Portfolio Selection TORIo Harry markowitz The Journal of finance, Vol. 7, No. 1.(Mar, 1952), pp. 77-91 Stable url: http://links.jstor.org/sici?sici=0022-1082%028195203%297%03a1%3c77903aps%3e2.0.c0%3b2-1 The 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 http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the 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 http://wwwjstor.org/journals/afina.html Each copy of any part of a JSTOR transmission must contain the same copy tice that appears on the screen or printed page of such transmission jStOR is an independent not-for-profit organization dedicated to creating and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor. org ttp://www.jstor.org Thu mar906:56:212006
PORTFOLIO SELECTION HARRY MARKOWITZ The Rand Corporation THE PROCESS OF SELECTING a portfolio may be divided into two stages The first stage starts with observation and experience and ends with beliefs about the future performances of available securities. The second stage starts with the relevant beliefs about future performances and ends with the choice of portfolio. This paper is concerned with the second stage. We first consider the rule that the investor does or should) maximize discounted expected, or anticipated, returns. This rule is re jected both as a hypothesis to explain, and as a maximum to guide in vestment behavior. We next consider the rule that the investor does(or should)consider expected return a desirable thing and variance of re- turn an undesirable thing. This rule has many sound points, both as a maxim for, and hypothesis about, investment behavior. We illustrate geometrically relations between beliefs and choice of portfolio accord ing to the expected returns-variance of returns ' rule One type of rule concerning choice of portfolio is that the investor does(or should) maximize the discounted (or capitalized) value of future returns. Since the future is not known with certainty, it must be"expected"'or"anticipated?returns which we discount. variations of this type of rule can be suggested. Following Hicks, we could let anticipatedreturns include an allowance for risk. 2 Or, we could let the rate at which we capitalize the returns from particular securities vary with risk. The hypothesis (or maxim) that the investor does(or should) maximize discounted return must be rejected. If we ignore market im- perfections the foregoing rule never implies that there versified portfolio which is preferable to all non-diversified portfolios. Diversi- fication is both observed and sensible: a rule of behavior which does not imply the superiority of diversification must be rejected both as a hypothesis and as a maxim This paper is based on work done Commission for Resea ouncil. It will be reprinted as Cowles Commission Paper Harvard University Press, 1938), pp 55-3 heory of Inwestment value(Cambridge,Mass See, for example, J B, Williams, T 2.J. R. Hicks, Value and Capital(New York: Oxford University Press, 1939), p. 126 Hicks applies the rule to a firm rather than a portfolio
The Journal of finance The foregoing rule fails to imply diversification no matter how the anticipated returns are formed; whether the same or different discount rates are used for different securities: no matter how these discount rates are decided upon or how they vary over time. The hypothesis implies that the investor places all his funds in the security with the greatest discounted value. If two or more securities have the same val ue, then any of these or any combination of these is as good as any We can see this analytically: suppose there are N securities; let fa be the anticipated return(however decided upon) at time t per dollar in vested in security i; let di be the rate at which the return on the i security at time t is discounted back to the present; let Xi be the rela- tive amount invested in security i. We exclude short sales, thus Xi> 0 for all i. Then the discounted anticipated return of the portfolio is R空ax X r,>dari is the discounted return of the ithsecurity,therefore R=EX, Ri where R, is independent of X Since X.>0 for all i and ZX:=1, R is a weighted average of R with the Xi as non-nega tive weights. To maximize R, we let X= 1 for i with maximum R, If several Ro 1,..., K are maximum then any allocation with maximizes R. In no case is a diversified portfolio preferred to all non- diversified portfoli It will be convenient at this point to consider a static model. In stead of speaking of the time series of returns from the i security of“ 'the fow of returns”,(r)fro the i security. The flow of returns from the portfolio as a whole 3. The results depend on the assumption that the anticipated returns and discount rates are independent of the particular investors portfolio 4. If short sales were allowed, an infinite amount of money would be placed in the
