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Seminal paper:

SHADWICK, William F., and Con KEATING, 2002. A Universal Performance Measure. *The Journal of Performance Measurement*, **6**(3).

- SORTINO, Frank, Gary MILLER, and Joeseph MESSINA, Short Term Risk-adjusted Performance: a Style Based Analysis, 1997.

‘It would be cumbersome to try to identify this with an acronym, therefore we propose calling it the omega return, omega being at the end of the Greek alphabet.’

‘The excess return that was actually earned on a risk adjusted basis is the difference between the omega return the portfolio manager earned and the omega return for the style benchmark. We will refer to this as the omega excess (W excess)."

‘Nevertheless, the best results over fifteen years were produced by the top Omega excess strategy, and the second best results by the top quartile of Omega excess funds’

‘We have presented a methodology for calculating risk-adjusted returns that we believe can be easily understood by most investors. While the methodology for calculating the omega return is fairly rigorous, anyone could understand the end result:’ - KUAN, Bernardo B., Research on the Excess Omega Return, December 1998.

‘In an effort to apply style analysis to downside risk, Sortino, Miller, and Messina [1997] developed a performance measure called the omega excess return. The purpose of this research project was to conduct an empirical investigation to compare the omega excess return with other performance measures. The methodology consisted of collecting monthly returns on twenty-eight mutual funds between 1978 and 1996. The Pension Research Institute Style Evaluator program was used to make the necessary calculations. The study indicates that the omega excess return was superior to other performance measures in predicting future performance. Also, selecting funds on the basis of the omega excess return proved to be a superior strategy to holding all of the funds or always investing in the fund with the highest return in the previous year’ - FARINELLI, Simone and Luisa TIBILETTI, Sharpe Thinking with Asymmetrical Preferences, 2002

‘The Omega Index (see Cascon et al. (2002) and the Upside Potential Ratio (see Sortino (2000) follow as special cases of the index Phi."

‘A special case of symmetrical preference to small and large deviations from the benchmark: the Omega Index.’ - CHEN, Peng, Barry FELDMAN and Chandra GODA, Portfolios with Hedge Funds and Other Alternative Investments, 2002.

‘The inadequacy of traditional approaches has led to the introduction of new methods, including the Stutzer (2001) performance index and the Omega measure of Keating and Shadwick (2002a and 2002b).’

‘The Omega measure provides a more complete way to compare assets with different return distributions.’ - The Finance Development Centre, Omega: A New Tool for Financial Analysis, 2002.

‘The Omega function for a returns distribution is a new tool for financial analysis using all the information in the distribution. Comparing the Omega functions for two or more assets, over a range of returns, ranks their performance and risk profiles without estimating any moments. The evolution of a manager's Omega function over time provides a complete picture of performance and risk. Omega functions reveal information invisible to mean/variance measures and can laed to significant improvements in portfolio values.

The construction of the Omega function can be motivated by considering the quality of a bet on a return above a given level r, which we regard as a loss threshold.’ - KEATING, Con and William F. SHADWICK, An Introduction to Omega, 2002.

‘A measure, known as Omega, which employs all the information contained within the returns series was introduced in a recent paperi. It can be used to rank and evaluate portfolios unequivocally. All that is known about the risk and return of a portfolio is contained within this measure. With tongue in cheek, it might be considered a Sharper ratio, or the successor to Jensen’s alpha.’ - KEATING, Con and William F. SHADWICK, A Universal Performance Measure, 2002.

‘We present a new approach to analysing returns distributions, the Omega function, which may be used as a natural performance measure. Analysis based on Omega is in the spirit of the downside, lower partial moment and gain-loss literatures. The Omega function captures all of the higher moment information in the returns distribution and also incorporates sensitivity to return levels. We indicate how this may be applied across a broad range of problems in financial analysis and apply it to a range of hedge fund style or strategy indices.’ - BACMANN, J.-F. and S. SCHOLZ, Alternative performance measures for hedge funds,
*AIMA Journal*, June 2003.

‘The Omega measure was introduced by Keating and Shadwick (2002)*. The main advantage is that this measure incorporates all the moments of the return distribution, including skewness and kurtosis. Moreover, in contrast to the Sharpe ratio, ranking is always possible, whatever the threshold.’

** Kazemi, Schneeweis and Gupta (2003) show that Omega (L)= C(L) / P(L), where C(L) is essentially the price of a European call option written on the investment and P(L) is essentially the price of a European put option written on the investment. They also provide a Sharpe-Omega ratio = (expected return – threshold) / put option price.* - BACMANN, Jean-François and Stefan SCHOLZ, Alternative Performance Measures for Hedge Funds,
*AIMA Journal*, June 2003.

‘The Omega measure suggested by Keating and Shadwick (2002) incorporates all the moments of the distribution as it is a direct transformation of it. This measure splits the return universe into two sub-parts according to a threshold. The “good” returns are above this threshold and the “bad” returns below. Very simply put, the Omega measure is defined as the ratio of the gain with respect to the threshold and the loss with respect to the same threshold.

The Omega function is defined by varying the threshold.’

‘In fact, the evaluation of an investment with the Omega function should be considered for thresholds between 0% and the risk free rate. Intuitively, this type of threshold corresponds to the notion of capital protection already advocated.’

‘Besides incorporating all the moments, the Omega function has two interesting properties. Firstly, when the threshold is set to the mean of the distribution, the Omega measure is equal to one. Secondly, whatever the threshold is, all investments may be ranked. In the context of the Sharpe ratio, the ranking is almost impossible for negative ratios.’

