CFA Institute

Market Index Futures Contracts: Some Thoughts on Delivery DatesAuthor(s): Dwight GrantSource: Financial Analysts Journal, Vol. 38, No. 3 (May - Jun., 1982), pp. 60-63Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4478548 .

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by Dwight Grant

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If there are no restrictions on short selling, no margin or performance bond re- quirements, no transactions costs and no taxes, then the prices of a three-month and a six-month futures contract in a market index will be determined by the price of the index and by the three-month and six-month rates of interest, respectively. The two contracts will possess identical market index hedging characteristics and will differ only inasmuch as the six-month contract possesses additional hedging potential with respect to the three-month forward interest rate. Consequently, the six-month contract can be duplicated by a combination of the three-month contract and a futures contract in the three-month forward interest rate.

Multiple contracts do not expand the hedging potential of the futures market and may, in fact, result in reduced marketability for each contract. On the other hand, transactions costs encourage overlapping contracts and taxes create a need for contracts that extend into the next year. Two one-year contracts with delivery dates in, say, January and July would satisfy these requirements.

IN AN ARTICLE in this Journal, Victor Niederhoffer and Richard Zeckhauser de- scribed the mechanics of a futures contract in

a stock market average or index and explored the potential uses of such a contract.' One charac- teristic of the market they did not discuss in detail was the number of contracts in a particular in- dex which will trade at one time. This question warrants attention.

Multiple contracts of the type proposed by the Kansas City Board of Trade and others are not necessary and may not be economical because they possess identical hedging characteristics with respect to stocks, and trading in multiple contracts will reduce the marketability of each contract. This conclusion can be proved under a set of specific assumptions; it also seems likely to be essentially valid when these assumptions are relaxed. To see why trading in more than one contract is largely redundant, we must first con- sider the pricing of futures contracts in a market index.

Contract Pricing2 Assume that there are no restrictions on short selling, no margin or performance bond re- quirements, no transactions costs and no taxes. Furthermore, assume that (1) there are two futures contracts in a market index with delivery dates three months and six months hence; (2) there are three-month and six-month Treasury bills; and (3) there is a futures market in three- month Treasury bills with delivery in three months. Under these conditions, the prices of the three-month and six-month futures contracts in the index will be determined by the price of the index and by the three-month and six-month rates of interest, respectively.

To see why this is so, consider an investor who buys the stocks that comprise the index, sells short an equivalent amount of the futures con- tract and holds his position for three months. He is exposed to no risk because gains or losses on his investment in the index are exactly offset by losses or gains in the futures contract. He must therefore earn the risk-free rate of return on his investment. The return he earns is the difference between the price at which he buys the index and

1. Footnotes appear at end of article.

Dwight Grant is Associate Professor of Finance at the Edwin L. Cox School of Business, Southern Methodist University.

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the price at which he sells the futures contract. Therefore the price of the futures contract must be the price of the index multiplied by one plus the rate of return on the three-month Treasury bill.3

The same conclusion can be derived by con- sidering the case in which an investor sells short the stocks in the index, invests the proceeds in Treasury bills, and purchases an equivalent long position in the futures market. In this case, his investment is zero, his risk is zero and, so, his profit must also be zero. This will be true only if the futures contract is priced as indicated above.

Using the six-month contract, we can deter- mine that the price of the six-month futures con- tract is equal to the price of the index multiplied by one plus the six-month Treasury bill rate. These two pricing relationships permit us to con- sider the hedging characteristics of the two futures contracts in the market index.

Hedging Characteristics The hedging characteristics of a futures contract

are determined by the relationship between its change in value and that of the asset being hedged. If an investor can take a position in a futures contract such that the change in value of his futures position is equal but opposite in sign to that of the asset, he has created a "perfect," or risk-free, hedge. This, of course, is not always possible or even desirable. An investor may not want to eliminate risk; he may often seek risk in pursuit of the attendant expected return. Nor may he be able to eliminate risk through hedg- ing, since factors other than the change in the value of the asset influence the value of the futures contract. With respect to the futures con- tract in the stock index, this "other factor" is the risk-free interest rate for the life of the contract. We discuss the full implication of this point below, but one effect is worth noting now: The futures contract in the market index can be a perfect hedge only if the trader's holding period corresponds to the delivery date of the futures contract; otherwise the futures contract will en- tail interest rate risk.

