We consider financial positions belonging to the Banach lattice of bounded measurable functions on a given measurable space. 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? Can the eligible asset be chosen in such a way that for every financial position the corresponding required capital is lower than if any other asset had been chosen? In contrast to most of the literature our discussion is not limited to convex or coherent acceptance sets and allows for eligible assets that are not necessarily bounded away from zero. This generality uncovers some unexpected phenomena and opens up the field for applications to acceptance sets based both on Value-at-Risk and on Tail Value-at-Risk .
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 .
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study capital requirements for
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the Banach lattice
spaces
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on a given measurable space. We discuss risk measures generated by
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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
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
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financial position? Can the
corresponding capital requirement dominates the capital requirement corresponding any other choice of the
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asset be chosen in such a way that for every financial position the corresponding required capital is lower than if any other asset had been chosen? In contrast to most of the literature our discussion is not limited to convex or coherent acceptance sets and allows for eligible assets that are not necessarily bounded away from zero. This generality uncovers some unexpected phenomena and opens up the field for applications to acceptance sets based both on Value-at-Risk and on Tail Value-at-Risk
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