Risk functionals with convex level sets

Author ORCID Identifier

Yunran Wei:https://orcid.org/0000-0002-9616-9454

Publication Title

Mathematical Finance

ISSN

09601627

E-ISSN

43978

Document Type

Article

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.

First Page

1337

Last Page

1367

Publication Date

10-1-2020

DOI

10.1111/mafi.12270

Keywords

backtestability, convex level sets, elicitability, Expected Shortfall, quantiles

Department

Department of Statistics and Actuarial Science

Link to Full Text

Share

COinS