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Global Financial Management

Forward and Future Contracts

Copyright 1997 by Campbell R. Harvey and Stephen Gray. All rights reserved. No part of this lecture may be reproduced without the permission of the authors.

Latest Revision: February 15, 1997

4.0  Overview:

This class provides an overview of forward and futures contracts. Forwards and futures belong
to the class of securities known as derivatives since their value is derived from the value of some other security. The price of a foreign exchange forward contract, for example, depends on the price of the underlying currency and the price of a pork belly futures contract depends on the price of pork bellies. Derivatives trade both on exchanges (where contracts are standardized) and over-
the-counter (where the contract specification can be customized). The focus of this class is on

(1) definitions and contract specifications of the major exchange-traded derivatives,
(2) the mechanics of buying, selling, exercising, and settling forward and futures contracts,
(3) derivative trading strategies including hedging, and
(4) the relationships between derivatives, the underlying security, and riskless bonds.
     In particular, it is possible to form combinations of derivatives and the underlying
     security that are riskless, providing a means of valuing derivatives.

4.1  Objectives

After completing this class, you should be able to:

• Determine the possible payoffs of forward and futures contracts.
• Understand the mechanics of buying, selling, exercising, and settling forward and futures contracts.
• Determine the possible payoffs of portfolios of futures, forwards, and the underlying asset.
• Examine market prices to determine whether arbitrage bounds are violated in futures and forward markets.
• Understand the directional effects of relevant variables on the value of derivative securities.
• Use standard valuation techniques to determine the price of forward and futures contracts.

4.2  Introduction

Despite the recent adverse press they have received, derivative securities provide a number of useful functions in the areas of risk management and investments. In fact, derivatives were originally designed to enable market participants to eliminate risk. A wheat farmer, for example, can fix a price for his crop even before it is planted, eliminating price risk. An exporter can fix a foreign exchange rate even before beginning to manufacture the product, eliminating foreign exchange risk. If misused, however, derivative securities are also capable of dramatically increasing risk.

This module focuses on the mechanics of forward and futures contracts. There is a particular emphasis on the interrelationship between the various contracts and the spot price of the underlying asset. The spot price is the price of an asset where the sale transaction and settlement is to occur immediately.

See Major Futures Exchanges

4.3  Forward Contracts

The Mechanics of Forward Contracts

A Forward Contract is a contract made today for delivery of an asset at a prespecified time in the future at a price agreed upon today. The buyer of a forward contract agrees to take delivery of an underlying asset at a future time, T, at a price agreed upon today. No money changes hands until time T. The seller agrees to deliver the underlying asset at a future time, T, at a price agreed upon today. Again, no money changes hands until time T.

A forward contract, therefore, simply amounts to setting a price today for a trade that will occur in the future. Example 4.4 illustrates the mechanics of a forward contract. Since forward contracts are traded over-the-counter rather than on exchanges, the example illustrates a contract between a user and a producer of the underlying commodity.

4.5  Valuation of Forward Contracts

Forward contracts can be valued by recognizing that, in many cases, forward markets are redundant. This occurs when the payoff from a forward contract can be replicated by a position
in (1) the underlying asset and (2) riskless bonds. Before illustrating this concept, we define the
cost of carry of the underlying commodity. This is the cost involved in holding a physical quantity
of the commodity. For wheat the cost of carry is a storage cost, for live hogs it consists of storage and feed costs, and for gold it consists of storage and security costs. Some commodities have a negative cost of carry. For example, holding a stock index provides the benefit of receiving dividends. In forward markets it is common to express the cost of carry as a continuously compounded annual rate, payable at inception. For example, if the cost of carry for wheat is reported to be 5%, this would mean that the cost of storing $100 of wheat for six months is
$100 (e^0.05(0.5)-1) = $2.53, payable immediately.

