Approximating Bond Spread Sensitivity with Duration
Summary
The document explains how to estimate the price effect of a credit-spread shock when only spread changes and bond duration or maturity are available. For a vanilla fixed-coupon bond without embedded options, it models cash flows as discounted using a risk-free rate plus a spread. Since both rates enter the discount factor in the same way, sensitivity to spread is comparable to sensitivity to the risk-free rate, allowing rate duration to serve as an approximate spread-sensitivity measure.
The proposed estimate uses duration multiplied by the spread change, based on a first-order Taylor approximation. The document cautions that this is approximate: rate and spread curves may use different interpolation methods or curve points. Its assumptions also limit applicability, especially for bonds with embedded optionality or more complex curve structures; it does not provide empirical validation or a full revaluation procedure.
Key ideas
- For a vanilla fixed-coupon bond without embedded options, spread and risk-free rates affect discounted cash flows in similar ways.
- Rate duration can approximate a bond’s sensitivity to a small spread change.
- The first-order estimate is duration multiplied by the spread change.
- Curve construction differences and embedded optionality can reduce the estimate’s accuracy.
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Full text
# Bond value as a function of spread change and duration/maturity
# Bond value as a function of spread change and duration/maturity
I am trying to calculate the change of value in a universe of bonds given a series of shocks to the credit spread of each bond. As a constraint, the initial dataset only contains the spread change for each individual bond and its duration/maturity.
Is there any simple way to approximate the change in value for each bond given this parameter set?
PS: I can also retrieve the initial bond price or other fields, but due to the size/heterogeneity of the portfolio this will substantially reduce the number of bonds with available data. Similarly, due to size/heterogeneity, I am looking for an approximate formula instead of a full revaluation.
## Answer by Kermittfrog (score 1, accepted)
https://quant.stackexchange.com/a/70811
You should be able to re-use the rate duration as a measure of spread sensitivity as well:
Let's assume a simple vanilla fixed-coupon-bearing bond w/o embedded optionalities that pays at coupon rate $c$ annually. Let's further assume a continuously compounded flat risk-free zero rate curve $r$ and a continuously compounded flat spread zero curve $s$. In this setting, assuming today $t_0=0$, today's present value of a risky cashflow at time $t$ equals
$$D(r,s,t)=e^{-(r+s)t}$$
The present value formula for the vanilla bond is a portfolio of discounted cash flows. As $r$ and $s$ are added in the discount factor formula, the derivatives (or sensitivity) of the term with respect to one or the other are identical.
Thus, in practice, for fixed-coupon bonds, you can re-use duration up to some interpolation issues for calculating the effect of a spread widening:
$$ PV(r,s+\epsilon)-PV(r,s)\approx\frac{\partial PV}{\partial s}\times\epsilon\approx\frac{\partial PV}{\partial r}\times\epsilon=\mathrm{Duration}\times\epsilon $$
where the first $\approx$ stems from the Taylor approximation and the second $\approx$ allows for the fact that the two derivatives are not exactly identical in practice, e.g. due to differences in curve interpolation mechanisms, different curve pillars or the like.
HTH?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.