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Omega and Sharpe Ratio Omega and Sharpe ratio Eric Benhamou∗1,2, Beatrice Guez1, and Nicolas Paris1 1A.I Square Connect 2Lamsade PSL November 26, 2019 Abstract return level. In an enlightening research, Keating and Shadwick (2002b) and Keating and Shadwick (2002a) Omega ratio, defined as the probability-weighted ra- introduce the Omega ! ratio, and claimed that this tio of gains over losses at a given level of expected universal performance measure, designed to redress return, has been advocated as a better performance the information impoverishment of traditional mean- indicator compared to Sharpe and Sortino ratio as it variance analysis would address these concern. They depends on the full return distribution and hence en- emphasize that the Omega metric has the great ad- capsulates all information about risk and return. We vantage over traditional measures to encapsulate all compute Omega ratio for the normal distribution and information about risk and return as it depends on show that under some distribution symmetry assump- the full return distribution, as well as to avoid looking tions, the Omega ratio is oversold as it does not pro- at a specific level as this measure is provided for all vide any additional information compared to Sharpe level, hence entitling each investor to look at his/her ratio. Indeed, for returns that have elliptic distribu- risk appetite level. Strictly speaking, the omega ra- tions, we prove that the optimal portfolio according to tio is defined as the probability-weighted ratio of gains Omega ratio is the same as the optimal portfolio ac- over losses at a given level of expected return. In fi- cording to Sharpe ratio. As elliptic distributions are a nancial words, this ratio determines the quality of the weak form of symmetric distributions that generalized investment bet relative to the return threshold. As nice Gaussian distributions and encompass many fat tail as it may sound, we argue that this is oversold, as for distributions, this reduces tremendously the potential a large class of returns distributions, that is distribu- interest for the Omega ratio. tions that are elliptic, we prove in this paper that the optimal portfolio according to the Omega ratio is also keywords: Omega ratio, Sharpe ratio, normal distri- the optimal portfolio according to the Sharpe ratio. In bution, elliptical distribution. other words, optimizing weights for a given portfolio of assets in order to get the optimal Omega ratio is 1 Introduction equivalent to optimizing the portfolio weights to find the optimal Sharpe ratio. As elliptic distributions are Omega ratio has been introduced on the pledge that a weak form of symmetric distributions that general- the Sharpe ratio and many performance measurement ized Gaussian distributions and encompass many fat arXiv:1911.10254v1 [q-fin.RM] 15 Oct 2019 and risk ratios rely on two excessive simplifications. tail distributions, this kills the potential interest of the The first one states that a limited numbers of statisti- Omega ratio. cal characteristics can fully describe returns. Typically mean and variance, or first and second statistical mo- ments for Sharpe or information ratio, mean and down- 2 Related Work side standard deviation for Sortino ratio. The second one states that this performance ratios should have a Omega ratio have been studied in many papers pro- ducing a vast literature on it. We will review here the ∗[email protected]/dauphine.eu. The authors would like to thank Francois Bertrand and Stephane main papers. Winton Research (2003) is an empirical Mysona for fruitful conversations about the Omega ratio. research work that looked at the historical performance 1 of CTAs hedge funds and that put light on Omega ra- at-Risk at different tolerance levels and has analytical tio. Passow (2004) looked at the analytical property closed-form expressions for commonly used distribu- and tractabilty of the Omega ratio for Johnson distri- tion like Normal, Log-normal, Student-t and Gener- butions. Kaffel and Prigent (2010) investigated perfor- alized Pareto distributions. They showed that under mance measurement for financial structured products, certain condition, a subset of GlueVaR risk measures thanks to the so called Sharpe-Omega ratio that is an fulfils the property of tail-subadditivity Sharma et al. extension of the Omega ratio. Their originality was (2016) worked on the threshold to be used in portfolio to compute downside risk measure using put option optimization with Omega ratio. In order to maximize volatility instead of historical volatility. This allows the Omega ratio for the overall portfolio, one needs to them to take account of the asymmetry of the return consider a threshold point to compute the Omega ratio probability distribution. They determined that the op- as optimizing at all thresholds is not realistic. They de- timal combination of risk free, stock and call/put in- cided to use the conditional value-at-risk at an α confi- struments with respect to this performance measure, dence level CVaRα of the