The essence of this article is that the three- month and six-month contracts in the market in- dex possess identical market index hedging characteristics. As trading vehicles, they differ only inasmuch as the six-month contract possesses additional hedging potential with respect to the three-month forward interest rate-i.e., the interest rate from the end of month three to the end of month six. Consequently, the

six-month futures contract in a market index can be duplicated by a combination of the three- month futures contract in the stock index and the futures contract in the three-month forward in- terest rate. One contract is, therefore, redundant. This conclusion follows from the pricing of the futures contracts and our discussion of the hedg- ing characteristics of futures.

The price of the three-month futures contract depends upon the price of the index and the three-month interest rate. The value of the futures contract changes when the value of the index changes and/or the three-month interest rate changes. The former change provides the futures contract with its hedging potential; the latter change keeps it from being a perfect hedge for the stock index.

The price of the six-month futures contract depends upon the price of the index and the six- month interest rate. The six-month interest rate can be thought of as a three-month rate and the implied three-month forward rate. Therefore the value of the six-month market futures contract changes when the value of the index changes, and/or the value of the three-month interest rate changes, and/or the value of the implied three- month forward interest rate changes.

Again, it is the first of these effects that pro- vides the six-month futures contract with its hedging potential. Because the effect is essentially identical to that which exists in the three-month contract, the two function identically as hedges for the market index, and one is sufficient for the investor who wishes to hedge (or to speculate in) market risk.

The other two factors that affect the six-month futures contract in the market index are (1) the change in the three-month rate, which is iden- tical to the effect found in the three-month market index futures contract and (2) the interest rate ef- fect caused by changes in the implied three- month forward rate, which is the same effect found in the futures contract in the three-month forward interest rate.

This same basic conclusion holds whether we consider two futures contracts or 10. Additional contracts add hedging potential with respect to interest rates, but not with respect to market risk. Inasmuch as there is already a well-developed futures market in interest rates, this characteristic is not sufficient reason for introducing multiple contracts in the market index.

Relaxing the Assumptions The conclusion reached depends upon the argu-

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ment that the prices of the futures contracts are determined by the price of the index and the ap- propriate interest rate. This result is premised on the arbitrage argument, which in turn depends upon the four assumptions made above-(1) no restrictions on short sales; (2) no margin re- quirements; (3) no transactions costs and (4) no taxes. We will relax these assumptions one at a time and consider their influences.

The owners of substantial amounts of equity are not permitted to sell equities short. These in- vestors cannot enforce the pricing relationship derived above when the market is overvalued rel- ative to the price of the market index futures con- tract, because to do so requires that equities be sold short and the market index purchased. These investors can, however, switch from long positions in the market to long positions in the futures with the effect of decreasing demand for equities and increasing the demand for futures without altering the risk of their portfolios. While this argument is not as strong as that based on riskless arbitrage, the switching opportunity is likely to keep prices close to those implied by arbitrage.

Futures markets do require a small margin pay- ment or performance bond. To the extent that the performance bond is an interest-bearing instru- ment, this does not undermine the arbitrage ar- gument. If the bond is cash, the interest foregone is a cost of arbitraging. Arbitrage will occur only if the gain is greater than this cost. Therefore, ac- tual prices of the market index and the futures contract and those implied by arbitrage may dif- fer by an amount determined by this cost.

There is another issue related to the margin or performance bond requirement. Gains and losses in futures positions are added to or subtracted from this margin as they occur; the position is marked to market. If the size of the cash margin requirement is uncertain, so too is the associated cost of arbitrage. This uncertainty or risk will also create a potential for a gap between market prices and those implied by the arbitrage argument.

These two assumptions-no restrictions on short sales and no margin requirements-do af- fect pricing when relaxed. Their effects would ap- pear to be small, but that is an empirical ques- tion. What is important for this analysis is that they do not create any apparent need for more than one futures contract in the market index. That is not the case when we relax the remain- ing two assumptions-no transactions costs and no taxes.

