apt套利模型Word格式.docx

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marketportfolioand02”,isthecovariancebetweenthereturnsontheith

assetandthemarketportfolio.（Ifarisklessassetdoesnotexist,pisthe

zero-betareturn,i.e.,thereturnonallportfoliosuncorrelatedwiththe

marketportfolio.）l

Thelinearrelationin

（1）arisesfromthemeanvarianceefficiencyofthe

marketportfolio,butontheoreticalgroundsitisdifficulttojustify

eithertheassumptionofnormalityinreturns（orlocalnormalityin

Wienerdiffusionmodels）orofquadraticpreferencestoguaranteesuch

efficiency,andonempiricalgroundstheconclusionsaswellasthe

*ProfessorofEconomics,UniversityofPennsylvania.Thisworkwassupported

byagrantfromtheRodneyL.WhiteCenterforFinancialResearchattheUniversity

ofPennsylvaniaandbyNationalScienceFoundationGrantGS-35780.

1SeeBlack[2]forananalysisofthemeanvariancemodelintheabsenceofariskless

asset.

341

Copyright!

CI1976byAcademicPress,Inc.

Allrightsofreproductioninanyformreserved.

342STEPHENA.ROSS

assumptionsofthetheoryhavealsocomeunderattack.2Therestrictiveness

oftheassumptionsthatunderliethemeanvariancemodelhave,however,

longbeenrecognized,butitstractabilityandtheevidentappealofthe

linearrelationbetweenreturn,Ei,andrisk,6,）embodiedin

（1）have

ensureditspopularity.Analternativetheoryofthepricingofriskyassets

thatretainsmanyoftheintuitiveresultsoftheoriginaltheorywas

developedinRoss[13,141.

Initsbarestessentialstheargumentpresentedthereisasfollows.

Supposethattherandomreturnsonasubsetofassetscanbeexpressed

byasimplefactormodel

$=zEi+pig+ci,

（2）

where8isameanzerocommonfactor,andCiismeanzerowiththe

vector（z）sufficientlyindependenttopermitthelawoflargenumbersto

hold.Neglectingthenoiseterm,Ei,asdiscussedinRoss[14]

（2）isa

statementthatthestatespacetableauofassetreturnsliesinatwodimensional

spacethatcanbespannedbyavectorwithelements6,）

（where0denotesthestateoftheworld）andtheconstantvector,

ecc（I,...,1）.

Step1.Formanarbitrageportfolio,7,ofallthenassets,i.e.,a

portfoliowhichusesnowealth,ne=0.Wewillalsorequirentobea

well-diversifiedportfoliowitheachcomponent,Q,oforderl/nin

（absolute）magnitude.

Step2.Bythelawoflargenumbers,forlargeIIthereturnonthe

arbitrageportfolio

（3）

Inotherwordstheinfluenceonthewell-diversifiedportfolioofthe

independentnoisetermsbecomesnegligible.

Step3.Ifwenowalsorequirethatthearbitrageportfolio,7,bechosen

soastohavenosystematicrisk,then

andfrom（3）

2SeeBlumeandFriend[3Jforarecentexampleofsomeoftheempiricaldifficulties

facedbythemeanvariancemodel.Foragoodreviewofthetheoreticalandempirical

literatureonthemeanvariancemodelseeJensen[6].

CAPITALASSETPRICING343

Step4.Usingnowealth,therandomreturnq.%hasnowbeenengineered

tobeequivalenttoacertainreturn,vE,hencetopreventarbitrarily

largedisequilibriumpositionswemusthaveV./Z=0.Sincethisrestriction

mustholdforall17suchthatve=-VP-=0,Eisspannedbyeandpor

Ei=p+A,&

（4）

forconstantspandX.Clearlyifthereisarisklessasset,pmustbeits

rateofreturn.Evenifthereisnotsuchanasset,pistherateofreturnon

allzero-betaportfolios,01,i.e.,allportfolioswithale=1andL@=0.

If01isaparticularportfolioofinterest,e.g.,themarketportfolio,!

x,,,,

withE,,,=a,,$,（4）becomes

Et=p+C-G,,-p）Pi.（5）

Condition（5）isthearbitragetheoryequivalentof

（1）andif8isa

marketfactorreturnthen&

willapproximatebi.Theaboveapproach,

however,issubstantiallydifferentfromtheusualmean-varianceanalysis

andconstitutesarelatedbutquitedistincttheory.Foronething,the

argumentsuggeststhat（5）holdsnotonlyinequilibriumsituations.but

inallbutthemostprofoundsortofdisequilibria.Foranother,themarket

portfolioplaysnospecialrole.

Thereare,however,someweakpointsintheheuristicargument.For

example,asthenumberofassets,n,isincreased,wealthwill,ingeneral,

alsoincrease.Increasingwealth,though,mayincreasetheriskaversionof

someeconomicagents.Thelawoflargenumbersimplies,inStep2.that

thenoiseterm,+,becomesnegligibleforlargen,butifthedegreeofrisk

aversionisincreasingwithnthesetwoeffectsmaycanceloutandthe

presenceofnoisemaypersistasaninfluenceonthepricingrelation.

