  Omega ratio, defined as the probability-weighted ratio of gains over losses
at a given level of expected return, has been advocated as a better performance
indicator compared to Sharpe and Sortino ratio as it depends on the full return
distribution and hence encapsulates all information about risk and return. We
compute Omega ratio for the normal distribution and show that under some
distribution symmetry assumptions, the Omega ratio is oversold as it does not
provide any additional information compared to Sharpe ratio. Indeed, for
returns that have elliptic distributions, we prove that the optimal portfolio
according to Omega ratio is the same as the optimal portfolio according to
Sharpe ratio. As elliptic distributions are a weak form of symmetric
distributions that generalized Gaussian distributions and encompass many fat
tail distributions, this reduces tremendously the potential interest for the
Omega ratio.
