Omega-CVaR portfolio optimization and its worst case analysis

Abstract

This paper presents a novel framework for optimizing portfolios using distribution dependent thresholds in Omega ratio to control the downside risk. Portfolios resulting from the maximization of the classical Omega ratio simultaneously maximize the probability of superior performance compared to a threshold point set by an investor and minimize the probability of a worse performance compared to the same threshold. However, there is no mandatory rule or mechanism to choose this threshold point in the Omega ratio optimization model yet. In this paper, we redefine the Omega ratio for a loss averse investor by taking the distribution dependent threshold point as the conditional value-at-risk at an $$\alpha $$ confidence level ( $$ {\mathrm{CVaR}_{\alpha }}$$ ) of the benchmark market. The $$\alpha $$ -value reflects the attitude of an investor towards losses. We then embed this new Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model in a robust portfolio optimization framework and present its worst case analysis under three uncertainty sets. The robustness is introduced both in the Omega measure and the $$ {\mathrm{CVaR}_{\alpha }}$$ measure. We show that the worst case Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ robust optimization models are linear programs for mixed and box uncertainty sets and a second order cone program under ellipsoidal sets, and hence tractable in all three cases. We conduct a comprehensive empirical investigation of the classical $$ {\mathrm{CVaR}_{\alpha }}$$ model, the STARR $$_{\alpha }$$ model, the Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model, and robust Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model under a mixed uncertainty set for listed stocks of the S&P 500. The optimal portfolios resulting from the Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model exhibit a superior performance compared to the classical $$ {\mathrm{CVaR}_{\alpha }}$$ model in the sense of higher expected returns, Sharpe ratios, modified Sharpe ratios, and lesser losses in terms of $${\mathrm{VaR}_{\alpha }}$$ and $$ {\mathrm{CVaR}_{\alpha }}$$ values. The robust Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model under mixed uncertainty set is shown to dominate the Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model in terms of all performance measures. Furthermore, both the Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ and robust Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model under a mixed uncertainty set yield significantly lower risk compared to STARR $$_{\alpha }$$ model in terms of $$\mathrm{CVaR}_{\alpha }$$ and variance values.