Duration and convexity explained for FRM candidates
Duration and convexity answer one question: if the yield changes, how far does the price move? They come up across FRM® Part I (valuation and risk models) and again in market risk, so it pays to know them well enough to compute them by hand. This guide builds them from the price formula. For where they sit in the syllabus, see how the FRM exam is structured, and the FRM landing page lists the other guides.
Starting point: price is a present value
A bond with cash flows CF(t) at times t, priced at yield y, is worth P = sum of CF(t) / (1 + y)^t. Price falls as yield rises, and the curve bends: it is steeper at low yields than at high ones.
Macaulay and modified duration
Macaulay duration is the present-value-weighted average time to receipt of the cash flows:
Macaulay duration = sum of t x PV(CF at t) / P
It is measured in years. A zero-coupon bond has a Macaulay duration equal to its maturity; a coupon bond has less, because some cash arrives earlier.
Modified duration converts that into a sensitivity:
Modified duration = Macaulay duration / (1 + y / m)
where m is the number of compounding periods a year (for annual compounding, divide by 1 + y). It is the proportional price fall for a small rise in yield: dP / P is approximately minus modified duration times dy.
DV01
DV01 (the price value of a basis point) is the money change in price for a one basis point move:
DV01 = modified duration x P x 0.0001
It is additive: the DV01 of a portfolio is the sum of the DV01s of its positions.
Convexity and the second-order estimate
Duration is the slope of the price-yield curve at one point. Convexity measures how the slope changes:
Convexity = (1 / P) x sum of t(t + 1) x CF(t) / (1 + y)t + 2
Adding it gives the two-term estimate:
dP / P is approximately minus modified duration x dy + 0.5 x convexity x dy^2
The second term is always positive for a plain bond, whichever way yields move, because dy is squared.
Why convexity helps the holder
Draw the curve and a straight tangent line (the duration estimate). The curve lies above the tangent on both sides. So when yields rise the price falls by less than duration predicts, and when yields fall it rises by more. Duration alone therefore understates the price after any move, and the gap grows with the square of the move. That is why convexity is valuable to a holder of a plain bond. Bonds with embedded options can behave differently: a callable bond can show negative convexity.
Worked example
An invented bond: face value 100, 5 years, 4% annual coupon, yield 5%. Computed in Python, with assertions that numerical differentiation of the price reproduces the duration and convexity.
| Measure | Value |
|---|---|
| Price | 95.67 |
| Macaulay duration | 4.62 years |
| Modified duration | 4.40 |
| Convexity | 24.47 |
| DV01 per 100 face | 0.0421 |
Now move the yield and compare the estimates with full repricing:
| Yield change | Full repricing | Duration only | Duration plus convexity |
|---|---|---|---|
| +1 point (to 6%) | 91.58 | 91.46 | 91.58 |
| -1 point (to 4%) | 100.00 | 99.88 | 100.00 |
| +2 points (to 7%) | 87.70 | 87.25 | 87.72 |
Check the +1 point row by hand: duration gives -4.40 x 0.01 = -4.40%, a fall of 4.21 to 91.46. The convexity term is 0.5 x 24.47 x 0.0001 = +0.12%, or 1.17 in price, which brings the estimate to 91.58. The duration-only estimate is too low in every row, as the curve-above-tangent argument says.
The usual mistakes
- Mixing modified and Macaulay duration. Price sensitivity uses modified duration. Using Macaulay overstates the move by the factor (1 + y / m).
- Forgetting the convexity term on large moves. For a few basis points it is negligible, but at 100 or 200 basis points it is the difference between a rough and a close answer.
- Sign errors. A yield rise lowers the price, so the duration term is negative. The convexity term is positive in both directions. Do not flip it when yields fall.
- Unit slips. Enter 1% as 0.01, not 1, and check how a source scales convexity.
- Treating duration as exact. It is a local estimate that assumes a parallel yield shift.
How the exam typically tests it
Typically you are asked for a price change from a given duration, convexity and yield move; for a DV01 from a price and duration; or to say which of two bonds has more convexity. Charta's practice questions for this topic follow the same pattern.
Common questions
Is duration the same as the maturity of a bond?
No. Maturity is when the last payment arrives. Macaulay duration is the weighted average time of all payments, so it is shorter than maturity for any coupon bond and equal to it only for a zero-coupon bond.
Which duration do I use to estimate a price change?
Modified duration, because it already contains the 1 / (1 + y / m) adjustment. Macaulay duration is a time measure, not a sensitivity.
Is convexity always positive?
For a plain bond with fixed cash flows, yes. Bonds with embedded options, such as callable bonds or mortgage-backed securities, can show negative convexity over some yield ranges.
Does DV01 change as yields change?
Yes. It is a local measure, so it shifts as price, yield and time to maturity change.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
The exam and the syllabus can change. Check GARP's current details before you book.