Risk Functionals With Convex Level Sets
Forthcoming in Mathematical Finance
39 Pages Posted: 20 Dec 2018 Last revised: 31 Jul 2020
Date Written: December 5, 2018
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 multi-dimensional 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 multi-dimensional elicitability.
Keywords: convex level sets, quantiles, expected shortfall, elicitability, backtestability
JEL Classification: C44, G32
Suggested Citation: Suggested Citation