We use the following notation, which is common in forward markets:

Both r and q are expressed as continuously compounded annual rates. Consider the strategy of:

• borrowing enough money to buy one unit of a commodity and to pay for the associated carrying costs through time T, and
• entering into a forward contract to sell the commodity at time T. The value of this position in terms of the initial (time 0) and terminal (time T) cash flows is tablulated in the following table.

Since this portfolio requires no initial cash outlay, the absence of arbitrage opportunities will ensure that the terminal payoff is also zero. Therefore, the futures contract can be valued as

The following example shows how arbitrage is possible if this pricing relation is violated.

Example 4.6: Forward arbitrage.

Suppose the spot price of wheat is 550 cents per bushel, the six-month forward price is 600, the riskless rate of interest is 5% p.a., and the cost of carry is 6% p.a. To execute an arbitrage, you borrow money, buy a bushel of wheat, pay to store it, and sell it forward. The cash flows are:

That is, it is possible to lock in a sure profit that requires no intial cash outlay.

4.7  Hedging With Forward Contracts

The primary motivation for the use of forward contracts is risk management. The wheat farmer in example 4.4 was able to eliminate price risk by selling his crop forward. Example 4.8 contains a more comprehensive example concerning foreign exchange risk management.

4.9 Futures Contracts

The Mechanics of Futures Contracts

A futures contract is similar to a forward contract except for two important differences. First, intermediate gains or losses are posted each day during the life of the futures contract. This feature is known as marking to market. The intermediate gains or losses are given by the difference between today's futures price and yesterday's futures price. Second, futures contracts are traded on organized exchanges with standardized terms whereas forward contracts are traded over-the-counter (customized one-off transactions between a buyer and a seller).

Example 4.10 illustrates the marking to market mechanics of the All Ordinaries Share Price Index (SPI) futures contract on the Sydney Futures Exchange. The SPI contract is similar to the Chicago Mercantile Exchange (CME) S&P 500 contract and the London International Financial Futures Exchange (LIFFE) FTSE 100 contract. The mechanics are the same for all of these contracts. Stock index futures were introduced in Australia in 1983 in the form of Share Price Index (SPI) futures which are based on the Australian Stock Exchange's (ASX) All Ordinaries Index which is the benchmark indicator of the Australian stock market. Users of SPI futures include major international and Australian banks, fund managers and other large investment institutions. SFE locals and private investors are also active participants in the market.

SPI futures have an underlying of A$25 x Index (ie., a SPI futures contract with a price of 2000.00 will have a contract value of A$50,000). The All Ordinaries Share Price Index (AOI) is a capitalisation weighted index and is calculated using the market prices of approximately 318 of the largest companies listed on the Australian Stock Exchange (ASX). The aggregate market value of these companies totals over 95% of the value of the 1,186 domestic stocks listed.

This concept of marking to market is standard across all major futures contracts. Contracts are marked to market at the close of trading each day until the contract expires. At expiration, there are two different mechanisms for settlement. Most financial futures (such as stock index, foreign exchange, and interest rate futures) are cash settled, whereas most physical futures (agricultural, metal, and energy futures) are settled by delivery of the physical commodity. Example 4.11illustrates cash settlement.

An alternative to cash settlement is physical delivery. Consider the SFE wool futures contract which requires delivery of 2500 kg of wool when the contract matures. Of course, there are different grades of wool, so a set of rules governing deliverable quality is required. These are detailed rules that govern the standard quality of the underlying commodity and a schedule of discounts and premiums for delivery of lower and higher quality respectively. Example 4.12 illustrates the rules governing deliverable quality for the SFE greasy wool futures contract.

Example 4.13 illustrates the process of physical delivery for the SFE greasy wool futures contract. The process is similar for most commodity futures contracts.

