Endogenous Default Boundaries and
Optimal Capital Structure
One limitation of the option-based pricing approach to modeling equity and corporate bonds is that it only gives us the prices of equity and bonds in a given capital structure. In essence, given a certain level of debt the models tell us how the firm’s asset value is split between debt and equity. By assumption, however, the total payoff to equity and debt does not depend on the level of debt and therefore the Modigliani-Miller proposition on capital structure holds. In practice, firms do worry about their capital structure and we need to develop models that explain this.
The most direct way of building models that describe optimal capital-structure choice is to introduce bankruptcy costs and tax advantage to issuing debt. The tax advantage favors higher debt levels, whereas bankruptcy costs push in the opposite direction, since these costs are lost to third parties (lawyers, accountants, courts, etc.) and therefore diminish the value left for equity and debt.
It is easy to build models which capture the trade-off between bankruptcy and tax shield advantages qualitatively. In this section we focus on a class of models which allows us to make more clear-cut quantitative predictions on the chosen level of leverage and on the effects on credit spread for corporate bonds.
The framework provides insights not only into the static choice of capital structure, but also gives us tools to analyze dynamic capital-structure choice, including important features such as maturity choice, callable debt, and strategic models of renegotiation. Including such dynamic features is important for our understanding of the term structure of credit spreads, which we will discuss in a later chapter.
The workhorse of this section is Brownian motion and various present-value functionals of Brownian motion which can be computed explicitly. All of the mathematical formulas that we need for this section are collected in Appendix B. In this chapter we write down every cash flow that has to be priced, but the reader will have to verify that the formulas are correct using the results in the appendix.