Constrained Optimization for Portfolio Construction

Abstract

This article deals with the relationship between conventional shrinkage approaches to the construction of the covariance matrix for portfolio optimization and the various types of constraints available in modern numerical algorithms for solving optimization problems. In particular, we consider the use of quadratic constraints on each part of the total risk (variance) measure, such as the systematic or specific risk associated with a factor risk model. By placing constraints on each part of the risk (perhaps in conjunction with constraints on the total risk), solutions are obtained that differ from the conventional constrained mean–variance solutions. We consider the use of this approach in the light of recent work focusing on portfolio optimization with alpha (expected return) terms that are cross-sectionally correlated with the risk factors of the model used to estimate the covariance matrix. To illustrate the practical value of this approach, using a well-documented set of alphas, we set out the results of a 13-year simulation exercise over the Russell 3000 Growth U.S. equity universe. The results, which can be of intuitive interest to investors, demonstrate how constraints that have the effect of shrinkage on the covariance matrix associated with the spanned part of the alpha will result in different portfolio allocations.