Convexity Adjustment: Futures Vs FRA

Imagine two bets that pay off based on the same interest rate, settled the same way in principle, except one of them pays you (or takes from you) cash every single day as rates wiggle, and the other settles only once, at the end. If rates and your daily cash balance happen to be correlated — and with interest-rate futures they mechanically are, because rising rates mean falling futures prices which means you're paying margin exactly when reinvesting that margin earns more — the two bets are not worth the same, even with identical terms. That gap is the convexity adjustment, and it exists purely because of when cash changes hands, not because of any difference in the underlying rate itself.

Where the gap comes from

A eurodollar-style future is quoted as $100 - r$ and margined daily: if rates rise, the future's holder pays variation margin and faces higher reinvestment rates on the cash they must post — a double hit correlated with the very rate the future tracks. An FRA settles once at maturity with no interim cash flows, so no such correlation exists. The standard approximation for the gap, using a single-factor short-rate model, is

$$\text{Futures rate} - \text{Forward rate} \approx \frac{1}{2}\,\sigma^2\, t_1\, t_2,$$

where $\sigma$ is the annualized volatility of the short rate, $t_1$ is the time until the futures contract's reference period starts, and $t_2$ is the time until it ends. In plain English: the adjustment grows with rate volatility and with how far out in time the contract sits — a 3-month rate starting tomorrow has almost no adjustment, while a rate starting in 8 years with high volatility can be off by tens of basis points.

Worked example 1 — a short-dated contract

Take a 3-month eurodollar future referencing the period from $t_1 = 2$ years to $t_2 = 2.25$ years, with short-rate volatility $\sigma = 1.2\%$ (0.012). Then $\frac{1}{2} \times (0.012)^2 \times 2 \times 2.25 = 0.5 \times 0.000144 \times 4.5 = 0.000324$, i.e. 3.24 basis points. So if the futures-implied rate quotes at 5.00%, the corresponding FRA/forward rate is roughly $5.00\% - 0.0324\% = 4.9676\%$ — small, but not zero, and every curve-building desk subtracts it before using eurodollar futures to bootstrap a forward curve.

Worked example 2 — a long-dated contract

Now take a contract 9 years out, $t_1 = 9$, $t_2 = 9.25$, same $\sigma = 1.2\%$: $0.5 \times 0.000144 \times 9 \times 9.25 = 0.5 \times 0.000144 \times 83.25 = 0.005994$, about 60 basis points. If the futures rate is 5.00%, the implied forward is $5.00\% - 0.60\% = 4.40\%$ — a 60-basis-point mispricing if you'd naively used the futures rate as the forward rate directly, easily large enough to flip the sign of a relative-value trade built on it.

The adjustment scales with $t_1 t_2$, a product of two times — roughly quadratic in how far out the contract sits, which is why the plot above (a quadratic shape) is the right intuition: the gap stays tiny for years, then accelerates.

What this means in practice

Curve-building desks routinely use eurodollar or SOFR futures for the front few years of a curve because they're the most liquid instruments available, but every one of those futures rates must be adjusted down before being treated as a forward rate — otherwise the bootstrapped curve is biased high, and every product priced off it (swaps, caps, swaptions) inherits that bias.