The Pricing of Risky Debt When Interest Rates are Stochastic

We derive a closed form solution for the valuation of risky discount debt when interest rates are stochastic. We extend Merton's [1974] approach to risky discount debt valuation under constant interest rates to the stochastic interest rate environment of Vasicek [1977]. Business risk is modelled as a geometric Brownian motion that is correlated with interest rates. We derive and explore the relationship between the credit premium and the term premium of corporate debt. In a banking environment, the results can also be applied to analyze (a) the allocation of capital to banking activities of varying credit risk and interest rate risk, (b) the measurement of relative capital adequacy when compared to peer banks, and (c) the interest rate risk-minimizing funding strategy.