Why do long-dated swaps need a convexity adjustment?

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AcadiFi · Accepted Answer

Convexity adjustment corrects for the fact that futures and FRAs have a linear payoff in rates, while bonds and swaps have a nonlinear, convex relationship between price and yield. For long tenors, Jensen's inequality makes the expected realized rate differ from the forward rate implied by the curve. The adjustment for a forward rate f_fwd over tenor T to implied forward start time t is approximately: f_adjusted = f_fwd - 0.5 × σ² × t × T, where σ is the forward rate volatility. Ignoring this over long tenors creates arbitrage. Example: a 10-year forward rate starting in 20 years, with σ = 1.2% (absolute), t = 20, T = 10. Adjustment = 0.5 × 0.0001440 × 20 × 10 = 1.44%. That's 144 basis points, enormous. A swap rate built from unadjusted forwards at this horizon would be meaningfully too high. ```mermaid flowchart LR A[Forward
Rate] --> B[Add
Convexity] A --> C[Unadjusted
Price] B --> D[Fair
Price] C -.->|Biased| D ``` In practice, dealers use SABR or HJM models calibrated to cap and swaption vols to produce a coherent set of adjusted forwards. For CMS swaps and constant-maturity products the convexity effect is even larger because the payoff references a swap rate observed in the future.

Why do long-dated swaps need a convexity adjustment?

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