Definition of continuously compounded yield for perpetual defaultable coupon bond

Question

In continuous-time asset pricing, the price of a defaultable perpetual coupon bond is given by $$P(V) = \frac{c}{r}\left[ 1- \left(\frac{V}{V_b}\right)^{-\gamma}\right] + (1-\alpha)V_b \left(\frac{V}{V_b}\right)^{-\gamma}$$

where $c$ is the coupon rate, $r$ is the interest rate, $V$ is the underlying asset (distributed as a GBM), $V_b$ is the default barrier, and $(1-\alpha)$ is the recovery rate at default.

How do I compute the continuously compounded yield $r^d$ for this asset?

With maturity and no default risk, it is usually defined from the formula $P_t = e^{- r^d(T-t)}$, but as it is a defaultable perpetual bond this formula does not apply.

Answer 1

You could equate the value function with an infinite series of discounted cash flows, discounted at the yield. Assuming a continuous coupon rate and a continuous yield $r^d$:

$$ r^d:P(V) \stackrel{!}{=} c\int_0^{\infty}e^{-r^dt}\mathrm{d}t=\frac{c}{r^d}\Rightarrow r^d=\frac{P(V)}{c} $$

In your equation, if the recovery rate at default $(1-\alpha)$ is zero, you'd arrive at the handy result:

$$ r^d=\frac{P(V)}{c}=\frac{1}{r}\left[1-\left(\frac{V}{V_b}\right)^{-\gamma}\right] $$