Convexity Adjustment
A convexity adjustment is a correction applied to a forward rate or expected value calculation to account for the convex (nonlinear) relationship between prices and interest rates, arising from the mathematical fact that the expected value of a convex function of a random variable is greater than the function evaluated at the expected value of that variable (Jensen's Inequality). It is essential in pricing interest rate derivatives, futures, and instruments whose payoffs are nonlinear functions of rates.
Key takeaways
- Jensen's Inequality states: E[f(X)] > f(E[X]) when f is a convex function — the convexity adjustment quantifies this gap.
- The most important application is the Eurodollar futures convexity adjustment: Eurodollar futures price systematically overestimates the forward rate due to daily mark-to-market and correlation between margin flows and discount factors.
- CMS (Constant Maturity Swap) rates require convexity adjustments because a CMS payment fixes the n-year swap rate at a future date, creating a convex payoff profile.
- The magnitude of the convexity adjustment grows with time horizon, rate volatility, and the degree of convexity in the price-rate relationship.
- In the Hull-White model, the Eurodollar futures convexity adjustment can be expressed as: CA ≈ ½ × σ² × T₁ × T₂, where T₁ is the futures expiration and T₂ is the payment date.
Explanation
The convexity adjustment arises whenever a market-quoted forward price or forward rate must be converted to an expectation under the appropriate probability measure for discounting. The core problem is that market prices reflect specific arbitrage relationships, but derivative payoffs are often nonlinear in the relevant rates or prices — creating a gap between the forward rate implied by futures prices and the true risk-neutral expected rate.
The canonical example is the Eurodollar futures contract. Eurodollar futures are marked to market daily, meaning gains and losses are settled in cash each day. Forward rate agreements (FRAs), by contrast, are settled at the beginning of the interest period. This settlement timing difference creates an asymmetry: when rates are high, the daily gains on a long Eurodollar futures position are reinvested at high rates; when rates are low, losses are funded at low rates. This asymmetry causes Eurodollar futures prices to be slightly higher than equivalent FRA prices — the futures rate is slightly lower than the forward rate. The convexity adjustment to convert from futures rate to forward rate is approximately:
CA ≈ σ² × T₁ × T₂ / 2
where σ is the annualized rate volatility, T₁ is the futures contract expiration (in years), and T₂ is the end of the interest period (T₁ + 0.25 years for quarterly contracts). For a 5-year Eurodollar futures contract with rate volatility of 1.5% per year: CA ≈ (0.015)² × 5 × 5.25 / 2 = 0.000225 × 5 × 5.25 / 2 ≈ 29.5 basis points. This represents the amount by which the Eurodollar futures rate overstates the forward LIBOR/SOFR rate.
For CMS products, the convexity adjustment addresses the swap rate's convexity. A CMS coupon payment references the n-year swap rate at a future date T. Because the swap rate is related to bond prices (which are convex functions of rates), its expected value under the forward measure differs from the forward swap rate by a convexity adjustment that depends on the swap rate's volatility and the payment timing. This adjustment can be several basis points for near-term CMS structures and tens of basis points for long-dated, high-volatility environments, materially affecting CMS pricing and hedging.
Formula
Eurodollar CA ≈ σ² × T₁ × T₂ / 2 | Jensen's Inequality: E[f(X)] > f(E[X]) for convex f
Example
A bank structures a 3-year CMS note that pays the prevailing 10-year swap rate quarterly. To price the first CMS coupon (paid in 3 months, referencing the 10-year swap rate in 3 months), the bank starts with the 3-month forward 10-year swap rate of 4.25%. The swaption market implies a normal vol of 80 bps (0.80%) for the 10-year rate. The convexity adjustment for a 3-month CMS coupon is approximately: CA ≈ ½ × σ_normal² × T = ½ × (0.0080)² × 0.25 = ½ × 0.000064 × 0.25 = 0.8 bps. Small for 3 months, but for the final CMS coupon paid in 3 years, the adjustment is approximately: CA ≈ ½ × (0.0080)² × (10 DV01 related term) × 3 years ≈ 15–20 bps, which is significant in terms of the bond's fair value and must be incorporated into the pricing.
Related terms
Arbitrage Basis Bond Central Limit Theorem Convexity Dv01 Eurodollar Futures Contract Hedging Interest Rate Jensens Inequality Libor