Portfolio Selection: Efficient Diversification of Investments 9780300191677

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COWLES FOUNDATION for Research in Economics at Yale University

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COWLES FOUNDATION For Research in Economics at Yale University

The Cowles Foundation for Research in Economics at Yale University, established as an activity of the Department of Economics in 1955, has as its purpose the conduct and encouragement of research in economics, finance, commerce, industry, and technology, including problems of the organization of these activities. The Cowles Foundation seeks to foster the development of logical, mathematical, and statistical methods of analysis for application in economics and related social sciences. The professional research staff are, as a rule, faculty members with appointments and teaching responsibilities in the Department of Economics and other departments. The Cowles Foundation continues the work of the Cowles Commission for Research in Economics founded in 1932 by Alfred Cowles at Colorado Springs, Colorado. The Commission moved to Chicago in 1939 and was affiliated with the University of Chicago until 1955. In 1955 the professional research staff of the Commission accepted appointments at Yale and, along with other members of the Yale Department of Economics, formed the research staff of the newly established Cowles Foundation. A list of Cowles Foundation Monographs appears at the end of this volume.

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Portfolio Selection E F F I C I E N T D I V E R S I F I C A T I O N OF INVESTMENTS

Harry M. Markowitz

NEW H A V E N AND LONDON, YALE UNIVERSITY PRESS

Copyright ). But w = Ar and exp^w) = A - expt(r), so w' = Ar — A • expt(r). But Ar — A- expt(r) = A[r - expt(r)], so w' = A[r — expt(r)]. Since r — expt(r) is what we refer to as r', we have w = Ar'y

as we sought to show.

If we square both sides of this equation, we find that (nO2 - A\r')\ This says that the random variable (w')2 is always equal to (is the same thing as) the random variable A2 • (r')2. Hence or, equivalently,

expt(H>')2 = expt(/*V2), expt(w')2 - A2 - expt(r')2.

But expt(n>')2 is the variance of w, and expt(r')2 is the variance of r; therefore var(w) - A2 - var(r).

Q.E.D.

COROLLARY: If r is a random variable generated by any wheel, A is any number, and w is the random variable which is inevitably A - r, then the standard deviation ofwisA times the standard deviation of r. PROOF: The theorem on variances states that var(w>) as A2 • var(r). Hence VvarOv) = A • VvarCr).

STANDARD DEVIATIONS AND VARIANCES

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But

Vvar(M>) = the standard deviation of M>, Vvar(r) = the standard deviation of r, from which the corollary follows immediately.

The theorem and corollary stated and proved above for a random variable can be rephrased to apply to past series and probability beliefs: 1. If series 1 is always A times as great as series 2, then the standard deviation of series 1 is A times as great as that of series 2; whereas the variance of series 2 is A2 times as great as the variance of series 1. 2. If r is an uncertain future outcome about which an individual holds a consistent set of probability beliefs, and if w is an outcome which the individual believes must inevitably be A times as great as r, then the standard deviation of w is A times as great as r; whereas the variance of w is A2 times as great as that of r. The proof for the case of random variable can be used, practically word for word, as a proof for the case of an uncertain event subject to a consistent set of probability beliefs. The proof for the case of a random variable becomes a proof for the case of a past series with little more than the substitution of the word "average" for the phrase "expected value." It makes sense to speak of "expected return per dollar invested" or "average return per dollar invested" or "standard deviation of return per dollar invested." But it does not make sense to speak of "variance of return per dollar invested." If the expected return from $1 bet on the spin of a wheel is .01, then the expected return from $10 is .1, the expected return from $100 is 1.0, and so on. In all cases the ratio expected total return amount bet is the same, .01. Similarly, the ratio standard deviation of total return amount bet remains constant as the size of bet varies. This is not true of variance. If the variance of a Si bet is, say, .001, then the variance of a $10 bet is .1, the variance of a $100 bet is 10, the variance of a $1000 bet is 1000. Depending on the amount bet, the ratio variance of total return amount bet may be .001, .01, .1, or 1.

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PORTFOLIO SELECTION

It is standard deviation rather than variance which is directly comparable with a past series or a set of probabilities. The high and low returns of the series in Figure 1 are about two standard deviations removed from the average of the series. If we change the unit of measurement from dollars to pennies (if we replace .5 by 50 on our graph), the high and low returns remain about two standard deviations removed from the average. The variance, however, goes from being smaller than the standard deviation (.127 vs. .357) to being larger than the standard deviation (1270 vs. 35.7). It is frequently more convenient to work with variance than with standard deviation. Relationships between securities and portfolios are generally simpler when expressed in terms of variances rather than standard deviations. The variance is less clumsy to deal with in most proofs. In computing efficient portfolios variance is used until the last, when variance is translated into standard deviation for the purpose of presentation. Of course, any relationship expressed in terms of variances can be translated into terms of standard deviation by substituting (standard deviation) 2 fo variance. We shall frequently express and prove relationships in terms of variances, leaving the translation to standard deviations for the reader. COVARIANCE AND CORRELATION

The definition of the covariance between two random variables is illustrated by the wheel in Figure 5. The outer ring of the wheel generates a random variable q. The second ring generates a random variable r. The third ring generates the random variable q' equal to q — expt(^). The fourth ring generates r equal to r —' expt(r). The entries on the fifth ring equal the product of the entries on the third and fourth rings. The fifth ring, in other words, generates the random variable q'r'. The covariance between q and r is defined to be the expected value of the fifth ring. The covariance between q and r, then, is the expected value of qr . It will be convenient to let cov(