PORTFOLIO SELECTION* HARRY MARKOWIT The Rand Cor poration 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 nd 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- to the " expected returns-variance of returns" ru One type of rule concerning choice of portfolio is that the investor does(or should) maximize the discounted (or capitalized) value of future returns. I 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 "anticipated"returns 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 is a diversified 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 Council. It will be reprinted as Cow les Commiss 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 J. R. Hicks, Value and Capital (New York: Oxford University Press, 1939), p. 126 licks applies the rule to a f her than a portfPORTFOLIO SELECTION* HARRYMARKOWITZ The Rand Corporation THEPROCESS 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.l Since the future is not known with certainty, it must be "expected" or "anticipatded7' returns which we discount. Variations of this type of rule can be suggested. Following Hicks, we could let "anticipated" returns 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 is a diversified 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 by the author while at the Cowles Commission for Research in Economics and with the financial assistance of the Social Science Research Council. It 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, Val~eand Capital (New York: Oxford University Press, 1939), p. 126. Hicks applies the rule to a firm rather than a portfolio