Coherent portfolio performance ratios

Omega Compatibility: A Meta-analysis

“…Furthermore, several generalizations of the Bernardo and Ledoit (2000) Gain-Loss ratio have been proposed in the literature (e.g. Keating and Shadwick (2002a,b); Kazemi et al (2004);; Zakamouline and Koekebakker (2009); Zakamouline (2014); Kroll et al (2021)), and each one has its own peculiarity and requires careful study in terms of coherence and compatibility with the Stochastic Dominance criteria (see Guo et al (2017Guo et al ( , 2019, and Kroll et al (2021) on coherent performance measures). Finally, it might be interesting to conduct a similar study of the advantages / disadvantages of such generalized performance measures.…”

“…As an example, Caporin et al (2018) show that we can find cases where there is clear dominance (SSD), but where the Omega for the two investments are strictly equal. An incompatibility between SSD and a risk measure, however, could not have been a great deal here since first, as recalled by Kroll et al (2021), FSD and SSD are rarely observed in real financial markets, and second, other performance measures, such as the Sharpe ratio, do suffer from the same problem. Nevertheless, this is important when using the Omega ratio, since writing that "Omega is compatible with SSD" may induce a misunderstanding and hide the fact that Omega may neglect the risk too much, might rely a lot on expected performance and, finally, may lead to computational economic difficulties and irrational decisions.…”

“…The interest in this field of research began with the seminal contribution of Sharpe (1966). An increasing number of studies appeared in the following decades (see the survey by Cogneau and Hübner (2009a,b), and Caporin et al (2014)) and still appear from time to time (see Kroll et al (2021); Kerstens et al (2022)).…”

“…This is shown, for example, by the following simple illustration, given by Hodges (1998): When we compare an investment A with potential outcomes such as {−1%; 1%; 5%} with probabilities {0.1; 0.45; 0.45}, to an investment B with potential outcomes of {−1%; 1%; 3%} with (the same) probabilities {0.1; .45; .45}, then the Sharpe ratio of B is higher than those of A (that is, 1.30 versus 1.16); but, obviously, A dominates B according to the First-order Stochastic Dominance criterion. 1 Furthermore, Kroll et al (2021) show that the original Sharpe ratio and some of its variants, as well as the Omega ratio, are not coherent performance measures, as they do not satisfy either the monotonicity axiom or the Portfolio Riskless Translation Invariance (PRTI) axiom, which are essential requirements for the separation between the optimal composition of the risky component of the portfolio and the subjective choice of splitting the portfolio into its risk-free and risky components.…”

An omega portfolio model with dynamic return thresholds

“…Theoretically, investors can set the threshold as any value based on their preferences and expectation. Previous research integrates a fixed threshold, such as the rate of return of a risk-free asset, the average expected return, or simply a zero (Avouyi-Dovi et al, 2004;Gilli et al, 2006;Mausser et al, 2006;Gilli et al, 2008;Kane et al, 2009;Gilli et al, 2011;Kapsos et al, 2014aKapsos et al, , 2014bKroll et al, 2021). However, the performance of the Omega portfolio is questionable if decision-makers randomly determine a return threshold without considering the dynamics of asset pricing.…”