25 markowitz

8/2/2019 25 Markowitz

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American Finance Association

Portfolio SelectionAuthor(s): Harry MarkowitzReviewed work(s):Source: The Journal of Finance, Vol. 7, No. 1 (Mar., 1952), pp. 77-91Published by: Blackwell Publishing for the American Finance AssociationStable URL: http://www.jstor.org/stable/2975974 .

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PORTFOLIO SELECTION*HARRYMARKOWITZThe Rand Corporation

THEPROCESSFSELECTINGportfolio aybe dividednto wo tages.The firsttagestartswith bservationnd experiencend endswithbeliefs bout the future erformancesf available securities. hesecond tage tartswith herelevant eliefsboutfutureerformancesandendswith hechoice fportfolio.hispaper sconcerned ith hesecond tage.Wefirstonsiderhe ule hat he nvestoroes or hould)maximize iscountedxpected,ranticipated,eturns.hisrule s re-jectedboth s a hypothesisoexplain,ndas a maximumoguide n-vestmentehavior.We next onsiderherule hat he nvestoroes orshould) onsiderxpected eturn desirable hingnd variance fre-turn nundesirablehing. hisrulehasmany oundpoints, oth s amaxim or, ndhypothesisbout, nvestmentehavior.We illustrategeometricallyelations etween eliefsndchoice fportfolioccord-ingtothe"expected eturns-variancefreturns" ule.Onetypeofrule oncerninghoice fportfolios thatthe nvestordoes (or should)maximize hediscountedor capitalized)value offutureeturns.1incethefutures not knownwith ertainty,tmustbe "expected" r"anticipated"eturns hichwe discount. ariationsof thistypeof rulecan be suggested. ollowing icks,we could et"anticipated" eturnsnclude n allowance or isk.2 r, we could ettherate at whichwe capitalize hereturnsrom articularecuritiesvarywith isk.The hypothesisor maxim)that the investor oes (or should)nmaximizeiscounted eturnmustberejected.fwe gnoremarketm-perfectionsheforegoingulenevermplies hatthere s a diversifiedportfolio hichs preferableo all non-diversifiedortfolios.iversi-fications bothobservednd sensible; ruleofbehaviorwhich oesnot mply he uperiorityfdiversificationustbe rejected oth s ahypothesisnd as a maxim.*Thispaper sbasedonwork onebythe uthor hile t theCowles ommissionorResearchnEconomicsndwith hefinancialssistancef theSocialScienceResearchCouncil.t willbe reprinteds Cowles ommissionaper,NewSeries, o. 60.1. See,for xample,.B.Williams,heTheoryf nvestmentalue Cambridge,ass.:Harvard niversityress, 938), p.55-75.2. J. R. Hicks,Value ndCapitalNewYork:Oxford niversityress, 939), . 126.Hicks pplies he ule o a firmatherhan portfolio.

77


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78 The Journal fFinanceThe foregoingulefails o imply iversificationo matter owtheanticipatedeturnsreformed; hetherhe ameordifferentiscountratesareusedfordifferentecurities; o matter owthesediscount

rates redecideduponor howtheyvaryovertime.3 he hypothesisimplies hat he nvestor lacesall hisfundsnthe security ith hegreatest iscountedalue. f twoor more ecuritiesave, he ameval-ue, then nyof theseoranycombinationf these s as goodas anyother.We canseethis nalytically:uppose herereN securities;etrB etheanticipatedeturnhoweverecided pon)at time perdollar n-vested n security; let d be the rateat which he returnn theOhsecurityt time is discountedack tothepresent;etXi be therela-tive mountnvestednsecurity.Weexdude hort ales, husXi 0for ll i. Then thediscountednticipatedeturnftheportfolios

oo NR=3 E di,t itXt=1 i=1= XiEdit rit)i=l t=l

RX E di rit s thediscountedeturnf heth security,hereforet=1R = YXiRiwhereRi is independentfXi. SinceXi 3 0 for ll iand2Xi = 1,R is a weightedverage fRXwith heXi as non-nega-tiveweights. o maximize , we letXi = 1 for withmaximum i.If severalRaa,a = 1, ..., K aremaximumhen nyallocationwith

