Pricing Contingent Convertible Bond with Idiosyncratic Risk

Abstract

We consider the optimal capital structure of a firm, including equity, a straight bond and a contingent convertible bond (CCB), under an incomplete market. The cash flow of the firm is an observable arithmetic Brownian motion, which is correlated with the market portfolio return. We derive semi-closed-form solutions of the utility-based prices of the equity and the CCB, while explicit equilibrium prices of all the securities are provided. Numerical calculations show that for a more risk-averse equity holder or a higher idiosyncratic risk of the cash flow, the agent chooses a higher leverage and increases the issued amount of the straight bond. However, for a more risk-averse CCB holder, the agent chooses a lower leverage and decreases the issued amount of the CCB. We analyze the risk premium of the CCB, which includes as a part the premium of the idiosyncratic risk. If the firm’s cash flow is not too strongly negatively correlated with the market portfolio return, the CCB enhances risk-prevention incentive of the equity holder. The more risk-averse the agent is, the stronger the risk-prevention incentive will be. Our model finds that the CCB can not only decease bankruptcy risk but also can significantly increase the total firm’s value.