A Generalization of the Mean-Variance Analysis
49 Pages Posted: 2 Apr 2008 Last revised: 6 Nov 2008
Date Written: October 28, 2008
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 generalizes not only the mean-variance utility of Tobin and Markowitz, but also the 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 the solution to the optimal capital allocation problem and derive an expression for a portfolio performance measure which generalizes 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.
Keywords: risk aversion, loss aversion, risk measure, partial moments of distribution, mean-variance utility, quadratic utility, certainty equivalent, risk premium, optimal capital allocation, portfolio performance evaluation
JEL Classification: D81, G11
Suggested Citation: Suggested Citation