First Stochastic Dominance and Risk Measurement

Farinelli and Tibiletti (2008) propose a general risk-reward performance measurement ratio. Due to its simplicity and generality, the F-T ratios have gained much attentions. F-T ratios are ratios of average gains to average losses with respect to a target, each raised by some power index. Omega ratio and Upside Potential ratio are both special cases of F-T ratios. In this paper, we establish the consistency of F-T ratios with respect to first-order stochastic dominance. It is shown that second-order stochastic dominance is not consistent to the F-T ratios. This point is illustrated by a simple example