Some Negative Results on the Existence of Comparative Statics Results in Portfolio Theory

Abstract

The purpose of this paper is to prove that the separation property is a necessary condition as well as a sufficient condition for the generalization of economist K.J. Arow's results to the case of many risky securities. It has been shown that given more than one risky security and a utility function which does not possess the separation property, it is always possible to pick probability distributions for the returns of the risky securities so that the directions of change which Arrow established for the single risky security case are reversed; that is, for some probability distributions of security returns, the total amount invested in risky securities decreases as wealth increases, and for other probability distributions of security returns, the proportion of wealth invested in risky securities increases as wealth increases. The article shows that there always exist probability distributions of security returns such that the amount invested in every risky security decreases as wealth increases.