4.14  Margin

Although futures contracts require no initial investment, futures exchanges require both the buyer and seller to post a security deposit known as margin. Margin is typically set at an amount that is larger than ususal one-day moves in the futures price. This is done to ensure that both parties will have sufficient funds available to mark to market. Residual credit risk exists only to the extent that (1) futures prices move so dramatically that the amount required to mark to market is larger than the balance of an individual's margin account, and (2) the individual defaults on payment of the balance. In this case, the exchange bears the loss so that participants in futures markets bear essentially zero credit risk. Margin rules are stated in terms of initial margin (which must be posted when entering the contract) and maintenance margin (which is the minimum acceptable balance in the margin account). If the balance of the account falls below the maintenance level, the exchange makes a margin call upon the individual, who must then restore the account to the level of initial margin before the start of trading the following day. Example 4.15 illustrates the margining procedure.

See more details on Margin

4.16  Valuation of Futures Contracts

Whereas the valuation of forward contracts is relatively straightforward, the marking to market feature complicates the valuation of futures contracts. The cash flows associated with forward and futures contracts are illustrated in the following table.

For both contracts, no money changes hands at the time the contract is initiated (time 0). For the forward contract, no money changes hand until the contract matures (time T). For the futures contract, money changes hands daily depending upon movements in the futures price.

In some circumstances, however, a futures contract is perfectly equivalent to a forward contract in which case the two contracts must have the same value. Since forward contracts are relatively easy to value using a no-arbitrage argument, this provides a convenient way of valuing a futures contract. In particular, if interest rates are constant (at a continuously compounded annual rate of r) over the life of the contract then the prices of the futures contract and the forward contract are identical.

This equivalence can be established by considering a roll-over strategy whereby at time 0 an investor purchases e^r futures contracts and invests FU₀ in a riskless bond where FU represents the futures price). At time 1 the profit (possibly negative) on the futures position is e^r(FU₁-FU₀), which he invests (or borrows) until maturity. At maturity, this has grown to e^r(T-1)e^r[FU₁-FU₀)] = e^rT(FU₁-FU₀).

At time 1 he increases his holding to e^2r contracts. At time 2 the profit (possibly negative) on this position is e^2r(FU₂-FU₁), which he invests (or borrows) until maturity. At maturity, this has grown to e^r(T-2)e^2r(FU₂-FU₁) = e^rT(FU₂-FU₁). At time 2 he increases his holding to e^3r and so on. At maturity, the total payoff on the futures position is:

where we note that FU_T = S_T. The payoff on the bond is FU₀ e ^rT. Therefore, the overall initial investment required for this strategy is $FU₀ and the overall payoff at time T is S₀e^rT.

Now consider the strategy of buying e^rT forward contracts on day o and investing $FO₀ in a riskless bond (where FO represents the price of a forward contract). The overall initial investment required for this strategy is $FO₀ and the overall payoff at time T is:

Since both of these strategies have the same payoff, they must cost the same. That is FO₀ = FU₀. The following table illustrates the cash flows associated with the two strategies.

4.17 Arbitrage Relationships

For the remainder of this module, we assume that interest rates are indeed constant over the period of the contract and hence the futures price equals the forward price. That is, we can consider the price and payoffs of a futures contract to be identical to those of a forward contract. This simplifies things because a forward contract has only a single payoff at maturity.

Consider, for example, the valuation of a futures contract on the S&P 500 stock index. This contract, which trades on the Chicago Mercantile Exchange (CME) entitles the buyer to receive the cash value of the S&P 500 stock index at the end of the contract period. There are always four contracts in effect at any one time expiring in March, June, September, and December. In contrast to the previous examples that involved a cost of carry, holding the S&P 500 index yields a benefit, in the form of dividends received, rather than a cost of carry. The result is that the value of an S&P 500 futures contract can be expressed as

where

This is the same as equation (1) except that "+q" has been replaced by "-d" as the cost of carry (storing wheat) has been replaced by a benefit (dividends). To see why this relationship must hold, consider the strategy of (1) borrowing e^-dTS₀ through time T, (2) using this to purchase e^-dT units of the index and reinvesting all dividends back into the index, and (3) selling a futures contract that matures at time T. If interest rates are constant, the futures contract is equivalent to a forward contract, which simplifies the analysis. In particular, the (equivalent) cash flows associated with this strategy are tabulated in the following table. Note that reinvestment of the dividends has resulted in the initial investment of e^-dT units of the index growing at a rate of d to amount to one unit by maturity.

