Portfolio SelectionHarry MarkowitzThe Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91.Stable URL: 1The Journal of Finance is currently published by American Finance Association.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected]://www.jstor.orgMon Sep 3 01:12:50 2007

PORTFOLIO SELECTION*HARRYMARKOWITZThe Rand CorporationTHEPROCESS OF SELECTING a portfolio may be divided into two stages.The first stage starts with observation and experience and ends withbeliefs about the future performances of available securities. Thesecond stage starts with the relevant beliefs about future performancesand ends with the choice of portfolio. This paper is concerned with thesecond stage. We first consider the rule that the investor does (or should)maximize discounted expected, or anticipated, returns. This rule is rejected both as a hypothesis to explain, and as a maximum to guide investment behavior. We next consider the rule that the investor does (orshould) consider expected return a desirable thing and variance of return an undesirable thing. This rule has many sound points, both as amaxim for, and hypothesis about, investment behavior. We illustrategeometrically relations between beliefs and choice of portfolio according to the "expected returns-variance of returns" rule.One type of rule concerning choice of portfolio is that the investordoes (or should) maximize the discounted (or capitalized) value offuture returns.l Since the future is not known with certainty, it mustbe "expected" or "anticipatded7'returns which we discount. Variationsof this type of rule can be suggested. Following Hicks, we could let"anticipated" returns include an allowance for risk.2 Or, we could letthe rate at which we capitalize the returns from particular securitiesvary with risk.The hypothesis (or maxim) that the investor does (or should)maximize discounted return must be rejected. If we ignore market imperfections the foregoing rule never implies that there is a diversifiedportfolio which is preferable to all non-diversified portfolios. Diversification is both observed and sensible; a rule of behavior which doesnot imply the superiority of diversification must be rejected both as ahypothesis and as a maxim.* This paper is based on work done by the author while a t the Cowles Commission forResearch in Economics and with the financial assistance of the Social Science ResearchCouncil. I t will be reprinted as Cowles Commission Paper, New Series, No. 60.1. See, for example, J. B. Williams, The Theory of Investment Value (Cambridge, Mass.:Harvard University Press, 1938), pp. 55-75.2. J. R. Hicks, V a l aend Capital (New York: Oxford University Press, 1939), p. 126.Hicks applies the rule to a firm rather than a portfolio.

The Journal of Finance78The foregoing rule fails to imply diversification no matter how theanticipated returns are formed; whether the same or different discountrates are used for different securities; no matter how these discountrates are decided upon or how they vary over time.3 The hypothesisimplies that the investor places all his funds in the security with thegreatest discounted value. If two or more securities have the same value, then any of these or any combination of these is as good as anyother.We can see this analytically: suppose there are N securities; let rit bethe anticipated return (however decided upon) at time t per dollar invested in security i; let djt be the rate at which the return on the ilksecurity at time t is discounted back to the present; let X i be the relative amount invested in security i . We exclude short sales, thus Xi 2 0for all i . Then the discounted anticipated return of the portfolio isxmRi di, T i t is the discounted return of the ithsecurity, thereforet-1R ZXiRi where Ri is independent of Xi. Since Xi 2 0 for all iand Z X i 1, R is a weighted average of Ri with the X i as non-negative weights. To maximize R, we let Xi 1 for i with maximum Ri.If several Ra,, a 1, . . . , K are maximum then any allocation withmaximizes R. In no case is a diversified portfolio preferred to all nondiversified poitfolios.I t will be convenient a t this point to consider a static model. Instead of speaking of the time series of returns from the ith security( r i l , ri2) . . . , rit, . . .) we will speak of "the flow of returns" (ri) fromthe ithsecurity. The flow of returns from the portfolio as a whole is3. The results depend on the assumption that the anticipated returns and discountrates are independent of the particular investor's portfolio.4. If short sales were allowed, an infinite amount of money would be placed in thesecurity with highest r .

