Abstract
We present two studies that contribute to the literature on data-driven and quantitative portfolio selection. In the ?rst study, we discuss the robustness and sparsity trade-o? in portfolio selection. A well-managed portfolio is crucial to an investor’s suc-cess. 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 proce-dures often su?er from being over-diversi?ed. 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 incorporat-ing an ellipsoidal uncertainty set and ?xed transaction costs to penalize an over-diversi?ed portfolio and promote sparsity. We explore several prop-erties 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 ?xed transaction cost induced portfolio. Moreover, we explicitly characterize how the number of traded assets changes by a sensi-tivity analysis. Our analytical results could help investors to maintain an appropriate trade-o? between robustness and sparsity and thus lead to a quantitative interpretation of the so-called diversi?cation paradox. In the second study, we study a robust sample average portfolio optimization problem under a piecewise linear risk function. It is formulated as a minimax problem and aims to obtain a robust portfolio based on the worst-case risk-return trade-o?. The key feature of our model is its capability in generalizing and extending a large class of one-period port-folio selection models in the literature. Our results can be summarized as follows. First, we demonstrate that there exists ubiquitous relationships between our model and sample average portfolio optimization (SAPO), distributionally robust portfolio optimization (DRPO), and regularization techniques. In particular, we prove that the di?erence of the risks com-puted by SAPO and DRPO is dominated by a regularization term over portfolio weights, and our model could ?ll the di?erence between SAPO and DRPO under certain ambiguity sets. Second, our model involves the maximization of a piece-linear convex function over a convex set, which is generally intractable. Therefore, we derive and work with a Lagrangian re-laxation upper bound by explicitly characterizing the impact of the param-eter space on the risk and return, and present a ?nancial modeling interpre-tation. Third, we extend our model by combining with LogDet covariance selection, and prove that the resulting portfolio mimics the James-Stein estimator on the mean of asset returns, thus demystifying the composition of the robust portfolio. Numerical tests are conducted to validate our the-oretical results and compare the in-sample and out-of-sample performance of various e?cient portfolios.