Robust and Sparse Portfolio Selection: Quantitative Insights and Efficient Algorithms

Abstract: We extend the classical mean-variance (MV) framework and propose a robust and sparse portfolio selection model incorporating an ellipsoidal uncertainty set to reduce the impact of estimation errors and fixed transaction costs to penalize over-diversification. In the literature, the MV model under fixed transaction costs is referred to as the sparse or cardinality-constrained MV optimization, which is a mixed integer problem and is challenging to solve when the number of assets is large. We develop an efficient semismooth Newton-based proximal difference-of-convex algorithm to solve the proposed model and prove its convergence to at least a local minimizer with a locally linear convergence rate. We explore properties of the robust and sparse portfolio both analytically and numerically. In particular, we show that the MV optimization is indeed a robust procedure as long as an investor makes the proper choice on the risk-aversion coefficient. We contribute to the literature by proving that there is a one-to-one correspondence between the risk-aversion coefficient and the level of robustness. Moreover, we characterize how the number of traded assets changes with respect to the interaction between the level of uncertainty on model parameters and the magnitude of transaction cost.