Pricing a Credit-Spread Bond with Flat Rates and Default Intensity

Summary

The note derives a simplified bond valuation under a flat risk-free rate and a constant default intensity. It links the credit spread to default intensity through the assumed recovery rate: intensity equals spread divided by one minus recovery. The dirty price is then formed from discounted coupon payments and principal, adjusted for survival, plus the expected recovery payment integrated over possible default times.

This gives a framework for revaluing a bond when rates and credit spreads change, provided the recovery assumption is specified. The setup is deliberately restrictive: it assumes flat rates and intensity, does not calibrate recovery or market price, and omits richer features such as changing credit quality or term structures. The formula describes a model-based approximation, not a universal relationship between spread and default risk.

Key ideas

• Under flat-rate and constant-intensity assumptions, credit spread is linked to default intensity through recovery.
• Coupon and principal payments are discounted by both the interest rate and default intensity.
• Expected recovery contributes an additional value based on default timing.
• The valuation depends on simplifying assumptions and requires a recovery-rate input.
• The model does not capture term structures or other complex credit dynamics.

Tags

# How to value a bond with a credit spread


# How to value a bond with a credit spread












Suppose I have a bond, with a face value of 95, a coupon of 2%, and a maturity of 50 years. Suppose the discount curve is flat at 2%, and there is no credit spread.

I am trying to calculate what happens when the credit/ interest rates change.

For example, how can I revalue the bond if the credit spread increases by 80bps, and interest rates increase to 3%?

Ideally I'd like to understand the theory behind it, as well as a method for calculating this (doesn't have to be exact, an approximation is fine).

## Answer by Kurt G. (score 2)

https://quant.stackexchange.com/a/73986

In a world with flat discount curve at interest rate $r$ and flat default intensity $\lambda$ the bond recovery $R$ is related to the credit spread $C$ by $$ \lambda=\frac{C}{1-R}\,. $$ If that bond has no market price and you are happy with those simplistic assumptions the dirty bond price is \begin{align} P&=\sum_{i=1}^n e^{-rt_i-\lambda t_i}K+e^{-rt_n-\lambda t_n}N+\int_0^{t_n}R\lambda e^{-(r+\lambda)s}\,ds\\ &=\sum_{i=1}^n e^{-rt_i-\lambda t_i}K+e^{-rt_n-\lambda t_n}N+R\lambda\frac{1-e^{-(r+\lambda)t_n}}{r+\lambda}\,. \end{align} Here $N$ is the face value and $K$ the cash amount of the bond coupon paid at time $t_i\,.$

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