‘That is why we advocate the use of new performance measures, namely the Omega measure and the Stutzer index. Moreover, these measures can be applied in order to generate a better asset allocation among hedge fund styles as recently shown by Bacmann and Pache (2003).’ - BACON, Carl, Monitoring and measuring performance and risk in the hedge fund world

"Omega returns a downside risk style adjusted return" - CASCON, A., C. KEATING and W. F. SHADWICK, The Omega Function, 2003.

‘In this paper we introduce a bijection between a class of univariate cumulative distribution functions and a class of monotone decreasing functions which we call Omega functions.’ - FARINELLI, Simone and Luisa TIBILETTI, Sharpe Thinking in asset ranking with a benchmark, 2003

‘The Omega Index (see Cascon et al. (2002)) and the Upside Potential Ratio (see Sortino (2000)) follow as special cases.’ - FAVRE-BULLE, Alexandre and Sébastien PACHE, The Omega Measure: Hedge Fund Portfolio Optimization, January, 2003.

‘It can be shown that investors care about all the moments of the distribution, which is of great importance when returns are not normally distributed. Omega is a new measure proposed by Keating and Shadwick (2002a, 2002b) that reflects all the statistical properties of the returns distribution, i.e. all the moments of the distribution are embodied in the measure. It requires no assumptions on the returns distribution or on the utility function of the investor.’

‘Omega is a function that takes into account all the moments of the returns distribution. Thus, it can be accommodated with any asset showing non-normally distributed returns, such as hedge funds, bonds or equity in illiquid markets. For the comparison of investment opportunities, the measure requires no assumption on the utility function. Moreover, even if the returns are normally distributed, omega can provide additional information to mean-variance analysis by incorporating the investor’s perception of loss and gain. Finally, omega presents a more accurate picture of the statistical properties of historical returns since it is directly computed from the distribution itself.’

‘However, we show that higher moments matter; as omega is the only measure embodying all the moments of the returns distribution, it produces markedly different results. Given that omega offers the most accurate definition of risk and reward, our results suggest that omega optimization provides enhanced capabilities for risk diversification and reward enhancement when returns are not normally distributed. If returns are normally distributed, omega may still contribute to the investment analysis by considering the specific threshold below which an investor considers a given return as a loss.’

‘The flexibility introduced in the analysis by the investor’s specific return level is a characteristic shared by both omega and the downside deviation measures." [...] "Thus, the mean-downside deviation framework fails to fully capture the investor’s preferences. We find that for identical return levels, omega optimization can provide markedly different results.’

‘Second, we assume that the return level of the omega function is exogenously defined, which does not exactly reflect the reality since the loss threshold is defined by the investor’s preferences.’ - KAZEMI, Hossein, Thomas SCHNEEWEIS and Raj GUPTA, Omega as a performance measure, June 2003.
- SORTINO, Frank A. and Bernardo KUAN, The Upside Potential Strategy: A Paradigm Shift in Performance Measurement,
*Senior Consultant*, Volume 6, No. 12, 2003.

‘The second step involves calculating the Omega excess return, which is a way to determine whether a manager outperformed a passive set of indexes. First, the manager’s style is replicated by a set of passive indexes called a style benchmark. Then the downside risk of the manager’s style benchmark is subtracted from the manager’s return, creating a risk-adjusted return. Similarly, a risk-adjusted return is calculated for the style benchmark. The difference between the two risk-adjusted returns is called the Omega excess return.’ - TILL, Hilary and David LEE, Risk and Performance Measurement For Alternative Investment, 2003.

Omega: ‘The ratio of the gain with respect to the threshold and the loss with respect to the same threshold.’ - Winton Capital Management, Assessing CTA Quality with the Omega Performance Measure, September 2003.

‘In this paper we use a recently introduced alternative performance measure called Omega (?) to assess Winton Capital Management’s performance in the context of the managed futures industry, using the track records of a number of top performing diversified CTAs.’

‘In this paper we used the new Omega performance measure to evaluate the results of the traditional Sharpe ratio rankings of CTAs. Our analyses have shown that radically different results are obtained when taking into account the non-normal features of CTA return distributions. This less idealised measure gives a much more accurate assessment of the risks involved in different trading programmes, and therefore offers a more reliable basis for comparison.’ - Winton Capital Management, Case Studies of CTA Assessment Using the Omega Performance Measure, October 2003.

‘In a recent paper we applied the new performance measure Omega to the assessment of CTA performance using a small sample of established managed futures advisors (Winton Capital Management 2003). We concluded that, owing to its superior handling of the higher moments of return distributions, Omega provides a superior return/risk measure to the more commonly used mean/variance metrics. In addition, the definition of a specific threshold value links this performance measure directly to the absolute demands of specific investor groups. In this paper, we follow up our initial assessment by focusing on specific case studies defined by the demands of pension funds and retail hedge fund products. In the final section of the paper we suggest an intuitive shortcut to Omega for similar return thresholds based on the Sortino ratio.’ - KAPLAN, Paul D. and James A. KNOWLES, Kappa: A Generalized Downside Risk-Adjusted Performance Measure, 2004.

‘The Sortino Ratio and the more recently developed Omega statistic are conceptually related “downside” risk-adjusted return measures, but appear distinct mathematically. We show that each of these measures is a special case of Kappa, a generalized risk-adjusted performance measure. A single parameter of Kappa determines whether the Sortino Ratio, Omega, or another risk-adjusted return measure is generated.’ - FREY, Robert J. On the Ω-Ratio