Mutual fund performance and seemingly unrelated as

MutualFundPerformanceandSeeminglyUnrelatedAssetsby*·Lubo·sP¶astorandRobertF.StambaughFebruary2001AbstractEstimatesofstandardperformancemeasurescanbeimprovedbyusingreturnsonassetsnotusedtode¯nethosemeasures.Alpha,theinterceptinaregressionofafund'sreturnonpassivebenchmarkreturns,canbeestimatedmorepreciselybyusinginformationinreturnsonnon-benchmarkpassiveassets,whetherornotonebelievesthoseassetsarepricedbythebenchmarks.Afund'sSharperatiocanbeestimatedmorepreciselybyusingreturnsonotherassetsaswellasthefund.Newestimatesoftheseperformancemeasuresforalargeuniverseofequitymutualfundsexhibitsubstantialdi®erencesfromtheusualestimates.JELClassi¯cations:G11,G12,C11Keywords:performanceevaluation,mutualfunds*GraduateSchoolofBusiness,UniversityofChicago(P¶astor)andtheWhartonSchool,UniversityofPennsylvaniaandtheNationalBureauofEconomicResearch(Stambaugh).ResearchsupportfromtheCenterforResearchinSecurityPricesandDimensionalFundAdvisorsisgratefullyacknowledged(P¶astor).Wearegratefultoseveralanonymousreferees,ChrisBlake,EugeneFama,WayneFerson,AnthonyLynch,AndrewMetrick,DeanPaxson,TobyMoskowitz,andseminarparticipantsattheFederalReserveBankofNewYork,NorthwesternUniversity,OhioStateUniversity,UniversityofChicago,UniversityofPennsylva-nia,UniversityofRochester,VanderbiltUniversity,the2000NBERSummerInstitute,the2000PortugueseFinanceNetworkConference,andthe2001AFAMeetingsforhelpfulcomments.Thispaperisbasedinpartontheauthors'earlierworkingpaper,\EvaluatingandInvestinginEquityMutualFunds.1.IntroductionAmutualfund'shistoricalperformanceisoftensummarizedbyanestimateofitsalphaoritsSharperatio.Alphaisde¯nedastheinterceptinaregressionofthefund'sexcessreturnontheexcessreturnofoneormorepassivebenchmarks,whiletheSharperatioisthefund'sexpectedexcessreturndividedbythestandarddeviationofthefund'sreturn.Thesemeasuresareusuallyestimatedwithhistoricalreturnsontheassetsthatde¯nethem.Thatis,alphaisestimatedusingexcessreturnsonthefundandthebenchmarks,andtheSharperatioisestimatedusingtheexcessreturnsonthefund.Thisstudydemonstratesthatanestimateofeitherperformancemeasurecantypicallybemademoreprecisebyusinghistoricalreturnson\seeminglyunrelatedassetsnotusedinthede¯nitionofthatmeasure.Alpha,forexample,isusuallyestimatedbyapplyingordinary-least-squares(OLS)totheregressionrA;t=®A+¯0ArB;t+²A;t;(1)whererA;tisthefund'sreturninmontht,rB;tisak£1vectorcontainingthebenchmarkreturns,and®Adenotesthefund'salpha.(Henceforthweuse\returnstodenoteratesofreturninexcessofarisklessinterestrateorpayo®sonzero-investmentspreadpositions.)Thechoiceofbenchmarksisoftenguidedbyapricingmodel,asinJensen's(1969)pioneeringuseoftheCapitalAssetPricingModel(CAPM)ofSharpe(1964)andLintner(1965)toinvestigatemutualfundalphasrelativetoasinglemarket-indexbenchmark.Otherstudies,beginningwithLehmannandModest(1987),examinefundalphaswithrespecttoasetofmultiplebenchmarksviewedastherelevantfactorsforpricinginamultifactormodel,suchastheArbitragePricingTheoryofRoss(1976).Asonespecialcase,assumethatthebenchmarkassetsusedtode¯nealphadoindeedexactlypriceotherpassiveassets.Considertheregressionofanon-benchmarkpassivereturnrn;tonthebenchmarkreturns,rn;t=®n+¯0nrB;t+²n;t;(2)wherethecorrelationbetween²A;tand²n;tispositive.Ifthebenchmarkspriceotherpassiveassets,then®n=0.NowsupposethatoverthesamesampleperiodusedtoobtaintheOLSestimateof®A,theOLSestimateof®nislessthanzero.Sincethetruevalueof®niszero,thenegativeestimateof®nisfullyattributedtosamplingerror.Giventhepositivecorrelationbetween²A;tand²n;t,theOLSestimateof®Aisexpectedtocontainnegativesamplingerroraswell,andthisadditionalinformationcanbeusedinestimating®A.1Asanotherspecialcase,assumethatthebenchmarksusedtode¯nealphahavenopricingability.Toseehownon-benchmarkassetsprovideadditionalinformationabout®Ainthiscase,considerafundwhoseavailablereturnhistoryisshorterthanthehistoriesofrn;tandrB;t.Theexplosivegrowthofthemutualfundindustryinrecentyearspresentsinvestorswithmanyfundsthathaverelativelyshorthistories.SupposethattheOLSestimateof®ncomputedforthesampleperiodofthefund'savailablehistoryislessthantheOLSestimateof®ncomputedforalongersampleperiod.Sincethelatterestimateismoreprecise,the¯rstestimateismorelikelytobelessthanthetrue(unknown)valueof®n.Giventhepositivecorrelationbetween²A;tand²n;t,thesamecanbesaidoftheOLSestimateof®Arelativetoitstruevalue,andthisinformationcanbeusedinestimating®A.Theadditionalinformationcomesnotthroughapricingmodel,asinthepreviouscase,butthroughthelongerhistoriesofthepassiveassetreturns.Inthetwospecialcasesdescribedabove,®nisassumedtobeeitherzeroorcompletelyunknown.Onemaywellpreferanintermediateversioninwhichthebenchmarksarebe-lievedtoberelevantforpricingotherpassiveassets,butnotwithouterror.Inthatgeneralcase,whichweimplementinaBayesianframework,non-benchmarkassetsplayarolethatcombinesaspectsofbothpreviouscases.Additionalinformationabout®Aisprovidedbytheextenttowhichtheshort-historyestimateof®ndi®ersfromzeroaswellasfromitslong-historyestimate.Ifthepriordistributionfor®nisconcentratedaroundzero,thenmostoftheadditionalinformationisextract