  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.
