Beyond mean-variance risk and performance measures

BEYONDMEAN-VARIANCE:RISKANDPERFORMANCEMEASURESFORPORTFOLIOSWITHNONSYMMETRICRETURNDISTRIBUTIONSHayneE.LelandHaasSchoolofBusinessUniversityofCalifornia,BerkeleyOctober25,1997RevisedOctober26,1998______________________________________________TheauthorthanksSteinarEkern,WayneFerson,MarkGrinblatt,RonaldKahn,MarkRubinstein,AamirSheikh,andDanStefekforhelpfuldiscussions.Errorsandomissionsremaintheauthor'sfault.AbstractMostpractitionersmeasureinvestmentperformancebasedontheCAPM,determiningportfolioalphasorSharpeRatios.ButthevalidityofthisanalysisrestsonthevalidityoftheCAPM,whichassumeseithernormallydistributed(andthereforesymmetric)returns,ormean-variancepreferences.Bothassumptionsaresuspect:evenifassetreturnswerenormallydistributed,thereturnsofoptionsordynamicstrategieswouldnotbe.Andinvestorsdistinguishupsidefromdownsiderisks,implyingskewnesspreference.Thishasledtotheadoptionofadhoccriteriaformeasuringriskandperformance,suchasValueatRiskandtheSortinoRatio.Weconsideraworldinwhichthemarketportfolio(butnotnecessarilyindividualsecurities)hasidenticallyandindependentlydistributed(i.i.d.)returns.Inthisworldthemarketportfoliowillbemean-varianceinefficientandtheCAPMalphawillmismeasurethevalueaddedbyinvestmentmanagers.Theproblemisparticularlysevereforportfoliosusingoptionsordynamicstrategies.Strategiespurchasing(writing)fairly-pricedoptionswillbefalselyaccordedinferior(superior)performanceusingtheCAPMalphameasure.WeshowhowasimplemodificationoftheCAPMbetacanleadtocorrectriskmeasurementforportfolioswitharbitraryreturndistributions,andtheresultingalphasofallfairly-pricedoptionsand/ordynamicstrategieswillbezero.Wediscussextensionswhenthemarketportfolioisnotassumedtobei.i.d.BEYONDMEAN-VARIANCE:RISKANDPERFORMANCEMEASURESFORPORTFOLIOSWITHNONSYMMETRICRETURNDISTRIBUTIONSI.IntroductionHowcanonedeterminewhetheraninvestmentmanagerhasaddedvaluerelativetorisk?Acorrectperformanceassessmentrequiresbothgoodtheory,todeterminethepropermeasureofrisk,andappropriatestatisticaltechniquestoquantifyriskmagnitudes.Thispaperfocusesonmeasuresofriskandtheirimplicationsforinvestmentperformanceevaluation.Whiletherehavebeensomenotablerecentadvancesinthetheoryofperformancemeasurement,mostpracticeisstillfirmlyrootedintheapproachoftheCapitalAssetPricingModel(CAPM).1IntheCAPMworld,theappropriatemeasureofriskofanyassetorportfoliopisgivenbyitsbeta:(1)]rVar[]r,rCov[=]r,rCov[]r,rCov[=mktmktpmktmktmktppβwhererpandrmktaretherandomreturnsontheportfoliopandonthemarket,respectively,andrfistheriskfreerateofinterest.Inequilibrium,allassetsandportfolioswillhavethesamereturnafteradjustmentforrisk,implying11Sharpe,Alexander,andBailey[1995]providesagoodoverviewofcurrentpracticeinChapter25.GrinblattandTitman[1989]reviewsomekeyissuesandprovideextensionsoftraditionalalphameasurement.GlostenandJagannathan[1994]provideanelegantandgeneralframework.Butapplicationsoftheirapproachrequiredassumptionssimilartoourframeworkbelow(lognormalindexreturnsandBlack-Scholesoptionpricing),whilerequiringgreatercomplexityofimplementation.(2))r-]r(E[+r=]rE[fmktpfpβSuperiorperformanceintheCAPMworldismeasuredbyalpha,whichistheincrementalexpectedreturnresultingfrommanagerialinformation(e.g.stockselectionormarkettiming).Thiscanberepresentedformallyas(3)r-)r-]r(E[-M]|rE[=]rE[-M]|rE[=ffmktpppppβαwhereE[rp|M]istheconditionalexpectedreturntotheportfoliogiventheinformationMusedbymanager.2IntheCAPMequilibrium,alphaswillbezerounlessamanagerhassuperiorinformation.Aportfoliowithpositivealphaoffersanexpectedreturninexcessofitsequilibriumrisk-adjustedlevelandinthissensehassuperiorperformance.322Measuringconditionalexpectationswhenmanagerialinformationisnotdirectlyobservedisanimportanteconometricchallenge.EarlyCAPM-basedstudies(e.g.Jensen[1969])regressedportfolioexcessreturnonmarketexcessreturn.Theconstanttermwasinterpretedasthealphaofinourequation(3),andtheslopecoefficientasbetainourequation(1).Roll[1978]indicatestheunreliabilityofalphameasureswhenthemarketportfolioproxyisnotmean-varianceefficient.FurtherdifficultiesinusingalphaasaperformancemeasurewhenmanagersareabletosuccessfullytimethemarketarediscussedbyDybvigandRoss[1984];theirresultsarecloselyrelatedtothenegativestatepricesobservedintheCAPMbyDybvigandIngersoll[1982].GrinblattandTitman[1989]proposetosolvetheproblembyusingpositiveperiod-weightingmeasures(i.e.statepricedensities),althoughtheirlaterempiricalstudy(GrinblattandTitman[1994])suggeststhismakeslittledifferenceforevaluatingmutualfundportfolios.FersonandSchadt[1996],whileretainingtheCAPMframework,arguethatbetashouldbeestimatedconditionallyonavectorofrelevantpublicly-availableinformationvariableswhichmaychangethroughthesampleperiod.3ArelatedbutnotidenticalperformancemeasureistheSharperatio(SR)ofaportfoliop,whereσpfppr-M]|rE[=SRTheSharperatioprovidesanappropriatemeasureofinvestorwelfarewhentheinvestorhasmean-variancepreferenceandinvestsintheportfolio(andperhapsarisk-freeasset)exclusively.Alpha,ontheotherhand,isameasureofperformancewhentheportfolioisasmallpartoftheinvestor'sentire(fully-diversified)portfo