An Empirical Study of Robust Mean-Variance Portfolios with Short Selling

Robust optimization is an effective tool for addressing the inevitable uncertainties in financial decision-making. In this paper, we incorporate robustness in the mean-variance framework for short-selling scenarios by considering the input parameters within an uncertainty set, constructed in two ways, viz., box (interval) and mixture (polytopic) uncertainty. Additionally, we use the $$l_1$$ l 1 norm constraint and impose bounds on the budget to induce sparsity and align with the institutional procedures for short selling. Both the uncertainty sets allow the robust problem to be formulated as a second-order cone programming problem, making it computationally tractable. We conduct a comprehensive empirical study of the proposed robust models over six global data sets, namely IBEX 35 (Spain), S&P Asia 50 (Asia), Hang Seng (Hong Kong), DAX 100 (Germany), CNX 100 (India) and FTSE 100 (UK). We compare these models with their respective nominal model and the traditional mean-variance model. Our findings demonstrate that the proposed model under the mixture uncertainty set consistently outperforms all other models across all data sets and over different market scenarios. Furthermore, both the robust models protect against extreme losses during the bearish phase and perform better than the nominal models. Sensitivity analysis conducted on the construction of these sets shows that altering the polytope yields minimal change, whereas expanding the interval of uncertainty diversifies the portfolio and further reduces risk.