When risk defies order: On the limits of fractional stochastic dominance

Abstract: Motivated by recent work on monotone additive statistics and questions regarding optimal risk sharing for return-based risk measures, we investigate the existence, structure, and applications of Meyer risk measures. Those are monetary risk measures consistent with fractional stochastic orders suggested by Meyer (1977a,b) as refinement of second-order stochastic dominance (SSD). These so-called $v$-SD orders are based on a threshold utility function $v$. The test utilities defining the associated order are those at least as risk averse in absolute terms as $v$. The generality of $v$ allows to subsume SSD and other examples from the literature. The structure of risk measures respecting the $v$-SD order is clarified by two types of representations. The existence of nontrivial examples is more subtle: for many choices of $v$ outside the exponential (CARA) class, they do not exist. Additional properties like convexity or positive homogeneity further restrict admissible examples, even within the CARA class. We present impossibility theorems that demonstrate a deeper link between the axiomatic structure of monetary risk measures and SSD than previously acknowledged. The study concludes with two applications: portfolio optimisation under a Meyer risk measure as objective, and risk assessment of financial time series data.