As with all other forms of futures contract, the fair price of short-term interest rate futures should preclude any arbitrage possibilities between the futures market and the underlying cash market. In the case of bank deposit interest rate futures, there should be no arbitrage possibilities between the forward interest rate implied by the future and the forward interest rate available on the appropriate type of bank deposit. For example, a three-month eurodollar futures contract that has 135 days to maturity should not provide any arbitrage possibility with the 135-day forward rate on a three-month eurodollar deposit.

Clearly, then, the first step in calculating the fair value is to calculate the 135-day forward rate on a three-month deposit. We will assume that the life of that deposit is actually 90 days, the current 135-day LIBOR is 10% p.a., the 225-day LIBOR is 10.25% p.a. and rates are quoted on an actual over 360-day basis.

The market practice for calculating the 90-day forward rate (referred to in futures market parlance as the 90-day forward-forward) is to use simple interest rates 10.24.

With the forward rate calculated as 10.24% and the market convention of quoting the futures price as 100 minus the forward rate, the theoretical futures price is 100 - 10.24 = 89.76.

In this example, the forward rate is actually below the 225-day rate. This result seems, at first sight, to be counter-intuitive. The forward rate should be such that a sum of money invested for the first time period at the current rate for that period, and the principal and interest reinvested for the second time period at the forward rate, will produce a total return equal to that earned by investing for the whole term at the longer-term rate of interest. If the 135-day rate is below the 225-day rate, it may be thought that the funds must be invested for the 90-day period beginning on day 136, at a rate above the 225-day rate in order for the combination of 135-day and 90-day investments to produce the same return as the 225-day investment. T answer is, not necessarily. The reason is that when the 135-day deposit is ro over, with interest, that interest is compounded. The interest rate on the 225- instrument has no compounding, but the interest on the comparable combination the 135-day and 90-day instruments is compounded after 135 days.

When the yield curve is positively sloped, the longer the time period before th notional roll over (135 days in this example), the greater will be the interest element that is being multiplied and thus the less likely it is that the forward rate will be above the longer-term deposit rate. When the yield curve is negatively sloped, the longer the period before the notional roll over, then the lower will become the forward rate.

Given that the forward deposit, and therefore the forward interest rate on which the future is based, does not yet exist, what is the arbitrage process that keeps th traded futures price at its fair price?

The answer is that forward rates can be locked i by the creation of forward deposits or loans.

By borrowing for 135 days and depositing for 225 days, the investor has created neutral position for the first 135 days (i.e. deposit and loans off-set each other) and net deposit position for the final 90 days. Thus a forward deposit has been create and the net return on that deposit will be the forward rate. Arbitragers will trad between these forward deposits and the futures contract, thereby keeping the future trading at close to its fair price. As there are bid—offer spreads in the money market and the futures markets, there will be an arbitrage-free channel similar to that found in other futures contracts.

It has given the 90-day forward rate 135 days hence as 10.24%. If the future were implying a yield of, say, 10.5%, arbitragers would set up a forward deposit by depositing for 225 days and borrowing for 135 days, thus being a net depositor for 90 days in 135 days‘ time. The arbitrager would then short the future to lock in a yield differential of 26 basis points per annum. If the future was quoted at, say, 10.0%, arbitragers would buy the cheap future, and set up a forward loan by borrowing for 225 days and redepositing the funds for 135 days. The result would be a net 90-day loan in 135 days‘ time hedged by a long futures position.

As forward interest rates reflect the markets’ consensus expectations of what interest rates will be in the future, money market interest rate futures contracts also reflect those same market expectations.

Futures prices and forward rate information

It was stated above that forward rates could not be directly observed in the marketplace. This is true; nevertheless, there is a good proxy for short-term forward rates in the form of the yield implicit in prices of money market interest rate futures. Here it shows a strip of three-month short Sterling futures with deliveries covering the period from late June 1991 to December 1992, a period of 18 months. The forward rates that are derived from these price quotations reflect the market’s current (June 1991) expectations of what the three-month bank deposit rates will be at each respective future point in time.

These rates will not be expected future spot rates, although such rates can be derived in a similar manner from Treasury bill futures contracts in the countries where these contracts exist.

Theoretically, the equality between forward rates and the rates implicit in futures prices depends upon the interest rate being constant. In such circumstances, there will be no margin payments, thus no margin funding cost or interest income, to complicate the comparison between the forward and the future. However, when interest rates are stochastic the futures and forward prices will differ; nevertheless, empirical studies find the rate implicit in short-term interest rate futures and forward rates to be virtually identical. Thus, the rates implicit in money market futures contracts may be taken as good proxies for forward rates.

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