KE Xaa = 1a=1maximizes . In no case is a diversifiedortfolioreferredo allnon-diversifiedortfolios.It willbe convenientt thispointto consider staticmodel. n-steadofspeaking f the time eries freturns rom he h security(ril,ri2, .. , rit, . .) we will speak of "the flowofreturns" ri) fromthe P security. he flow freturnsromheportfolios a whole s

3. The results epend ntheassumptionhattheanticipatedeturnsnd discountrates re ndependentf heparticularnvestor'sortfolio.4. If short aleswere llowed,n infinitemount fmoneywouldbe placed nthesecurity ith ighest.


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Portfolio election 79R = 2X,Xr. s nthedynamicase if he nvestor ished omaximize"anticipated"eturnrom heportfolioewould laceallhisfundsnthat ecurity ithmaximumnticipatedeturns.There s a rulewhich mplies oth hat he nvestorhould iversifyandthatheshouldmaximize xpected eturn.herule tates hat heinvestoroes or should)diversifyis fundsmong ll those ecuritieswhich ivemaximumxpected eturn.he lawof argenumbers illinsurehat he ctualyield ftheportfolio illbe almost he ame sthe xpected ield.5hisrule s a special ase ofthe xpectedeturns-variance freturnsule tobe presentedelow). t assumes hat hereisa portfoliohich ives othmaximumxpectedeturnndminimumvariance,ndit commendshisportfolioothe nvestor.This presumption,hatthe aw of argenumberspplies o a port-folio fsecurities,annotbeaccepted.Thereturnsromecuritiesretoo ntercorrelated.iversificationannot liminatell variance.Theportfolio ithmaximumxpected eturns notnecessarilyheone withminimumariance. here s a rate t which he nvestorangain xpectedeturnytakingn variance,rreduce ariance ygiv-ingupexpected eturn.

We saw thattheexpected eturnsranticipatedeturns ule s in-adequate.Let us nowconsiderheexpected eturns-variancefre-turnsE-V) rule. t willbenecessaryofirst resent few lementaryconcepts nd results f mathematicaltatistics.We willthenshowsome mplicationsftheE-V rule.Afterhiswewilldiscusstsplausi-bility.In ourpresentatione try oavoidcomplicated athematicaltate-mentsndproofs. sa consequencepricespaid ntermsfrigorndgenerality.he chiefimitationsromhissource re (1) we do notderiveour results nalyticallyor then-securityase; instead,wepresenthem eometricallyor he3and4 securityases; 2)weassumestatic robabilityeliefs.n a general resentatione mustrecognizethat heprobabilityistributionfyields fthevarious ecuritiess afunctionftime.Thewriterntends opresent,nthefuture,hegen-eral,mathematicalreatmenthich emoveshese imitations.

We will need the followinglementaryoncepts nd results fmathematicaltatistics:LetY bea randomariable,.e., variablewhose alue sdecided ychance.Suppose,for implicityf exposition,hatY can takeon afinite umberfvaluesyi,y2, .. , YN.Let theprobabilityhatY =5. Williams,p. cit., pp. 68, 69.


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80 The Journal fFinanceyi,bePi; thatY = y2bep2etc.Theexpected alue or mean)ofY isdefinedobe

E=Plyl+p2y2+. * *+PNYNThe variance fY isdefinedo beV = p1 y1-E) 2+ P2 y2-E) 2+. . +PN (YN-E) 2

Visthe verage quared eviation fY fromtsexpectedalue.V isacommonlysedmeasure fdispersion. thermeasures fdispersion,closely elated o V arethestandard eviation,- -VV and theco-efficientfvariation,lE.Supposewehave a number f randomvariables:R1, . . ,R,. If R isa weightedum linear ombination)ftheRiaLiRi a2R2 . . . + a.R.

