Meyer risk measures

Risk measures summarize the risk profile of financial positions in a single metric, which allows their comparison and supports investment decisions. When they respect common stochastic orders such as second-order stochastic dominance (SSD), they rest on sound conceptual grounds. Since SSD can be overly conservative and rigid, we study Meyer risk measures. Those are monetary risk measures that are consistent with stochastic orders induced by a threshold utility function $v$, where the defining test utilities are at least as risk averse as $v$. The flexibility in choosing $v$ allows to encompass a wide range of stochastic orders, including SSD. Our contribution is threefold. First, we show that Meyer risk measures whose threshold utility $v$ is CARA admit a lower envelope representation via adjusted risk measures, and analyse how they extend and compare to classical risk measures. Second, we show that monotone additive statistics, recently introduced in decision theory, are Meyer risk measures under mild conditions. Third, impossibility results for the situation beyond the CARA class establish that monetary risk measures compatible to the associated stochastic orders either fail to exist or cannot satisfy key properties simultaneously. Throughout, we highlight practical implications, for example to portfolio selection and risk management.