Appendix 3A provides an arbitrage argument to show that, when the risk-free interest rate is constant and the same for all maturities, the forward price for a contract with a certain delivery date is the same as the futures price for a contract with the same delivery date. The argument in Appendix 3A can be extended to cover situations where the interest rate is a known function of time.
When interest rates vary unpredictably (as they do in the real world), forward and futures prices are in theory no longer the same. The proof of the relationship between the two is beyond the scope of this book. However, we can get a sense of the nature of the relationship by considering the situation where the price of the underlying asset, S, is strongly positively correlated with interest rates. When S increases, an investor who holds a long futures position makes an immediate gain because of the daily settlement procedure. Since increases in S tend to occur at the same time as increases in interest rates, this gain will tend to be invested at a higher-than-average rate of interest. Similarly, when S decreases, the investor will make an immediate loss. This loss will tend to be financed at a lower-than-average rate of interest. An investor holding a forward contract rather than a futures contract is not affected in this way by interest rate movements. It follows that, ceteris paribus, a long futures contract will be more attractive than a long forward contract. Hence, when 5 is strongly positively correlated with interest rates, futures prices will tend to be higher than forward prices. When S is strongly negatively correlated with interest rates, a similar argument shows that forward prices will tend to be higher than futures prices.
The theoretical differences between forward and futures prices for contracts which last only a few months are in most circumstances sufficiently small to be ignored. As the life of the contracts increase these differences become greater. In practice, there are a number of factors, not reflected in theoretical models, that may cause forward and futures prices to be different. These factors include taxes, transaction costs, and the treatment of margins. Also, in some instances, futures contracts are more liquid and easier to trade than are forward contracts. Despite all these points, in most circumstances it is reasonable to assume that forward and futures prices are the same. This is the assumption that will be made throughout this book. The symbol F will be used to represent both the futures price and the forward price of an asset.
Some empirical research that has been carried out comparing forward and futures contracts is listed at the end of this chapter. Cornell and Reinganum studied forward and futures prices on the British pound, Canadian dollar, German mark, Japanese yen, and Swiss franc between 1974 and 1979. They found very few statistically significant differences between the two prices. Their results were confirmed by Park and Chen who as part of their study looked at the British pound, German mark, Japanese yen, and Swiss franc between 1977 and 1981.
French studied copper and silver during the period 1968 to 1980. The results for silver show that the futures price and the forward price are significantly different (at the 5% confidence level) with the futures price generally above the forward price. The results for copper are less clear cut. Park and Chen looked at gold, silver, silver coin, platinum, copper, and plywood between 1977 and 1981. Their results are similar to those of French for silver. The forward and futures prices are significantly different with the futures price above the forward price. Rendleman and Carabini studied the Treasury bill market between 1976 and 1978. They also found statistically significant differences between futures and forward prices.
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