16 - Portfolio Management: The Mean-Gini Approach

Summary

INTRODUCTION

One of the most frequently used methods for ranking uncertain rates of return on assets is the mean-variance approach. The essential idea is to represent the distribution of the rate of return (or return, for short) on an asset by two summary statistics: the mean (indicating the reward) and the variance (indicating variability). The use of summary statistics definitely simplifies the ranking problem. However, the mean-variance analysis may lead to unjustified conclusions. Two assets may have the same mean return and the corresponding variance for one may be higher than that of the other, yet the former may be preferred to the latter by some risk-averse individuals (see Rothschild and Stiglitz 1970; see also Giora and Levy 1969). The stochastic dominance approach does not lead to such unwarranted conclusions. It then becomes worthwhile to develop an alternative approach that summarizes the distributions in terms of two statistics and that retains stochastic dominance efficiency.

In this chapter, we present an alternative approach for comparing returns on assets, building on the theory of stochastic dominance. It uses the mean and the Gini evaluation function as the summary statistics of the distribution of the return. The Gini evaluation function is based on the well-known Gini index, which is used extensively as a measure of income inequality. Use of the Gini evaluation function as a summary statistic of a risky investment allows the derivation of necessary conditions for stochastic dominances (Yitzhaki 1982; Shalit and Yitzhaki 1984).