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How Convexity Estimates Yield-Driven Changes in Bond Duration

Article Quant Q&A · Author: Jerome Zerbib
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Summary

The note derives how bond convexity can help estimate the change in modified duration after a yield move. Starting with the definitions of duration and convexity as price sensitivities, it rewrites convexity in terms of duration and the rate at which duration changes with yield. This leads to an approximation for the duration change: the difference between squared duration and convexity, multiplied by the yield change.

A default-free annual coupon bond example compares the approximation with a direct recalculation after a yield increase; the two estimates are close. The derivation treats a portfolio as though it were one bond whose risk depends on a single yield. That assumption may not fit a portfolio exposed to multiple curve points or other yield factors, and the result is an approximation for a specified yield move rather than an exact repricing method.

Key ideas

  • Convexity is the normalized second derivative of bond price with respect to yield.
  • The yield sensitivity of duration can be expressed using squared duration minus convexity.
  • Multiplying that sensitivity by a yield change gives an approximate change in duration.
  • The portfolio-as-one-bond assumption limits the formula when several yield risk factors matter.

Tags

Fixed incomeDerivatives pricingRisk management
Full text
# Duration and Convexity


# Duration and Convexity












I am searching to estimate the evolution of my portfolio duration following a yield increase/decrease. Can i use the convexity? I mean IR delta x (- convexity) = Duration delta

Is it correct?

Thanks a lot !

## Answer by Sharad (score 2)

https://quant.stackexchange.com/a/59487

Yes, you can use convexity although the formula you have is not quite correct. Think of the portfolio as a single bond with price $P(y)$, where $y$ is the yield of the portfolio (we're making the assumption that the duration hedging of the portfolio is based on a single risk variable, the yield to maturity of the portfolio). Then, we have the usual definitions for modified duration $D$ and convexity $C$: $$ D = -\frac{1}{P}\frac{dP}{dy} $$ $$ C = \frac{1}{P}\frac{d^2P}{dy^2} $$ We can rewrite the expression for $C$ in terms of $D$: $$ \begin{align} C &= \frac{1}{P}\frac{d}{dy} \left[ \frac{dP}{dy} \right] \\ &= \frac{1}{P}\frac{d}{dy} \left[ -PD \right] \\ &= D^2 - \frac{dD}{dy} \end{align} $$ This suggests that for a given change in yield $\Delta y$, we can approximate the change in duration, $\Delta D$, by: $$ \Delta D \approx (D^2 - C)\Delta y $$

Example. Consider a default-free bond with a face of 100, a coupon of 6%, a yield of 5% and a term of 10 years. Assume annual compounding. Then, we can directly calculate $D = 7.52$ and $C = 72.17$. If yields increase by 25bps, then direct calculation shows the new duration $D' = 7.48$. On the other hand, using our formula above gives: $$ \Delta D \approx (7.52^2 - 72.17)*(0.25/100) = -0.04 $$

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.