We study the existence of portfolios of traded assets making a given financial institution pass some pre-specified (internal or external) regulatory test. In particular, we are interested in the existence of optimal portfolios , i.e. portfolios that allow to pass the test at the lowest cost, and in their sensitivity to changes in the underlying capital position. This naturally leads to investigate the continuity properties of the set-valued map associating to each capital position the corresponding set of optimal portfolios. We pay special attention to inner semicontinuity, which is the key continuity property from a financial perspective. This  property is always satisfied if the test is based on a polyhedral risk measure such as Expected Shortfall, but it generally fails , even in a convex world, if we depart from polyhedrality . In this case, the optimal portfolio map may even fail to admit a continuous selection. Our results have applications to capital adequacy, pricingand hedging, and capital allocation . In particular, we allow for regulatory tests designed to capture systemic risk .
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 .
R
We study the existence of portfolios of traded assets making a given financial institution pass some
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
R
(internal or external) regulatory test. In particular, we are interested in the existence of optimal portfolios , i.e. portfolios
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
R
at the lowest cost, and in
by raising the least amount of capital. We study the existence and uniqueness of such optimal portfolios as well as
R
investigate
investigating
R
inner
lower
A
"stability"
D
such as Expected Shortfall,
R
, even in a convex world, if
once
R
. In this case, the optimal portfolio map may even fail to admit a continuous selection. Our results have applications to capital adequacy, pricingand
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,
R
. In particular, we allow for regulatory tests designed to capture systemic risk
problems