Efficient Portfolio Selection with Quadratic and Cubic Utility

Abstract

Decisions about investment, or portfolio selection, are regarded as choices among alternative probability distributions of returns, where the optimal choice is determined by maximization of the expected value of an investor's utility function.' In the real world, investors' utility functions and investment probability distributions of returns may assume highly complex or irregular forms. However, most theoretical discussions of choice under risk have dealt with relatively simple forms, for example, quadratic utility functions and normal probability distributions, in order to make more manageable the description and testing of investment decision rules.2 This paper presents optimal efficiency criteria for portfolio selection when the utility function is quadratic in money returns and for a variety of kinds of information about the distribution of returns. In addition, it provides an optimal criterion for cubic utility. By efficiency criteria we mean conditions for dominance, or preference among risks, which apply to all investors whose utility functions are of a given general class (e.g., quadratic), independent of specific individual tastes or specific parameters of the utility function. Our main conclusions are: first, that the common procedures and criteria for quadratic utility, of which the simple mean-variance criterion is the best known and most widely used,3 are insufficient and may be improved considerably. The criteria given in the following section are all weaker sufficient conditions for dominance, relative to the mean-variance criterion, and thus they are more effective. Second, we claim that a cubic utility may be preferable, in some respects, to the quadratic form and is also amenable to a complete efficiency analysis, with some interesting implications. * The authors wish to thank Merton Miller and A. Beja for valuable comments and criticism on a first draft.