INTRODUCTION

In financial economics we are often confronted with the necessity of ordering distributions in terms of a decision-maker's preference, where the distributions considered are usually those of random returns on various financial assets. In other words, we need to make a prediction about a decision-maker's preference between given pairs of uncertain alternatives, without having any knowledge about the decision-maker's utility function. (All utility functions considered in this chapter are von Neumann–Morgenstern utility functions. See Chapter 3, for a full discussion.) Several approaches to this and related issues have appeared in the literature. One of the most important approaches of this type involves a comparison of means and variances of the distributions under consideration (see Tobin 1958, 1965; Markowitz 1959). If the expected utility function is of quadratic type—for instance, if it is of the type mean minus a positive multiple of variance—then a decision-maker, who is a risk-averter, will unambiguously prefer the risky asset that has lower variance, given that the assets under consideration have equal mean or prefer the one with higher mean when the variances of the assets are the same.

But a decision-maker's preferences need not be presented by a quadratic utility function. There are many other forms of utility functions that can be used for ranking risky assets. However, there is no guarantee that the ranking of two different assets by two different utility functions will be the same. Therefore, a natural question here is: how do you rank alternative risky assets when the utility function is unknown?