Deriving Risky Duration for a Continuously Paying Defaultable Bond

Summary

The document explains the formula (1 − exp(−sT))/s by modeling a bond that pays continuously at a unit rate until maturity or default. With zero interest rates, no recovery, and a flat default intensity equal to the spread, the probability of surviving to time t is exp(−st). Integrating this survival probability from zero to maturity gives the stated present value.

It defines risky duration here as the derivative of that present value with respect to spread and gives the resulting expression, including its small-spread approximation. The formula therefore applies to this specific continuous-payment setup, rather than directly describing a conventional zero-coupon bond or a standard coupon schedule. Its assumptions are restrictive: the model omits recovery and discounting and equates spread with default intensity. The answer also frames the setup as a simplifying interpretation, so applying the formula to other bond structures requires adapting the cash flows and credit assumptions.

Key ideas

• The formula comes from integrating survival probabilities for a bond that pays continuously until default or maturity.
• A flat default intensity makes survival to time t equal to an exponential function of spread and time.
• The setup assumes zero interest rates, no recovery, and equality between spread and default intensity.
• Risky duration is defined as the derivative of the modeled present value with respect to spread.

Tags

# Risky duration formula for what kind of bond?


# Risky duration formula for what kind of bond?












In a documentation, there is the following formula for "zero interest rate risky duration" of a bond: $\frac{1-exp(-s \cdot T)}{s}$, where $s$ is spread, $T$ time until maturity.

What type of bond (zero-coupon, floating rate, etc.) is this formula valid for and how is it derived?

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/26218

Here is my understanding of your question, I might have oversimplified your problem, and made some hypothesis that were not yours.

Assuming we are talking about a bond paying $1$ each day until $T$ or default event if occuring before $T$.

Let's write the risky coupon bond payment in a continous time manner:

$$\int_0^T \mathbb{1}_{\tau>t} dt$$

with zero interest rate, NPV is given by (assuming flat intensity)

$$NPV = \int_{0}^{T} \mathbb{E}[\mathbb{1}_{\tau>t}]dt = \int_{0}^{T} e^{-st}dt=\frac{1-e^{-sT}}{s}$$

We focus on risky duration as derivative of the NPV with respect to spread.

Here, we are in a continuous time framework where we assumed no recovery in NPV, so intensity and spread are the same

$$\text{risky duration}=\frac{dNPV}{ds}=\frac{(1+sT)e^{-sT}-1}{s^2}=-\frac{T^2}{2}+O_{s\to 0}(sT^3)]$$

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