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There is a very important difference between interest rate forwards and futures that matters for bond math. This has to do with the timing of the gains and losses on the contracts. Suppose that in June 2014 a hedge fund asks a commercial bank for pricing on a 60 x 63 OTC forward contract on 3-month LIBOR – these exist and are called forward rate agreements (FRAs). That FRA is essentially a bet on the level of 3-month LIBOR in June 2019. The bank in turn observes that June 2019 Eurodollar futures are trading at price index of 94.00, implying a futures rate of 6.00%. (In this market, the price index is 100 minus the rate.) If the bank sets the cost and risk of hedging the OTC forward over the five years at 4 basis points, would the bank show the hedge fund a bid FRA rate of 5.98% and an ask rate of 6.02%? That means the bank would be willing to “buy” LIBOR in the future, paying 5.98%, or to “sell” LIBOR, receiving 6.02%.

The answer is no, those forward rates are too high. An adjustment factor first must be subtracted from the futures rate. The amount of that adjustment is a subject of advanced bond math theory and goes well beyond this book. To do justice to this problem, we really would need to study theoretical term structure models that often are named after their developers (e.g., the Flo-Lee model, the Heath- Jarrow-Morton model, the Hull-White model, the Black-Derman-Toy model). A problem is that each model gives a somewhat different adjustment factor.

Equation 8.1 is an example of the relationship between the forward and futures rates based on the Ho-Lee model.


The rates are stated for continuous compounding. Variance is the square of the standard deviation of daily changes in the reference rate (3-month LIBOR), and Time 1 and Time2 are the years to the forward dates (for a 60 x 63 forward, 5.00 and 5.25 years). Suppose the standard deviation is 0.012 and the observed futures rate is 6.00% (annualized for a periodicity of 4). Equation 2.3 from Chapter 2 converts that rate to 5.9554% for continuous compounding.

The Ho-Lee adjusted forward rate is 5.7664%.

Equation 2.4 converts that rate back to a conventional quote for 3-month LIBOR of 5.8082%.

Therefore, given these assumptions, the adjustment factor is 19.2 basis points for the 5-year forward (0.06000 – 0.05808 = 0.00192). For the same standard deviation, the adjustment is only 3.3 basis points for a 2-year forward but 74.8 basis points for a 10-year forward. The longer the time frame, the more significant is the adjustment factor.

Why is the forward interest rate lower than the otherwise comparable futures rate? The key idea is that the gains and losses on an OTC forward contract are realized in a lump sum at the future delivery date. In contrast, the gains and losses on an exchange-traded futures contract are realized day by day over the lifetime of the transaction. The salient feature of the futures market is daily mark-to-market valuation and settlement into a margin account. That allows for the potential to invest gains and perhaps the need to finance losses. Usually there is no persistent pattern of correlation between market interest rates and the payoffs on a commodity, stock index, or foreign exchange futures contract. However, one-to-one correlation between gains and losses and changes in market rates is the essence of an interest rate futures contract, such as the one on 3-month LIBOR.

Suppose the commercial bank market maker is pricing the bid side of the 60 x 63 FRA. That is, the bank commits to “buy” LIBOR on a forward basis, paying a preset “price.” The risk is that LIBOR turns out to be less than that fixed rate, so the hedge is to go long June 2019 Eurodollars futures. The problem is that gains occur on days when the futures price rises and the rate goes down; losses occur on days when the price falls and the rate rises. Systematically, the bank hedging its risk on the OTC forward gets to invest when rates are lower and might have to finance losses when rates are higher. The opposite scenario plays out when the bank is pricing the ask side of the FRA. The futures hedge is to go short; thereby the bank is able to invest gains when rates rise (and the futures price falls) and to finance losses when rates fall (and the futures price rises).

Both circumstances of hedging the OTC contract with exchange-traded futures have the effect of dragging the forward rate down. On the bid side, the hedge is “bad,” so the pay-fixed rate on the FRA is lowered relative to the futures rate as compensation. On the ask side, the hedge is “good” and the received-fixed rate is lowered due to market competition. How much lower depends on the anticipated volatility of rates and the time frame. That is evident in the Ho-Lee adjustment factor shown in equation 8.1. A more developed model also could include a term for the correlation between the short-term rate driving the gains and losses on the futures contract (e.g., 3-month LIBOR) and the long-term rate representing the time to the delivery date on the forward contract.

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