thenR isalsoa random ariable.Forexample 1,maybe thenumberwhich urns p on onedie;R2, hatofanother ie,and R thesumofthesenumbers.n this asen = 2,a, = a2 = 1).It willbe importantorus to knowhow the expected alue andvariance ftheweightedum R) arerelated o theprobabilityis-tribution f the R1, . . ,R,n.We state these relationsbelow; we referthereader oanystandardext or roof.6The expected alueof a weightedum s theweightedumof theexpected values. I.e., E(R) = alE(Ri) + a2E(R2) + . . . + anE(Rn)The variance f weightedum s not s simple. o expresst wemustdefine covariance." he covariancefR1andR2 iso12=E I [R1-E (R1) [R2-E (R2) I Ii.e.,theexpected alueof (thedeviationfR1fromtsmean)times(thedeviationfR2fromtsmean)]. n generalwe definehecovari-ance between i andRj ascrijE I [R -E (Ri)] [R -E (Rj) I }

O'imaybe expressedn terms fthe familiarorrelationoefficient(pi2). The covarianceetween i andRj isequalto[(their orrelation)times thestandard eviation fRj) times thestandard eviation fRj)]T

6. E.g.,J.V.Uspensky,ntroductionoMathematicalProbabilityNewYork:McGraw-Hill,1937), hapter, pp.161-81.


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Portfolio election 8iThevariance fa weightedum s

N N NI

V(R) = a2 V (Xi) +2 ? aia jaiji=1 i=1 t>1Ifweusethefact hatthevariance fRi is oi thenV R) = aiajxij

LetRibe thereturnnthe th security.etAuie theexpected alueofRi; oij,bethe ovarianceetween i andRj (thus ii is thevarianceofRi). LetXi be thepercentagefthe nvestor'sssetswhich real-located o the 1k security.heyield R) onthe portfolios a whoie s

R-=RiXiTheRi (andconsequently) are consideredo be random ariables.7TheXi arenotrandom ariables, utarefixed ythe nvestor.incetheXi arepercentages ehave2Xi= 1. In ouranalysiswewillex-cludenegative aluesoftheXi (i.e.,short ales); thereforei I 0 forall i.ThereturnR) on theportfolios a wholes a weightedumofran-domvariableswhere he nvestoranchoose heweights). romourdiscussionfsuchweightedumswe see thattheexpected eturnfrom heportfolios a whole s

NE = 1and thevariances

V= oEijxixi-1 j-17. I.e., we assumethatthe nvestor oes (andshould)act as ifhe had probability eliefsconcerninghesevariables. n generalwe wouldexpectthat the nvestor ould tellus,for

anytwoevents A and B), whether epersonally onsideredA more ikely hanB, B morelikely hanA, or bothequally ikely. f the nvestorwere onsistentn hisopinions n suchmatters, e wouldpossessa system fprobability eliefs.We cannotexpectthe investorto be consistentn everydetail. We can, however, xpecthis probabilitybeliefsto beroughly onsistent n importantmatters hat have been carefully onsidered.We shouldalso expectthathewillbase his actionsupontheseprobability eliefs-even though heybe in part subjective.Thispaperdoesnot consider hedifficultuestionofhow investors o (or should)formtheirprobability eliefs.


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82 The Journal fFinanceForfixed robabilityeliefspi, aij) the nvestor as a choice fvari-ous combinationsf E and V dependingn his choiceof portfolioX1, .. ,XN.Suppose hatthesetof all obtainableE, V) combina-tionswere s inFigure . TheE- V rule tates hatthe nvestor ould(or should)want oselect neofthose ortfolioshich ive rise othe(E, V) combinationsndicateds efficientnthefigure;.e.,thosewithminimum for iven or more ndmaximum for ivenV or ess.There re techniquesy whichwecan compute he setofefficientportfoliosnd efficientE, V) combinationsssociatedwithgivenAi

