A Generalisation of the Mean‐Variance Analysis

TLDR

The paper studies a decision maker whose kinked utility at a reference point and distorted probabilities lead to a mean‑partial‑moments utility that extends both mean‑variance and mean‑semivariance frameworks. The authors approximate the expected utility with mean and partial moments, derive a small‑risk premium, and solve the optimal capital allocation problem, yielding a generalized Sharpe/Sortino performance measure. The analysis reveals loss aversion, uncertainty aversion in gains and losses, and that skewness preferences exert a first‑order influence on risk measurement even at low risk levels.

Abstract

Abstract In this paper we consider a decision maker whose utility function has a kink at the reference point with different functions below and above this reference point. We also suppose that the decision maker generally distorts the objective probabilities. First we show that the expected utility function of this decision maker can be approximated by a function of mean and partial moments of distribution. This ‘mean‐partial moments’ utility generalises not only mean‐variance utility of Tobin and Markowitz, but also mean‐semivariance utility of Markowitz. Then, in the spirit of Arrow and Pratt, we derive an expression for a risk premium when risk is small. Our analysis shows that a decision maker in this framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Finally we present a solution to the optimal capital allocation problem and derive an expression for a portfolio performance measure which generalises the Sharpe and Sortino ratios. We demonstrate that in this framework the decision maker's skewness preferences have first‐order impact on risk measurement even when the risk is small.