Cox, Ingersoll and Ross (1981) and Jarrow and Oldfield (1981) 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 exasperated. Not only do the forward rates they 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. If this were also true for Eurodollar futures, this convexity effect would partially offset the margining effect identified by Cox, Ingersoll and Ross (1981) and Jarrow and Oldfield (1981). But it is not true of Eurodollar futures. Contract specifications for Eurodollar futures 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. Eurodollar futures rates should diverge slightly from corresponding forward rates for FRAs. Burghardt and Hoskins (1995) have called the effect a convexity bias. Note that convexity bias is due to a combination—interaction—of both
For short-dated Eurodollar futures—those out to a year or eighteen months, the effect is hardly noticeable, 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
where
For example, if 3-month Libor has a standard deviation of 0.012 (120 basis points), the three-year Eurodollar futures rate will be
or 7 basis points higher than the corresponding FRA rate. 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 the 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 widely used for hedging swaps and other fixed income derivatives, but convexity bias has rendered Eurodollar rates a poor benchmark for pricing other instruments. Today, the Libor-swap curve has replaced Eurodollar rates as a benchmark.
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the
margining effect identified by Cox, Ingersoll and Ross (
