Before revision:  We discuss risk measures generated by general acceptance sets allowing for capital injections to be invested in a pre-specified eligible asset with an everywhere positive payoff. Risk measures play a key role when defining required capital for a financial institution. We address the three critical questions: when is required capital a well-defined number for any financial position? When is required capital a continuous function of the financial position?
After revision:  We discuss risk measures generated by general acceptance sets and general positive eligible (or "reference") assets, which include defaultable bonds, options, or limited liability assets. Since the payoff of these assets is not bounded away from zero the resulting capital requirements cannot be transformed into cash-invariant risk measures by a simple change of numeraire. However, extending the range of eligible assets is important because, as exemplified by the recent financial crisis, the existence of default-free securities may not be a realistic assumption to make. We study finiteness and continuity properties of capital requirements in this general context. We apply the results to capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability, the two most important acceptability criteria in practice. Finally, we prove that it is not possible to choose the eligible asset so that the financial position?