Catastrophe bonds feature full collateralization of the underlying risk transfer, and thus abandon the insurance principle of economizing on collateral through diversification. We examine the theoretical foundations beneath this paradox, finding that fully collateralized instruments have important uses in a risk transfer market when insurers cannot contract completely over the division of assets in the event of insolvency, and, more generally, cannot write contracts with a full menu of state-contingent payments. In this environment, insureds have different levels of exposure to an insurer's default. When contracting constraints limit the insurer's ability to smooth out such differences, catastrophe bonds can be used to deliver coverage to those most exposed to default. We demonstrate how catastrophe bonds can improve welfare in this way by mitigating differences in default exposure, which arise with: (1) contractual incompleteness, and (2) heterogeneity among insureds, which undermines the efficiency of the mechanical pro rata division of assets that takes place in the event of insurer insolvency.