  In a capital adequacy framework, risk measures are used to determine the
minimal amount of capital that a financial institution has to raise and invest
in a portfolio of pre-specified eligible assets in order to pass a given
capital adequacy test. From a capital efficiency perspective, it is important
to identify the set of portfolios of eligible assets that allow to pass the
test by raising the least amount of capital. We study the existence and
uniqueness of such optimal portfolios as well as their sensitivity to changes
in the underlying capital position. This naturally leads to investigating the
continuity properties of the set-valued map associating to each capital
position the corresponding set of optimal portfolios. We pay special attention
to lower semicontinuity, which is the key continuity property from a financial
perspective. This "stability" property is always satisfied if the test is based
on a polyhedral risk measure but it generally fails once we depart from
polyhedrality even when the reference risk measure is convex. However, lower
semicontinuity can be often achieved if one if one is willing to focuses on
portfolios that are close to being optimal. Besides capital adequacy, our
results have a variety of natural applications to pricing, hedging, and capital
allocation problems.
