A-critique-of-the-asset-pricing-theory’s-tests

JournalofFinancialEconomics4(1977)129-176.(0North-HollandPublishingCompanyACRITIQUEOFTHEASSETPRICINGTHEORY’STESTSPartI:OnPastandPotentialTestabilityoftheTheory*RichardROLL*lUniversityofCalifornia,LosAngeles,CA90024,U.S.A.ReceivedJune1976,revisedversionreceivedOctober1976Testingthetwo-parameterassetpricingtheoryisdifficult(andcurrentlyinfeasible).Duetoamathematicalequivalencebetweentheindividualreturn/beta’linearityrelationandthemarketportfolio’smean-varianceefficiency,anyvalidtestpresupposescompleteknowledgeofthetruemarketportfolio’scomposition.Thisimplies,interalia,thateveryindividualassetmustbeincludedinacorrecttest.Errorsofinferenceinduciblebyincompletetestsarediscussedandsomeambiguitiesinpublishedtestsareexplained.Ifthehornhonksandthemechanicconcludesthatthewholeelectricalsystemisworking,heisindeeptrouble...Pirsig(1974)1.IntroductionandsummaryThetwo-parameterassetpricingtheoryistestableinprinciple;butargumentsaregivenherethat:(a)Nocorrectandunambiguoustestofthetheoryhasappearedintheliterature,and(b)thereispracticallynopossibilitythatsucha*ThisisPartIofathree-partstudy.Parts11andIIIaresummarizedintheintroductionhere.butwillappearinlaterissues.Acopyofthecompletepapercanbeobtainedbywritingtheauthorat:GraduateSchoolofManagement,UniversityofCalifornia,LosAngeles,CA90024.USA.**ThispaperwaswrittenwhiletheauthorwasattheCentrcd’EnseignementSup&ieurdesAfl’aires.France.EugeneFama,MichaelC.Jensen,JohnB.Long,Jr.,StephenRossandBrunoH.SolnikprovidedmanyusefulcommentsandPatriciaPorterprovidedexcellentsecretarialservice.Whilethepaperwasbeingwritten,Famapointedoutthathisnewbook(1976)containssomeofthesameanalysisandconclusions.NewpapersbyStephenRoss(forthcoming)andJohnB.Long(1976)containresultsemphasized,andformerlybelievedtohavebeendiscovered,here.Thereaderwillbeabletoverify,however,thatmostofthismaterialisnon-redundant.Totheauthorscriticisedhem:thesepapersweresingledoutbecausetheyarethebestandmostwidelyreadonthesubject.Ihavewrittensomepapersinthisareatooandhavetaughtthesubjecttoanumbcr,ofunsuspectingstudents.So,theabsenceofdetailedself-criticismshouldbeattributedtothegreaterimportanceoftheotherpapersanddoesnotimplyanypersonalprescicncc.Nonewaspresent.130R.Roll,Critiqueofassetpricingtheorytests-Itestcanbeaccomplishedinthefuture.Thisbroadindictmentofoneofthethreefundamentalparadigmsofmodernfinancewillundoubtedlybegreetedbymycolleagues,asitwasbyme,withscepticismandconsternation.Thepurposeofthispaperistoeliminatethescepticism.(Noreliefisofferedfortheconsternation.)Herearethepaper’sconclusions:(1)Thereisonlyasingletestablehypothesisassociatedwiththegeneralizedtwo-parameterassetpricingmodelofBlack(1972).Thishypothesisis:‘themarketportfolioismean-varianceefficient’.(2)Allotherso-calledimplicationsofthemodel,thebestknownbeingthelinearityrelationbetweenexpectedreturnand‘beta’,followfromthemarketportfolio’sefficiencyandarenotindependentlytestable.Thereisan‘ifandonlyif’relationbetweenreturn/betalinearityandmarketportfoliomean-varianceefficiency.(3)Inanysampleofobservationsonindivjdualreturns,regardlessofthegener-atingprocess,therewillalwaysbeaninfinitenumberofex-postmean-varianceefficientportfolios.Foreachone,thesample‘betas’calculatedbetweenitandindividualassetswillbeexactlylinearlyrelatedtothein-dividualsamplemeanreturns.Inotherwords,ifthebetasarecalculatedagainstsuchaportfolio,theywillsatisfythelinearityrelationexac+whetherornotthetruemarketportfolioismean-varianceefficient.(Thesamepropertiesalsoholdexante,ofcourse).Theseresultsareimpliedinearlierliterature[e.g.,Ross(1972)],butIdonotbelievethattheirfullconsequenceshavebeenadequatelyexploredpreviously.Someoftheseconsequencesare:(4)Thetheoryisnottestableunlesstheexactcompositionofthetruemarketportfolioisknownandusedinthetests.Thisimpliesthatthetheoryisnottestableunlessallindividualassetsareincludedinthesample.(5)Usingaproxyforthemarketportfolioissubjecttotwodifficulties.Firsttheproxyitselfmightbemean-varianceefficientevenwhenthetruemarketportfolioisnot.Thisisarealdangersinceeverysamplewilldisplayefficientportfoliosthatsatisfyperfectlyallofthetheory’simplications.Forexample,supposethereexist1000assetsbutonly500areusedinthesample.Forthesample,therewillexistwell-diversifiedportfoliosofthe500assetsthatseemtobereasonableproxiesforthemarketandforwhichobservedreturnsareexactlylinearilyrelatedcross-sectionallytoobservedbetas.Ontheotherhand,thechosenproxymayturnouttobeinefficient;butobviously,thisaloneimpliesnothingaboutthetruemarketportfolio’sefficiency.Further-more,mostreasonableproxieswillbeveryhighlycorrelatedwitheachotherandwiththetruemarketwhetherornottheyaremean-varianceefficient.Thishighcorrelationwillmakeitseemthattheexactcompositionisun-important,whereasitcancausequitedifferentinferences.R.Roll,Critiqueofassetpricingtheorytests-I131(6)Asacaseinpoint,adetaileddiscussionisprovidedofthepapersbyFamaandMacBeth(1973).Black,Jensen,andScholes(1972)andBlumeandFriend(1973),inthecontextoftheirrejectionoftheSharpe-Lintnermodel.ItisshownthattheirtestsresultsarefullycompatiblewiththeSharpe-Li