Elicitability of Return Risk Measures

Abstract: Informally, a risk measure is said to be elicitable if there exists a suitable scoring function such that minimizing its expected value recovers the risk measure. In this paper, we analyze the elicitability properties of the class of return risk measures (i.e., normalized, monotone and positively homogeneous risk measures). First, we provide dual representation results for convex and geometrically convex return risk measures. Next, we establish new axiomatic characterizations of Orlicz premia (i.e., Luxemburg norms). More specifically, we prove, under different sets of conditions, that Orlicz premia naturally arise as the only elicitable return risk measures. Finally, we provide a general family of strictly consistent scoring functions for Orlicz premia, a myriad of specific examples and a mixture representation suitable for constructing Murphy diagrams.