Asset Proportions in Optimal Portfolios

Abstract

The paper is concerned with conditions under which the proportion of a given asset in the optimal portfolio of a risk averse agent is at least as large as some given proportion. The paper provides a condition that is necessary and sufficient for such a result to hold. The analysis is then confined to portfolios in which the distributions of assets differ by either a first-degree stochastic dominance shift or by a mean-preserving shift. Examples are provided to show that under some conditions a risk averter may invest a smaller proportion of his wealth in the dominating asset than in the dominated asset. The paper then provides conditions that are necessary and sufficient for a risk averter to invest more in the dominating asset. The question of whether risk-averse agents should or should not choose diversified portfolios (as opposed to specialized ones) has been analysed quite extensively in the literature. Examples of such works are Samuelson (1967), Brumelle (1974), Hadar and Russell (1974), Russell and Seo (1979), Hadar and Seo (1980), and MacMinn (1984). What has not been examined at all (except for very special cases) is the question of the optimal proportions of the various assets in a diversified portfolio.' In this paper we present conditions which imply that the optimal amount invested in a particular asset is at least as large as some given proportion of the available fund. In the case of two-asset portfolios, one may especially want to know which of the two assets represents more than half the investible fund. Indeed, we find it convenient to start the analysis with the two-asset case. While the extension to n-asset protfolios is essentially straightfoward, it is presented in a subsequent section. Focusing first on the two-asset case will make it easier for the reader to see the main logical steps in the proofs of the theorems. Some of our analysis is facilitated by the use of a particular subset of risk averters. This subset has the property that if its members behave in a particular way, then all risk averters behave in that way. We refer to such a subset as a representative set of all risk averters.2 The usefulness of the representative set derives from the fact that its members have utility functions of a very simple form (consisting of two linear pieces), and its applicability to the problems on hand derives from the fact that any concave function can be approximated by a linear combination of members of the representative set. In the next section we present some formal results demonstrating the equivalence of unanimity among members of the representative set and unanimity among all risk averters in certain types of choice situations. These theorems are then used in Section 3 in which we present the results about optimal asset proportions.