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Convexity

Convexity is the curvature of the price-yield relationship, the second-order term that duration alone misses. Because a bond's price-yield curve bends upward, a straight-line duration estimate overstates the price fall when yields rise and understates the gain when yields fall. Adding the convexity term sharpens the estimate. For an option-free bond convexity is always positive, which works in the holder's favour.

Try it yourself

Duration & Convexity

Duration draws a straight tangent to the price-yield curve. Convexity adds the curvature the tangent misses, so it tracks the true price more closely as the yield moves.

$30$56$81$106$131$1570%4%8%12%16%20%Yield (%)Price ($)$100.00True priceDuration (tangent)Duration + convexity
Price
$100.00
Macaulay dur.
8.11 yr
Modified dur.
7.72 yr
Convexity
75.00
At Δy = +1.0% (new yield 6.0%)
True new price$92.64
Duration estimate$92.28error -0.36
Duration + convexity$92.65error +0.01
Adding the ½·C·Δy² term shrinks the error: convexity captures the curve the tangent misses, and because the price-yield curve bows toward the origin it always sits above the tangent.

Why it matters

Duration is the tangent line to the price-yield curve; convexity is how fast that line curves away from the true price. For small yield moves the line and the curve nearly coincide, so duration is enough. For large moves the curve sits above the line on both sides, so the real price is always better for the holder than duration predicts: a bit more gain when yields drop, a bit less pain when they rise. That favourable asymmetry is positive convexity.

Formulas

Convexity (annual compounding)
C=1P∑t=1nt(t+1) CFt(1+y)t+2C = \frac{1}{P}\sum_{t=1}^{n} \frac{t(t+1)\,CF_t}{(1+y)^{t+2}}
Measured in years squared. PP is the bond price, yy the yield, CFtCF_t the cash flow at time tt. Hull (2022), Ch. 4 (Interest Rates).
Second-order price change
ΔPP≈−Dmod Δy+12 C (Δy)2\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2
When convexity is positive, as it is for any option-free bond, the term 12C(Δy)2\tfrac{1}{2} C (\Delta y)^2 is positive for either direction of Δy\Delta y, so it lifts the estimate on both sides.

Worked examples

Scenario

3-year bond, face A$100, 5% annual coupon, YTM 5% (par); its modified duration is 2.72. Find its convexity and re-estimate a 100 bp yield rise.

Solution

Convexity C=1P∑t(t+1)CFt/(1+y)t+2=10.21C = \tfrac{1}{P}\sum t(t+1)CF_t/(1+y)^{t+2} = 10.21. Duration alone predicts −2.72%-2.72\%; the convexity term adds +12×10.21×(0.01)2=+0.05%+\tfrac{1}{2} \times 10.21 \times (0.01)^2 = +0.05\%, giving −2.67%-2.67\%. The actual reprice is −2.67%-2.67\%, so the two-term estimate is almost exact while duration alone overstated the loss.

Scenario

Apply a ±200\pm 200 bp shock to the same bond. Show the convexity asymmetry.

Solution

Yields down 200 bp lift the price by +5.66%+5.66\%; yields up 200 bp cut it by only −5.25%-5.25\%. The gain beats the loss by about 0.4%. That asymmetry is the convexity term +12×10.21×(0.02)2=+0.20%+\tfrac{1}{2} \times 10.21 \times (0.02)^2 = +0.20\%, added on both sides. Positive convexity always favours the bondholder.

Common mistakes

  • ✗Convexity is a negligible detail. For small moves, yes, but for large yield shocks and for portfolio risk management it matters. Traders pay up for convexity (barbell versus bullet portfolios) precisely because of the favourable asymmetry.
  • ✗Once you know duration you can ignore convexity. Duration is a straight line; the real price-yield curve bends. Without convexity the estimate drifts off for any sizeable yield move.
  • ✗All bonds have positive convexity. Option-free bonds do. Bonds with embedded short options, such as callable bonds and mortgage-backed securities, can show negative convexity over some yield ranges, which hurts the holder.

Revision bullets

  • •Convexity is the curvature of the price-yield curve (second order)
  • •ΔP/P≈−Dmod Δy+12C(Δy)2\Delta P / P \approx -D_{\text{mod}}\,\Delta y + \tfrac{1}{2} C (\Delta y)^2
  • •For positive convexity, the term lifts the estimate either way (it uses (Δy)2(\Delta y)^2)
  • •Duration overstates losses and understates gains; convexity corrects both
  • •Option-free bonds have positive convexity, favouring the holder
  • •Callable bonds and MBS can have negative convexity

Quick check

For an option-free bond, adding the convexity term to a duration estimate of the price change always

Which security is most likely to display negative convexity?

Connected topics

Yield CurvesBond PriceDuration

Sources

  1. Hull (2022), Ch. 4
    Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
    Introduces convexity as the second-order correction to the duration price estimate.
  2. Fabozzi (2021), Ch. 5
    Fabozzi, Frank J. Bond Markets, Analysis, and Strategies. 10th ed. MIT Press, 2021. ISBN 978-0-262-04627-3.
    Covers convexity measurement, positive versus negative convexity, and callable-bond behaviour.
  3. Tuckman and Serrat (2022)
    Tuckman, Bruce and Angel Serrat. Fixed Income Securities: Tools for Today's Markets. 4th ed. Wiley, 2022. ISBN 978-1-119-83555-0.
    Treats convexity, its hedging implications, and the negative convexity of mortgage-backed securities.
How to cite this page
Dr. Phil's Quant Lab. (2026). Convexity. Derivatives Atlas. https://phucnguyenvan.com/concept/convexity
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