Before revision:  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 .
After revision:  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 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 .