On The Equivalence Of The Mean Variance Criterion And Stochastic Dominance Criteria

TL;DR

This paper investigates the conditions under which the Mean-Variance Criterion (MVC) aligns with stochastic dominance rules and expected utility maximization, revealing limitations in their equivalence across different distributions and utility functions.

Contribution

It clarifies the precise distributional conditions for MVC's equivalence to SSDR and challenges the assumption that MVC universally maximizes expected utility for quadratic utility functions.

Findings

MVC is equivalent to SSDR under symmetric Elliptical distributions.

MVC does not always coincide with SSDR for Skew-Normal distributions.

Approximately quadratic utility functions are limited; some common forms do not approximate quadratic utility well.

Abstract

We study the necessary and sufficient conditions under which the Mean-Variance Criterion (MVC) is equivalent to the Maximum Expected Utility Criterion (MEUC), for two lotteries. Based on Chamberlain (1983), we conclude that the MVC is equivalent to the Second-order Stochastic Dominance Rule (SSDR) under any symmetric Elliptical distribution. We then discuss the work of Schuhmacher et al. (2021). Although their theoretical findings deduce that the Mean-Variance Analysis remains valid under Skew-Elliptical distributions, we argue that this does not entail that the MVC coincides with the SSDR. In fact, generating multiple MV-pairs that follow a Skew-Normal distribution it becomes evident that the MVC fails to coincide with the SSDR for some types of risk-averse investors. In the second part of this work, we examine the premise of Levy and Markowitz (1979) that "the MVC deduces the maximization of the expected utility of an investor, under any approximately quadratic utility function, without making any further assumption on the distribution of the lotteries". Using Monte Carlo Simulations, we find out that the set of approximately quadratic utility functions is too narrow. Specifically, our simulations indicate that $\log{(a+Z)}$ and $(1+Z)^a$ are almost quadratic, while $-e^{-a(1+Z)}$ and $-(1+Z)^{-a}$ fail to approximate a quadratic utility function under either an Extreme Value or a Stable Pareto distribution.