Elicitable distortion risk measures: A concise proof

Abstract

Elicitability has recently been discussed as a desirable property for risk measures. Kou and Peng (2014) showed that an elicitable distortion risk measure is either a Value-at-Risk or the mean. We give a concise alternative proof of this result, and discuss the conflict between comonotonic additivity and elicitability.

Risk functionals with convex level sets

“…In the literature of risk measures, many results on the characterization of elicitable risk measures are obtained via characterizing the CxLS property in dimension one; see Weber (2006), Bellini and Bignozzi (2015), and Delbaen, Bellini, Bignozzi, and Ziegel (2016) for convex risk measures, Ziegel (2016) for coherent risk measures, Kou and Peng (2016) and Wang and Ziegel (2015) for distortion risk measures, Liu and Wang (2020) for tail risk measures, and Fissler, Hlavinová, and Rudloff (2019a, b) for set‐valued risk functionals. As far as we are aware of, studies dedicated to CxLS in the multidimensional setting are not found in the literature, although multidimensional elicitability has been a popular topic; see Lambert, Pennock, and Shoham (2008), Fissler and Ziegel (2016), Nolde and Ziegel (2017), and Acerbi and Szekely (2017) for multidimensional elicitability and their statistical implications.…”

Commonotonicity and time-consistency for Lebesgue-continuous monetary utility functions

“…Among the many papers on these utility functions, we could refer the reader to the cited papers and to e.g. Bellini et al [3], Bellini et al [4], Wang and Ziegel [19], Weber [20] and Ziegel [22].…”

Multivariate extensions of expectiles risk measures

“…It is also examined in a purely mathematical framework in [34]. The elicitability of risk measures is further analyzed in [3] and [35]. Expectiles, as risk measures, were the subject of the works presented in [16], [5], and [4].…”