Risk, Return, Skewness and Preference

Abstract

This paper considers choice between individual projects and shows that when the choice set includes arbitrary distributions, then any assumed relationship between expected utility theory and general moment preferences for individual decision makers is theoretically unsound. In particular, a risk averse investor with any common utility function may, when choosing between two positive return opportunities, prefer the project simultaneously having a lower mean, higher variance, and lower positive skewness. Moreover, the decision maker can prefer opportunities with higher variance even when the opportunities are continuous, unimodal, and arbitrarily visually and statistically close to the normal distribution in shape. Our conclusions hold for any decision maker with a utility function whose derivatives alternate in sign being strictly positive or negative (i.e., we exclude the uninteresting cases of quadratic and cubic utilities). The method of analysis is based upon the theory of Tchebychev systems of functions which deals with the expected value of [utility] functions of stochastic variables with known moments. Although we focus on the first three moments, the results, as presented here, apply to all higher moments as well. It is also shown that there can be extremely large deviations between the certainty equivalents of distributions having the same moments, so this result is also pertinent to practical decision analysts as well. The paper demonstrates that the properties of utility functions have implications which are much more subtle than previously recognized for evaluating distributions in terms of their moments.