Portfolio Selection and Security Prices - Pye, Gordon (The Review of Economics and Statistics, 1967)
Following the lead of Markowitz,' the portfolio selection problem has usually taken the following form. An investor's utility for his portfolio is assumed to depend only on the mean and variance of its return over the following period. The investor has in mind means and variances for the returns on all the available securities. The problem is to determine the optimal proportions of these securities in his portfolio. Markowitz, himself, was largely concerned with calculating the dominant set of portfolios (i.e., those with minimum variance for given mean and maximum mean for given variance). Later Tobin 2 studied for two securities, one risky and the other riskless, how the optimal proportion of risky security would change with the mean and variance of its return. He also showed that the mean-variance approach of Markowitz would be consistent with the expected utility principle if the investor had a quadratic utility function or if he considered only a two parameter family of probability distributions. Recently Arrow 3 has reformulated the problem studied by Tobin. He lets the investor decide how to allocate a given amount of wealth between a risky and a riskless security so as to maximize his expected utility. Among other things he shows that the amount invested in the risky security will rise or fall with increasing wealth as minus U/U' (U being the investor's utility function) falls or rises with increasing wealth. In particular, for a quadratic U the amount invested in the risky security will necessarily decrease which is unrealistic. Thus, the mean-variance approach appears unduly restrictive for the portfolio problem, unless restrictions are placed on the form of the probability distribution. The restrictions require that the returns on the securities have a multivariate distribution such that a linear combination of the returns has a two parameter distribution. The multivariate normal is an important example of such a multivariate probability distribution. However, it may not be easy to find others when the returns are not independent. In this paper, the problem of portfolio selection is formulated so that it becomes formally identical to the traditional consumer theory for certain commodities. This permits all of the well-known results of this theory to be applied to the portfolio problem. It also permits the direct use of the traditional models for the price determination of certain commodities for determining the prices of securities. Investors are assumed to follow the expected utility principle and to form their own probability beliefs.