Groundlearn / Guides / Duration and convexity
Duration and convexity: how good is the approximation?
A ten-year bond repriced at four yield shocks, comparing the exact price change with the duration estimate and the duration-plus-convexity estimate.
The two formulas
Duration is a first-order estimate of how a bond's price changes when its yield changes. Convexity adds the second-order term, which corrects for the fact that the price-yield curve is bent, not straight.
Modified duration is Macaulay duration divided by (1 + y), where y is the yield per period. Convexity is the second derivative of price with respect to yield, divided by price. Both are measured with Δy as a decimal, so 100 bp is 0.01.
Worked example
A 10-year bond pays a 5% annual coupon and yields 5%, so it prices at par, 100.00. Its measures are:
- Macaulay duration: 8.1078 years
- Modified duration: 7.7217
- Convexity: 75.00
For a +100 bp shock the duration estimate is −7.7217 × 0.01 × 100 = −7.72. The convexity term adds ½ × 75.00 × 0.01² × 100 = +0.37, giving −7.35. Repricing the bond at 6% gives the exact figure. The table compares all four shocks (price change per 100 of par):
| Yield shock | Exact | Duration only | Duration + convexity |
|---|---|---|---|
| +100 bp | −7.36 | −7.72 | −7.35 |
| −100 bp | +8.11 | +7.72 | +8.10 |
| +200 bp | −14.05 | −15.44 | −13.94 |
| −200 bp | +17.06 | +15.44 | +16.94 |
What the table shows
- Duration alone is symmetric. It gives the same size of gain for a fall in yields as the loss for a rise. The real price moves are not symmetric.
- Duration overstates losses and understates gains. For a +200 bp shock it says −15.44 while the true change is −14.05. For −200 bp it says +15.44 while the true change is +17.06.
- Convexity closes most of the gap. Adding it moves both estimates close to the exact figures, and the remaining error grows with the size of the shock (third-order terms).
- Positive convexity is good for the holder. For the same duration, a more convex bond loses less when yields rise and gains more when they fall. That is why investors pay for it.
Common exam traps
- Mixing percent and decimal: 100 bp is 0.01 in the formula, so (Δy)² is 0.0001, not 1.
- Using Macaulay duration where the question gives modified duration, or the reverse. Modified is the one that links price to yield directly.
- Forgetting the ½ in front of the convexity term.
- Assuming callable bonds or mortgage-backed securities have positive convexity. Embedded options can make convexity negative, so the duration-plus-convexity estimate is built differently (effective duration and effective convexity, from repricing in a model).
- Treating duration as a time measure only. It is both a weighted-average time to cash flows and a price sensitivity.
Educational content only. This is not financial, investment, legal or regulatory advice. Figures are illustrations, not market data. Check the official source for your own exam or regulator before relying on any rule described here.
Practise it in Cornerstone
Snapshot cards and five-question sessions for CFA® and FRM® candidates, free on Android and offline. Cornerstone is an independent study aid.
About CornerstoneMore guides
- Value at Risk: the square-root-of-time ruleScale a one-day VaR to ten days with a worked example, and see where the rule breaks.
- Initial margin and the uncleared margin rules, explainedWho has to post it, how it is sized, and why the phase-in dates matter.
- Valuing an interest rate swap: a three-year worked exampleDiscount factors, the par swap rate and the value of a 4% swap, step by step.
- Expected shortfall versus VaR: what changes in the tailWhy regulators moved from 99% VaR to 97.5% expected shortfall, with normal and fat-tailed numbers.