Convexity bias is a difference in the convexity in the economic benefit of holding futures vs. forwards in a given underlier. When convexity bias exists, the result is a divergence in the prices of the respective futures and forwards.

Cox, Ingersoll and Ross (1981) and Jarrow and Oldfield (1981) first suggest that daily margin payments on futures may cause forward and futures prices to diverge. If there is a correlation between daily futures prices and interest rates, one party to a futures contract will tend to receive margin payments on days when interest rates rise and make margin payments on days when interest rates decline. On average, she will invest the margin payments she receives at interest rates that are higher than those at which she finances the margin payments she makes. The other party will experience the opposite situation. This should cause a divergence in forward and futures prices. The effect should increase with the maturity of contracts and with the standard deviation of a future’s price.

Empirical studies by Cornell and Reinganum (1981), French (1983) and Park and Chen (1985) confirm a modest effect in gold, silver, silver coins, platinum, copper and plywood prices but fail to find one for various currencies.

For Eurodollar futures (and other Eurocurrency futures) the effect is magnified by a second issue. Not only do the forward rates these contracts are linked to correlate highly with the overnight rates at which margin payments would typically be financed or invested, but there is an issue with how those margin payments are calculated.

A forward on a deposit has convexity. Its market value rises more for a given decline in the forward rate than it would decline for the same sized rise in the forward rate. Eurocurency futures do not have convexity. Contract specifications for Eurodollar futures, for example, set the daily margin payment at USD 25 per basis point move in the futures rate. While this margining formula couldn’t be simpler, by construction, it deprives Eurodollar futures of the convexity possessed by the forward rate agreements (FRAs) they are intended to mimic.

Accordingly, Eurodollar futures rates should diverge from corresponding FRA rates for two reasons:

- the margining effect identified by Cox, Ingersoll and Ross (1981) and Jarrow and Oldfield (1981), and
- the unique margining formula for Eurodollar futures (and other Eurocurrency futures).

Burghardt and Hoskins (1995) studied the latter effect. They called it a **convexity bias**, although the name reasonably encompasses both effects.

For short-dated Eurodollar futures—those out to a year or eighteen months, the effect is hardly noticeable, perhaps a basis point or less. For longer-dated futures, convexity bias can be more pronounced, causing Eurodollar futures rates to exceed corresponding forward rates by ten basis points or more at the longest maturities. The actual magnitude depends on the level and volatility of interest rates. Hull[1] provides the following approximation

forward rate = futures rate – σ^{2}*t*_{1}*t*_{2}/2

[1]

where

- σ is the standard deviation of the underlying interest rate;
*t*_{1}is the time (in years) until maturity of the futures contract; and*t*_{2}is the time (in years) until maturity of the underlying loan.

For example, if 3-month Libor has a standard deviation of 0.012 (120 basis points), the three-year Eurodollar futures rate will be higher than the corresponding FRA rate by

0.012^{2}(3)(3.25)/2 = 0.0007

[2]

or 7 basis points. The approximation assumes both the futures and FRA rates are continuously compounded. Hull cites no source and offers no justification for the formula other than to say it is based on the Ho-Lee interest rate model.

Prior to the early 1990s, traders were unaware of the convexity bias. Eurodollar futures were used to hedge interest rate swaps, which were priced using Eurodollar futures rates as if they were forward rates. This caused swap rates to be higher than they should have been. The situation started to change during the early 1990s, as awareness of convexity bias spread. Financial engineering models were developed to calculate the convexity bias, which was then taken into account whenever swaps were priced or hedged with Eurodollar futures.

Eurodollar futures are still used for hedging swaps and other fixed income derivatives, but convexity bias has rendered Eurodollar rates a poor benchmark for pricing other instruments. By the early 2000s, the Libor-swap curve replaced Eurodollar rates as a benchmark.

## Notes

- [1] Hull, John (2002).
*Futures, Options and Other Derivatives, Fifth Edition*, Upper Saddle River: Prentice-Hall, p. 111.

## References

- Burghardt, Galen and Bill Hoskins (1995). A question of bias,
*Risk*, 8 (3), 63-70. - Cornell, Bradford and Marc R. Reinganum (1981). Forward and futures prices: Evidence from the foreign exchange markets,
*Journal of Finance*, 36 (12), 1035-1045. - Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross (1981). The relation between forward prices and futures prices,
*Journal of Financial Economics*, 9, 321-346. - French, Kenneth R. (1983). A comparison of futures and forward prices,
*Journal of Financial Economics*, 12, 311-342. - Jarrow, Robert A. and George S. Oldfield (1981). Forward contracts and futures contracts,
*Journal of Financial Economics*, 9, 373-382. - Park, Hun Y. and Andrew H. Chen (1985). Differences between futures and forward prices: A further investigation of the mark-to-market effects,
*Journal of Futures Markets*, 5 (1), 77-88.