Why do Eurodollar futures (and now SOFR futures) need a convexity adjustment when used for swap pricing?

The convexity adjustment is one of the trickiest concepts in fixed-income derivatives. Here's the intuition and the math.

Why Futures Rates != Forward Rates:

Futures contracts are marked to market daily, meaning gains and losses are settled in cash every day. Forward rate agreements (FRAs) settle only at maturity. This daily settlement creates a systematic bias:

• When rates rise, futures positions lose money, and you reinvest those losses at higher rates
• When rates fall, futures positions gain money, but you reinvest gains at lower rates

This asymmetry hurts the long position — you get cash when reinvestment rates are bad and lose cash when rates are good. To compensate, futures rates must be higher than the equivalent forward rate.

The Adjustment Formula:

Forward Rate = Futures Rate - (1/2) x sigma^2 x T1 x T2

Where:

• sigma = volatility of the short rate
• T1 = time to futures expiration
• T2 = time to end of the futures underlying period

Worked Example:

Suppose a 2-year SOFR futures contract implies a rate of 4.50%, rate volatility is 1.2% (0.012), T1 = 2.0, T2 = 2.25.

Convexity adjustment = 0.5 x (0.012)^2 x 2.0 x 2.25 = 0.5 x 0.000144 x 4.5 = 0.000324 = 3.24 bps

Forward rate = 4.50% - 0.0324% = 4.4676%

Key Points for FRM:

• The adjustment grows with the square of maturity — negligible for short-dated contracts, significant for 5+ year maturities
• Higher volatility increases the adjustment
• SOFR futures replaced Eurodollar futures post-LIBOR transition, but the convexity issue is identical
• Ignoring this adjustment overestimates swap fixed rates

Master this topic with our FRM Part I practice problems in the question bank.