Coherent Diversification Measures in Portfolio Theory: An Axiomatic Foundation

Abstract

This paper provides an axiomatic foundation of the measurement of diversification in a one-period portfolio theory under the assumption that the investor has complete information about the joint distribution of asset returns. Four categories of portfolio diversification measures can be distinguished: the law of large numbers diversification measures, the correlation diversification measures, the market portfolio diversification measures and the risk contribution diversification measures. We offer the first step towards a rigorous theory of correlation diversification measures. We propose a set of nine desirable axioms for this class of diversification measures, and name the measures satisfying these axioms coherent diversification measures that we distinguish from the notion of coherent risk measures. We provide the decision-theoretic foundations of our axioms by studying their compatibility with investors’ preference for diversification in two important decision theories under risk: the expected utility theory and Yaari’s dual theory. We explore whether useful methods of measuring portfolio diversification satisfy our axioms. We also investigate whether or not our axioms have forms of representation.

Keywords: portfolio theory, portfolio diversification, preference for diversification, correlation diversification, expected utility theory, dual theory

JEL Classification: D81, G1, G11