Before revision:  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.
After revision:  In particular, we are interested in the existence of optimal portfolios , i.e. portfolios 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.