Portfolio Selection R=EX/. 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 and 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 ariance, 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 variance 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 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 some implications of the E-V rule. After this we will discuss its plausi- pility. In our presentation we try to avoid complicated mathematical state- 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 static 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 nathematical statistics Let y be a random variable i.e., a variable whose value is decided by chance. Suppose, for simplicity of exposition, that r can take on a finite number of values y WN. Let the probability that YPortfolio Selection 79 R = ZX,r,. 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 and 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.5 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 variance. 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 some implications of the E-V rule. After this we will discuss its plausi￾bility. In our presentation we try to avoid complicated mathematical state￾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 static 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: Let Y be a random variable, i.e., a variable whose value is decided by chance. Suppose, for simplicity of exposition, that Y can take on a finite number of values yl, yz, . . . ,y,~. Let the probability that Y = 5. U'illiams, op. cit., pp. 68, 69