Geometric Characterization of Maximum Diversification Return Portfolio via Rao’s Quadratic Entropy

Abstract

Diversification return has been well studied in finance literature, mainly focusing on the various sources from which it may be generated. The maximization of diversification return, in its natural form, is often handed over to convex quadratic optimization for its solution. In this paper, we study the maximization problem from the perspective of Rao's quadratic entropy (RQE), which is closely related to the Euclidean distance matrix and hence has deep geometric implications. This new approach reveals a fundamental feature that the maximum diversification return portfolio (MDRP) admits a spherical embedding with the hypersphere having the least volume. This important characterization extends to the maximum volatility portfolio, the long-only MDRP, and the ridge-regularized MDRP. RQE serves as a unified formulation for diversification return related portfolios and generates new portfolios that are worth further investigation. As an application of this geometric characterization, we develop a computational formula for measuring the distance between a new asset and an existing portfolio that has the hyperspherical embedding. Numerical experiments demonstrate the developed theory.