Bond Convexity

Convexity is the second-order correction to duration, capturing the curvature of the price-yield relationship that duration alone misses.

1. The idea

The bond price-yield relationship is convex (curves upward), not linear, so the duration-only approximation systematically underestimates price for both yield increases and decreases. Convexity $C$ captures this curvature as the second derivative of price with respect to yield, scaled by price.

Adding the convexity term produces a second-order Taylor approximation: $Δ P / P \approx - D_{m o d} Δ y + \frac{1}{2} C (Δ y)^{2}$ . Since the convexity term is always non-negative (quadratic), it improves the approximation in the same direction for both up and down yield moves — which is why positive convexity is a desirable, valuable property for a bondholder.

Higher convexity is generally attractive: for the same duration, a more convex bond loses less when yields rise and gains more when yields fall, which is why callable bonds (negative convexity from the call feature) trade cheap relative to comparable option-free bonds.

2. Why it matters

Convexity distinguishes bonds with identical duration but different risk profiles, and it is central to pricing embedded options like callability and to explaining hedging error from duration-only hedges.

3. Core mathematics

Second-order price approximation

\frac{Δ P}{P} \approx - D_{m o d} Δ y + \frac{1}{2} C (Δ y)^{2}

Convexity definition

C = \frac{1}{P} \frac{d ^{2} P}{d y ^{2}}

4. How to recognize it

Think of this when…

'The duration approximation understated the price change' — asking why (convexity)
Comparing two bonds with equal duration but different convexity
Callable bond pricing (negative convexity near the call price)

5. Worked example

Worked example

A bond has $D_{m o d} = 8$ , convexity $C = 100$ , price $1000$ . Yields fall by 2% (200bp). Approximate the new price including convexity.

1
Compute the duration term.
$- D_{m o d} Δ y = - 8 (- 0.02) = 0.16$
2
Compute the convexity correction.
$\frac{1}{2} (100) (- 0.02)^{2} = \frac{1}{2} (100) (0.0004) = 0.02$
3
Sum and apply to price.
$Δ P / P \approx 0.16 + 0.02 = 0.18 \Rightarrow P_{n e w} \approx 1180$
Answer: Approximate new price ≈ $1180 (18% increase)

6. Common mistakes

Treating convexity as negative or subtracting it — the correction term is always added (it's a squared term with a positive coefficient for option-free bonds)
Forgetting the factor of $\frac{1}{2}$ in the convexity term
Ignoring convexity entirely for large yield moves, where the duration-only approximation is materially wrong

7. Quick check

The convexity correction term in the price approximation is:

Between two bonds with equal duration, the one with higher convexity is:

8. Practice

2 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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