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 be able to do so at the lowest possible cost and to identify the corresponding optimal portfolios. We study the existence and uniqueness of such optimal portfolios as well as their stability with respect to perturbations of the underlying capital position. This behavior is naturally linked to the continuity properties of the set-valued map that associates to each capital position the corresponding set of optimal portfolios. Upper semicontinuity can be established under fairly natural assumptions. Lower semicontinuity is more elusive. While it is always satisfied in a polyhedral setting, it generally fails otherwise, even when the reference risk measure is convex. However, lower semicontinuity can often be established for portfolios that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problem.