Examples 4.18 and 4.19 illustrate how to execute a riskless arbitrage if this equality does not hold.

Once again, since this strategy requires no initial cash outlay, the cash flow at maturity must also be zero or an arbitrage opportunity exists. In particular, if F > S₀ e^(r-dT) the strategy of buying the index and selling the futures generates an arbitrage profit. Conversely, if F < S₀ e^(r-dT) the strategy of selling the index and buying the futures generates an arbitrage profit.

Examples 4.18 and 4.19 illustrate how to execute a riskless arbitrage if this equality does not hold.

4.20  Hedging with Futures

In this section, we examine how three common business risks - interest rate risk, stock market risk, and foreign exchange risk - can be hedged in a practical setting. In each case, we describe the nature of the risk and illustrate, through a series of practical examples, how the risk can be managed.

4.21  Hedging Interest Rate Risk

There are two primary interest rate futures contracts that trade on US exchanges. The Eurodollar Futures Contract trades on the Chicago Mercantile Exchange and the US T-Bill Futures Contract trades on the Chicago Board of Trade.

The Eurodollar contract is the more successful and heavily traded contract. At any point in time, the notional loan amount underlying outstanding Eurodollar futures contracts is in excess of $4 trillion. This contract is based on LIBOR (London Interbank Offer Rate), which is an interest rate payable on Eurodollar Time Deposits. This rate is the benchmark for many US borrowers and lenders. For example, a corporate borrower may be quoted a rate of LIBOR+200 basis points on a short-term loan. Eurodollar time deposits are non-negotiable, fixed rate US dollar deposits in banks that are not subject to US banking regulations. These banks may be located in Europe, the Carribean, Asia, or South America. US banks can take deposits on an unregulated basis through their international banking facilities. LIBOR is the rate at which major money center banks are willing to place Eurodollar time deposits at other major money center banks. Corporations usually borrow at a spread above LIBOR since a corporation's credit risk is greater than that of a major money center bank. By convention, LIBOR is quoted as an annualized rate based on an actual/360-day year (i.e., interest is paid for each day at the annual rate/360).

Example 4.22 demonstrates how interest is calculated on a LIBOR loan according to the conventions.

The Eurodollar futures contract is based on a 3-month $1 million Eurodollar time deposit. It is cash settled, so no actual delivery of the time deposit occurs when the contract expires. Delivery months are March, June, September, and December. The minimum price move is $25 per contract which is equivalent to 1 basis point: (.0001/4)1,000,000=25. The futures price at expiration (time T) is determined as F_T = 100-LIBOR. Prior to expiration, the futures price implies the interest rate that can be effectively locked in for a 3-month loan that begins on the day the contract matures. Settlement of the Eurodollar futures contract is illustrated in Example 4.23.

Example 4.24 contains a detailed illustration of how the Eurodollar futures contract can be used to hedge interest rate risk.

4.25  Hedging Market Risk

Another source of risk that an individual or organization may wish to hedge is stock market risk. For example, a person nearing retirement may wish to hedge the value of the equities component of his retirement fund against a stock market crash before he retires. A fund manager, who believes he can pick winners among individual stocks may wish to hedge market-wide movements. The dominant stock market index futures contract is the S&P 500 futures contract. This contract trades on the Chicago Mercantile Exchange and has delivery months March, June, September, and December. The underlying quantity is $500 times the level of the S&P 500 index. The minimum price move is 0.05 index points, which is $25 per contract. Example 4.26 illustrates the settlement mechanics for the S&P 500 contract.

