Omega-CVaR Portfolio Optimization and Its Worst Case Analysis

Abstract

This paper presents a novel framework for optimizing portfolios using stochastic 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; until now it has been left to an investor to decide an appropriate threshold point. In this paper, we redefine the Omega ratio for a loss averse investor by taking the stochastic threshold point as the conditional value-at-risk at an alpha confidence level (CVaR) of the benchmark market. We call this model the Omega-CVaR model. The alpha-value reflects the attitude of an investor towards losses; higher values of indicate a higher loss averse attitude of an investor. We then embed the Omega-CVaR model in a robust portfolio optimization framework and present its worst case analysis under three uncertainty sets, namely mixed, box, and ellipsoidal, for discrete distribution under consideration. The robustness is introduced both in the Omega measure as well as the CVaR measure by respectively maximizing and minimizing their worst scenarios over the uncertainty sets. We show that the worst case Omega-CVaR robust optimization models are linear programs for mixed and box uncertainty sets and a second order cone program under the ellipsoidal set, and hence tractable in all three cases. We conduct a comprehensive empirical investigation to analyze the performance of the Omega-CVaR portfolios and robust Omega-CVaR portfolios under a mixed uncertainty set for listed stocks of the S&P 500 index as of June 2015. We perceive that the optimal portfolios resulting from the Omega-CVaR model exhibit a superior performance compared to the classical CVaR model in the sense of higher expected returns, Sharpe ratios, modified Sharpe ratios, and lesser losses in terms of VaR and CVaR values. Also, robust Omega-CVaR model under mixed uncertainty set is shown to dominate the Omega-CVaR model in terms of all performance measures. Furthermore, the classical CVaR model, the Omega-CVaR model and its robust Omega-CVaR model under mixed uncertainty sets have significantly less losses in terms of CVaR values compared to a naive 1/m portfolio strategy of constituents of the S&P 500 index.