Portfolio rankings with skewness and kurtosis

In this paper we discuss the issue of portfolio ranking for a rational risk averse investor with and without the option to buy a risk free asset. We find that in the former case the use of Sharpe, Omega, Sortino, and Kappa rankings are all justified although they follow from different definitions. We also find that for portfolios with Gaussian distributed returns these rankings, as well as the Stutzer ranking, are equivalent to each other. Finally we prove that without a risk free asset all the above rankings are incompatible with being a rational risk averse investor and a different ranking is required. We propose an exact analytical formula as well as an approximate formula for practical use.