Example 4.27 contains a detailed illustration of how the S&P 500 futures contract can be used to hedge stock market risk.

4.28  Hedging Foreign Exchange Risk

Another source of risk that an individual or organization may wish to hedge is foreign exchange risk. For example, a person who will be travelling overseas in the coming months may wish to hedge the value of the amount of money he intends to spend abroad against a devaluation of his domestic currency relative to the foreign currency. An exporter who sells goods overseas on credit may wish to hedge against a devaluation of the foreign currency in which payment occurs.

A number of foreign currency futures contracts trade on the International Monetary Market division of the Chicago Mercantile Exchange. The currencies on which contracts are based, and the underlying notional amount are listed in the following Table. Delivery months for all contracts are March, June, September, and December. Prices are quoted as US dollars per unit of foreign currency. For example, if one Swiss franc buys 69.15 US cents, the price will be quoted as 0.6915.

Example 4.29 contains a detailed illustration of hedging exchange risk.

4.30  Basis Risk

There is no such thing as a perfect hedge. You can never completely eliminate a cash position's risk. Consider a holder of Q Treasury bonds maturing in 2004 with a coupon rate of 8%. Assume that the holder of bonds believes that bond prices are going to fall. To hedge his risk, the person shorts an equivalent amount of futures contracts for Treasury bonds. At a later date, the person will close out both its bond and futures positions. At the close, the firm will receive B_T per bond sold in the regular spot or cash market. The futures price is F₀ at the time the futures are sold short, and its price at the closeout is F_T. Prior to the closeout, both B_T and F_T are uncertain, although F₀ is known. The usual computation of the funds that the person will have at closeout is:

From the above equation, the net revenue from the hedge position is composed of (1) a certain component that depends upon the futures price at the time of the hedge (F₀) and (2) an uncertain component that depends upon the difference between the price received for bonds in the spot market and the futures price at closeout (B_T-F_T). The difference between the spot and the futures price is called the basis. Thus, uncertainty about the net hedged revenue arises if there is uncertainty about the basis. To quote Holbrook Working, "hedging is speculation in the basis".

There are many reasons for the basis to be uncertain.

• First, the good or instrument being hedged may be different from the good or instrument for which there is a futures contract. This would be the case if a corporate bond offering is hedged with Treasury bond futures; basis risk arises due to the uncertainty of the yield differential at the time the hedge is lifted.
  
• Second, in commodity futures, there is basis risk due to locational differentials. For example, a cattle farmer in Texas who hedges with a cattle futures contract that calls for delivery in Omaha has the uncertainty of the closeout differential between the Texas steer price and the Omaha steer price. This is called locational basis risk. This is usually an important factor in agricultural contracts. The risk is compounded by the fact that the seller usually has the option of where delivery is made.
  
• The third type of basis risk arises because the seller of the futures contract often has the option to choose the quality of the goods or financial instrument delivered. For example, the Treasury bond futures market calls for delivery of any U.S. Treasury bond that is not callable within 15 years. Since there are many instruments that are candidates for delivery, the hedge has the risk of fluctuations in the yield spread between the instrument hedged and the instrument ultimately delivered.
  
• Fourthly, with most futures contracts, the seller has the choice of the date of delivery within the delivery month. This choice is an uncertain value and thus contributes to basis risk.
  
• Finally, the mark to market aspect of futures results in hedging risk. The uncertainty is about the amount of interest earned or forfeited due to the daily transfers of profits and losses. In fact, the equations for net revenue are not exactly right due to the omission of interest earned (lost) on futures profits (losses).

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4.31  The Volatility of Futures

A common mistake made is to assume that futures are much more volatile than stocks. Percentage changes of futures prices are generally less volatile than the percentage changes of a typical stock. Annualized standard deviations for most futures contracts are in the 15-20% range whereas a typical stock's is about 30%.

