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Omega Compatibility: A Meta-analysis

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Abstract

The Omega performance measure introduced by Keating and Shadwick (An introduction to Omega. AIMA Newsletter, 2002a; J Perform Meas 6(3):59-84, 2002b) is widely used in asset allocation and performance measurement. We contribute to the debates around this measure by focusing on its relation and compatibility with Second-order Stochastic Dominance, introducing two conditions of compatibility: the Non-Strict Dominance Compatibility and the Strict Dominance Compatibility conditions. We show that Omega is compatible with the First-order Stochastic Dominance criterion when using the Non-Strict Dominance Compatibility condition (as already shown), but also in the sense of the Strict Dominance Compatibility condition. We also prove again that Omega is compatible with Second-order Stochastic Dominance when using the Non-Strict Dominance Compatibility condition, but only under some conditions on the threshold used in the computation of the Omega measure, as usual. However, we finally also show that Omega is not compatible (i.e. incompatible) with Second-order Stochastic Dominance criterion when using the Strict Dominance Compatibility condition. We further provide a critical meta-analysis that separates good from approximate statements when comparing the views and results provided in many articles on the topic and point out that the use of Omega in asset selection and optimal asset allocation may entail real computational economics issues and may lead to unreasonable financial decisions. Finally, trying to avoid further disputes, ill-posed optimization procedures, and ultimately incorrect economic decisions in computational financial applications, we recall the main potential drawbacks of Omega that, in our opinion, mainly lies in its incompatibility with the Second-order Stochastic Dominance criterion under the Strict Dominance Compatibility condition.

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Notes

  1. See other examples in Hodges (1998), Weston and Copeland (1998); see also Homm and Pigorsch (2012a, 2012b) for further discussions.

  2. We should mention here that some authors, in another setting, also present some justifications of the use of Omega in the context of behavioural finance, following the results of Tversky and Kahneman (1992), applied to portfolio management (see Bernard & Ghossoub, 2010; Zakamouline, 2014; Zakamouline & Koekebakker, 2009). In particular, Zakamouline (2014) shows that the Kappa measures (which include the ratio Sharpe-Omega as a special case) correspond to performance measures based on a combination of piece-wise linear and power utility functions. We thank an anonymous referee for highlighting this point.

  3. This equality is true under the assumption that returns are characterized by a density with a finite mean, as in this case \(\lim _{r \rightarrow -\infty } rF(r)=0\), where F(.) is the Cumulative Distribution Function of the returns r. We thank the second anonymous referee for highlighting the fact that in the case of the Cauchy distribution, since the mean is not finite, the notation will not be appropriate.

  4. We thank Jean-Luc Prigent for discussions on the first point and an anonymous referee for highlighting the second point.

  5. Available upon request.

  6. We thank an anonymous referee for furthermore highlighting this point.

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Funding

Carole Bernard acknowledges funding from the FWO.

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Authors and Affiliations

  1. Department of Accounting, Law and Finance, Grenoble Ecole de Management, Grenoble, France

    Carole Bernard

  2. Department of Economics and Political Sciences, Vrije Universiteit Brussel (VUB), Brussel, Belgium

    Carole Bernard

  3. Department of Statistical Sciences, University of Padova, Padua, Italy

    Massimiliano Caporin

  4. emlyon business school (AIM QUANT Research Center) and Univ. La Reunion, Saint-Denis, France

    Bertrand Maillet

  5. Variances, Paris, France

    Bertrand Maillet

  6. School of Finance, Institute of Big Data Research, Southwestern University of Finance and Economics (SWUFE), Chengdu, China

    Xiang Zhang

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We thank Michele Costola for previous collaborations on this topic, and Jean-Luc Prigent for numerous discussions and positive suggestions when writing an early draft of this article. We also sincerely appreciate the constructive remarks by the participants to the FEM2021 conference (Paris, June 2021), as well as the Editor of ANOR in charge and the two anonymous referees for their fair comments and recommendations. Carole Bernard acknowledges FWO for financial support (FWOAL942). Massimiliano Caporin acknowledge financial support from the Italian Ministry of University and Research project PRIN2017 HiDEA: Advanced Econometrics for High Frequency Data (Grant Agreement No. 2017RSMPZZ). Resources linked to this article are available on: www.performance-metrics.eu. The usual disclaimer applies.

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Bernard, C., Caporin, M., Maillet, B. et al. Omega Compatibility: A Meta-analysis. Comput Econ 62, 493–526 (2023). https://doi.org/10.1007/s10614-022-10306-x

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