On Consistency of the Omega Ratio with Stochastic Dominance Rules

Abstract

Omega ratios have been introduced in [1] as a performance measure to compare the performance of different investment opportunities. It does not have some of the drawbacks of the famous Sharpe ratio. In particular, it is consistent with first order stochastic dominance. Omega ratios also have an interesting relation to expectiles, which found increasing interest recently as risk measures. There is some confusion in the literature about consistency with respect to second order stochastic dominance. In this paper, we clarify this and extend it to a consistency result with respect to stochastic dominance of order 1 + γ recently introduced in [2] and generalizing the classical concepts of stochastic dominance of first and second order. Several examples illustrate the usefulness of this result. Finally, some consistency results for even more general stochastic dominance rules are shown, including the concept of -almost stochastic dominance introduced in [3].

Dispersive orderings induced by differences of inter risk measures

“…the ratio of the upside of X to the downside. Some basic properties of the Omega ratio of X are as follows (see [3] and [12]). For , denote and .…”

A State-of-the-Art Fund Performance Index: Higher-Order Omega and Its Consistency with Almost Stochastic Dominance

“…According to the dominance rules, investors will choose one risky asset over the other according to the ranking order. However, it is well-known that the Sharpe ratio is not consistent with the FSD (Klar and Müller 2018). The investors could make an irrational decision, i.e., choose the investment with a smaller return by maximizing the Sharpe ratio.…”

“…Their findings were supported by testing fund data. However, Klar and Müller (2018) claimed that there was a consistent relationship between Omega function and FSD, but no consistency with SSD. In addition, they also used expectiles (Bellini et al 2016) to prove that the Omega index is consistent with fractional stochastic order (1 + γ), which was proposed by Muller et al (2017).…”

Stochastic orders and measures of skewness and dispersion based on expectiles

“…Research which treats expectile-based measures and related stochastic orderings includes Bellini (2012); Bellini et al (2014Bellini et al ( , 2018a; Klar and Müller (2019); Klar (2020, 2021); Arab et al (2021). In particular, Klar (2021, 2020) introduced expectile-based measures of skewness which possess quite promising properties.…”