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 .
We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of num\'{e eligible asset is a general defaultable bond. In this paper we fill this gap allowing for general eligible assets. We provide a variety of finiteness and continuity results for general risk measures, as well as dual representations for the convex case. We apply our results to risk measures based on Value-at-Risk and Tail Value-at-Risk on L^p spaces, as well as to shortfall risk measures based on utility functions on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. In important cases, we provide characterizations of cash subadditivity for general risk measures and show that, when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasi-convex and show that cash subadditivity is only compatible with continuous pricing rules .
R
general capital requirements
risk measures
D
that
R
eligible, or reference, asset
eligible asset
R
Financial positions are modeled as elements belonging to an ordered topological vector space. The payoff of the
Most of the literature has focused on cash-additive risk measures, for which the
R
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
a risk-free bond, on the grounds that the general case can be reduced to the
R
capital requirements by a simple change of numeraire unless the payoff of the
case by a change of num\'{e
R
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
a general defaultable bond. In this paper we fill this gap allowing for general
R
when the eligible asset has a payoff with "interior-like" qualities, paying particular
for general risk measures, as well as dual representations for the convex case. We apply our results to risk measures based on Value-at-Risk and Tail Value-at-Risk on L^p spaces, as well as to shortfall risk measures based on utility functions on Orlicz spaces. We pay special
R
case where the underlying space of positions is a Fr\'{e
property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. In important cases, we provide characterizations of cash subadditivity for general risk measures and show that, when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasi-convex and show that cash subadditivity is only compatible with continuous pricing rules