Maximum risk diversification for portfolio selection

We consider a new measure of diversification for a portfolio of risky assets, and we address the problem of finding portfolios with maximum diversification, possibly with the addition of return constraints. The diversification measure is based on a convexity gap between the risk of a convex combination of assets and the convex combination of their risks. We first provide some theoretical results on this diversification measure, and we establish connections with the Herfindahl index for risk. Then we formulate the portfolio diversification model for several risk measures. Finally, we provide some preliminary computational results.