# Elicitability of Return Risk Measures

## TL;DR

This paper investigates the elicitability of return risk measures, providing dual representations, axiomatic characterizations of Orlicz premia, and constructing scoring functions to evaluate their performance.

## Contribution

It introduces new axiomatic characterizations of Orlicz premia as the only elicitable return risk measures and develops scoring functions for their assessment.

## Key findings

- Orlicz premia are uniquely elicitable among return risk measures
- Dual representation results for convex and geometrically convex measures
- A family of scoring functions for evaluating Orlicz premia

## 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.