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

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.

Keywords: Capital Adequacy Tests; Value-At-Risk; Surplus-Invariance; Conicity; Positive Homogeneity; Numeraire-Invariance

JEL Classification: D81, G18, G28, G32, K23