马克维茨投资组合选择

PortfolioSelectionHarryMarkowitzTheJournalofFinance,Vol.7,No.1.(Mar.,1952),pp.77-91.StableURL:=0022-1082%28195203%297%3A1%3C77%3APS%3E2.0.CO%3B2-1TheJournalofFinanceiscurrentlypublishedbyAmericanFinanceAssociation.YouruseoftheJSTORarchiveindicatesyouracceptanceofJSTOR'sTermsandConditionsofUse,availableat://@jstor.org.:252007PORTFOLIOSELECTION*HARRYMARKOWITZTheRandCorporationTHEPROCESSOFSELECTINGaportfoliomaybedividedintotwostages.Thefirststagestartswithobservationandexperienceandendswithbeliefsaboutthefutureperformancesofavailablesecurities.Thesecondstagestartswiththerelevantbeliefsaboutfutureperformancesandendswiththechoiceofportfolio.Thispaperisconcernedwiththesecondstage.Wefirstconsidertherulethattheinvestordoes(orshould)maximizediscountedexpected,oranticipated,returns.Thisruleisrejectedbothasahypothesistoexplain,andasamaximumtoguideinvestmentbehavior.Wenextconsidertherulethattheinvestordoes(orshould)considerexpectedreturnadesirablethingandvarianceofreturnanundesirablething.Thisrulehasmanysoundpoints,bothasamaximfor,andhypothesisabout,investmentbehavior.Weillustrategeometricallyrelationsbetweenbeliefsandchoiceofportfolioaccordingtotheexpectedreturns-varianceofreturnsrule.Onetypeofruleconcerningchoiceofportfolioisthattheinvestordoes(orshould)maximizethediscounted(orcapitalized)valueoffuturereturns.lSincethefutureisnotknownwithcertainty,itmustbeexpectedoranticipatded7'returnswhichwediscount.Variationsofthistypeofrulecanbesuggested.FollowingHicks,wecouldletanticipatedreturnsincludeanallowanceforrisk.2Or,wecouldlettherateatwhichwecapitalizethereturnsfromparticularsecuritiesvarywithrisk.Thehypothesis(ormaxim)thattheinvestordoes(orshould)maximizediscountedreturnmustberejected.Ifweignoremarketimperfectionstheforegoingruleneverimpliesthatthereisadiversifiedportfoliowhichispreferabletoallnon-diversifiedportfolios.Diversificationisbothobservedandsensible;aruleofbehaviorwhichdoesnotimplythesuperiorityofdiversificationmustberejectedbothasahypothesisandasamaxim.*ThispaperisbasedonworkdonebytheauthorwhileattheCowlesCommissionforResearchinEconomicsandwiththefinancialassistanceoftheSocialScienceResearchCouncil.ItwillbereprintedasCowlesCommissionPaper,NewSeries,No.60.1.See,forexample,J.B.Williams,TheTheoryofInvestmentValue(Cambridge,Mass.:HarvardUniversityPress,1938),pp.55-75.2.J.R.Hicks,Val~eandCapital(NewYork:OxfordUniversityPress,1939),p.126.Hicksappliestheruletoafirmratherthanaportfolio.78TheJournalofFinanceTheforegoingrulefailstoimplydiversificationnomatterhowtheanticipatedreturnsareformed;whetherthesameordifferentdiscountratesareusedfordifferentsecurities;nomatterhowthesediscountratesaredecideduponorhowtheyvaryovertime.3Thehypothesisimpliesthattheinvestorplacesallhisfundsinthesecuritywiththegreatestdiscountedvalue.Iftwoormoresecuritieshavethesamevalue,thenanyoftheseoranycombinationoftheseisasgoodasanyother.Wecanseethisanalytically:supposethereareNsecurities;letritbetheanticipatedreturn(howeverdecidedupon)attimetperdollarinvestedinsecurityi;letdjtbetherateatwhichthereturnontheilksecurityattimetisdiscountedbacktothepresent;letXibetherelativeamountinvestedinsecurityi.Weexcludeshortsales,thusXi20foralli.ThenthediscountedanticipatedreturnoftheportfolioisRi=xmdi,Titisthediscountedreturnoftheithsecurity,thereforet-1R=ZXiRiwhereRiisindependentofXi.SinceXi20foralliandZXi=1,RisaweightedaverageofRiwiththeXiasnon-negativeweights.TomaximizeR,weletXi=1foriwithmaximumRi.IfseveralRa,,a=1,...,KaremaximumthenanyallocationwithmaximizesR.Innocaseisadiversifiedportfoliopreferredtoallnondiversifiedpoitfolios.Itwillbeconvenientatthispointtoconsiderastaticmodel.Insteadofspeakingofthetimeseriesofreturnsfromtheithsecurity(ril,ri2)...,rit,...)wewillspeakoftheflowofreturns(ri)fromtheithsecurity.Theflowofreturnsfromtheportfolioasawholeis3.Theresultsdependontheassumptionthattheanticipatedreturnsanddiscountratesareindependentoftheparticularinvestor'sportfolio.4.Ifshortsaleswereallowed,aninfiniteamountofmoneywouldbeplacedinthesecuritywithhighestr.PortfolioSelection7