Surplus-Invariant, Law-Invariant, and Conic Acceptance Sets Must be the Sets Induced by Value-at-Risk

TL;DR

This paper characterizes the form of capital adequacy acceptance sets that are surplus-invariant, law-invariant, and conic, showing they must be based on Value-at-Risk criteria, which has implications for regulatory testing.

Contribution

It proves that such acceptance sets are necessarily those defined by Value-at-Risk, extending the understanding of regulatory capital tests under specific invariance properties.

Findings

Acceptance sets must be based on Value-at-Risk.

The result holds under surplus-invariance, law-invariance, and conicity.

Numeraire-invariance leads to the same characterization.

Abstract

The regulator is interested in proposing a capital adequacy test by specifying an acceptance set for firms' capital positions at the end of a given period. This set needs to be surplus-invariant, i.e., not to depend on the surplus of firms' shareholders, because the test means to protect firms' liability holders. We prove that any surplus-invariant, law-invariant, and conic acceptance set must be the set of capital positions whose value-at-risk at a given level is less than zero. The result still holds if we replace conicity with numeraire-invariance, a property stipulating that whether a firm passes the test should not depend on the currency used to denominate its assets.