The Valuation of Corporate Securities with Finite Maturity Debt

We study the valuation of corporate securities, i.e., debt and equity, when the debt issued by the firm has a finite maturity date. The equity value has two components: the first corresponds to its value if default is restricted to the maturity date, the second is a premium for early default, prior to that date. Default is optimal when the underlying cash flow generated by the activities of the firm hits an endogenous default boundary, which solves a nonlinear Volterra integral equation. We derive a new valuation formula for debt, which depends on the local time of the cash flow process along the default boundary and is parameterized by the delta of the debt (its derivative with respect to cash flow) along that boundary. We show the debt delta satisfies a linear Volterra integral equation of the first kind. An algorithm for numerical implementation is provided. A sensitivity analysis is carried out in the context of numerical examples to illustrate the behaviors of equity, debt, the default boundary and the debt delta. We find the equity (debt) value can decrease (increase) with the maturity date, the default boundary can display non-monotone decreasing-increasing behavior with respect to time, and the debt delta can increase with cash flow in a neighborhood of the default boundary.