Omega and Sharpe ratio

TL;DR

This paper analyzes the Omega ratio, showing that under symmetric elliptic distributions, it offers no advantage over the Sharpe ratio in portfolio optimization.

Contribution

It proves that for elliptic distributions, the Omega ratio and Sharpe ratio lead to the same optimal portfolio, reducing the Omega ratio's practical usefulness.

Findings

Omega ratio equals Sharpe ratio under elliptic distributions.

Omega ratio does not provide additional information beyond Sharpe ratio in symmetric cases.

The result limits the practical interest of Omega ratio for many distributions.

Abstract

Omega ratio, defined as the probability-weighted ratio of gains over losses at a given level of expected return, has been advocated as a better performance indicator compared to Sharpe and Sortino ratio as it depends on the full return distribution and hence encapsulates all information about risk and return. We compute Omega ratio for the normal distribution and show that under some distribution symmetry assumptions, the Omega ratio is oversold as it does not provide any additional information compared to Sharpe ratio. Indeed, for returns that have elliptic distributions, we prove that the optimal portfolio according to Omega ratio is the same as the optimal portfolio according to Sharpe ratio. As elliptic distributions are a weak form of symmetric distributions that generalized Gaussian distributions and encompass many fat tail distributions, this reduces tremendously the potential interest for the Omega ratio.