v

/ a~~~~~ffa;'nableE,V combinations

\ / ~~~~~~~~~~~~~fficient\ /"~ E,Vcombinations

EFIG. 1and aio.Wewillnotpresenthese echniquesere.Wewill,however,illustrate eometricallyhe nature fthe efficienturfaces orcasesinwhichN (thenumberfavailable ecurities)s small.The calculation fefficienturfacesmight ossibly e ofpracticaluse.Perhaps here reways,bycombiningtatisticalechniquesndthe udgment fexperts,o form easonable robabilityeliefsii,oij). We could use thesebeliefs o compute heattainable fficientcombinationsf E, V). The investor, eing nformedfwhat E, V)combinations ere ttainable,ould tatewhich e desired.We couldthen ind heportfoliohich avethisdesired ombination.


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Portfolio election 83Two conditions-ateast-mustbesatisfiedeforet wouldbe prac-ticaltouseefficienturfacesn themanner escribedbove.First, heinvestormustdesire o act accordingo theE-V maxim. econd,wemust e ableto arrive t reasonable i ando,j.We willreturno thesemattersater.Letusconsiderhe aseof hreeecurities.n the hreeecurityaseourmodel educes o

1) E =EXil43 32) V=iXjaiji=l j=l

33) EXi=1i-14) XiO for i=1,2,3.

From (3) we get3') X3=1-X1-X2Ifwe substitute3') inequation1) and 2)wegetE andV as functionsofX1 andX2.For examplewefind

It) E = 3 +Xl (Al -A3) + X2 (A2 - A)The exactformulasre not too importantere that fV is given e-low).8Wecansimplywrite

a) E -E (X1, X2)b) V = V(XI, X2)C) XI,>0, X2)(), 1iI- X1- X2)>'0

By usingrelationsa), (b), (c), we can workwithtwodimensionalgeometry.The attainable et of portfoliosonsists f all portfolios hichsatisfyonstraintsc) and (3') (or equivalently3) and (4)). The at-tainable ombinationsfX1,X2arerepresentedythetrianglebc nFigure . Anypoint othe eft ftheX2axis s notattainable ecauseitviolates he conditionhatX1 0. Anypointbelow heX1axisisnot attainablebecauseit violatesthe conditionhatX2 0. Any

8. V =X (oi - 2I1a + a8) + X2(o22 - 2o2a + 8T3) + 2XtX2(ff,2 - I13 -28 + o8B)+ 2X1 (ff18 ag) + 2X2(9a2a - a,3) + Ta8


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84 The Journal f Financepoint bove the ine 1 - X - X2 0) is not attainable ecause tviolates heconditionhatX3 = 1 - X -X2 0.We definen isomneanurve o be the setofall points portfolios)with given xpected eturn. imilarlynisovarianceine sdefinedobe theset ofall points portfolios)ith given ariance freturn.An examinationf heformulaeor andV tells s the hapes f heisomeannd sovarianceurves. pecificallyhey ellus that ypically9the somean urves rea systemfparallel traightines;the sovari-ancecurvesre a systemf oncentricllipses seeFig. 2). Forexample,if 2 ; A3 equation1' can be writtenn the familiarormX2 = a +bX1; pecifically1)

X2=E-3 AlTA X1.A2- A3 Al2 3Thusthe lopeof the someanineassociatedwith = Eo is - (1 -A3)/(2 - A3) its ntercept s (E0 - A3)/(Ab2 - 3). If we changeE wechange he nterceptut nottheslopeofthe somean ine.Thiscon-firmshecontentionhatthe somean inesform system fparallellines.

