Worst-case analysis of Omega-VaR ratio optimization model

Abstract

The Omega ratio, a performance measure that separately considers upside and downside deviations from a fixed threshold, improves the Sharpe ratio by incorporating the higher-order moments. In this paper, we analyse the performance of a robust optimization model based on maximizing the worst-case of Omega ratio by taking its threshold point as the robust value of a defined percentile of the underlying loss distribution. To this aim, the threshold point is computed as the worst-case of the value-at-risk at a particular confidence level. We formulate robust model of the proposed strategy under two cases of uncertainty sets, the mixture distribution uncertainty and the box uncertainty. We show that, in the first case, the problem reduces to a second-order cone program (SOCP) and, in the second one, to a semi-definite program (SDP), hence tractable in both the cases. We conduct a comprehensive empirical investigation of the proposed models over six data sets across the globe, namely BSE 100 (India), FTSE 100 (UK), Hang Seng (Hong Kong), S&P Asia 50 (Asia), Dow Jones Industrial Average (USA), and IBEX (Spain). We compare our models with the three variants of the Omega ratio model, one its robust variant taking worst-case conditional value-at-risk as threshold point, and two of its nominal variants using value-at-risk and conditional value-at-risk as threshold points, respectively. We find that the proposed model under mixture distribution uncertainty exhibits a better performance over most of the data sets and scenarios than its CVaR-based robust counterpart. Under the box set, the proposed model performs similar or generates mixed results compared to its CVaR-based robust counterpart model. We also note that both the proposed robust models save investors against the risk of losses over the bearish phase of market in comparison to their nominal counterparts. Finally, on comparing the proposed model under mixture distribution uncertainty with the box uncertainty, the former model is found to be more suitable for optimistic investors, whereas the later strategy is more ideal for pessimistic investors.