SOFR Futures Convexity Adjustment: Forward vs Futures Rates Explained
Table of Contents
When constructing yield curves from SOFR futures or valuing interest rate swaps, you cannot use futures rates directly as forward rate estimates. Daily settlement creates a systematic bias that requires correction. This guide explains the convexity adjustment — why futures rates are higher than forward rates — and the related timing adjustment for nonstandard payment timing.
This article covers the futures-to-forward rate adjustment — a bias caused by daily settlement. This is NOT bond convexity, which describes price-yield sensitivity. Despite sharing the word “convexity,” these are entirely different concepts.
What Is the Convexity Adjustment?
The convexity adjustment is a correction applied to interest rate futures rates to convert them into forward rates suitable for yield curve construction and swap pricing.
Futures rates are systematically higher than the equivalent forward rates due to daily settlement (mark-to-market). The convexity adjustment subtracts this bias to recover the true forward rate for discounting and valuation.
This matters because derivatives pricing models — including swap valuation and swaption pricing — require forward rates, not futures rates. Using unadjusted futures rates introduces systematic error that compounds across the yield curve.
Why the Convexity Adjustment Exists
The adjustment exists because of an asymmetry created by daily settlement. Consider two traders:
- Trader A (futures): Enters a futures contract where gains/losses are settled daily
- Trader B (forward): Enters an equivalent forward contract with settlement only at maturity
When interest rates rise, Trader A receives cash immediately through the daily settlement process. This cash can be reinvested at the now-higher interest rates. When rates fall, Trader A pays cash, but the cost of financing that payment is lower because rates have fallen.
This creates an asymmetric advantage for the short position in futures (the position that profits when rates rise). To compensate, futures rates must be set higher than forward rates. The difference is the convexity adjustment.
SOFR Futures Convexity Adjustment Formula
For SOFR and Eurodollar futures, a practical approximation of the convexity adjustment is:
Where:
- σ — volatility of the forward rate (in decimal form, e.g., 0.012 for 120 bps)
- T1 — time to futures maturity (in years)
- T2 — time to end of the rate period (in years)
This is a simplified approximation using absolute (basis point) volatility. For CME 3-month SOFR futures, the quote is based on 100 minus the compounded SOFR over the reference quarter. This approximation works well for practical curve construction but is not a contract-exact model.
SOFR Futures Convexity Adjustment Example
In practice, traders apply convexity adjustments when building yield curves from CME’s 3-Month SOFR futures (SR3) contracts. The adjustment matters most for deferred contracts with longer maturities.
Suppose you’re constructing a yield curve and observe:
- CME SR3 December 2028 futures price: 95.50 (implied rate: 4.50%)
- Forward rate volatility from swaptions market: 120 bps (0.012 in decimal)
- T1 = 2.00 years (time to December 2028 contract expiry)
- T2 = 2.25 years (time to end of the 3-month reference period)
Step 1: Calculate the convexity adjustment
Adjustment = ½ × (0.012)2 × 2.00 × 2.25
Adjustment = 0.5 × 0.000144 × 4.50 = 0.000324 = 3.24 basis points
Step 2: Calculate the forward rate
Forward Rate = 4.50% – 0.0324% = 4.4676%
The adjustment is 3.24 bps for this 2-year maturity. For the SR3 December 2031 contract (5-year maturity), the adjustment would be roughly 15-20 bps — significant for swap pricing.
Consider two SR3 contracts observed on the same date:
| Contract | Time to Maturity | Convexity Adjustment |
|---|---|---|
| SR3 September 2026 (front month) | 3 months | ~0.1 bp (negligible) |
| SR3 December 2028 | 2 years | ~3.2 bps |
| SR3 December 2031 | 5 years | ~18 bps |
This illustrates why the adjustment is often ignored for front-month contracts but critical for deferred maturities used in curve construction.
What Is the Timing Adjustment?
The timing adjustment corrects for situations where a market variable is observed at one time but the payment occurs at a different (usually later) time.
When a payoff depends on a variable V observed at time T, but payment is made at time T* (where T* > T), the expected value of V must be adjusted to account for the correlation between V and interest rates over the delay period.