Similarly,ya somewhatesssimple pplicationf nalytic eome-try,we can confirmhe contentionhatthe sovarianceinesformfamilyfconcentricllipses. he "center" fthesystems thepointwhichminimizes . Wewill abel this ointX.Its expectedeturnndvariancewewill abelE and V.Variancencreasess youmove wayfromX.Moreprecisely,f oneisovariance urve, 1, ies closer o Xthan nother,2, henC1 s associatedwith smaller ariance hanC2.Withtheaid of theforegoingeometricpparatus et us seektheefficientets.X,the center f the system f sovariancellipses,mayfalleitherinside routside he ttainableet.Figure illustratescase nwhichXfallsnside he ttainableet. nthis ase: Xis efficient.ornootherportfolioas a V as lowas X; thereforeoportfolioan have eithersmallerV (with he ameorgreater ) orgreater with hesameorsmallerV. No point portfolio) ith xpected eturn less thanEis efficient.orwehaveE > E and V < V.Considerllpointswith given xpectedeturn; i.e.,allpoints nthe someanine associatedwithE. Thepointof the someanineatwhichV takeson ts eastvalue s thepoint twhich he someanine9. The isomean "Ccurves"re as described bove exceptwhen I,u = ,U2= IA3. In thelatter case all portfolios ave the same expectedreturn nd the investor hoosestheonewithminimum ariance.As to theassumptions mplicitnourdescriptionf the sovariance urves eefootnote12.


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Portfolio election 85is tangentoan isovarianceurve.We callthispointX(E). IfweletE vary,X(E) traces uta curve.Algebraiconsiderationswhichwe omit ere) how s that his urveisa straightine.Wewill all t the riticaline1.The criticalinepassesthrough for his ointminimizes for ll pointswith (Xi, X2) = E.As wegoalong ineither irectionromX,V increases. he segmentof thecriticalinefromX to thepointwhere he criticalinecrosses

