Lesson 6 of 8
Convexity
Duration is a straight line drawn on a curve. Convexity measures the curve, and it explains why duration understates every price move in both directions.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- The relationship between a bond's price and its yield is a curve, and duration is the straight line touching it at one point.
- Because the curve bends away from the line in both directions, duration understates gains and overstates losses.
- The error is negligible for small rate moves and becomes material past about two percentage points.
The duration table in the last lesson had one feature worth stopping on. Every estimate was too low: too low on the way down and too low on the way up. A one-sided error across four cases is not rounding. It is the shape of the relationship.
Price against yield for the five-year 5 percent bond
- Actual price
Show the numbers
| Actual price | |
|---|---|
| 1.78% | $1,152.75 |
| 2.78% | $1,102.31 |
| 3.78% | $1,054.65 |
| 4.78% | $1,009.58 |
| 5.78% | $966.95 |
| 6.78% | $926.58 |
| 7.78% | $888.36 |
Why the line sits below the curve
Price is one divided by a growing power of one plus the yield. Halving a divisor doubles a quotient; doubling a divisor only halves it. That asymmetry is inherent in division, and it makes the price curve bend upwards at both ends.
A tangent line matches the curve exactly at the point it touches and falls below it everywhere else. So duration alone always predicts a price lower than the real one, whichever way rates move. A bondholder gains more than duration says when rates fall and loses less than it says when they rise.
Measuring the bend
Convexity is the second-order term. For the five-year bond it is 24.05, and the correction it supplies is half the convexity multiplied by the square of the rate change.
Squaring the rate change is the whole story. At a one point move the square is 0.01 and the correction is small. At three points the square is 0.09, nine times larger, and the correction is no longer optional.
Where the bend gets large
Convexity grows with maturity and shrinks with coupon size, for the same reason duration does: both measure how far out the value sits. A thirty-year zero-coupon bond has a great deal of it. A two-year bond with a large coupon has very little.
Positive convexity is a favourable property for a holder, and the market knows that, so a bond with more of it tends to be priced with a slightly lower yield. There is no free asymmetry available here.
Where it reverses
Some bonds have negative convexity over part of their range. A callable bond is the clearest case: as rates fall, the issuer's right to redeem early caps how far the price can rise, so the curve flattens instead of bending upwards. The holder keeps the downside bend and loses the upside one. Recognising that shape matters more than computing the number.
Check yourself
Rates move and duration alone predicts the new price. In which direction is the estimate wrong?
- Too high when rates rise, too low when they fall
- Too low in both directions
- Too high in both directions
Show the answer
Answer: B.
The price curve bends away from the tangent line on both sides, so the straight-line estimate sits below the actual price whichever way rates move.The convexity correction is proportional to what?
- The rate change
- The square of the rate change
- The square root of the rate change
Show the answer
Answer: B.
It is half the convexity multiplied by the rate change squared, which is why a three point move produces nine times the correction of a one point move.Why does a callable bond have negative convexity over part of its range?
- Because its coupon is variable
- Because the issuer's right to redeem caps how far the price can rise as rates fall
- Because it has a shorter duration
Show the answer
Answer: B.
The call option belongs to the issuer. As rates fall and the bond would otherwise rise, redemption becomes likely, which flattens the upper part of the price curve.Sources
- Damodaran, Investment Valuation: the second-order price sensitivity to yield and the limits of a duration-only estimate.
- US Department of the Treasury, Daily Treasury Par Yield Curve Rates, session of 11 September 2026: the 4.78 percent yield at which the tangent is taken.
What this leaves out
- The bond is invented so that every row can be recomputed. Only the starting yield is a real published figure.
- Both duration and convexity assume the whole yield curve shifts in parallel. Real curves twist, and neither measure sees that.
- The rate moves shown here are arithmetic scenarios, not predictions that rates will move by any amount.
- Nothing here is advice about any security.