We discuss finiteness (effectiveness), continuity (robustness) and optimality (efficiency) results for capital requirements , or risk measures, defined for financial positions belonging to an ordered topological vector space. Given a set of acceptable financial positions and a pre-specified traded asset ( the eligible asset ), the associated capital requirement for a financial position represents to the amount of capital that needs to be raised and invested in the eligible asset to make the position acceptable. Our abstract approach allows to provide results that are applicable to a whole range of spaces of financial positions commonly used in the literature. Moreover, it allows to unveil the key properties of acceptance sets and of eligible assetsdriving the effectiveness and the robustness of the associated capital requirements. In particular, if the underlying space is a Fr\'echet lattice, we provide new finiteness results for required capital, as well as a simplified proof of the Extended Namioka-Klee theorem for risk measures . As an application we present a comprehensive treatment of finiteness and continuity for capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability .
We discuss general capital requirements representing the minimum amount of capital that a financial institution needs to raise and invest in a pre-specified eligible, or reference, asset to ensure it is adequately capitalized. Financial positions are modeled as elements belonging to an ordered topological vector space. The payoff of the eligible asset is assumed to be an arbitrary non-zero positive element, thus allowing for a wide range of choices. In the context of function spaces, these general capital requirements cannot be transformed into cash-additive capital requirements by a simple change of numeraire unless the payoff of the eligible asset is bounded away from zero. This excludes the possibility of choosing a defaultable security as the eligible asset which, given the potential unavailability of risk-free assets, constitutes an important gap in the existing theory of capital requirements. This paper fills this gap and provides a detailed analysis of the interplay between acceptance sets and eligible assets. We provide a variety of finiteness and continuity results when the eligible asset has a payoff with "interior-like" qualities, paying particular attention to the case where the underlying space of positions is a Fr\'{e . As an application we provide a complete characterization of finiteness and L^p-continuity for quantile-based capital requirements, the most important types of capital requirements encountered in practice .
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finiteness (effectiveness), continuity (robustness) and optimality (efficiency) results for capital requirements , or risk measures, defined for financial positions
general capital requirements representing the minimum amount of capital that a financial institution needs to raise and invest in a pre-specified eligible, or reference, asset to ensure it is adequately capitalized. Financial positions are modeled as elements
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Given a set of acceptable financial positions and a pre-specified traded asset (
The payoff of
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), the associated capital requirement for a financial position represents to the amount of capital that needs to be raised and invested in the eligible asset to make the position acceptable. Our abstract approach allows to provide results that are applicable to a whole range of spaces of financial positions commonly used in the literature. Moreover, it allows to unveil the key properties of
is assumed to be an arbitrary non-zero positive element, thus allowing for a wide range of choices. In the context of function spaces, these general capital requirements cannot be transformed into cash-additive capital requirements by a simple change of numeraire unless the payoff of the eligible asset is bounded away from zero. This excludes the possibility of choosing a defaultable security as the eligible asset which, given the potential unavailability of risk-free assets, constitutes an important gap in the existing theory of capital requirements. This paper fills this gap and provides a detailed analysis of the interplay between
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of eligible assetsdriving the effectiveness and the robustness of the associated capital requirements. In particular, if the underlying space is a Fr\'echet lattice, we provide new finiteness results for required capital, as well as a simplified proof of the Extended Namioka-Klee theorem for risk measures
eligible assets. We provide a variety of finiteness and continuity results when the eligible asset has a payoff with "interior-like" qualities, paying particular attention to the case where the underlying space of positions is a Fr\'{e
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present a comprehensive treatment
provide a complete characterization
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continuity for capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability
L^p-continuity for quantile-based capital requirements, the most important types of capital requirements encountered in practice