马克维茨投资组合选择.docx

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马克维茨投资组合选择

马克维茨投资组合选择

PortfolioSelection

HarryMarkowitz

TheJournalofFinance,Vol.7,No.1.（Mar.,1952）,pp.77-91.

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252007

PORTFOLIOSELECTION*

HARRYMARKOWITZ

TheRandCorporation

THEPROCESSOFSELECTINGaportfoliomaybedividedintotwostages.

Thefirststagestartswithobservationandexperienceandendswith

beliefsaboutthefutureperformancesofavailablesecurities.The

secondstagestartswiththerelevantbeliefsaboutfutureperformances

andendswiththechoiceofportfolio.Thispaperisconcernedwiththe

secondstage.Wefirstconsidertherulethattheinvestordoes（orshould）

maximizediscountedexpected,oranticipated,returns.Thisruleisrejected

bothasahypothesistoexplain,andasamaximumtoguideinvestment

behavior.Wenextconsidertherulethattheinvestordoes（or

should）considerexpectedreturnadesirablethingandvarianceofreturn

anundesirablething.Thisrulehasmanysoundpoints,bothasa

maximfor,andhypothesisabout,investmentbehavior.Weillustrate

geometricallyrelationsbetweenbeliefsandchoiceofportfolioaccording

tothe"expectedreturns-varianceofreturns"rule.

Onetypeofruleconcerningchoiceofportfolioisthattheinvestor

does（orshould）maximizethediscounted（orcapitalized）valueof

futurereturns.lSincethefutureisnotknownwithcertainty,itmust

be"expected"or"anticipatded7'returnswhichwediscount.Variations

ofthistypeofrulecanbesuggested.FollowingHicks,wecouldlet

"anticipated"returnsincludeanallowanceforrisk.2Or,wecouldlet

therateatwhichwecapitalizethereturnsfromparticularsecurities

varywithrisk.

Thehypothesis（ormaxim）thattheinvestordoes（orshould）

maximizediscountedreturnmustberejected.Ifweignoremarketimperfections

theforegoingruleneverimpliesthatthereisadiversified

portfoliowhichispreferabletoallnon-diversifiedportfolios.Diversification

isbothobservedandsensible;aruleofbehaviorwhichdoes

notimplythesuperiorityofdiversificationmustberejectedbothasa

hypothesisandasamaxim.

*ThispaperisbasedonworkdonebytheauthorwhileattheCowlesCommissionfor

ResearchinEconomicsandwiththefinancialassistanceoftheSocialScienceResearch

Council.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.

78TheJournalofFinance

Theforegoingrulefailstoimplydiversificationnomatterhowthe

anticipatedreturnsareformed;whetherthesameordifferentdiscount

ratesareusedfordifferentsecurities;nomatterhowthesediscount

ratesaredecideduponorhowtheyvaryovertime.3Thehypothesis

impliesthattheinvestorplacesallhisfundsinthesecuritywiththe

greatestdiscountedvalue.Iftwoormoresecuritieshavethesamevalue,

thenanyoftheseoranycombinationoftheseisasgoodasany

other.

Wecanseethisanalytically:

supposethereareNsecurities;letritbe

theanticipatedreturn（howeverdecidedupon）attimetperdollarinvested

insecurityi;letdjtbetherateatwhichthereturnontheilk

securityattimetisdiscountedbacktothepresent;letXibetherelative

amountinvestedinsecurityi.Weexcludeshortsales,thusXi20

foralli.Thenthediscountedanticipatedreturnoftheportfoliois

Ri=xm

di,Titisthediscountedreturnoftheithsecurity,therefore

t-1

R=ZXiRiwhereRiisindependentofXi.SinceXi20foralli

andZXi=1,RisaweightedaverageofRiwiththeXiasnon-negative

weights.TomaximizeR,weletXi=1foriwithmaximumRi.

IfseveralRa,,a=1,...,Karemaximumthenanyallocationwith

maximizesR.Innocaseisadiversifiedportfoliopreferredtoallnondiversified

poitfolios.

Itwillbeconvenientatthispointtoconsiderastaticmodel.Instead

ofspeakingofthetimeseriesofreturnsfromtheithsecurity

（ril,ri2）...,rit,...）wewillspeakof"theflowofreturns"（ri）from

theithsecurity.Theflowofreturnsfromtheportfolioasawholeis

3.Theresultsdependontheassumptionthattheanticipatedreturnsanddiscount

ratesareindependentoftheparticularinvestor'sportfolio.

4.Ifshortsaleswereallowed,aninfiniteamountofmoneywouldbeplacedinthe

securitywithhighestr.

PortfolioSelection79

R=ZX,r,.Asinthedynamiccaseiftheinvestorwishedtomaximize

"anticipated"returnfromtheportfoliohewouldplaceallhisfundsin

thatsecuritywithmaximumanticipatedreturns.

Thereisarulewhichimpliesboththattheinvestorshoulddiversify

andthatheshouldmaximizeexpectedreturn.Therulestatesthatthe

investordoes（orshould）diversifyhisfundsamongallthosesecurities

whichgivemaximumexpectedreturn.Thelawoflargenumberswill

insurethattheactualyieldoftheportfoliowillbealmostthesameas

theexpectedyield.5Thisruleisaspecialcaseoftheexpectedreturnsvariance

ofreturnsrule（tobepresentedbelow）.Itassumesthatthere

isaportfoliowhichgivesbothmaximumexpectedreturnandminimum

variance,anditcommendsthisportfoliototheinvestor.

Thispresumption,thatthelawoflargenumbersappliestoaportfolio

ofsecurities,cannotbeaccepted.Thereturnsfromsecuritiesare

toointercorrelated.Diversificationcannoteliminateallvariance.

Theportfoliowithmaximumexpectedreturnisnotnecessarilythe

onewithminimumvariance.Thereisarateatwhichtheinvestorcan

gainexpectedreturnbytakingonvariance,orreducevariancebygiving

upexpectedreturn.

Wesawthattheexpectedreturnsoranticipatedreturnsruleisinadequate.

Letusnowconsidertheexpectedreturns-varianceofreturns

（E-V）rule.Itwillbenecessarytofirstpresentafewelementary

conceptsandresultsofmathematicalstatistics.Wewillthenshow

someimplicationsoftheE-Vrule.Afterthiswewilldiscussitsplausibility.

Inourpresentationwetrytoavoidcomplicatedmathematicalstatements

andproofs.Asaconsequenceapriceispaidintermsofrigorand

generality.Thechieflimitationsfromthissourceare

（1）wedonot

deriveourresultsanalyticallyforthen-securitycase;instead,we

presentthemgeometricallyforthe3and4securitycases;

（2）weassume

staticprobabilitybeliefs.Inageneralpresentationwemustrecognize

thattheprobabilitydistributionofyieldsofthevarioussecuritiesisa

functionoftime.Thewriterintendstopresent,inthefuture,thegeneral,

mathematicaltreatmentwhichremovestheselimitations.

Wewillneedthefollowingelementaryconceptsandresultsof

mathematicalstatistics:

LetYbearandomvariable,i.e.,avariablewhosevalueisdecidedby

chance.Suppose,forsimplicityofexposition,thatYcantakeona

finitenumberofvaluesyl,yz,...,y,~L.ettheprobabilitythatY=

5.U'illiams,op.cit.,pp.68,69.

80TheJournalofFinance

yl,bepl;thatY=y2bepzetc.Theexpectedvalue（ormean）ofYis

definedtobe

ThevarianceofYisdefinedtobe

VistheaveragesquareddeviationofYfromitsexpectedvalue.Visa

commonlyusedmeasureofdispersion.Othermeasuresofdispersion,

closelyrelatedtoVarethestandarddeviation,u=.\/Vandthecoefficient

ofvariation,a/E.

Supposewehaveanumberofrandomvariables:

R1,...,R,.IfRis

aweightedsum（linearcombination）oftheRi

thenRisalsoarandomvariable.（ForexampleR1,maybethenumber

whichturnsupononedie;R2,thatofanotherdie,andRthesumof

thesenumbers.Inthiscasen=2,a1=a2=1）.

Itwillbeimportantforustoknowhowtheexpectedvalueand

varianceoftheweightedsum（R）arerelatedtotheprobabilitydistribution

oftheR1,...,R,.Westatetheserelationsbelow;werefer

thereadertoanystandardtextforproof.6

Theexpectedvalueofaweightedsumistheweightedsumofthe

expectedvalues.I.e.,E（R）=alE（R1）+aZE（R2）+...+a,E（R,）

Thevarianceofaweightedsumisnotassimple.Toexpressitwemust

define"covariance."ThecovarianceofR1andRzis

i.e.,theexpectedvalueof[（thedeviationofR1fromitsmean）times

（thedeviationofR2fromits