A Generalisation of the Mean-Variance Analysis

37 Pages Posted: 21 Oct 2009

See all articles by Valeriy Zakamulin

Valeriy Zakamulin

University of Agder - School of Business and Law

Steen Koekebakker

School of Business and Law at the University of Agder

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.

Suggested Citation

Zakamulin, Valeriy and Koekebakker, Steen, A Generalisation of the Mean-Variance Analysis. European Financial Management, Vol. 15, Issue 5, pp. 934-970, November 2009. Available at SSRN: https://ssrn.com/abstract=1491790 or http://dx.doi.org/10.1111/j.1468-036X.2009.00483.x

Valeriy Zakamulin (Contact Author)

University of Agder - School of Business and Law ( email )

Service Box 422
Kristiansand, N-4604
Norway
+47 38141039 (Phone)

HOME PAGE: http://vzakamulin.weebly.com/

Steen Koekebakker

School of Business and Law at the University of Agder ( email )

Servicebox 422
Kristiansand, N-4604
Norway
+47 38 14 15 31 (Phone)

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