This paper provides a formal model of contingent convertible bonds (CCBs), a new instrument offering potential value as a component of corporate capital structures for all types of firms, as well as being considered for the reform of prudential bank regulation following the recent financial crisis. CCBs are debt instruments that automatically convert to equity if and when the issuing firm or bank reaches a specified level of financial distress. CCBs have the potential to avoid bank bailouts of the type that occurred during the subprime mortgage crisis when banks could not raise sufficient new capital and bank regulators feared the consequences if systemically important banks failed. While qualitative discussions of CCBs are available in the literature, this is the first paper to develop a formal model of their properties. The paper provides analytic propositions concerning CCB attributes and develops implications for structuring CCBs to maximize their general benefits for corporations and their specific benefits for prudential bank regulation.