Weighted Elastic Net Penalized Mean-Variance Portfolio Design and Computation

It is well known that the out-of-sample performance of Markowitz's mean-variance portfolio criterion can be negatively affected by estimation errors in the mean and covariance. In this paper we address the problem by regularizing the mean-variance objective function with a weighted elastic net penalty. We show that the use of this penalty can be motivated by a robust reformulation of the mean-variance criterion that directly accounts for parameter uncertainty. With this interpretation of the weighted elastic net penalty we derive data driven techniques for calibrating the weighting parameters based on the level of uncertainty in the parameter estimates. We test our proposed technique on US stock return data and our results show that the calibrated weighted elastic net penalized portfolio outperforms both the unpenalized portfolio and uniformly weighted elastic net penalized portfolio. This paper also introduces a novel Adaptive Support Split-Bregman approach which leverages the sparse nature of $\ell_{1}$ penalized portfolios to efficiently compute a solution of our proposed portfolio criterion. Numerical results show that this modification to the Split-Bregman algorithm results in significant improvements in computational speed compared with other techniques.