  We study capital requirements for bounded financial positions defined as the
minimum amount of capital to invest in a chosen eligible asset targeting a
pre-specified acceptability test. We allow for general acceptance sets and
general eligible assets, including defaultable bonds. Since the payoff of these
assets is not necessarily bounded away from zero the resulting risk measures
cannot be transformed into cash-additive risk measures by a change of
numeraire. However, extending the range of eligible assets is important
because, as exemplified by the recent financial crisis, assuming the existence
of default-free bonds may be unrealistic. We focus on finiteness and continuity
properties of these general risk measures. As an application, we discuss
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 there is no optimal choice of the eligible asset. Our
results and our examples show that a theory of capital requirements allowing
for general eligible assets is richer than the standard theory of cash-additive
risk measures.
