Portfolio selection with marginal risk control

Abstract

Marginal risk that represents the risk contribution of an individual asset is an important criterion in portfolio selection and risk management. In the literature, however, the measure of marginal risk has been only employed in ex post analysis of a portfolio policy, and the control of marginal risk is achieved usually via position diversification that simply imposes upper bounds on the portfolio position without considering the effect of correlations of asset returns in risk diversification. We investigate in this paper a new optimal portfolio selection problem with direct (relative) marginal risk control in the mean-variance framework, accounting for the correlations of asset returns. The resulting optimization model, however, is a notorious non-convex quadratically constrained quadratic programming problem. By exploiting the special structure of the problems, we propose an efficient branch-and-bound solution method to achieve a global optimality in which convex quadratic relaxation subproblems with second-order cone constraints are formulated to generate a tight lower bound. Empirical study shows that the model with marginal risk control is a suitable analytical tool for active portfolio risk management and demonstrates several preferable features of this new model to the traditional mean-variance model in risk management. The method is tested and compared with the commercial global optimization solver BARON for portfolio optimization problems with up to hundreds of assets and tens of marginal risk constraints.