Why do Eurodollar futures rates need a convexity adjustment when used to build a forward rate curve, and how is it calculated?

The convexity adjustment corrects for a systematic bias in Eurodollar (and SOFR) futures rates relative to the true forward rates implied by the zero curve. This bias arises from the daily mark-to-market (margining) feature of futures contracts, which creates an asymmetric payoff that favors the short.\n\nWhy the Bias Exists:\n\nFutures contracts are settled daily. When rates rise, the short profits and receives margin cash that can be reinvested at the now-higher rates. When rates fall, the short loses and must post margin, but borrowing costs are lower. This positive correlation between futures P&L and reinvestment rates creates a systematic advantage for the short, pushing futures rates above true forward rates.\n\n`mermaid\ngraph TD\n A[\"Rates Rise\"] --> B[\"Short profits on futures\"]\n B --> C[\"Reinvest gains at
HIGHER rates\"]\n D[\"Rates Fall\"] --> E[\"Short loses on futures\"]\n E --> F[\"Finance losses at
LOWER rates\"]\n C --> G[\"Net advantage to short
= convexity bias\"]\n F --> G\n G --> H[\"Futures rate > Forward rate
Adjustment needed\"]\n`\n\nThe Convexity Adjustment Formula:\n\nA widely used approximation:\n\nConvexity Adjustment = 0.5 x sigma^2 x T1 x T2\n\nWhere:\n- sigma = annualized volatility of the forward rate\n- T1 = time to futures expiration (years)\n- T2 = time to end of the futures period (years)\n\nWorked Example:\n\nCampbell Trading is building a zero curve from SOFR futures. Consider the 5-year futures contract:\n\n- Futures rate: 4.25%\n- Time to expiration (T1): 5.0 years\n- Futures period end (T2): 5.25 years\n- Rate volatility (sigma): 1.20% (120 bps annualized)\n\nConvexity adjustment:\n= 0.5 x (0.012)^2 x 5.0 x 5.25\n= 0.5 x 0.000144 x 26.25\n= 0.001890 = 18.9 bps\n\nAdjusted forward rate: 4.25% - 0.189% = 4.061%\n\nKey Observations:\n\n1. Grows with maturity squared: The adjustment is negligible for short-dated contracts (< 2 years) but becomes large for longer tenors. At 10 years, adjustments can exceed 60 bps.\n\n2. Volatility sensitivity: The adjustment scales with sigma-squared, so it doubles when volatility increases by ~41%.\n\n3. Always subtract: Futures rates always exceed forward rates (for positive correlation between rates and margin flows), so the adjustment is always subtracted.\n\n| Tenor | Futures Rate | Convexity Adj. | Forward Rate |\n|---|---|---|---|\n| 1Y | 4.52% | 0.4 bps | 4.516% |\n| 3Y | 4.38% | 3.4 bps | 4.346% |\n| 5Y | 4.25% | 18.9 bps | 4.061% |\n| 7Y | 4.15% | 37.0 bps | 3.780% |\n| 10Y | 4.08% | 75.6 bps | 3.324% |\n\nWithout the convexity adjustment, a zero curve built from long-dated futures will be systematically too high, leading to mispriced swaps and swaptions. This is one reason dealers prefer swap rates over futures for the long end of the curve.\n\nPractice convexity adjustment problems in our FRM question bank.