InSectionIwewillpresentanexampleofamarketwherethisoccurs.

Furthermore,evenifthenoisetermcanbeeliminated,itisnotatall

obviousthat（5）musthold,sincethedisequilibriumpositionofoneagent

mightbeoffsetbythedisequilibriumpositionofanother.3

InRoss[13],however,itwasshownthatif（5）holdsthenitrepresents

anEorquasi-equilibrium.Theintentofthispaperistosupplytherigorous

analysisunderlyingthestrongerstabilityargumentsabove.InSectionIt

wewillpresentsomeweaksufficientconditionstoruleouttheabove

exceptions（andtheexampleofSectionI）andwewillproveageneral

versionofthearbitrageresult.Section11alsoincludesabriefargument

ontheempiricalpracticalityoftheresults.Amathematicalappendix

3Greenhasconsideredthispointinatemporaryequilibriummodel.Essentialli

hearguesthatifsubjectiveanticipationsdiffertoomuch,thenarbitragepossibilities

willthreatentheexistenceofequilibrium.

344STEPHENA.ROSS

containssomesupportiveresultsofasomewhattechnicalandtangential

nature.SectionIIIwillbrieflysummarizethepaperandsuggestfurther

generalizations.

1.ACOUNTEREXAMPLE

Inthissectionwewillpresentanexampleofamarketwherethe

sequenceofequilibriumpricingrelationsdoesnotapproachtheone

predictedbythearbitragetheoryasthenumberofassetsisincreased.

Thecounterexampleisvaluablebecauseitmakesclearwhatsortof

additionalassumptionsmustbeimposedtovalidatethetheory.

Supposethatthereisarisklessassetandthatriskyassetsareindependently

andnormallydistributedas

where

and

5i=Ei+E”f,（6）

E{q=0,

E（Q）=u2.

Thearbitrageargumentwouldimplythatinequilibriumallofthe

independentriskwoulddisappearand,therefore,

Ei=sp,（7）

Assume,however,thatthemarketconsistsofasingleagentwitha

vonNeumann-Morgensternutilityfunctionoftheconstantabsoluterisk

aversionform,

U（z）=-exp（--AZ）.（8）

Lettingwdenotewealthwiththerisklessassetasthenumeraire,andCYthe

portfolioofriskyassets（i.e.,0~~istheproportionofwealthplacedinthe

ithriskyasset）andtakingexpectationswehave

=-exp（--Awp）E{exp（--Awol[Z-p.e]））

=-exp（--Awp）{exp（--Awol[E-p.e]+（c~~/~）（Aw）~（~oL））}.（9）

Thefirst-orderconditionsatamaximumaregivenby

CAPITALASSETPRICING345

Iftherisklessassetisinunitsupplythebudgetconstraint（Walras’Law

forthemarket）becomes

11’=f@&

M+’1=（l//W）i（Ei-p）+1,

i-li=l

（11）

Theinterpretationofthebudgetconstraint（11）dependsonthe

particularmarketsituationwearedescribing.Suppose,first,thatweare

addingassetswhichwillpayarandomtotalnumeraireamount,Zi.

Ifpiisthecurrentnumerairepriceoftheassetthen

Normalizingallriskyassetstobeinunitsupplywemusthave

andthebudgetconstraintsimplyassertsthatwealthissummedvalue,

IfweletFidenotethemeanofFiandc2,itsvariance,then（10）canbe

solvedforpias

pi=（l/p）{C,-AC2）.

Asaconsequence,theexpectedreturns,

Ei=&

/pi=p（c’J（Ti-Ac2）},

willbeunaffectedbychangesinthenumberofassets,n,fori<

n,and

needbearnosystematicrelationtopasnincreases.Thisisaviolation

ofthearbitragecondition,（7）.Notice,too,thataslongasC+isbounded

aboveAc2,wealthandrelativeriskaversion,Aw,areunboundedinn.

AnalternativeinterpretationofthemarketsituationwouldbethatasII

increasesthenumberofriskyinvestmentopportunitiesoractivitiesis

beingincreased,butnotthenumberofassets.Inthiscasewealth,w,would

simplybethenumberofunitsoftherisklessassetheldandwouldremain

constantasnincreased.Thequantitiesaiwnowrepresenttheamountof

therisklessholdingsputintotheithinvestmentopportunityandforthe

marketasawholewemusthave

346STEPHENA.ROSS

Furthermore,iftherandomtechnologicalactivitiesareirreversible,

theneach01~30.From（10）itfollowsthat

Ei-p>

O

fEj-p=5IEi-p）=u2（Aw）2iyi<

054u..

i=li=li=l

Hence,asn---fco,thevectorEapproachestheconstantvectorwith

entriespinabsolutesum（theZ1norm）whichisaverystrongtypeof

approximation.Underthissecondinterpretation,then,th