Portfolio Selection with Robust Estimates of Risk

Mean-variance portfolios constructed using the sample mean and covariance matrix of asset returns perform poorly due to estimation error. Moreover, it is commonly accepted that estimation error in the sample mean is much larger than in the sample covariance matrix. For this reason, recent research has focused on the minimum-variance portfolio, which relies only on estimates of the covariance matrix and thus usually performs better out-of-sample. But even minimum-variance portfolios are quite sensitive to estimation error and have unstable weights that fluctuate substantially over time. Jagannathan and Ma (2003) show that imposing shortselling constraints can help alleviate this difficulty. In this paper, we propose a class of portfolio policies that have better stability properties than the traditional minimum-variance portfolio. The proposed policies are based on certain robust estimators of risk and can be computed by solving a single nonlinear program, where estimation and portfolio optimization are performed in one step. We show analytically that the portfolio weights of the resulting policies are less sensitive to changes in the distributional assumptions than those of the traditional minimum-variance policy. Moreover, our numerical results on simulated and empirical data confirm that the proposed policies are more stable (even in the presence of constraints) and that they preserve (or slightly improve) the already relatively high out-of-sample Sharpe ratio of the minimum-variance policy.