There is no reason that the futures should be played in a high risk manner by a large investor. Of course, if the futures investor does not have enough capital (5-8 times margin), then he is required to play with considerable leverage or not at all. Before taking great leverage, the small investor should consider looking at a smaller contract (grain on CBT is 5,000 bushels whereas Mid-America contract is 1,000 bushels).

The effect of leverage is to increase volatility. Borrowing to meet the margin requirements will increase gains but also increase losses. Setting aside larger amounts of capital which are invested in a safe asset will decrease the volatility.

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4.32  Trading the Contracts

The futures exchanges provide detailed information on each of their contracts. You should obtain this information before trading. It is especially important to know the details of the delivery procedure (if you are going to make delivery or take delivery). WWWFinance contains contract specifications and general information on over 200 futures and options contracts traded worldwide. In addition, a serious trader would be in contact with the exchange to confirm the most recent set or specifications.

4.33  Risk in the Futures Markets

As we have already seem, one the most important applications of the futures is for hedging. Futures contracts were initially introduced to help farmers that did not want to bear the risk of price fluctuations. The farmer could short hedge in March (agree to sell his crop) for a September delivery. This effectively locks in the price that the farmer receives. On the other side, a cereal company may want to guarantee in March the price that it will pay for grain in September. The cereal company will enter into a long hedge.

There are a number of important insights that should be reviewed. The first is that we should be careful about what we consider the investment in a futures contract. It is unlikely that the margin is the investment for most traders. It is rare that somebody plays the futures with a total equity equal to the margin. It is more common to invest some of your capital in a money market fund and draw money out of that account as you need it for margin and add to that account as you gain on the futures contract. It is also uncommon to put the full value of the underlying contract in the money market fund. It is more likely that the futures investor will put a portion of the value of the futures contract into a money market fund. The ratio of the value of the underlying contract to the equity invested in the money market fund is known as the leverage. The leverage is a key determinant of both the return on investment and on the volatility of the investment. The higher the leverage -- the more volatile are the returns on your portfolio of money market funds and futures. The most extreme leverage is to include no money in the money market fund -- only commit your margin.

The second important insight had to do with hedging with futures contracts. The concept of basis risk was introduced. It is extremely unlikely that you can create a perfect hedge. A perfect hedge is when the loss on your cash position is exactly offset by the gain in the futures position. We suggested some reasons why it is unlikely that we can construct a perfect hedge.

• The most obvious case is when you are trying to hedge a cash position with futures positions in different instruments. This is the case that we introduced in one of the first lectures when we hold the Ginnie Mae security and want to hedge this security with a combination of T-Bonds and Euros. It is unlikely, however, that at the expiration of the futures contract, the cash price of the T-Bond and Euros will equal the Ginnie Mae. This is the basis risk.
  
  
• A second type of basis risk arises out of the quality option. We discussed this in terms of food and financial instruments. If you are a farmer and want to lock in the price for your crop of wheat, you may use a futures contract that may call for delivery of a number of different types of wheat. Similarly, in the T-Bond and T-Note contracts, there are a whole range of instruments that are available for delivery. This difference will induce basis risk.
  
  
• Third, there is a timing option. The futures contract is different from an options contract. Most futures call for delivery within the contract month. It is unclear when the short will deliver the goods. This uncertainty leads to basis risk.
  
  
• The fourth type of risk is locational basis risk. This is mainly applicable to agricultural commodities. There could be a difference the cash price of the good that you are selling (cattle) and the futures price at a different location.
  
• The last type of uncertainty is linked to the uncertain interest rate flows from the money you make in excess of the margin.

4.34  More on Hedging

Since hedging is such an important application of futures contracts, we have provided more examples of hedging Some textbook examples which do the hedging incorrectly are included, to show you some of the common pitfalls involved with hedging.

Summary of Important Formulas

The price of a forward contract when there is a cost of carry q. When interest rates are constant, the same relationship holds for a futures contract.

The price of a forward contract when there is a dividend benefit d. When interest rates are constant, the same relationship holds for a futures contract.