Portfolio selection R=EXri. as in the dynamic case if the investor wished to maximize anticipated"return from the portfolio he would place all his funds in that security with maximum anticipated returns There is a rule which implies both that the investor should diversify d that he should maximize expected return. The rule states that the investor does(or should) diversify his funds among all those securities which give maximum expected return. The law of large numbers will insure that the actual yield of the portfolio will be almost the same as the expected yield. This rule is a special case of the expected returns- variance of returns rule(to be presented below). It assumes that there is a portfolio which gives both maximum expected return and minimum variance, and it commends this portfolio to the investor This presumption, that the law of large numbers applies to a port folio of securities, cannot be accepted. The returns from securities are too intercorrelated Diversification cannot eliminate all varianc The portfolio with maximum expected return is not necessarily the one with minimum variance. There is a rate at which the investor can gain expected return by taking on variance, or reduce variance by giv ing up expected return We saw that the expected returns or anticipated returns rule is in Adequate. Let us now consider the expected returns--variance of re turns(E-v)rule It will be necessary to first present a few elementary concepts and results of mathematical statistics. We will then show bility: plications of the E-V rule. After this we will discuss its plausi In our presentation we try to avoid complicated mathematical ments and proofs. As a consequence a price is paid in terms of rigor and generality. The chief limitations from this source are (1)we do not derive our results analytically for the n-security case; instead, we present them geometrically for the 3 and 4 security cases; (2 )we assume tatic probability beliefs. In a general presentation we must recognize that the probability distribution of yields of the various securities is a function of time. The writer intends to present, in the future, the gen eral. mathematical treatment which removes these limitations We will need the following elementary concepts and results of mathematical statistics cK Let Y be a random variable, 1. e, a variable whose value is decided by ance. Suppose, for simplicity of exposition, that. y can take on a finite number of values y1, 22, . .. yN. Let the probability that Y 5. Williams, op cit, pp. 68, 69
The Journal of finance yi, be Pi; that Y y2 be P2 etc. The expected value (or mean)of Y is E=p1y1+Py+.十力 The variance of y is defined to be V=力(y1-B)2+P2(y2-E)2+,,+PN(yN-E)2 V is the average squared deviation of Y from its V is a commonly used measure of dispersion. Other measures of closely related to V are the standard deviation, o=vV Suppose we have a number of random variables: Rl,..., Rn. If R is a weighted sum (linear combination) of the R R=aRi+agR2+.+a,R then r is also a random variable. (For example Rl, may be the number which turns up on one die; R2, that of another die, and r the sum of these numbers. In this case n= 2, a1=a?= 1) It will be important for us to know how the expected value and tribution of the plighted sum(R)are related to the probability dis- riance of the wei Ri,..., R,. We state these relations below we refer the reader to any standard text for proof. 6 The expected value of a weighted sum is the weighted sum of the expected values. I.e., E(R)=aE(R1)+ a2E(R2)+..+anE(Rn) The variance of a weighted sum is not as simple To express it we must define"covariance. The covariance of R, and Ro is σ12=E{[R1-E(R1)][R2-E(R2)]} i. e. the expected value of [(the deviation of Ri from its mean) times (the deviation of R2 from its mean)]. In general we define the covari ance between R, and R, as σ;=E{[R;-E(R)][R;-E(R)1} oi; may be expressed in terms of (pi). The covariance between R; and R, is equal to [(their correlation) times(the standard deviation of Ri) times(the standard deviation of ]: 订-pij0;子 lily(New York: McGra
Portfolio selection The variance of a weighted sum is V(R)=∑a2V(X小+2 aiajoij If we use the fact that the variance of R, is o then V(R)=∑∑ Let R: be the return on the i "security. Let u be the expected value of R,; oi, be the covariance between R, and R, (thus oi is the variance of Ri. Let X, be the percentage of the investor's assets which are al located to the i security. The yield(R)on the portfolio as a whole is R=∑RX The R,(and consequently R)are considered to be random variables. 7 The Xi are not random variables, but are fixed by the investor. Since the X: are percentages we have EXi=1. In our analysis we will ex clude negative values of the X(i.e, short sales); therefore Xi20 for ll The return(R)on the portfolio as a whole is a weighted sum of ran- dom variables(where the investor can choose the weights). From our discussion of such weighted sums we see that the expected return E from the portfolio as a whole is X and the variance is XiX any two events(a and B), whe to be consistent in every detail considered. We should be in part subjective
