General Acceptance Sets, Risk Measures and OptimalCapital Injections

We consider nancial positions belonging to the Banach lattice of bounded measurable functionson a given measurable space. We discuss risk measures generated by general acceptance sets allowingfor capital injections to be invested in a pre-specied eligible asset with an everywhere positive payo.Risk measures play a key role when dening required capital for a nancial institution. We addressthe three critical questions: when is required capital a well-dened number for any nancial position?When is required capital a continuous function of the nancial position? Can the eligible asset bechosen in such a way that for every nancial position the corresponding required capital is lower thanif any other asset had been chosen? In contrast to most of the literature our discussion is not limitedto convex or coherent acceptance sets and allows for eligible assets that are not necessarily boundedaway from zero. This generality uncovers some unexpected phenomena and opens up the eld forapplications to acceptance sets based both on Value-at-Risk and on Tail Value-at-Risk.[...]