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The Three moment CAPM: Theoretical Foundations and an Asset Pricing Models

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Niveau: Supérieur, Doctorat, Bac+8
The Three-moment CAPM: Theoretical Foundations and an Asset Pricing Models Comparison ? Emmanuel Jurczenkoy Bertrand Mailletz April 2000 - P r e l im in a r y d ra f t ; n o t t o b e q u o t e d - Abstract The purpose of this article is to present the Three Moment Capital Asset Pricing Model and some recent extensions. The traditional CAPM is based on several restrictive hypotheses. In particular, the normality of the return distribution and a “small” risk are supposed to be valid. These hypotheses are subject to two traditional criticisms: one is tied to their theoretical foundations; the other is related to their inadequation with the stylised facts highlighted in empirical studies. Following the seminal work of Krauss and Litzenberger (1976) and more recently those of Simaan (1993) and Gamba and Rossi (1998), we extend the Sharpe- Lintner-Mossin framework to incorporate the e?ect of skewness on asset valuation. Using a two-fund monetary separation theorem, we develop an exact three-moment capital asset pricing relation. We then present a three mutual fund theorem, which allows to extend the three-moment CAPM in a market without riskless asset. Finally, we put into perspective the link between some multifactor models and this revisited three-moment CAPM. Keywords: CAPM, Asset Pricing Models, Return Densities.

  • increasing absolute risk

  • sharpe-lintner-mossin capm

  • utility function

  • density return

  • selection criterion

  • rational investor

  • asset pricing

  • no risk-free


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TheThree-momentCAPM:TheoreticalFoundationsandanAssetPricingModelsComparison¤EmmanuelJurczenkoyBertrandMailletzApril2000-Preliminarydraft;nottobequoted-AbstractThepurposeofthisarticleistopresenttheThreeMomentCapitalAssetPricingModelandsomerecentextensions.ThetraditionalCAPMisbasedonseveralrestrictivehypotheses.Inparticular,thenormalityofthereturndistributionanda“small”riskaresupposedtobevalid.Thesehypothesesaresubjecttotwotraditionalcriticisms:oneistiedtotheirtheoreticalfoundations;theotherisrelatedtotheirinadequationwiththestylisedfactshighlightedinempiricalstudies.FollowingtheseminalworkofKraussandLitzenberger(1976)andmorerecentlythoseofSimaan(1993)andGambaandRossi(1998),weextendtheSharpe-Lintner-Mossinframeworktoincorporatethee¤ectofskewnessonassetvaluation.Usingatwo-fundmonetaryseparationtheorem,wedevelopanexactthree-momentcapitalassetpricingrelation.Wethenpresentathreemutualfundtheorem,whichallowstoextendthethree-momentCAPMinamarketwithoutrisklessasset.Finally,weputintoperspectivethelinkbetweensomemultifactormodelsandthisrevisitedthree-moment.MPACKeywords:CAPM,AssetPricingModels,ReturnDensities.J.E.L.Classi…cation:G.110,G.120.¤WethankThierryChauveauforhelpfulcommentswhenpreparingthispaper.Erryorsorweaknessofthispreliminarydraftrestunderourresponsability.czenTkoE@AaMol.-coEmS.AC8o0r5re9spdounCdiNngRSau-thUonri:veEr.siJtuyrcozfenPkaori,sM1SPEa-nTtEAoMn-,S1o0r6b-o1n1n2e.BdE-dmelailH:ôepjituarl-756z47ParisCedex13FRANCE.Tel.:(331)44078269/70(facsimile).A.A.TAEdAviMsor-s(EASBAN8-0A59MRduOCGrNoRuSp).-EU-nmivaeilr:sibtymaoifllePta@riusni1v-Ppaarnitsh1é.forn.-Sorbonne,ESCPand1
TheThree-momentCapitalAssetPricingModel:TheoreticalFoundationsandanAssetPricingModelsComparison1IntroductionThevalidityoftheSharpe-Lintner-Mossin-TreynorCapitalAssetPricingModel(CAPM)hasbeenquestionedbyseveralempiricaltests.Thismodelremains,nevertheless,oneofthemostimportantcontributionofthemodern…nancetheory,asemphasizedbyJagannathanandWang(1996)forinstance.Thismodelisationofthe…nancialmarketequilibriumisbasedonseveralrestrictivehypotheses.Twoofthemconcernthenormalityofthereturndistributionandthecharacteristicsoftheagentspreferences.Thelatterisnecessarytolegitimisethisformalizationoftheinvestorsoptimisationinariskysituation.Underthe…rsthypothesis,thereturndistributioniscompletelyde…nedbyitstwoinitialmoments.Underthesecond,itisalsopossibletoconsiderthattheexpectedutilityfunctionisbasedonlyonthemeanandthevarianceofthereturns.Thesehypothesesaresubjecttotwotraditionalcriticisms:the…rstoneistiedtothetheoreticalfoundationsofthisapproach;thesecondisbasedontheirinadequacywiththestylizedfactshighlightedinempiricalstudies.Thereturnnormalityhypothesisimpliesindeedthattheinvestorcanlosemorethanhisinitialwealthandthequadraticutilityfunctiondoesnotcorrespondtotherationalagentbehaviouralcharacteristics(seePratt,1964).Moreover,theapproximationdoneinconsideringonlythetwoinitialmomentsofthereturndensityiseconomicallyjusti…edbytheexistenceofa“small”absoluteriskinthesenseofSamuelson(1970)orbyasmallrelativeriskinthesenseofTsiang(1972).Thehypothesisofnormalityofreturnsdensities1isclearlyrejectedaccordingtotheresultsofmanyempiricalstudies(seeforinstanceMandelbrot,1997).Theworldwidesuccessofderivativesmarketsandtheiruseinthecontextofcovered(oruncovered)operationsleadstoconvexpay-o¤functionswhicharehardlycompatiblewiththetraditionalGaussianorlog-normalreturndistributions.Activeportfoliosmanagementandtheexistenceofhedgefundshighlightthesameproblem.Such…nancialstrategies,commonlyusedinthemarket,clearlyindicatethatinvestorscareforhighermomentsthanthe…rsttwo.Moreover,theexistenceoftheseconvexpay-o¤functionsdoesnotsupportthe“smallrisk”hypothesis.These“heroic”hypotheses,aswellasmanyempiricaldi¢culties,haveledseveralauthorstorejecttheexistenceofalinearrelationbetweenthesystematicriskandthereturnofanasset(seeFamaandFrench,1992).AlternativeapproacheshavebeenproposedafterthisrelativefailureoftheoriginalCAPM.Amongthemainpossibilitiesinvestigated,thesixfollowingextensionscanbedistinguished(withoutpretencetoexhaustivity).Di¤erentdensityprobabilitieshavebeensubstitutedtotheGaussiandistribution(seeHarveyandZhou,1993).Atime-1Whichcanbeconteredbythelognormalityargument.Nevertheless,thelognormalhy-pothesishassomeothersdrawbacks.2