The Evolution of Portfolio Rules and the Capital A

TheEvolutionofPortfolioRulesandtheCapitalAssetPricingModelEmanuelaSciubba¤FacultyofEconomicsandPoliticsUniversityofCambridgeJuly1998AbstractTheaimofthispaperistotesttheperformanceofthestandardversionofCAPMinanevolutionaryframework.Weimagineahet-erogeneouspopulationoflong-livedagentswhoinvesttheirwealthaccordingtodi¤erentportfoliorulesandweaskwhatisthefateofthosewhohappentobehaveasprescribedbyCAPM.Inacompletesecurities’marketwithaggregateuncertainty,weprovethattraderswhoeither“believe”inCAPManduseitasaruleofthumb,orareen-dowedwithgenuinemean-variancepreferences,undersomeveryweakconditions,vanishinthelongrun.Weshowthatasu¢cientcondi-tiontodriveCAPMormeanvariancetraders’wealthsharestozeroisthataninvestorendowedwithalogarithmicutilityfunctionentersthemarket.We…nallychecktherobustnessofourresultsallowingfordi¤erentkindsofheterogeneityamongtraders.JELClassi…cationC61,D81,G11KeywordsEvolution;portfoliorules;CAPM;Kellycriterion.AddressforCorrespondenceEmanuelaSciubba,LucyCavendishCollege,CB30BUCambridge,UK.E-mail:[e-mail]¤IamdeeplyindebtedtoLucaAnderliniforhishelpfulguidance.Ialsobene…tedfromdiscussionwithRobertEvansandPeterSorensen.UsefulcommentscamefromparticipantstotheSt.John’sTheoryWorkshops,UniversityofCambridge,andtotheThirdInternationalConferenceon“ComputationinEconomicsandFinance”,StanfordUniversity.Allremainingerrorsaremine.11Introduction1.1MotivationAmajorpartoftheresearchin…nancialeconomicsisdirectedtowardsim-provingourunderstandingofhowinvestorsmaketheirportfoliodecisionsandhenceofhowassets’pricesaredetermined.Manycapitalasset-pricingmodelshavebeenputforthintheliterature.Inparticular,mean-varianceanalysisandtheSharpe-Lintner-MossinCAPM1arewidelyviewedasoneofthe“majorcontributionsofacademicresearchinthepostwarera”[Jagan-nathanandWang(1996),p.4].Overthepasttwodecadesanumberofstudieshaveexaminedtheem-piricalperformanceofCAPM,invariablyprovidingstrongevidenceofitsinabilitytoexplain(andthereforetopredict)thebehaviourof…nancialmar-kets2.Nevertheless,“[i]nspiteofthelackofempiricalsupport,theCAPMisstillthepreferredmodelforclassroomuseinMBAandothermanagerial…nancecourses.InawayitremindsusofcartooncharacterslikeWileE.Coyotewhohavetheabilitytocomebacktooriginalshapeafterbeingblowntopiecesorhammeredoutofshape”[JagannathanandWang(1996),p.4].InthispaperweareafterWileE.Coyoteonceagain,butwithanewdevice.Infact,econometricianshaveempiricallyrejecteditspredictionsand…nancialtheoristshavecriticiseditsrestrictiveassumptions,butnoonetoourknowledgehasstudiedCAPMinanevolutionaryframework.Thefocusofourpaperisto…llthisgapintheliteratureand,inparticular,totesttheperformanceofthestandardversionofCAPMinanevolutionarysetting.Conventional…nancialtheoryshowsthat,underwell-knownassumptions,CAPMstemsfromrationalbehaviour.However,arecentstrandofliteratureonevolutionandmarketbehaviourstressesthatrationalityisneithersu¢-cientnoranecessaryconditionforsurvival.ThereforeaninterestingquestiontoaskiswhetherCAPMprescribesabehaviourwhichcanbeconsidered“…t”inanevolutionarysense.Weimagineaheterogeneouspopulationoflong-livedagentswhoinvestaccordingtodi¤erentportfoliorulesandweaskwhatistheasymptoticmar-ketshareofthosewhohappentobehaveasprescribedbyCAPM.Namely1SeeSharpe(1964),Lintner(1965)andMossin(1996).2TheempiricalliteratureonCAPMissovastthatwewillnotattemptareviewhere.Forarecentaccount,see,foristance,JagannathanandWang(1996)andFamaandFrench(1996a,1996b).2weaimatdetectingtheasymptoticpropertiesofthewealthsharesoftradersthateither“believe”inCAPManduseitasaruleofthumbfortheirport-foliodecisions,ordisplaygenuinemean-variancebehaviour.Ourresultssug-gestthatthereareseveralcircumstancesofeconomicinterestinwhichtheirwealthsharewillconvergealmostsurelytozero.Asu¢cientconditiontodriveCAPMtraderstoextinctionisthataninvestorendowedwithaloga-rithmicutilityfunctionentersthemarket.WebelievethatthisisaninterestingresultnotonlybecauseitprovesthatCAPMisnotrobustinanevolutionarysense,butalsobecauseittrig-gersonceagainthedebateonthenormativeappealanddescriptiveappealoflogarithmicutilityapproachasopposedtomean-varianceapproachin…-nance.SinceaseminalarticlebyKelly(1956),several…nancialeconomistsandappliedprobabilistshavebeendebatingwhethermaximisingalogarith-micutilityfunctionis“morerational”forarationaltrader.Thedebateoriginatesfromthedissatisfactionwiththemean-varianceapproachwhichfailstosingleoutauniqueoptimalportfolio.Infact,thechosenmixbetweentheriskfreeassetandthemarketportfoliodependsoneachinvestor’sde-greeofriskaversion.Severalauthors3havearguedthatarationallongruninvestorshouldmaximisetheexpectedgrowthrateofhiswealthshareand,therefore,shouldbehaveasifhewereendowedwithalogarithmicutilityfunction4(thesocalledKellycriterion).Thisyieldsauniquesolutiontotheoptimalportfolioproblem.ThisclaimhasbeenopposedbyMertonandSamuelson(1974)andGoldman(1974).Inparticular,MertonandSamuelson’scritiquestressedtheobviouscon-tradictionwhichliesinarguingthatrationaltradersshouldmaximiseautil-ityfunctionwhichisdi¤erentfromtheirown5.Theevolutionaryframeworkadoptedinthispape