Why do we need a convexity adjustment and how does it improve the duration-based price estimate?

Duration gives a straight-line estimate of price changes, but bond prices actually move along a curve. Convexity captures that curvature and corrects duration's error.

Why Duration Alone Isn't Enough:

Modified duration assumes a linear relationship:

%ΔP ≈ -ModDur x Δy

But the actual price-yield relationship is a convex curve. Duration overestimates price declines when yields rise and underestimates price gains when yields fall. The error grows as the yield change increases.

The Full Price Change Formula:

%ΔP ≈ (-ModDur x Δy) + (0.5 x Convexity x (Δy)^2)

The convexity term is always positive (for option-free bonds), meaning it always adds to the price estimate — which is correct because the actual curve lies above the duration tangent line.

Worked Example:

Pelham Financial holds a bond with:

• Modified duration: 7.25
• Convexity: 62.8
• Current price: $1,034.50

Scenario: Yields rise by 150 bps (+0.015)

Duration-only estimate:

%ΔP ≈ -7.25 x 0.015 = -10.875%

Estimated price: $1,034.50 x (1 - 0.10875) = $921.99

With convexity adjustment:

%ΔP ≈ -7.25 x 0.015 + 0.5 x 62.8 x (0.015)^2

%ΔP ≈ -0.10875 + 0.5 x 62.8 x 0.000225

%ΔP ≈ -0.10875 + 0.00707 = -10.168%

Estimated price: $1,034.50 x (1 - 0.10168) = $929.30

The convexity adjustment recovers $7.31 of the overestimated loss.

Scenario: Yields fall by 150 bps (-0.015)

Duration-only: %ΔP ≈ +10.875% → $1,147.02

With convexity: %ΔP ≈ +10.875% + 0.707% = +11.582% → $1,154.33

Convexity adds $7.31 to the gain estimate as well, correctly capturing the asymmetry.

Key Insights:

• Convexity is always positive for option-free bonds
• Higher convexity is desirable — it means bigger gains when yields fall and smaller losses when yields rise
• Convexity matters most for large yield changes; for small changes (10-20 bps), duration alone is adequate

For more on managing interest rate risk, check our CFA Level I Fixed Income course.