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, pricing and hedging, and capital allocation. In particular, we allow for regulatory tests designed to capture systemic risk.
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