On the Robustness and Sparsity Trade-Off in Mean-Variance Portfolio Selection

A well-managed portfolio is crucial to an investor's success. Robustness against parameter uncertainty and low trading costs are two desired properties when constructing a portfolio. Robust optimization techniques have been applied to improve the stability of a portfolio under parameter uncertainty. However, portfolios generated from robust procedures often suffer from being over-diversified. Hence, an investor has to hold a multitude of assets and pay a large amount of transaction costs. In this paper, we extend the classical mean-variance framework by incorporating an ellipsoidal uncertainty set and fixed transaction costs which penalize an over-diversified portfolio and promote sparsity. We explore several properties of the optimal portfolio under this model. In particular, we show that it can be approximated by a linear combination of three benchmark portfolios, including the mean-variance portfolio, the minimum-variance portfolio, and a fixed transaction cost induced portfolio. Moreover, we explicitly characterize how the number of traded assets changes by a sensitivity analysis. Our analytical results could help investors to maintain an appropriate trade-off between robustness and sparsity and thus lead to a quantitative interpretation of the so-called diversification paradox