Inferring Yield for a Defaultable Perpetual Coupon Bond

Summary

The document addresses how to express the yield of a defaultable perpetual coupon bond when the usual maturity-based yield formula does not apply. It starts with a bond value that depends on the issuer’s asset value relative to a default barrier, the coupon and interest rates, and the recovery rate at default. The proposed approach equates the bond’s value with the present value of an infinite stream of continuous coupon payments discounted at a constant yield.

Under that setup, the value is coupon divided by yield, so the implied continuously compounded yield is the bond price divided by the continuous coupon rate. In the special case of zero recovery at default, substituting the stated price formula produces a simplified expression involving the interest rate and the asset-to-barrier ratio. This is a yield convention based on coupon cash flows; the document does not derive a full default-adjusted discounting model or discuss how the inferred yield changes with risk assumptions.

Key ideas

• A perpetual bond has no maturity date, so a standard maturity-based yield expression is unsuitable.
• The proposed yield equates bond value to the present value of continuous coupons discounted indefinitely.
• Under this convention, continuously compounded yield equals bond price divided by the coupon rate.
• The document gives a simplified expression for the special case of zero recovery at default.

Tags

# Definition of continuously compounded yield for perpetual defaultable coupon bond


# Definition of continuously compounded yield for perpetual defaultable coupon bond












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 by Kermittfrog (score 2, accepted)

https://quant.stackexchange.com/a/73406

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:

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.