Duration and convexity explained: how far a bond price moves
We know that a bond's price falls when yields rise. The practical question is by how much. Duration answers it to a first approximation, and convexity corrects the answer. Together they are how fixed income investors measure and manage interest rate risk. If bond pricing itself is unfamiliar, read the bond pricing guide first.
Macaulay duration: the weighted time to payment
A bond pays its cash flows at different times. Macaulay duration is the average of those times, in years, with each one weighted by its share of the bond's present value. A bond that pays everything at the end, like a zero-coupon bond, has a Macaulay duration equal to its maturity. A bond that pays coupons along the way gets some of its value back sooner, so its duration is shorter than its maturity.
Three rules follow, and the exam asks about all of them:
- A higher coupon means shorter duration, because more of the value arrives early.
- A longer maturity means longer duration.
- A lower yield means longer duration, because distant cash flows are discounted less heavily.
Modified duration: the sensitivity
Macaulay duration is a time. The number you use to estimate a price change is modified duration, which divides it by one plus the yield per period:
Modified duration = Macaulay duration ÷ (1 + y)
where y is the yield per coupon period. The estimated percentage price change for a small change in yield is:
ΔPrice ÷ Price ≈ −modified duration × Δyield
Consider a 10-year bond with a 4% coupon paid semiannually, priced to yield 5%. Its price is 92.21, its Macaulay duration is 8.26 years and its modified duration is 8.05. If yields rise by one percentage point, duration alone estimates a fall of 8.05%.
The actual repricing gives a fall of 7.68%. Duration overstated the loss. Why?
Why duration is only an estimate
Duration treats the relationship between price and yield as a straight line. The true relationship is a curve that bends upward, convex to the origin. A straight line drawn tangent to a curve that bends away from it will always be below the curve, so duration alone underestimates the price after any change in yield, up or down:
- When yields rise, the real price falls by less than duration predicts.
- When yields fall, the real price rises by more than duration predicts.
Convexity measures that bend, and adding it corrects the estimate:
ΔPrice ÷ Price ≈ −modified duration × Δy + ½ × convexity × (Δy)2
For the same bond, convexity is about 77. A one-point rise in yield gives −8.05% + ½ × 77.1 × 0.012 × 100 = −7.67%. That is within a hair of the exact −7.68%. A one-point fall gives +8.44%, against an exact +8.45%.
Duration for small moves, convexity for big ones
For a change of a quarter of a percentage point, the correction is tiny: duration alone estimates −2.01% against an exact −1.99%. The bigger the move, the more convexity matters. You can see this with the bond calculator: change the yield shift and compare the three estimates.
Duration compared across bonds
Because duration measures sensitivity, it lets you compare the interest rate risk of bonds with different coupons and maturities. A portfolio's duration is the weighted average of the durations of its bonds. A manager who expects rates to fall lengthens duration to gain more; one who expects them to rise shortens it to lose less.
Mistakes to avoid
- Using Macaulay duration in the price formula. The estimate uses modified duration. Macaulay is in years; modified is the percentage sensitivity.
- Forgetting the period. For a semiannual bond the yield in the modified duration formula is the half-year yield, and the duration itself is converted to years.
- Applying convexity with the wrong sign. The convexity term is always added: it increases the price whichever way yields move.
- Treating duration as exact. It is a first-order estimate, good for small changes in yield.
- Assuming all convexity is positive. A callable bond can show negative convexity when yields fall far enough that it is likely to be called.
Where this is tested
Duration and convexity sit in the fixed income topic, and they come back in portfolio management when a bond portfolio is matched to a liability. The outcomes below set out the calculations you are expected to do.