Risk functionals with convex level sets

Abstract

We analyze the “convex level sets” (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability, and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the multidimensional setting, with a special focus on signed Choquet integrals, a class of risk functionals that are generally not monotone or convex. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one‐dimensional signed Choquet integrals, and that of all two‐dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that under some continuity assumption, a comonotonic‐additive coherent risk measure is co‐elicitable with Value‐at‐Risk if and only if it is the corresponding Expected Shortfall. The new findings generalize several results in the recent literature, and partially answer an open question on the characterization of multidimensional elicitability.

Distortion Riskmetrics on General Spaces

“…Nevertheless, there are several additions to the existing literature. The similarity of this paper with Wang et al (2020) and the new results of this paper are summarized in Table 2.…”

“…constants (Liu, F. et al, 2020), and this includes all law-invariant convex risk measures in Föllmer and Schied (2016) and all law-invariant deviation measures in Grechuk et al (2009), as well as the classic mean variance and mean standard deviation principles in insurance. We already mentioned that characterization and various properties of distortion riskmetrics are studied on L ∞ by Wang et al (2020). As a follow-up of the previous work, the main purpose of this paper is to extend the domain of distortion riskmetrics to more general spaces, including L p -spaces for p ∈ [1, ∞).…”

“…Historically, such functionals, assuming monotonicity, were studied by Yaari (1987) in the economic literature and by Denneberg (1994) and Wang et al (1997) in the actuarial literature. More recently, for nonmonotone h, Wang et al (2020) called the functional in (1.1) a signed Choquet integral on the space L ∞ of bounded random variables. To be precise, a signed Choquet integral refers to the right-hand side of (1.1).…”

“…Notably, as a special example, the interquantile range (see Table 1) is not monotone, convex, or L p -continuous, but it is a popular measure of dispersion in statistics, and it belongs to the class of distortion riskmetrics. Finally, we extend distortion riskmetrics to a multidimensional setting, where the concepts of elicitability and convex level sets have been popular recently; see Fissler and Ziegel (2016), Frongillo and Kash (2018) and Wang and Wei (2020).…”

“…Most results in this paper are similar to those in the literature in terms of both statements and proofs, and our findings that these results hold on general spaces are not surprising. However, most of the results in previous literature on L ∞ , especially those in Wang et al (2020), may not be convenient to directly use in practice where most applications require results on more general spaces of random variables. As such, more general results are in need, and this paper can be viewed as a convenient toolkit for future studies and applications of distortion riskmetrics.…”