X2 \ Direction ofincreasing*\\ \ \ isomean lines--

~~~~~~isovariance curves\ \m \ \ efficientortfolios\\ attainable et

c b

ft~~~ _\\

c \b XI\ \\ \ \ X*directionf ncreasingdependson It, ,u.#3

FIG. 2theboundaryf he ttainableet spartof he fficientet.Therest ftheefficientetis (inthecase illustrated)hesegmentftheab linefrom tob.b s thepoint fmaximumttainable . In Figure , Xliesoutside he admissiblerea but thecriticalinecuts theadmissiblearea.The efficientine begins t theattainable ointwithminimumvarianceinthis aseon theabline). t moves oward until tinter-sects hecriticaline,moves long hecriticalineuntil t intersectsboundarynd finally oves long heboundaryo b.Thereadermay


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a ~~~~~~~~~~~~~~~increasing

-.0 ~ ~ ~ -

I--,

FIG. 3

Xw|efficient portfolios

FIG. 4


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Portfolio election 87wish to constructnd examine he followingther ases: (1) X liesoutside he ttainable et and thecriticalinedoes not cuttheattain-able set. n thiscase there s a security hich oes not nter nto nyefficientortfolio.2) Two securitiesave the ame isi.n this asetheisomeanines reparallel o a boundaryine. t mayhappen hattheefficientortfolio ithlmaximum is a diversifiedortfolio.3) A casewhereinnly neportfolios efficient.Theefficientet n the4 securityase s,as in the3 securityndalsotheN securityase,a series fconnectedine egments. tone endoftheefficientet s thepoint f minimumariance; t the other nd sa point fmaximumxpected eturn'0see Fig. 4).

Nowthatwe have seen thenature f the set of efficientortfolios,it snotdifficulto seethenature f he etof fficientE, V) combina-tions.n the hreeecurityaseE = ao + aiX, + a2X2s a plane;V =bo+ b,Xj+ b2X2 bl2XlX2 b,,X + b,2X2 is a paraboloid."Asshown nFigure ,the ection f heE-plane ver heefficientortfolioset s a series f connectedinesegmen'ts.he section f theV-parab-oloidoverthe efficientortfolioet is a series fconnectedarabolasegments.fweplottedV against for fficientortfolios e wouldagainget series fconnectedarabola egmentssee Fig. 6). This re-sultobtains or nynumber fsecurities.

Various easons ecommendheuse of theexpected eturn-varianceofreturnule, oth s a hypothesisoexplainwell-establishednvest-mentbehavior nd as a maxim oguideone'sownaction.The ruleserves etter, e will ee,as an explanation f, ndguide o,"invest-ment" s distinguishedromspeculative" ehavior.4

10. Justs weused he quation = 1toreduce hedimensionalityn thethreei= 1securityase,wecanuse it torepresenthefour ecurityasein3 dimensionalpace.Eliminating4wegetE = E(X,,X2,X8),V = V(X,, X2, X8). Theattainableet srep-resented,n hree-space,y he etrahedronith ertices0,0,0), (0, , 1), 0, 1, ), (1,0,0),'representingortfoliosith, espectively,4 = 1,X3 = 1, X2 = 1, XI = 1.Let s12s e thesubspace onsistingfall pointswithX4 = 0. Similarly e can defineSaX ... , aa tobethe ubspace onsistingfall pointswithXi = 0, i $ a,, .. , aa. Foreach ubspaceal,.... , aa we can define critical ine la, .... aa. This ine s the ocus fpoints where minimizesfor ll pointsn al,... Xaa with he ame as P. If a pointis n al, . . . , aa and s efficienttmust e on a,, . . ., aa. The efficientetmaybe tracedoutby startingt thepoint fminimumvailablevariance,moving ontinuouslylongvariousa, .. . , aa accordingo definiteules, ndingn a pointwhich ivesmaximum.As nthe wodimensionalase thepointwithminimumvailable ariancemaybe n theinteriorf he vailableetorononeof tsboundaries.ypically eproceedlong givencriticalineuntil itherhis ine ntersectsne ofa larger ubspace rmeets boundary(and simultaneouslyhecriticalineofa lower imensionalubspace).n either f thesecases he fficientine urnsndcontinueslong henew ine.The efficientine erminateswhen pointwithmaximum is reached.11. See footnote.


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E

/ I I _Xsetofefficienta portfolios

X2FIG. S

v

efficientE,Vcombinations

EFIG. 6


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Portfolio election 89Earlierwe rejected heexpected eturns ule n thegrounds hat tnever mplied he superiorityfdiversification.he expected eturn-variance freturn ule, ntheother and, mplies iversificationorwiderange fpi,o-j.ThisdoesnotmeanthattheE-V ruleneverm-plies he uperiorityf n undiversifiedortfolio.t is conceivable hatone securitymight ave an extremelyigher ield nd ower ariancethan ll other ecurities;omuch o thatoneparticular ndiversifiedportfolio ouldgive maximum and minimum . But for large,presumablyepresentativeange fAi,rij heE- V rule eads o efficientportfolioslmost ll ofwhich re diversified.Not onlydoes theE-V hypothesismply iversification,t implies

