Convexity

Bonds & Fixed Income
Updated Apr 2026 • Has calculator

The second-order measure of a bond's price sensitivity to yield changes, capturing the curvature that modified duration misses.

What is Convexity?

Convexity measures the curvature of the price-yield relationship. While modified duration approximates the bond price change as a straight line, the actual relationship is curved — a bond gains more in price when yields fall by 1% than it loses when yields rise by 1%. Positive convexity, which all standard bonds exhibit, is therefore beneficial to bondholders. Convexity is added to the duration approximation for large yield moves: ΔPrice ≈ −ModDur × ΔY × Price + 0.5 × Convexity × (ΔY)² × Price. Callable bonds can exhibit negative convexity near the call price.

Formula

Convexity = [Σ t(t+1)·CF_t / (1+r)^(t+2)] / (Price × freq²)

Worked Example

Worked example — Hypothetical 8% Annual Coupon Bond

5-year maturity, YTM = 8%

Step 1  Face: $1,000 | Annual coupon: $80 | YTM: 8% | Maturity: 5 yrs
Step 2  Convexity = Σ t(t+1)×CF_t/(1.08)^(t+2) / Price
Step 3  Convexity ≈ 21.04 years²
Step 4  For a 200-bps yield rise: ΔPrice ≈ −3.99×0.02 + 0.5×21.04×0.02² = −6.77%'
Step 5  → Convexity adds ~0.42% correction to the duration-only estimate

Source: CFA Institute — Fixed Income Analysis, 3rd ed., Ch. 5 (2023-01-01)

Calculate Convexity

Par value repaid at maturity

Total annual coupon in dollars

Annual YTM used as the discount rate

Remaining years until maturity

1 = annual, 2 = semi-annual

Convexity

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Not investment advice.

How to Interpret Convexity

< 10
< 10: Low convexity — short maturity or high-coupon bond
10 – 30
10–30: Moderate — typical 5–10 year investment-grade bond
30 – 100
30–100: High — long-duration bond; large positive convexity benefit
> 100
> 100: Very high — zero-coupon or very long maturity bond

📚 Bond Risk — Complete the path

  1. Macaulay Duration
  2. Modified Duration
  3. Effective Duration
  4. Convexity
  5. Duration Price Approximation
Sources
  1. CFA Institute — Fixed Income Analysis, 3rd ed., Ch. 5
  2. Investopedia — Convexity