Chapter 9 - On ranking schemes and portfolio selection

Abstract

This chapter discusses different criteria for ranking portfolios, including the Sharpe ratio, the Sortino ratio, the kappa ratio, the Omega ratio, and the Stutzer index. It is proved that in a world where portfolio returns are Gaussian distributions, all of the above ranking systems are equivalent in the sense that although they produce different numbers, they will produce the same ranking order. The chapter also proves that all of the above ranking systems implicitly assume a non-natural utility function that attributes the same utility to any positive return (utility equals to +1) and to all negative returns (utility equals to -1). It proposes a more natural utility function from which a different ranking system for Gaussian portfolios is derived, which is not equivalent to the Sharpe ratio or any of the other rankings considered. Using the Berry–Esseen theorem, it is proved that the ranking system is applicable to portfolios with non-Gaussian returns under the condition that one plans to hold the portfolio for a sufficiently long time. The chapter describes how to apply the findings to Markowitz’ Modern Portfolio Theory.