Consistent mean-variance preferences

Mean-variance utility functions exhibiting a certain set of properties underpin a large body of financial and economic theories. This paper provides a firm choice-theoretic foundation for such a function. Under the assumption that preferences over distributions are utility-representable, we show that the preferences can be represented by a differentiable mean-variance utility function if and only if the preference functional is Lp-Fréchet differentiable (for) and the local utility function is quadratic for all distributions. Assuming the conditions for such a mean-variance utility function, we further identify easily interpretable necessary and sufficient conditions on the preferences for each of the properties that the mean-variance utility function is commonly assumed to exhibit in applications of the mean-variance approach. In the light of the characterizations, it is also shown that the apparent inconsistency demonstrated by Borch in a mean-variance model can be ruled out by appropriate restrictions on the mean-variance utility function. © Oxford University Press 2010. All rights reserved.