Stochastic Dominance among Log-Normal Prospects

Abstract

THE MEAN-VARIANCE EFFICIENCY ANALYSIS introduced by Markowitz [16] and Tobin [24] is a valid decision rule either for the case in which the utility function is quadratic or if the returns are normally distributed and risk-aversion is assumed.2 The notion of stochastic dominance has recently been developed by Quirk and Saposnik [18], Hadar and Russell [7], Hanoch and Levy [9], and Rothschild and Stiglitz [19, 20]. We say that prospect F dominates prospect G (FDG) within a class of utility functions U, if for every utility function u, EFu(x) ? EGU(X), and if for at least one utility function, uo C U, the inequality is sharp. Obviously, the criterion for dominance is a function both of the assumptions about class U and of the information available on the probability distributions of returns. The mean-variance rule is clearly a special case of the above definition: If EF(x) ?EG(x) and aF(X) < JG(X) (with at least one strong inequality), and x is normally distributed, then one canl conclude that prospect F dominates prospect G within U2, where U2 is the class of all non-decreasing concave titility functions. In this paper we shall apply the concept of stochastic dominance to the comparison of log-normally distributed prospects. This application is useful especially for comparison between prospects taken from the stock market, provided that the investment horizon is not very short; if it is, returns, being the cumulative products of random variables, tend to distribute log-normally (see Cootner [3]). This technique also sheds light on the role of the geometric mean criterion raised by Latane [14], the average comnpounded return rule. used by Hakansson [8], and Samuelson's [21, 22, 23] analysis refuting Latane's claim regarding the role of the geometric mean criterion. We shall demonstrate that for log-normal prospects, the mean-variance rule, as Feldstein [5] pointed out correctly, might lead to paradoxical results. However, it is shown in this paper that the meanvariance rule in the above case-though not optimal-is not invalid. It is shown that it is a sufficient but not a necessary rule, and hence can lead to a relatively large efficient set. Nevertheless, by making some modifications in the conventional mean-variance rule, we establish an optimal decision rule for the lognormal case, when risk-aversion is assumned. In Section 2 we apply stochastic dominance technique to log-normal prospects. In Section 3 we assumiie a very long investment horizon and analyze the relation-