We study capital requirements for financial positions belonging to spaces of bounded measurable functions . We allow for 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 corresponding capital requirement dominates the capital requirement corresponding any other choice of the eligible asset. Our examples and results on finiteness and continuity show that a theory of capital requirements allowing for general eligible assets is richer than that of cash-invariant capital requirements .
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 .
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financial positions belonging to spaces of bounded measurable functions
bounded financial positions defined as the minimum amount of capital to invest in a chosen eligible asset targeting a pre-specified acceptability test
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positive eligible (or "reference") assets, which include defaultable bonds, options, or limited liability assets
eligible assets, including defaultable bonds
A
necessarily
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capital requirements
risk measures
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cash-invariant
cash-additive
D
simple
A
assuming
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securities may not be a realistic assumption to make. We study
bonds may be unrealistic. We focus on
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capital requirements in this general context. We apply the results to
these general risk measures. As an application, we discuss
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it is not possible to choose the eligible asset so that the corresponding capital requirement dominates the capital requirement corresponding any other
there is no optimal
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examples and results on finiteness and continuity
results and our examples
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that of cash-invariant capital requirements
the standard theory of cash-additive risk measures