If there were no transactions costs, there would

be no advantage to even one futures contract in the market index. An investor could costlessly ad- just his market risk just as he would with a futures contract, by buying or selling short the appropri- ate stocks. Transactions costs create a need for at least one contract. But if there is only one futures contract, it becomes a very short-term contract as delivery date approaches. Anyone wishing to take a longer-term hedge will incur extra costs in open- ing and closing a position in the short-term con- tract and then reopening a position in the new contract. Indeed, if most positions at the delivery date had to be reopened, this might disrupt the smooth functioning of the market. Transaction costs therefore create a demand for two contracts that overlap, hence accommodate longer-term hedges and the orderly rolling of positions from expiring to continuing contracts.

Niederhoffer and Zeckhauser have suggested that there may be tax advantages to being able to hedge market positions when the investor does not want to incur the market risk but wants to postpone his gain for tax purposes. Most of these situations would be accommodated if there were always a contract outstanding with a delivery date in the next calendar year. This also will re- quire two contracts.

Conclusion Multiple contracts do not expand the hedging potential of the futures market. Furthermore, the fewer the number of contracts, the greater the marketability of the futures. On the other hand, transactions costs encourage overlapping con- tracts and taxes create a need for contracts that extend into the next year. Based on these arguments, a futures market should offer, for ex- ample, two one-year contracts with delivery dates in January and July. In this case, there would always be a contract of at least six months' dura- tion, there would be a tax hedge for virtually the entire year, and there would be a minimum dif- fusion of trading among contracts.

Existing futures markets are characterized by substantial trading and open interest in six or more contracts. We have argued that two con- tracts are sufficient. The explanation for this seeming conflict lies in the pricing of the market index futures contracts as compared with the pricing of other types of futures. New informa- tion will influence expectations concerning future stock prices and will be reflected in the current price. The futures prices are the spot prices scaled by interest rates. Consequently, new information with respect to the index has the same effect, ex-

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cept for an arbitrary scale factor, on all futures contracts.

This is not the case with many other types of futures contracts. For example, rosy predictions in May concerning the likely size of September's wheat crop might have a substantial effect on the price of wheat futures for delivery in September, but much less impact on the futures price for de- livery in June. The prices of these contracts are influenced by factors that are to some degree in- dependent of each other, and the multiple futures contracts provide a hedging potential with re- spect to these independent risks. In the absence of such independent risks, any case for multiple contracts must, as we have argued above, be based on institutional considerations such as transactions costs and taxes. M

Footnotes

1. Niederhoffer and Zeckhauser have discussed the details of the futures contract in the Value Line Composite Index proposed by the Kansas City Board of Trade. For a complete description, see their article. The more important details of trading in such a contract are: (1) The price of the contract will be a multiple of the value of a widely based index of the price of common stocks. The multiple will be chosen to provide a desired market value for the contract-$50,OOO is one value suggested. (2) The futures market will not be a delivery market, but rather a cash settlement market. Instead of the in- dex being delivered, all contracts will be settled by cash payments determined by the value of the in- dex on the delivery date. This is feasible because the price of the index is not subject to price manipulation. (3) The margin or performance bond requirements will be limited to four to eight per cent of the market value of the contract, depending upon whether the transaction is by a hedger or speculator. Victor Niederhoffer and Richard Zeckhauser, "Market Index Futures Contracts," Financial Analysts Journal, January/February 1980, 49-55.

2. A more mathematical presentation of the arguments is presented in an appendix.

3. The length of the holding period does not influence the conclusion as long as there are assets with maturities that match the holding period.

Appendix Let PO be the price of the index; P3f and P6f be the prices of the futures contracts deliverable in three and six months; and R3 and R6 be one plus the rates of return on three and six-month Treasury bills. The implied forward rate is r6 (R6 = R3r6). The dollar return from selling short the index, in- vesting the proceeds in Treasury bills and buy- ing the futures contract in the index must be zero. There is no capital invested and there is no risk. Therefore:

PoR3 - P3f =

P3f = POR3 . (1)

Similarly:

P6f = POR3r6 4 (2)

Taking the total derivative of Equations (1) and (2) we have:

dP3f = R3dPo + PodR3, (la)

dP6f = r6(R3dPO + PodR3) + POR3dr6 (2a)

Substituting (la) into (2a) we can write:

dP6f = r6dP3' + POR3dr6

The change in the value of the six-month contract is equal to the change in the value of the three- month contract, scaled by a constant, plus a variable that depends on the change in the im- plied three-month forward rate. Thus the six- month contract has no hedging potential with respect to the index, beyond that which is em- bodied in the three-month contract. The converse is, of course, also true.

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