How does the convexity adjustment improve on duration-based price estimates?

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AcadiFi · Accepted Answer

The second-order Taylor expansion of bond price with respect to yield gives: %ΔPrice ≈ −Duration × Δy + ½ × Convexity × (Δy)² The first term is the linear approximation; the second is the convexity adjustment that captures the curvature of the price-yield relationship. Convexity is always positive for option-free bonds, so the adjustment adds to price both when yields fall (more upside) and subtracts less when yields rise (less downside than duration predicts). For the 30-year Treasury, D = 18, C = 420, Δy = +0.0075: - Duration estimate: −18 × 0.0075 = −13.50% - Convexity adjustment: ½ × 420 × (0.0075)² = 0.5 × 420 × 0.00005625 = +1.18% - Total estimate: −13.50% + 1.18% = −12.32% So the duration-only estimate overstates the loss by 118 bps of price. For a large yield move like 75bps on a long-duration bond, ignoring convexity materially misstates the P&L. ```mermaid flowchart LR A[Yield Change] --> B[First Order
Duration] A --> C[Second Order
Convexity] B --> D[Linear
Estimate] C --> E[Curvature
Correction] D --> F[Total Price
Change] E --> F ``` **When convexity matters most**: 1. Large yield moves (>25bps) — convexity scales with (Δy)² while duration is linear 2. Long-duration bonds — convexity rises roughly with maturity² 3. Zero-coupon bonds — highest convexity for given duration 4. Callable and MBS — negative convexity regions require care For small moves (5-10bps), duration alone is fine for trading desks. For portfolio rebalancing after regime changes (100+ bps), always use the full second-order approximation. Market convention: 'convexity' often reported as annualized %; the formula uses the raw number ×0.5 × (Δy)². Check the scaling in your data source.

How does the convexity adjustment improve on duration-based price estimates?

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