Corporate Bond Valuation and Hedging with Stochastic Interest Rates and Endogenous Bankruptcy

This paper analyzes corporate bond valuation and optimal call and default rules when interest rates and firm value are stochastic. It then uses the results to explain the dynamics of hedging. Bankruptcy rules are important determinants of corporate bond sensitivity to interest rates and firm value. Although endogenous and exogenous bankruptcy models can be calibrated to produce the same prices, they can have very different hedging implications. We show that empirical results on the relation between corporate spreads and Treasury rates provide evidence on duration and find that the endogenous model explains the empirical patterns better than typical exogenous models. Corporate bonds are standard investment instruments, yet the embedded options they contain are quite complex. Most corporate bonds are callable and call provisions interact with default risk. In any case, corporate bond investors face the problem of managing interest rate and credit risk simultaneously. This paper examines the valuation and risk management of callable defaultable bonds when both interest rates and firm value are stochastic and when the issuer follows optimal call and default rules. To our knowledge, this is the first model of couponbearing corporate debt that incorporates both stochastic interest rates and endogenous bankruptcy. Existing models either treat interest rates as constant or impose exogenous default rules. These assumptions can significantly impact bond pricing and hedging. Yield spreads can be sensitive to interest rate levels, volatility, and correlation with firm value. Spreads are also sensitive to assumptions about the bankruptcy process. Some exogenous bankruptcy specifications produce negative spreads. Even when they guarantee positive spreads, exogenous default models can have hedging implications that are very different from those of endogenous default models. Working with a general Markov interest rate process, we develop analytical results about the existence and shape of optimal call and default boundaries. Then we numerically study the dynamics of hedging, using the results on exercise boundaries to explain patterns in bond duration and sensitivity to firm value. Finally, we link duration to the slope coefficient in a regression of changes in yield spreads on changes in interest rates and find that the endogenous bankruptcy model seems to explain empirical patterns in the spread-rate relation better than typical exogenous bankruptcy models. To clarify the interaction between call provisions and default risk, we model the callable defaultable bond together with its pure callable and pure defaultable counterparts. We view each of the three bonds as a host bond minus a call option on that host bond. The call options differ only in their strike prices. The strike of the pure call is the provisional call price. The strike of pure default option is firm value. The strike of the option to call or default is the minimum of the two.