteninmodelswithoverlappinggenerations,nobequestmotive,andincompletemarketsTheOGgetsridoftheindividualÞrstorderconditionthatremovesbubbles,andtheincompletemarketsgetsridofthearbitrageopportunityThepossibilityofbubblesÞguresintheevaluationofvolatilitytestsΛ=e−δtuc(c,l)dΛ=−δΛdte−δt·uccdcucldlucccdcuclldluccldcdl¸dΛΛ=−δdt·uccucdcucluccdlucccucdcucllucdlucclucdcdl¸AftermultiplicationbydPPonlythedcanddltermswillhaveanythingleft,soEtµdpp¶Dpdt−rftdt=EtµdppdΛΛ¶=uccucEtµdppdc¶uclucEtµdppdl¶or,Et(Ri)−Rf≈uccuccovt(Ri,c)ucluccovt(Ri,l)thisisyourÞrstviewofamultifactormodel,onewithmultiplebetasorfactorsontherighthandsideOfcourse,thereisnothingdeepaboutmultiplefactors—thesamemodelisexpressedwiththesingleΛontherighthandsideButtheremaybemoreeconomicintuitioninhavingthecandlseparatelyratherthancombiningthetwointoΛ=E(elnmlnR)>eE(lnm)E(lnR)>E(lnm)E(lnR)−E(lnm)>E(lnR)IfyouincreaseleverageαinR=(−α)RfαRmyouincreasemeanandvolatilityIfRcangetanywherenearzero,lnRgoesoffto∞Thus,increasingαeventuallyleadstoadecreaseinElnRForexample,ifreturnsarenormal,thenE(R)=eE(lnR)σ(R)lnE(R)=E(lnR)σ(R)E(lnR)=lnE(R)−σ(R)E(lnR)=lnhαE(Rm)(−α)Rfi−ασ(Rm)Asαincreases,thesecondtermeventuallydominatesProblemsforChapter(a)pt=EtXβjµctjct¶−γctjptct=EtXβjµctjct¶−γIfγ=,pc=β(−β)=δwhereβ=(δ)(b)Ifγ<,thenariseinctjraisesptIfγ>,however,ariseinctjlowersptAnypieceofnewshastwopossibleeffects:cashßowsanddiscountratesInthiscasethediscountraterisesfasterthanthepayoffs,sothepriceactuallydeclines(a)TheÞrstorderconditionsarect−c∗=EtRβ(ct−c∗)withR=r,andhencect=Et(ct)Iteratethetechnologyforward,kt=R(Rktit)it=RktRititkt=RktRitRititRkt=ktR·itRitRit¸βkt=ktβhitβitβitiContinuingandwiththetransversalityconditionlimT→∞βTktT=,andi=e−ckt∞Xj=βjetj=∞Xj=βjctjTakingexpectations,kt∞Xj=βjEtetj=∞Xj=βjEtctjIntuitively,thepresentvalueoffutureconsumptionmustequalwealthplusthepresentvalueoffutureendowment(laborincome)Thejcomesfromthetiming,alasstandardinthemacroliteratureandnationalincomeaccountsIfyouadoptthemorecommonÞnancetimingconventionkt=(r)(ktit)yougetmorenaturalpresentvalueformulaswithβjNow,substitutetheÞrstorderconditioninthebudgetconstraint(productionpossibilityfrontierifyouwanttheGeneralEquilibriuminterpretation)kt∞Xj=βjEtetj=∞Xj=βjct=β(−β)ct=R(−R)ct=R−ct=ctrct=rktr∞Xj=βjEtetjConsumptionequalstheannuityvalueofwealth(capital)rktplusthepresentvalueoffuturelaborincome(endowment)ThisisthepermanentincomehypothesisItisnota“partialequilibrium”result—itisageneralequilibriummodelwithlineartechnologyandanendowmentincomeprocessNowtotherandomwalkinconsumptionJustquasiÞrstdifference,andusekt−kt=rktit,ct=rktr³βetβEtetβEtet´ct−=rkt−r³βet−βEt−etβEt−et´ct−ct−=r(kt−kt−)ct−ct−=r(rkt−et−−ct−)ct−ct−=rhrkt−et−−rkt−−r³βet−βEt−etβEt−et´ict−ct−=ret−r³βetβEtetβEtet´−³rr´³βet−βEt−etβEt−et´ct−ct−=ret−r³βetβEtetβEtet´−r³et−βEt−etβEt−et´ct=ct−(Et−Et−)rβ∞Xj=βjetjConsumptionisarandomwalkChangesinconsumptionequaltheinnovationinthepresentvalueoffutureincomeBobHall()noticedtherandomwalknatureofconsumptioninthismodel,andsuggestedtestingitbyrunningregressionsof∆ctonanyvariableattimet−ThispaperwasawatershedItistheÞrst“Eulerequation”testofamodelnoteitdoesnotrequirethefullmodelsolutiontyingtheshocksin∆cttofundamentaltasteandtechnologyshocks—thesecondterminourrandomwalkequationTheHansenSingleton()Eulerequationtestsgeneralizetononquadraticutility,randomassetreturnsforwhichitisimpossibletofullysolvethemodelTechnicaldetails:IhaveassumednofreedisposalyoufollowtheÞrstorderconditionsevenifpasttheblisspointIfyoucanfreelydisposeofconsumption,thenyouwillalwaysendupattheblisspointc∗soonerorlater(ThankstoAshleyWangforpointingthisoutHansenandSargent’streatmentsofthisproblemdealwiththeblisspointissue)Bytheway,thealgebraismucheasierifyouuselagoperators,iewritect=rktrβEt£(−βL−)−et¤Butifyouknowhowtodothat,you’veprobablyseenthismodelbefore(b)ct=rktr∞Xj=βjEtetj=rktrβ∞Xj=βjρjet=rktrβ−βρetct=ct−(Et−Et−)rβ∞Xj=βjetj=ct−rβ∞Xj=βjρjεt=ct−rβ−βρεtThetopequationdoeslooklikeaconsumptionfunction,butnoticethattheparameterrelatingconsumptionctoincomeedependsonthepersistenceofincomeeItisnota“psychologicallaw”oraconstantofnatureIfthegovernmentchangespolicysothatincomeismoreunpredictable(ieitgetsridofthepredictablepartofrecessions),thenthiscoefficientdeclinesdramaticallyTheincomecoefficientisnot“policyinvariant”ThisisthebasisofBobLucas()dramaticdeconstructionofKeynesianmodelsbasedonconsumptionfunctionsthatwereusedforpolicyexperimentsInbothequations,youseethatconsumptionrespondsto“permanentincome”andthatasshocksgetmore“permanent”—asρrises—consumptionmovesmore(c)RwastherateofreturnontechnologyDespitethesymbol,itisnot(yet)theinterestrate—theequilibriumrateofreturnononeperiodclaimstoconsumptionThatremainstobeprovedThelogicis,ÞrstÞndc,thenpricethingsfromtheequilibriumconsumptionstreamTobepreciseandpedantic,calltheriskfreerateRf,andRft=Etµβu(ct)u(ct)¶=βEtµct−c∗ct−c∗¶=βµct−c∗ct−c∗¶=β=RNow,thefunstuffWecanapproachthepriceoftheconsumptionstreambybruteforce,pt=Et∞Xj=mt,tjctj=Et∞Xj=βjc∗−ctjc∗−ctctj=Et∞Xj=βjc∗ctj−ctjc∗−ct=∞Xj=βjc∗ct−Et³ctj´c∗−ct=∞Xj=βjc∗ct−ct−vart(ctj)c∗−ctct=ctrβ−βρεtct=ctrβ−βρ(εtεt)ctj=ctrβ−βρ(εtεtj)Et(ctj)=ct(ofcourse)vart(ctj)=jµrβ−βρ¶σεpt=∞Xj=βjct(c∗−ct)−j³rβ−βρ´σεc∗−ct=∞Xj=βjct−j³rβ−βρ´σεc∗−ct=∞Xj=βjct−∞Xj=jβj³rβ−βρ´σεc∗−ct∞Xj=jβj=β(β−)pt=β−βct−β(−β)³rβ−βρ´σεc∗−ct=r−rct−r³−r´³rβ−βρ´σεc∗−ctpt=rct−β(−βρ)c∗−ctσεWowTheÞrsttermistheriskneutralprice—thevalueofaperpetuitypayingc(Don’tforgetEt(ctj)=ct)ThesecondtermisariskcorrectionItlowersthepriceIfσεishigh—morerisk—thepriceislowerIfρishigh—morepersistentconsumption—thepriceislowerNow,thehardterm—theeffectofconsumptionAttheblisspoint,theconsumerisashappyascanbe,andmarginalutilityfallstozeroHence,theconsumerisinÞnitelyriskaverse(u(c)u(c)risestoinÞnity)Thereisnoconsumptionyoucangivehimtocompensateforrisk,sincehe’sattheblisspointNosurprisethatthepricegoesoffto−∞hereAsconsumptionrisestowardstheblisspoint,theconsumergetsmoreandmoreriskaverse(uisconstant,uisfalling),sothepricedeclinesAbovetheblisspoint,theconsumervaluesconsumptionnegatively,sothepriceishigherthantheriskneutralversionThisfeature—thatriskaversionrisesasconsumptionrises—isobviouslynotagoodoneQuadraticutilityisbestusedasalocalapproximationFindac∗thatgivesasensibleriskaversion,andthenmakesurethemodeldoesn’tgettoofaraway!ThequestionsayspriceasafunctionofeandkI’mcurioushowIevergotthat,sinceitseemsamuchmorenaturalfunctionofccisafunctionofeandk,ofcourse,butsubstitutingthatindoesnotseemveryeasyThisisnotonlyahistoricallyimportantmodel,itintroducesaveryimportantmethodEvaluatinginÞnitesumsasinthelastproblemisahugepainInmostmodels,conditioninginformationisafunctionofonlyafewstatevariables,xtEverythingyoucouldwanttoknowaboutthecurrentstateoftheeconomy,andtheconditionaldistributionofeverythingyoucouldwanttoknowinthefutureiscontainedinthestatevariablesHence,prices(atleastproperlyscaled)havetobeafunctionofthestatevariablesInsteadofsolvingforpintermsofahugeinÞnitesum,youcansolvethefunctionalequationp(x)=Etmt,t(xt,xt)(p(xt)dt)Herewego(a)FromthebasicÞrstordercondition,pbt=Etβu(ct)u(ct)=Etβ∆c−γtpb(∆ct=h)=βπh→hh−γβπh→ll−γpb(∆ct=l)=βπl→hh−γβπl→ll−γ"pb(∆ct=h)pb(∆ct=l)#="πh→hπh→lπl→hπl→l#"βh−γβl−γ#pb=πxTheriskfreerateisofcourseRf=pb(b)Theconsumptionstream:pt=Ethβ∆c−γt(ptct)iptct=βEt·∆c−γtµptct¶¸Solvethisasafunctionalequation,asexplainedaboveFindpcinthehstateandinthelstate(functionsfromtwopointstothereallineareeasytodetermine—youjustÞndthevaluesatthetwopoints)pc(h)=βπh→hh−γµpc(h)¶βπh→ll−γµpc(l)¶"pc(h)pc(l)#=β"πh→hh−γπh→ll−γπl→hh−γπl→ll−γ#Ã"#"pc(h)pc(l)#!pc=βπ∗(pc)pc=(−βπ∗)−βπ∗WecanÞndreturnsfromRt=ptctptctctctNotewhenpcisconstant,RisjustaconstanttimesconsumptiongrowthYouneedaverysmallpcbeforeRismuchdifferentfromconsumptiongrowthConditionallyexpectedreturnsfollowfromtheprobabilities(c)StartwiththecalibrationIt’smostnaturaltotakethetwopointstobeequallyaboveandbelowthemean,h=x,l=−xandequalprobabilitiesThen,youwant(x)(−x)=xx=ie,x=HerearemyresultsInstateTostatehlγ=bondpriceRfpcRhlγ=bondpriceRfpcRhlThemajorfailingistheequitypremiumThemeanstockreturnisalmostexactlythesameastheriskfreerateAlso,stockreturnsareperfectlycorrelatedwithconsumptiongrowthThestandarddeviationofstockreturnsisabout,notaboutTheSharperatiohE(R)−Rfiσ(R)iswaytoolow(d)Togetserialcorrelationinconsumptiongrowth,Itriedπoftheformπ="θ−θ−θθ#Now,E(dct|dct=h)=(θ)∗(gx)(−θ)∗(g−x)=gθxE(dct|dct=l)=(−θ)∗(gx)(θ)∗(g−x)=g−θxHerearemyresultsforapositiveserialcorrelationhlγ=,θ=pbRfpcRhlρ(∆ct,∆ct−)ThemainreasonIputthisinatthisstageistogetvariationinpriceswiththeinitialstateInthepreviouscase,theworldlooksthesamefromanystartingdate,sothereisnovariationinprices(exante)Theinterestrateandstockreturnarehigherfromthehighstate,becauseexpectedfutureconsumptiongrowthishigherHigherreturnmeanslowerpriceorpcProblemsforChapterTheabsenceofarbitrageimpliestheLOOP,butnotviceversaNA→LOOPSupposetheabsenceofarbitrageholds,butnottheLOOPLetz=axbyIfp(z)>ap(x)bp(y),however,theportfolioz−(axby)isanarbitrageIndiscountfactorlanguage,ifthereisanm>,thenthereisanmTheLOOPtheoremspeciÞesanminX,howeverGivenanm,wecanconstructanminXbyx∗=proj(m|X)LOOP→NAIndiscountfactorlanguage,imagineacompletemarketwithadiscountfactorthatisnegativeinsomestateofnatureThisgeneratesasetofpricesthatobeythelawofoneprice,butleavearbitrageopportunitiesThecorrespondingsetofpricesandpayoffsareacounterexampleinportfoliolanguageThedangerofapplyingtheLOOPornoarbitrageinasampleisthatyoutypicallydon’tseeallofthepossiblerealizationsinanyÞnitesampleForexample,acorporateyearbondandagovernmentyearbondwillhaveidenticalpayoffsinanysampleinwhichthecorporationdoesnotdefault,butthecorporatebondwillhavealowerpriceThislookslikeaviolationofthelawofonepriceHansenJagannathanboundswithpositivitytypicallyshow“arbitragebound”limitsontheriskfreerate,whichcomefromsamplesinwhichonesecuritydominatesanotherThesearbitrageboundsdisappearifonepositsadistributioninwhichitisalwayspossibleforeachsecuritytounderperformtheother(a)R−isadiscountfactorItisnotnecessarilyinthepayoffspace,sincethatspaceisconstructedoflinearcombinationsoftheassetsx∗istheuniquediscountfactorinthepayoffspace,butnottheonlydiscountfactorOften,R>,ieforlimitedliabilitysecuritieslikestocksInthiscase,R−isalwayspositive,butsoisRE(R)Securitiesdonothavetobelimitedliability,soingeneralR−canbenegativeThebiggesttroublewiththisdiscountfactoristhatitcanbeinÞniteifR=canhappen,inwhichcasetheexpectationmaynotbedeÞned(Itmaybeoutofthesetofrandomvariableswithsecondmoments)(b)TheÞrstorderconditionsareEµαRR¶=λThus,m=λαRisadiscountfactorIngeneral,youcan’tsolvetheÞrstorderconditionsforαanalyticallyAnothermorebeautifulwaytodothisWeknowthateverypayoffinXcanbepricedbyadiscountfactorm>StatetheproblemasmaxE(ln(R))st=E(mR)maxXπiln(Ri)st=XπimiRitheÞrstorderconditions—chooseRiineachstatei—areπiRi=λπimiRi=λmiPluggingthisintotheconstraint,youÞndλ=Thus,wehaveproved:theinversereturnoftheportfoliothatmaximizeslnreturnsisequaltoadiscountfactor(c)ThelatterapproachisaquickwaytodothisincontinuoustimeAdiscountfactorisaprocesΛthatpricespayoffsatanydateThus,considerthe“growthoptimaltradingstrategy”—thevalueprocessVthatmaximizesmaxE·lnµVTV¶¸stVΛ=EtVTΛTJustasbefore,wehaveVTV=ΛΛTThisholdsatanyhorizon,soVtisanumeraire—apriceprocesssuchthatforanysecuritypricedbyΛ,ptVt=EtZ∞s=DtsVtsdsProblemsforChapterYouhavetoÞndequationsthatexpresstherightanglesinthepictureRightanglesmeansorthogonalwithsecondmomentnorm,sowewanttoprovethatthelinefromanyRetoitsprojectiononRe∗liesatrightanglestoRe∗,

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