Robust minimum variance portfolio with L-infinity constraints

Portfolios selected based on the sample covariance estimates may not be stable or robust, particularly so in situations with a large number of assets. The l1 or l2 norm constrained portfolio optimization method has been used as a robust method to control the sparsity or to shrink the estimated weights of assets. In this paper, we propose to add an additional l∞ norm constraint or to add a pairwise l∞ norm constraint in the l1 norm constrained minimum-variance portfolio (MVP) problem. The l∞ constraint controls the largest absolute component of the weight vector and the pairwise l∞ constraint encourages retaining the cluster structure of highly correlated assets in MVP optimization. By simulation study and analysis of empirical data, we find that the proposed portfolios often have better out-of-sample performance in terms of Sharpe ratios, variances and turn-overs than existing popular portfolio strategies including the l1 norm constrained MVP, l2 norm constrained MVP and the 1/N portfolio. In addition, we provide moment shrinkage interpretations of the new strategies and an upper bound of errors in the approximation of the empirical optimal portfolio risk based on the theoretical optimal portfolio risk.