The Journal of finance For fixed probability beliefs (ui, o the investor has a choice of vari- ous combinations of E and v depending on his choice of portfolio X1,..., XN. Suppose that the set of all obtainable (e, n) combina tions 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 with minimum 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 efficient rtfolios and efficient(E, v) combinations associated with given u attainable E, V combinations and oii. We will not present these techniques here. We will, however illustrate geometrically the nature of the efficient surfaces for cases in which N(the number of available securities) is small The calculation of efficient surfaces might possibly be of practical use. Perhaps there are ways, by combining statistical techniques and the judgment of experts, to form reasonable probability beliefs (ui, a. We could use these beliefs to compute the attainable efficient combinations of (E, v). The investor, being informed of what(E, V combinations were attainable, could state which he desired. We could then find the portfolio which gave this desired combination
Portfolio selection Two conditions--at least--must be satisfied before it would be prac tical to use efficient surfaces in the manner described above first, the investor must desire to act according to the e-v maxim. Second, we must be able to arrive at reasonable ui and o. We will return to these matters later Let us consider the case of three securities. In the three security case our model reduces to 1)E X 2)V XiX 4)X;≥0for 1,2,3 From(3)we get 3′)X3 If we substitute(3) in equation(1 )and (2)we get E and V as functions of XI and X2. For example we find 1)E=3+x1(1-3)+X2(2-43) The exact formulas are not too important here(that of V is given be low). We can simply write a) E=E(X1, X, b) V=V(XI, x2) c)x1>0,X2≥0,1-X1-X2≥0 By using relations(a),(b),(c), we can work with two dimensional The attainable set of portfolios consists of all portfolios which satisfy constraints(c) and (3)(or equivalently ( 3)and(4)). The at- tainable combinations of X1, X2 are represented by the triangle abc in Figure 2. Any point to the left of the X2 axis is not attainable because it violates the condition that X1>0. Any point below the Xi axis is not attainable because it violates the condition that x2> 0. any +σx)+(四n-20+aa)+2X1xn(o-0a-02a+a) +2x1(:-d)+2x303-3)+m
The Journal of finance point above the line(1- X1-X2=0) is not attainable because it violates the condition that X3=1-X1-X220 We define an isomean curve to be the set of all points (portfolios) with a given expected return. Similarly an isovariance line is defined to be 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 the the isomean curves are a system of parallel straight lines; the cally% isomean and isovariance curves. Specifically they tell us ance curves are a system of concentric ellipses(see Fig. 2). For example if un us equation 1 can be written in the familiar form X2=a+ 6X1; specifically(1) E X. Thus the slope of the isomean line associated with E= Eo is-(1 u3)/(u2-u3)its intercept is(E0-43)/(42-43). If we change E we change the intercept but not the slope of the isomean line. This con firms the contention that the isomean lines form a system of parallel Similarly by a somewhat less simple application of analytic ge try, we can confirm the contention that the isovariance lines form a family of concentric ellipses. The "center"of the system is the point which minimizes V. We will label this point X Its expected return and variance we will label E and v. variance increases as you move away from X. More precisely, if one isovariance curve, C1, lies closer to X than and nother, C?, then C1 is associated with a smaller variance than C2 With the aid of the foregoing geometric apparatus let us seek the efficient sets X, the center of the system of isovariance ellipses, may fall either inside or outside the attainable set. Figure 4 illustrates a case in which X falls inside the attainable set In this case: X is efficient. For no other portfolio has a V as low as X; therefore no portfolio can have either smaller v (with the same or greater E)or greater E with the same or smaller V. No point (portfolio) with expected return E less than E is efficient For we have e>E and v v. ts with a the isomean line associated with E. The point of the isomean line at which v takes on its least value is the point at which the isomean line 9. The isomean "curves"are as described above except when 4=4= 43. In the As to the assumptions implicit in our description of the isovariance curves see footnote
is tangent to an isovariance curve. We call this point X(E). If we let E vary, X(e) traces out a curve Algebraic considerations(which we omit here)show us that this curve is a straight line. We will call it the critical line I. The critical line passes through X for this point minimizes V for all points with E(X,, X2)=E As we go along l in either direction from X,V increases. The segment of the critical line from X to the point where the critical line crosses isomean lines Isovarlance curves direction of increasing E A. Ha FIG. 2 the boundary of the attainable set is part of the efficient set. The rest of the efficient set is ( the case illustrated)the segment of the ab line from d to b. b is the point of maximum attainable E. In Figure 3, X lies outside the admissible area but the critical line cuts the admissible area. The efficient line begins at the attainable point with minimum variance(in this case on the ab line). It moves toward b until it inter sects the critical line, moves along the critical line until it intersects a boundary and finally moves along the boundary to b. The reader may
