Robust Portfolio Optimization via Linear Deviation Risk Measures

This paper examines the three deviation risk measures based portfolio optimization (PO) models within the robust framework. The three deviation risk measures are taken as Semi Mean Absolute Deviation (SemiMAD), Deviation Conditional Value-at-Risk (DCVaR), and Deviation MiniMax (DMM). We frame robustness in the PO models with respect to underlying probability distribution belonging to three most common uncertainty sets, namely, the mixed, box, and ellipsoidal. The robust counterparts of SemiMAD, DCVaR, and DMM PO models result in linear programs under mixed and box uncertainty whereas the models are second order cone program under ellipsoidal uncertainty. By testing the models on eight global data sets, we find that robust PO models outperform their respective nominal models in terms of risk and financial ratios, which contradict those studies emphasizing the under-performance of robust counterparts. Among the robust PO models, box uncertainty yields least risk as compared to mixed and ellipsoidal uncertainty, and hence box uncertainty is most suitable for risk-averse investors. Further findings reveal that robust model of DCVaR results in less risky portfolios than the other two robust models for the case of mixed and box uncertainty, while for the case of ellipsoidal, robust model of SemiMAD performs well in terms of risk. In terms of reward, a robust model of DMM generates highest mean return under the box and ellipsoidal uncertainty sets. Consequently, we setup an educated fusion between the three PO models with the three uncertainty sets that helps investor to take better investment decisions under the robust optimization framework.