Mean-Variance Framework and Diversification Objective: Theoretical and Empirical Implications

Abstract

We analyze the introduction of a diversification constraint into the portfolio optimization program. We show that such a constraint is equivalent to an unconstrained portfolio optimization program with a change of the sample covariance matrix by another matrix obtained as the sum of the sample one with a weighting of the identity matrix. Using a parameter uncertainty approach, we provide an optimal weighting parameter for the identity matrix, and then an optimal diversification constraint to set. We also give a theoretical explanation of the empirical evidence showing the superiority (resp. the poor performance) of the short sale constrained global minimum variance portfolio (resp. the short sale constrained minimum variance portfolio) in comparison to their unconstrained counterparts. It appears that the short sale constraint actually hurts the efficiency of the sample mean. We also justify the investment strategy consisting in a linear combination between the tangency, the global minimum variance and the equally weighted portfolios, and propose a family of optimization strategies which produce portfolios with realistic short and long positions, with a better performance than their short sale constrained counterparts. We show the relevance of such an approach by running an empirical study in the US market.

Keywords: Covariance Matrix, Diversification Constraint, Mean-Variance Portfolio, Parameter Uncertainty, Shrinkage

JEL Classification: G11, G12, C11