the right ind" fdiversificationor he righteason." headequacyofdiversifications notthought y investorso depend olely n thenumber fdifferentecurities eld.Aportfolioith ixty ifferentail-way ecurities,or xample, ouldnotbe as welldiversifieds the amesize portfolio ith omerailroad,omepublicutility,mining, arioussort of manufacturing,tc. The reason s that it is generallymorelikely or irms ithin he ame ndustryo dopoorly t the ametimethanfor irmsndissimilarndustries.Similarlyntryingo makevariance mall t s notenouglho investinmany ecurities.t is necessaryo avoid nvestingnsecurities ithhigh ovariancesmong hemselves. e should iversifycross ndus-triesbecausefirmsn differentndustries,speciallyndustries ithdifferentconomicharacteristics,ave lower ovariances han firmswithinn industry.The concepts yield" and "risk" appearfrequentlyn financialwritings. sually f the term"yield"werereplacedby "expectedyield"or"expected eturn,"nd "risk"by"variance freturn,"ittlechange fapparentmeaning ouldresult.Variances a well-knowneasure fdispersionbout theexpected.Ifinstead fvariance he nvestor as concerned ith tandard rror,a' = VV, orwith hecoefficientfdispersion,yE,his choicewouldstill ieinthe etofefficientortfolios.Suppose n nvestor'iversifiesetweenwoportfoliosi.e., fheputssomeofhismoneynoneportfolio,herest fhismoneyntheother.Anexample fdiversifyingmong ortfoliossthebuyingf he haresoftwodifferentnvestmentompanies).f the twooriginal ortfolioshave equalvariance hen ypicallyl2hevariance ftheresultingcom-pound)portfolio illbe lessthanthevariance feither riginal ort-12. In nocasewillvariance e increased.heonly ase nwhich ariance illnotbedecreaseds if thereturn rom othportfoliosreperfectlyorrelated.o draw he so-varianceurves s ellipsest sbothnecessarynd ufficiento assume hatnotwo istinctportfoliosaveperfectlyorrelatedeturns.


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go TheJournal fFinancefolio. his s llustratedy Figure . To interpretigure we note hata portfolioP) which s builtoutoftwoportfolios' = (X, X2) andP (X1, X ) is of the formP = XP + (1 - X)P = (XX +(- X)X7,XXI+ (1 - )X2). P is on thestraightine connectingP' andP".The E- Vprinciplesmore lausible s a rule ornvestmentehavioras distinguishedrompeculativeehavior. he thirdmoment"33 of

X2a

X ;~~~~~~~~~sovarionee

c bXIFIG. 7

theprobabilityistributionfreturns rom heportfolio aybe con-nectedwith propensityogamble. orexamplefthe nvestormaxi-mizesutilityU) which epends nE andV(U = U(E, V),dU/1E >0, dUl/E < 0) hewill never cceptan actuariallyair'4bet. But if13. IfR is a randomariablehat akes n a finiteumberfvalues l,. . . , r. withnprobabilitiesi, . , pnespectively,ndexpectedalueE, thenMs = pi(ri E)3

t=114. One nwhich he mountained ywinninghe et imes he robabilityfwinning


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Portfolio election 91U = U(E, V,M3) and if69U/0M3 0 then here resomefairbetswhichwouldbe accepted.Perhaps-for greatvariety finvestingnstitutionshich on-sidery'ield o be a good thing;risk, bad thing;gambling,o beavoided-E, V efficiencys reasonables a workingypothesisndaworking axim.Twousesof theE-V principleuggesthemselves. emight seitin theoreticalnalyses r wemight se it in the actual selection fportfolios.In theoreticalnalyseswe mightnquire, or xample, bout thevarious ffectsf a changen the beliefs enerallyeld abouta firm,ora generalhangenpreferences toexpectedeturnersus arianceofreturn,r a changenthesupply f a security.n ouranalyses heXimight epresentndividualecuritiesrtheymight epresentggre-gates uch s, say,bonds, tocks nd realestate."To use theE-V rule ntheselectionf ecurities emusthavepro-cedures or indingeasonableui nd aiq.Theseprocedures,believe,shouldcombine tatisticalechniquesnd the udgment fpracticalmen.Myfeelings that he tatisticalomputationshould e usedtoarrive t a tentativeet ofA,i nd -ij.Judgmenthould henbe usedin ncreasingrdecreasingome ftheseAiandaojon thebasis offac-torsor nuancesnot taken nto accountby the formalomputations.Using hisrevised et ofAiand aii, the setofefficient, V combina-tions ouldbe computed,he nvestorould elect hecombinationepreferred,nd theportfolio hich averiseto thisE, V combinationcouldbe found.One suggestion s to tentative i, aij is to use the observedAi,aifor omeperiod f thepast. believe hatbettermethods, hich akeintoaccountmore nformation,an be found. believe hatwhat sneeded s essentially"probabilistic"eformulationfsecuritynaly-sis. willnotpursue his ubject ere, or his s "anothertory."t isa story f which have readonly he first ageof the firsthapter.In thispaperwehaveconsideredhesecond tage ntheprocess fselecting portfolio.hisstage tartswith he relevant eliefsboutthe securitiesnvolvednd endswith heselectionf a portfolio.Wehave not consideredhefirsttage:theformationf the relevant e-liefs n thebasisofobservation.15. Caremust e used nusingnd nterpretingelationsmong ggregates. e cannotdealherewith heproblemsndpitfallsf ggregation.

25 markowitz

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