Portfolio Selection79R ZX,r,. As in the dynamic case if the investor wished to maximize"anticipated" return from the portfolio he would place all his funds inthat security with maximum anticipated returns.There is a rule which implies both that the investor should diversifyand that he should maximize expected return. The rule states that theinvestor does (or should) diversify his funds among all those securitieswhich give maximum expected return. The law of large numbers willinsure that the actual yield of the portfolio will be almost the same asthe expected yield.5This rule is a special case of the expected returnsvariance of returns rule (to be presented below). I t assumes that thereis a portfolio which gives both maximum expected return and minimumvariance, and it commends this portfolio to the investor.This presumption, that the law of large numbers applies to a portfolio of securities, cannot be accepted. The returns from securities aretoo intercorrelated. Diversification cannot eliminate all variance.The portfolio with maximum expected return is not necessarily theone with minimum variance. There is a rate a t which the investor cangain expected return by taking on variance, or reduce variance by giving up expected return.We saw that the expected returns or anticipated returns rule is inadequate. Let us now consider the expected returns-variance of returns (E-V) rule. I t will be necessary to first present a few elementaryconcepts and results of mathematical statistics. We will then showsome implications of the E-V rule. After this we will discuss its plausibility.In our presentation we try to avoid complicated mathematical statements and proofs. As a consequence a price is paid in terms of rigor andgenerality. The chief limitations from this source are (1) we do notderive our results analytically for the n-security case; instead, wepresent them geometrically for the 3 and 4 security cases; (2) we assumestatic probability beliefs. In a general presentation we must recognizethat the probability distribution of yields of the various securities is afunction of time. The writer intends to present, in the future, the general, mathematical treatment which removes these limitations.We will need the following elementary concepts and results ofmathematical statistics:Let Y be a random variable, i.e., a variable whose value is decided bychance. Suppose, for simplicity of exposition, that Y can take on afinite number of values yl, yz, . . . , y, .Let the probability that Y 5. U'illiams, o p . cit., pp. 68, 69.

T h e Journal of Finance80yl, be pl; that Ydefined to be y2 be pz etc. The expected value (or mean) of Y isThe variance of Y is defined to beV is the average squared deviation of Y from its expected value. V is acommonly used measure of dispersion. Other measures of dispersion,closely related to V are the standard deviation, u .\/V and the coefficient of variation, a/E.Suppose we have a number of random variables: R1, . . . , R,. If R isa weighted sum (linear combination) of the Rithen R is also a random variable. (For example R1, may be the numberwhich turns up on one die; R2, that of another die, and R the sum ofthese numbers. In this case n 2, a1 a2 1).It will be important for us to know how the expected value andvariance of the weighted sum (R) are related to the probability distribution of the R1, . . . , R,. We state these relations below; we referthe reader to any standard text for proof.6The expected value of a weighted sum is the weighted sum of theexpected values. I.e., E(R) alE(R1) aZE(R2) . . . a,E(R,)The variance of a weighted sum is not as simple. To express it we mustdefine "covariance." The covariance of R1 and Rz is i.e., the expected value of [(the deviation of R1 from its mean) times(the deviation of R2 from its mean)]. In general we define the covariance between Ri and R as jE ( [Ri -E (Ri) I [Ri - E (Rj)I fiuij may be expressed in terms of the familiar correlation coefficient(pij).The covariance between Ri and R j is equal to [(their correlation)times (the standard deviation of Ri) times (the standard deviation ofRj)l:Uij PijUiUj6. E.g., J. V. Uspensky, Introduction to mathematical Probability (New York: McGrawHill, 1937), chapter 9, pp. 161-81.

Portfolio SelectionThe variance of a weighted sum isIf we use the fact that the variance of Ri is uii thenLet Ri be the return on the iN" security. Let pi be the expected vaIueof Ri; uij, be the covariance between Ri and R j (thus uii is the varianceof Ri). Let X i be the percentage of the investor's assets which are allocated to the ithsecurity. The yield (R) on the portfolio as a whole isThe Ri (and consequently R) are considered to be random variables.'The X i are not random variables, but are fixed by the investor. Sincethe X i are percentages we have ZXi 1. In our analysis we will exclude negative values of the Xi (i.e., short sales); therefore X i 0 forall i.The return (R) on the portfolio as a whole is a weighted sum of random variables (where the investor can choose the weights). From ourdiscussion of such weighted sums we see that the expected return Efrom the portfolio as a whole is and the variance is7. I.e., we assume that the investor does (and should) act as if he had probability beliefsconcerning these variables. I n general we vouldexpect that the investor could tell us, forany two events (A and B), whether he personally considered A more likely than B, B morelikely than A, or both equally likely. If the investor were consistent in his opinions on suchmatters, he would possess a system of probability beliefs. We cannot expect the investorto be consistent in every detail. We can, however, expect his probability beliefs to beroughly consistent on important matters that have been carefully considered. We shouldalso expect that he will base his actions upon these probability beliefs-even though theybe in part subjective.This paper does not consider the difficult question of how investors do (or should) formtheir probability beliefs.

