Abstracts

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified reference asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the reference 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 numeraire. However, discounting does not work in all financially relevant situations, for instance, if the eligible asset is a general defaultable bond. We provide finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on Value-at-Risk and Expected Shortfall, as well as to shortfall risk measures. We pay special attention to the property of cash sub-additivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash sub-additivity and show that when the reference asset is a defaultable bond, cash sub-additivity is the exception rather than the rule. Finally, we consider the situation where the reference 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 sub-additivity is only compatible with continuous pricing rules.