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Convexity

Convexity is the curvature correction to the straight-line price estimate that duration provides. The true relationship between a bond’s price and its yield is curved, bowing toward the origin, so duration alone, a tangent line, understates the price for large yield moves. Adding a convexity term captures this curvature and improves the estimate, raising the predicted price gain when yields fall and softening the predicted loss when yields rise. Because the price-yield curve is convex, the duration approximation always errs in the investor’s favour, and convexity is itself a desirable property that investors will pay for.

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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 draws a straight line tangent to a curve, and like any tangent it drifts away from the true curve as you move further along. The actual price-yield relationship curves, so for a big yield change duration alone misses the mark. The good news is that the curvature works for the bondholder. When yields fall, prices rise by more than duration predicts, and when yields rise, prices fall by less. Convexity measures that bend and adds it back, so the bigger the move, the more it matters.

Formulas

Price change with duration and convexity
ΔPP≈− Dmod Δy+12 C (Δy)2\frac{\Delta P}{P} \approx -\,D_{\mathrm{mod}}\,\Delta y + \tfrac{1}{2}\,\mathcal{C}\,(\Delta y)^{2}
The first term is the duration straight line. The second adds half the convexity C\mathcal{C} times the squared yield change, which is always positive and so corrects the duration estimate upward.

Worked examples

Scenario

A bond has modified duration Dmod=7D_{\mathrm{mod}}=7 and convexity C=90\mathcal{C}=90. Yields rise sharply by 200 basis points, that is Δy=0.02\Delta y = 0.02. Compare the duration-only estimate with the duration-plus-convexity estimate.

Solution

Duration alone gives −7×0.02=−0.14-7\times 0.02 = -0.14, a fall of 14 percent. The convexity term adds 12×90×(0.02)2=12×90×0.0004=0.018\tfrac{1}{2}\times 90\times (0.02)^2 = \tfrac{1}{2}\times 90\times 0.0004 = 0.018, that is 1.8 percent. The combined estimate is −0.14+0.018=−0.122-0.14 + 0.018 = -0.122, a fall of about 12.2 percent. Convexity shows the loss is smaller than duration alone implies.

NoteFor a large move the convexity term is meaningful. The duration-only figure overstated the loss because it ignored the upward bend of the price-yield curve.

Common mistakes

  • ✗Duration alone is enough for any yield change. Duration is accurate only for small moves. For large changes the curvature of the price-yield relationship makes the convexity term necessary.
  • ✗Convexity makes the duration estimate worse. Convexity refines the estimate. Because the curve bows toward the investor, adding convexity corrects duration’s error and brings the estimate closer to the true price.
  • ✗Convexity is a disadvantage to the bondholder. Positive convexity is desirable. It means prices rise more when yields fall and fall less when yields rise, so investors pay for higher convexity.
  • ✗Convexity matters equally for small and large yield moves. The convexity term scales with the square of the yield change, so it is negligible for tiny moves and grows in importance as the move gets larger.

Revision bullets

  • •Convexity is the curvature correction to duration’s straight line
  • •The true price-yield relationship is curved, not linear
  • •Duration alone understates the price for large yield moves
  • •The convexity term scales with the square of the yield change
  • •Positive convexity helps the investor and is a property worth paying for

Quick check

Why does the duration approximation understate a bond’s price after a large change in yields?

The convexity adjustment to a bond’s estimated price change is

Connected topics

Bond BasicsBond PricingBond YieldYield CurveDuration

Sources

  1. Brailsford, Heaney & Bilson (2015), Ch. on interest-rate risk
    Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
    Introduces convexity as the second-order correction to the duration price estimate.
  2. Bodie, Kane & Marcus (2021), Ch. 16
    Bodie, Z., Kane, A., & Marcus, A. J. Investments. 12th ed. McGraw-Hill Education, 2021.
    Explains convexity, why duration understates price for large moves, and its value to investors.
How to cite this page
Dr. Phil's Quant Lab. (2026). Convexity. Derivatives Atlas. https://phucnguyenvan.com/concept/im-convexity
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