82The Journal of FinanceFor fixed probability beliefs (pi, oij) the investor has a choice of various combinations of E and V depending on his choice of portfolioX I , . . . , X N . Suppose that the set of all obtainable (E, V) combinations were as in Figure 1. The E-V rule states that the investor would(or should) want to select one of those portfolios which give rise to the(E, V) combinations indicated as efficient in the figure; i.e., those withminimum V for given E or more and maximum E for given V or less.There are techniques by which we can compute the set of efficientportfolios and efficient (E, V) combinations associated with given piattainableE, V combinationsand oij. We will not present these techniques here. We will, however,illustrate geometrically the nature of the efficient surfaces for casesin which N (the number of available securities) is small.The calculation of efficient surfaces might possibly be of practicaluse. Perhaps there are ways, by combining statistical techniques andthe judgment of experts, to form reasonable probability beliefs (pi,a i j ) . We could use these beliefs to compute the attainable efficientcombinations of (E, V). The investor, being informed of what (E, V)combinations were attainable, could state which he desired. We couldthen find the portfolio which gave this desired combination.

Portfolio Selection83Two conditions-at least-must be satisfied before it would be practical to use efficient surfaces in the manner described above. First, theinvestor must desire to act according to the E-V maxim. Second, wemust be able to arrive a t reasonable pi and uij. We will return to thesematters later.Let us consider the case of three securities. In the three security caseour model reduces to4)Xi Ofori l,2,3.From (3) we get3')X s 1-XI--XzIf we substitute (3') in equation (1) and (2) we get E and V as functionsof X1 and Xz. For example we find1')E' 3 x1(111- 3 ) x2 (112- 113)The exact formulas are not too important here (that of V is given below).8 We can simply writea)E E (XI, Xdb)V V (Xi, Xz)By using relations (a), (b), (c), we can work with two dimensionalgeometry.The attainable set of portfolios consists of all portfolios whichsatisfy constraints (c) and (3') (or equivalently (3) and (4)). The attainable combinations of XI, X2 are represented by the triangle abc inFigure 2. Any point to the left of the Xz axis is not attainable becauseit violates the condition that X1 3 0. Any point below the X1 axis isnot attainable because it violates the condition that Xz 3 0. Any

84The Journal of Financepoint above the line (1 - X1 - Xz 0) is not attainable because itviolates the condition that X3 1 - XI - Xz 0.We define an isomean curve to be the set of all points (portfolios)with a given expected return. Similarly an isovariance line is defined tobe the set of all points (portfolios) with a given variance of return.An examination of the formulae for E and V tells us the shapes of theisomean and isovariance curves. Specifically they tell us that typicallygthe isomean curves are a system of parallel straight lines; the isovariance curves are a system of concentric ellipses (see Fig. 2). For example,if 2 p3 equation 1' can be written in the familiar form X2 abX1; specifically (1) Thus the slope of the isomean line associated with E Eois -(pl j 3)/(. 2- p3) its intercept is (Eo - p3)/(p2 - p3). If we change E wechange the intercept but not the slope of the isomean line. This confirms the contention that the isomean lines form a system of parallellines.Similarly, by a somewhat less simple application of analytic geometry, we can confirm the contention that the isovariance lines form afamily of concentric ellipses. The "center" of the system is the pointwhich minimizes V. We will label this point X. Its expected return andvariance we will label E and V. Variance increases as you move awayfrom X. More precisely, if one isovariance curve, C1, lies closer to Xthan another, Cz, then C1 is associated with a smaller variance than Cz.With the aid of the foregoing geometric apparatus let us seek theefficient sets.X, the center of the system of isovariance ellipses, may fall eitherinside or outside the attainable set. Figure 4 illustrates a case in whichXfalls inside the attainable set. In this case: Xis efficient. For no otherportfolio has a V as low as X; therefore no portfolio can have eithersmaller V (with the same or greater E) or greater E with the same orsmaller V. No point (portfolio) with expected return E less than Eis efficient. For we have E E and V V.Consider all points with a given expected return E; i.e., all points onthe isomean line associated with E. The point of the isomean line atwhich V takes on its least value is the point at which the isomean line pz pa I n the9. The isomean "curves" are as described above except whenlatter case all portfolios have the same expected return and the investor chooses the onewith minimum variance.As to the assumptions implicit in our description of the isovariance curves see footnote12.

