Elicitable Risk Measures

A statistical functional is elicitable if it can be defined as the minimizer of a suitable expected scoring function (see Gneiting (2011), Ziegel (2013) and the references therein). With financial applications in view, we suggest a slightly more restrictive definition than Gneiting (2011), and we derive several necessary conditions. For monetary risk measures, we show that elicitability leads to a subclass of the shortfall risk measures introduced in Follmer and Schied (2002). In the coherent case, we show that the only elicitable risk measures are the expectiles. Further, we provide an alternative proof of the result in Ziegel (2013) that the only coherent comonotone elicitable risk measure is the expected loss