Consistent measures of risk

In this paper we compare overall as well as downside risk measures with respect to the criteria of first and second order stochastic dominance. While the downside risk measures, with the exception of tail conditional expectation, are consistent with first order stochastic dominance, overall risk measures are not, even if we restrict ourselves to two-parameter distributions. Most common risk measures preserve consistent preference orderings between prospects under the second order stochastic dominance rule, although for some of the downside risk measures such consistency holds deep enough in the tail only. In fact, the partial order induced by many risk measures is equivalent to sosd. Tail conditional expectation is not consistent with respect to second order stochastic dominance.