Portfolio Selection85Ais tangent to an isovariance curve. We call this point X(E). If we letE vary, X(E) traces out a curve.Algebraic considerations (which we omit here) show us that this curveis a straight line. We will call it the critical line I. The critical line passesthrough X for this point minimizes V for all points with E(X1, X z ) E.As we go along l in either direction from X, V increases. The segmentof the critical line from X to the point where the critical line crossesh*direction of increasing Edepends on p,. p:. p3FIG. 2the boundary of the attainable set is part of the efficient set. The rest ofthe efficient set is (in the case illustrated) the segment of the 3 linefrom d to b. b is the point of maximum attainable E. In Figure 3, X liesoutside the admissible area but the critical line cuts the admissiblearea. The efficient line begins at the attainable point with minimumvariance (in this case on the Z line). It moves toward b until it intersects the critical line, moves along the critical line until it intersects aboundary and finally moves along the boundary to b. The reader may

efficient portfolios

Portfolio Selection87wish to construct and examine the following other cases: (1) X liesoutside the attainable set and the critical line does not cut the attainable set. In this case there is a security which does not enter into anyefficientportfolio. (2) Two securities have the same pi. In this case theisomean lines are parallel to a boundary line. I t may happen that theefficient portfolio with maximum E is a diversified portfolio. (3) A casewherein only one portfolio is efficient.The efficientset in the 4 security case is, as in the 3 security and alsothe N security case, a series of connected line segments. At one end ofthe efficient set is the point of minimum variance; a t the other end isa point of maximum expected returnlo (see Fig. 4).Now that we have seen the nature of the set of efficient portfolios,it is not difficult to see the nature of the set of efficient (E, V) combinations. In the three security case E a0 alXl a2X2 is a plane; V bo blX1 hX2 b12XlX2 b1lx: B X ;is a paraboloid.ll Asshown in Figure 5, the section of the E-plane over the efficient portfolioset is a series of connected line segments. The section of the V-paraboloid over the efficient portfolio set is a series of connected parabolasegments. If we plotted V against E for efficient portfolios we wouldagain get a series of connected parabola segments (see Fig. 6). This result obtains for any number of securities. Various reasons recommend the use of the expected return-varianceof return rule, both as a hypothesis to explain well-established investment behavior and as a maxim to guide one's own action. The ruleserves better, we will see, as an explanation of, and guide to, "investment" as distinguished from ('speculative" behavior.10. Just as we used the equation5Xi I to reduce the dimensionality in the threei 1security case, we can use it to represent the four security case in 3 dimensional space.Eliminating X, we get E E(X1, Xz, Xs), V V(X1, Xz, Xs). The attainable set is represented, in three-space, by the tetrahedron with vertices (O,0, O), (0,0, I), (0,1, O), (1,0, O),representing portfolios with, respectively, X4 1, Xs 1, Xz 1, XI 1.Let sisa be the subspace consisting of all points with X4 0. Similarly we can definesol, . ,aa to be the subspace consisting of all points with Xi 0, i # a ., . . , aa. Foreach subspace sol, . . , aa we can define a critical lilze lal, . aa. This line is the locus ofpoints P where P minimizes V for all points in sol, . , aa with the same E as P. If a pointis in s,l, . , aa and is efficient it must be on lal, . . , aa. The efficient set may be tracedout by starting a t the point of minimum available variance, moving continuously alongvarious lal, . . , aa according to definite rules, ending in a point which gives maximum E.As in the two dimensional case the point with minimum available variance may be in theinterior of the available set or on one of its boundaries. Typically we proceed along a givencritical line until either this line intersects one of a larger subspace or meets a boundary(and simultaneously the critical line of a lower dimensional subspace). In either of thesecases the efficient line turns and continues along the new line. The efficient line terminateswhen a point with maximum E is reached.11. See footnote 8. . .