This arises in structured products, certain exotic derivatives, and any instrument where rate observation and payment timing are decoupled.
Timing Adjustment Formula
Where:
- ρVR — correlation between V and the forward rate RF
- σV — volatility of the variable V (proportional/lognormal)
- σR — volatility of the forward rate (proportional/lognormal, not absolute bps)
- RF — forward interest rate for the period T to T* (in decimal)
- T* – T — delay between observation and payment (in years)
- m — compounding frequency
The timing adjustment formula uses proportional (lognormal) volatility, where σR is a percentage of the rate level. Do not use absolute basis point volatility here — the RF term already incorporates the rate level.
Direction of adjustment: When the correlation ρVR is positive (V tends to move with rates), the adjustment reduces the expected value. When correlation is negative, the adjustment increases the expected value.
| Scenario | Timing Adjustment Needed? |
|---|---|
| Standard swap — rate observation and payment follow natural timing | No — natural accrual period aligns observation with payment |
| In-arrears swap — rate observed at payment date | Yes — rate observed for previous period but paid at period end |
| Constant Maturity Swap (CMS) — swap rate used for payment | Yes — requires both convexity and timing adjustments |
| Delayed settlement derivative | Yes — if payment occurs after natural maturity |
Convexity Adjustment vs Bond Convexity
Despite sharing the word “convexity,” these are completely different concepts. Understanding the distinction is critical.
Futures Convexity Adjustment
- Corrects rates (futures → forward)
- Caused by daily settlement
- Affects yield curve construction
- Adjustment grows with maturity squared
- Measured in basis points
- Relevant for: SOFR futures, swap pricing
Bond Convexity
- Describes price sensitivity
- Second derivative of price w.r.t. yield
- Affects bond valuation accuracy
- Benefits bondholders (gains > losses)
- Measured in years squared
- Relevant for: Bond trading, duration hedging
For a detailed explanation of bond price-yield convexity, see our Bond Convexity guide.
How to Apply Convexity Adjustments
When constructing a yield curve from SOFR futures, follow these steps:
- Obtain SOFR futures rates from CME or your data provider
- Estimate forward rate volatility — use swaption-implied volatilities or historical estimates. Longer-dated futures require model/market-implied adjustments when volatility is material.
- Calculate the adjustment for each futures contract using T1 and T2
- Subtract the adjustment from each futures rate to get forward rates
- Bootstrap the zero curve from the adjusted forward rates
Common Mistakes
Practitioners frequently make these errors when applying convexity adjustments:
1. Confusing with bond convexity — Bond convexity is about price sensitivity; futures convexity adjustment is about rate bias. They are unrelated concepts that happen to share a name.
2. Ignoring the adjustment for long-dated futures — The adjustment grows with T1 × T2. For near-dated front-month contracts, it’s negligible. For 2+ year maturities, it can be 5-15+ basis points — material for swap pricing.
3. Using the wrong volatility — The formula requires forward rate volatility, not spot rate volatility or historical price volatility. Use swaption-implied volatilities for the relevant tenor.
4. Forgetting the direction — Futures rates are higher than forward rates. Subtract the adjustment from the futures rate to get the forward rate.
5. Confusing volatility units — The simplified formula uses absolute volatility in decimal form (e.g., 0.012 for 120 bps). If using lognormal percentage volatility, you must convert: multiply by the rate level. Using the wrong units will produce adjustments off by orders of magnitude.
Limitations
The convexity adjustment formula shown is a practical approximation that works well for most applications but has important limitations:
- Requires accurate volatility estimate — The adjustment depends on σ2, so volatility errors are amplified
- Assumes constant volatility — Real volatility is stochastic; more sophisticated models (e.g., Vasicek, Hull-White) account for this
- Small for short maturities — Often ignored for futures under 1 year where the adjustment is under 1 bp
- SOFR vs Eurodollar nuances — SOFR futures settle on compounded overnight rates, not simple forward LIBOR. The approximation is practical but not contract-exact
Frequently Asked Questions
Disclaimer
This article is for educational and informational purposes only and does not constitute investment advice. The convexity adjustment formulas shown are practical approximations; actual adjustments in professional settings may use more sophisticated models. Always consult with qualified professionals before making trading decisions.
