Modified Duration and Convexity for Estimating Bond Price Changes

Summary

The document clarifies how duration measures approximate a bond’s price response to a yield change. Modified duration is the appropriate first-order estimate for an infinitesimal yield move and works well for small changes. For a larger move, such as one percentage point, the linear estimate can become inaccurate because bond prices respond nonlinearly to yield changes.

Convexity is the second-order term in the Taylor expansion of price with respect to yield and improves the estimate when yield changes are larger. The question reports that ordinary duration seemed to approximate price gains after a yield drop better than modified duration did across several coupon and yield combinations. The response says this can occur because duration is somewhat larger, making the estimated gain larger, but offers no theoretical basis for choosing it in that way. The apparent fit may be coincidental; the discussion provides no broad empirical test beyond the questioner’s examples.

Key ideas

• Modified duration estimates bond price sensitivity to very small yield changes.
• The linear modified-duration estimate loses accuracy as the yield move grows.
• Convexity adds a second-order adjustment that can improve price-change estimates.
• Ordinary duration is not a theoretically justified substitute for modified duration in this calculation.
• An apparent match from using duration may be coincidental.

Tags

# duration and modified duration


# duration and modified duration












By modelling duration and modified duration in Excel, I found that modified duration approximates bond price change well when there is a 1% increase in yield, while duration is a good approximation when there is a 1% decrease in yield. I checked this with about 10-15 couples of coupon rates and YTM, and it seems it works always. Is this true? If yes, what is the reason behind it?

## Answer by Alex C (score 3)

https://quant.stackexchange.com/a/37613

Modified duration is the right concept to use to estimate change in price in response to an infinitesimal change in yield.

It works very well for a small change in yield (say a few basis points). However with a bigger yield change it gets less accurate, arguably with a 1% change in yield it is no longer satisfactory.

What to do? Modified duration is the first term in a Taylor series expansion. To increase accuracy you should use the second term also, known as Convexity.

Using Duration instead of Modified Duration may work to some extent as you say, but it has no sound theoretical justification. Duration is a number slightly larger than modified duration, so I can see that it makes the predicted price increase for a 1% yield drop a little bigger, but if the amount is just right that is more or less a coincidence. You are using the wrong method to reach an apparently good result.

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.