benchmark market. They ar- is not necessarily increasing and concave as opposed to gue that this α-value reflects the attitude of an investor traditional optimal Sharpe ratio portfolio for the same towards losses. Like in Kapsos et al. (2014), this for- instruments. Similarly, Gilli et al. (2011) studied port- mulation can be cast as a linear program for mixed and folios using the Omega function, looking at their empir- box uncertainty sets and a second order cone program ical performance, especially the effects of allowing short under ellipsoidal sets, and hence becomes tractable. positions. Their originality was to consider short po- They showed that the optimal portfolios resulting from sition which is traditionally ignored. They found that the Omega-CVaRα model exhibit a superior perfor- overall, short positions can improve risk-return charac- mance compared to the classical CVaRα model in the teristics of a portfolio but mitigated this findings with sense of higher expected returns, Sharpe ratios, modi- the constraints involved in short positions that often fied Sharpe ratios, and lesser losses in terms of VaRα carries additional constraints in terms of transactions and CVaRα values. Guo et al. (2016) worked on the costs and liquidity. property for one asset to have a higher Omega ratio Bertrand and luc Prigent (2011) analyzed the per- than a second one. They showed that second-order formance of two main portfolio insurance methods, the stochastic dominance (SSD) and/or second-order risk OBPI and CPPI strategies, using downside risk mea- seeking stochastic dominance (SRSD) alone for any two sures, thanks to Omega measure. They showed that prospects is not sufficient to imply that the Omega CPPI strategies perform better than OBPI. Kapsos ratio of one asset is always greater than that of the et al. (2014) looked at the maximum Omega ratio. other one. Indeed, they proved that the second-order They established that it can be computed as a linear stochastic dominance only implies higher Omega ratios program optimization problem. While the Omega ra- only for thresholds that are between the mean of the tio is theoretically a nonconvex function, Kapsos et al. smaller-return asset and the mean of the higher-return (2014) l showed that this can be reformulated as a con- asset. When considering first-order stochastic domi- vex optimization problem that can be solved thanks to nance, the restriction on the thresholds does not apply a linear program. This convex reformulation for Omega and first-order stochastic dominance implies preference ratio maximization can be seen as a direct analogy of the corresponding Omega ratios for any threshold. of the mean-variance framework for the Sharpe ratio Krezolek and Trzpiot (2017) introduced an extension maximization and paved the way for our work that of Omega ratio called GlueVaR risk measure and illus- will show the strong connection between Omega and trated this on metals market investments. GlueVaR Sharpe ratio. van Dyk et al. (2014) provided a nice risk measures combine Value-at-Risk and Tail Value- empirical research on the difference of ranking between at-Risk at different tolerance levels and have analytical Sharpe and Omega ratio. They compared the ranking closed-form expressions for the most frequently used of 184 international long/short (equity) hedge funds, distribution functions in many applications, i.e. Nor- over the period January 2000 to December 2011 using mal, Log-normal, Student-t and Generalized Pareto their monthly returns. They concluded that Omega distributions. Metel et al. (2017) is an illuminating ratio does indeed provide useful additional informa- paper as it is the first to notice the correspondence be- tion to investors compared to the one only provided tween Sharpe and Omega ratio under jointly elliptic by Sharpe ratio alone. Belles-Sampera et al. (2014) distributions of returns. Compared to our work, their generalized Omega ratio in a so called GlueVaR risk proof is more convoluted and does not emphasize the measure. It combines Value-at-Risk and Tail Value- important fact that elliptic distributions satisfy some 2 symmetry properties that validates the proof. Rambo 4 Elliptical distributions and Vuuren (2017) ranked fund returns and compared results obtained with those obtained from the Sharpe It is well known that a normal distribution is fully char- ratio over two periods: 2001 to 2007 and 2008 to 2013. acterized by its first and second moments. It is less well They found that Omega ratio provides far superior known or at least less emphasized that the normal dis- rankings. Guo et al. (2018) investigated whether there tribution also has a very nice property in terms of sym- is any Sharpe or Omega ratio rule that prove that metry with respect to its first and second moments. If one asset outperforms another one. They found that one plots iso-density curves in two or three dimension Sharpe ratio rule is not able to detect preference under for the multi variate normal, one would obtain ellip- general strong dominance cases.
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