vefficientE, V eombinofionrE

Portfolio Selection89Earlier we rejected the expected returns rule on the grounds that itnever implied the superiority of diversification. The expected returnvariance of return rule, on the other hand, implies diversification for awide range of pi, aij. This does not mean that the E - V rule never implies the superiority of an undiversified portfolio. I t is conceivable thatone security might have an extremely higher yield and lower variancethan all other securities; so much so that one particular undiversifiedportfolio would give maximum E and minimum V. Rut for a large,presumably representative range of pi, aij the E- V rule leads to efficientportfolios almost all of which are diversified.Not only does the E - V hypothesis imply diversification, it impliesthe "right kind" of diversification for the "right reason.'' The adequacyof diversification is not thought by investors to depend solely on thenumber of different securities held. A portfolio with sixty different railway securities, for example, would not be as well diversified as the samesize portfolio with some railroad, some public utility, mining, varioussort of manufacturing, etc. The reason is that it is generally morelikely for firms within the same industry to do poorly a t the same timethan for firms in dissimilar industries.Similarly in trying to make variance small it is not enough to investin many securities. It is necessary to avoid investing in securities withhigh covariances among themselves. We should diversify across industries because firms in different industries, especially industries withdifferent economic characteristics, have lower covariances than firmswithin an industry.The concepts "yield" and "risk" appear frequently in financialwritings. Usually if the term "yield" were replaced by "expectedyield" or "expected return," and "risk" by "variance of return," littlechange of apparent meaning would result.Variance is a well-known measure of dispersion about the expected.If instead of variance the investor was concerned with standard error,or, with the coefficient of dispersion, a/E, his choice woulda Tvstill lie in the set of efficient portfolios.Suppose an investor diversifies between two portfolios (i.e., if he putssome of his money in one portfolio, the rest of his money in the other.An example of diversifying among portfolios is the buying of the sharesof two different investment companies). If the two original portfolioshave equal variance then typically12the variance of the resulting (compound) portfolio will be less than the variance of either original port12. In no case will variance be increased. The only case in which variance will not bedecreased is if the return from both portfolios are perfectly correlated. To draw the isovariance curves as ellipses it is both necessary and sufficient to assume that no two distinctportfolios have perfectly correlated returns.

The Journal of Finance90folio. This is illustrated by Figure 7. To interpret Figure 7 we note thata portfolio iP) which is built out of two portfolios P' (x:,x:)andP" (xi::is of the form P XP' (1 - h ) " (AX:(1 - X)XI , AX: (1 - x)x:'). P is on the straight line connectingP' and P".The E-V principle is more plausible as a rule for investment behavioras distinguished from speculative behavior. The third moment13M 8 ofxi') the probability distribution of returns from the portfolio may be connected with a propensity to gamble. For example if the investor maximizes utility ( U ) which depends on E and V(U U(E, V ) ,d U/aE 0, aU/dE 0) he will never accept an actuarially fair14 bet. But if13. If R is a random variable that takes on a finite number of values 71,. . . ,m withprobabilities *I,. . . , gn respectively, and expected value E, then 2*i(ri- El3i l14. One in which the amount gained by winning the bet times the robabilitvof winning

Port)olio Selection9IU U(E, V, Mg) and if d U / d M 3 # 0 then there are some fair betswhich would be accepted.Perhaps-for a great variety of investing institutions which consider yield to be a good thing; risk, a bad thing; gambling, to beavoided-E, V efficiency is reasonable as a working hypothesis and aworking maxim.Two uses of the E-V principle suggest themselves. We might use itin theoretical analyses or we might use it in the actual selection ofportfolios.In theoretical analyses we might inquire, for example, about thevarious effects of a change in the beliefs generally held about a firm,or a general change in preference as to expected return versus varianceof return, or a change in the supply of a security. In our analyses theX i might represent individual securities or they might represent aggregates such as, say, bonds, stocks and real estate.15To use the E-V rule in the selection of securities we must have procedures for finding reasonable pi and aij. These procedures, I believe,should combine statistical techniques and the judgment of practicalmen. My feeling is that the statistical computations should be used toarrive at a tentative set of pi and aij. Judgment should then be usedin increasing or decreasing some of these pi and uij on the basis of factors or nuances not taken into account by the formal computations.Using this revised set of pi knd uij, the set of efficient E, V combinations could be computed, the investor could select the combination hepreferred, and the portfolio which gave rise to this E, V combinationcould be found.One suggestion as to tentative pi, aij is to use the observed pi, aiifor some period of the past. I believe that better methods, which takeinto account more information, can be found. I believe that what isneeded is essentially a "probabilistic" reformulation of security analysis. I will not pursue this subject here, for this is "another story." It isa story of which I have read only the first page of the first chapter.In this paper we have considered the second stage in the process ofselecting a portfolio. This stage starts with the relevant beliefs aboutthe securities involved and ends with the selection of a portfolio. Wehave not considered the first stage: the formation of the relevant beliefs on the basis of observation.15. Care must be used in using and interpreting relations among aggregates. We cannotdeal here with the problems and pitfalls of aggregation.

It will be convenient at this point to consider a static model. In- stead of speaking of the time series of returns from the ithsecurity (ril, ri2) . . . ,rit, . . .) we will speak of "the flow of returns" (ri) from the ithsecurity. The flow of returns from the portfolio as a whole is 3.