# Generalised mean-risk preferences

**Daniel Schoch**

In this paper, published in the * Journal of Economic Theory*, Daniel Schoch (School of Economics, Malaysia Campus) axiomatises mean-risk preferences for a larger class of deviation measures. The first result extends Grechuk et al's case (Risk Anal. 2012) when the decision maker has mean-risk preferences with some unknown deviation risk measure. The second and major result covers the case when the deviation measure is given. This generalises Löffler's result (JET 1996) to a broad class of risk measures.

Modern portfolio theory has been introduced in the 1950s by Markowitz and Tobin, both receiving the Nobel Memorial Prize for Economics for their contribution. It made finding the optimal portfolio analytically tractable by standard stochastic techniques. Its main assumption is that the decision maker's preference depends only on the average return, as a measure of yield, and the stochastic variance, or its square root, the standard deviation, taken as a measure of risk. These so-called mean-variance preferences have been widely applied to the study of financial markets.

There is, however, a problem with the interpretation of variance or standard deviation as a measure of risk. Variance treats all deviations from the mean equally, whether they are positive gains or negative shortfalls. In particular, for every portfolio we can construct a "stochastic dominant" portfolio which yields better outcomes in every situation, but has a larger variance. To account for the principle of stochastic dominance, Postmodern portfolio theory has come up with alternative measures of risks, mostly monetary measures of downside risk, expected shortfall and (conditional) value at risk, which are not only used by investment companies, but found their ways into banking regulations. One closely linked class which drew recent attention are deviation measures. They were originally studied in the early 1900s by the German mathematician Hermann Minkowski, who is famous for putting the special theory of relativity of his student Einstein in its present four-dimensional form. He got the beautiful insight that each deviation measure corresponds to a convex body, eg the standard deviation to the ball, and vice versa.

Such a multitude is appreciated in the study of behavioral finance. If every investor was maximizing risk-variance preferences, then, so the standard theory implies, they would all invest their money into the same risky market portfolio. This is clearly not what we see. It is therefore interesting to modify the mean-variance approach by substituting deviation risk measures for the (square root of) variance. To test the theory if a certain decision maker maximizes generalised mean-risk preferences, we have to find the behavioral principles, or axioms, which constitute the theory. This has been achieved for the classical case of mean-variance preferences by Andreas Löffler (JET 1996) and for a large class of mean-risk preferences with deviation measures by Grechuk et al. (Risk Anal. 2012).

*Journal of Economic Theory*, "Generalized mean risk preferences", by Daniel Schoch www.sciencedirect.com/science/article/pii/S0022053116301053

*Image credit: https://de.wikipedia.org/wiki/Datei:Sao_Paulo_Stock_Exchange.jpg*

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