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
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
(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.
After completing this class, you should be able to:
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
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.
|Example 4.4: Forward contract mechanics.
A wheat farmer has just planted a crop that is expected to yield 5000 bushels. To eliminate the risk of a decline in the price of wheat before the harvest, the farmer can sell the 5000 bushels of wheat forward. A miller may be willing to take the other side of the contract. The two parties agree today on a forward price of 550 cents per bushel, for delivery five months from now when the crop is harvested. No money changes hands now. In five months, the farmer delivers the 5000 bushels to the miller in exchange for $27,500. Note that this price is fixed and does not depend upon the spot price of wheat at the time of delivery and payment.
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 (e0.05(0.5)-1) = $2.53, payable immediately.
We use the following notation, which is common in forward markets:
|St||The spot price of the underlying commodity at time t|
|S0||the spot price now, which is known.|
|ST||The spot price at maturity of the contract and is not known when the contract is entered into.|
|r||The riskless rate of interest from now until maturity of the contract,|
|q||The cost of carry of the underlying commodity|
|F||The forward price for delivery at time T.|
Both r and q are expressed as continuously compounded annual rates. Consider the strategy of:
Arbitrage relationship between spot and forward contracts
|Position||Initial Cash Flow||Terminal Cash Flow|
|Buy one unit of commodity||
|Pay Cost of Carry||
-S0 (eqT - 1)
|Enter 6-month forward sale||0||F - ST|
|Net Portfolio Value||0||F - S0 e(q+r)T|
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
F = S0 e(q+r)T
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:
|Position||Initial Cash Flow||Terminal Cash Flow|
|Buy one unit of commodity
Pay Cost of Carry
Enter 6-month forward sale
-550 (e0.06x0.5 - 1)
600 - ST
|Net Portfolio Value||0||600-550 e(0.06+0.05)0.5 = 18.90|
That is, it is possible to lock in a sure profit that requires no intial cash outlay.
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.
|Example 4.8: Forward contacts and risk management.
XYZ is a multinational corporation based in the US. Its manufacturing facilities are located in Pittsburgh and hence its labor and manufacturing costs are incurred in US dollars (USD). A large fraction of its sales, however, are made to German customers who pay for the goods in Deutschemarks (GDM).
There is a six-month lead time between the placement of a customer order and delivery of the product. XYZ's cost of production is 80% of the sale price. Suppose XYZ receives a $1MM GDM order and that the current USD/GDM exchange rate is 0.60 (i.e. 1 GDM = 0.60 USD). The cost of production of this order is $480,000 (0.60 x $1MM x 0.80). The exchange rate six months from now is, of course, uncertain in which case XYZ is exposed to exchange rate risk.
If the exchange rate stays at 0.60, then XYZ will convert the 1MM GDM to $600,000 and earn a 25% profit on the $480,000 cost of production. If, however, the exchange rate falls to 0.40 six months from now, XYZ will convert the 1MM GDM to only $400,000, registering a loss on the sale.
Conversely, if the exchange rate rises to 0.80 six months from now, XYZ will convert the 1MM GDM to $800,000, registering a very large profit on the sale. Whereas XYZ are very good at manufacturing and marketing their product, they have no expertise in forecasting exchange rate movements. Therefore, they want to avoid the exchange rate risk inherent in this transaction (i.e., the risk that they do everything right and then lose money on the sale, solely because exchange rates move against them). They can do this by selling forward 1MM GDM. This involves entering a contract today with, say, an investment bank under which XYZ agrees to deliver 1MM GDM six months from now in exchange for a fixed number of US dollars. This rate of exchange is the six-month forward rate. Suppose the six-month forward rate is 0.62 (which is set according to market expectations and relative interest rates as described below). Then, when XYZ receives 1m GDM from its customer, they deliver it to the investment bank in exchange for $620,000 (locking in a profit) regardless of whether the exchange rate happens to be 0.40 or 0.80 at that time.
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.
|Example 4.10: Marking to market.
Suppose an Australian futures speculator buys one SPI futures contract on the Sydney Futures Exchange (SFE) at 11:00am on June 6. At that time, the futures price is 2300. At the close of trading on June 6, the futures price has fallen to 2290 (what causes futures prices to move is discussed below).
Underlying one futures contract is $25 x Index, so the buyer's position has changed by $25(2290-2300)=-$250. Since the buyer has bought the futures contract and the price has gone down, he has lost money on the day and his broker will immediately take $250 out of his account. This immediate reflection of the gain or loss is known as marking to market.
Where does the $250 go? On the opposite side of the buyer's buy order, there was a seller, who has made a gain of $250 (note that futures trading is a zero-sum game - whatever one party loses, the counterparty gains). The $250 is credited to the seller's account. Suppose that at the close of trading the following day, the futures price is 2310. Since the buyer has bought the futures and the price has gone up, he makes money. In particular, $25(2310-2290)=+$500 is credited to his account. This money, of course, comes from the seller's account.
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.
|Example 4.11: Cash settlement.
Suppose the SPI futures contact price was 2350 at the close of trading on the day before expiration and 2360 at the close of trading on the expiration day. Settlement simply involves a payment of $25(2360-2350) = $250 from the seller's account to the buyer's account. The expiration day is treated just like any other day in terms of standard marking to market.
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 4.12: Deliverable quality:
Greasy wool futures.
Delivery must be made at approved warehouses in the major wool selling centres throughout Australia. For wool to be deliverable, it must possess the relevant measurement certificates issued by the Australian Wool Testing Authority (AWTA) and appraisal certificates issued by the Australian Wool Exchange Limited (AWEX). In particular, it must be good topmaking merino fleece with average fibre diameter of 21.0 microns, with measured mean staple strength of 35 n/ktx, mean staple length of 90mm, of good colour with less than 1.0% vegetable matter. Because any particular bale of wool is unlikely to exactly match these specifications, wool within some prespecified tolerance is deliverable. In particular, 2,400 to 2,600 clean weight kilograms of merino fleece wool, of good topmaking style or better, good colour, with average micron between 19.6 and 22.5 micron, measured staple length between 80mm and 100mm, measured staple strength greater than 30 n/ktx, less than 2.0% vegetable matter is deliverable. Premiums and discounts for delivery that does not match the exact specifications of the underlying contract are fixed on the Friday prior to the last day of trading for all deliverable wools above and below the standard, quoted in cents per kilogram clean.
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.
|Example 4.13: Physical delivery.
Suppose the greasy wool futures contact price was 700 cents at the close of trading on the expiration day. Settlement involves physical delivery, from the seller of the futures contract to the buyer, of the underlying quantity of wool (2,500 kilograms) on the business day following the expiration day. Delivery, therefore, involves the seller delivering 2,500 kg of wool to the buyer, in return for a payment of A$17,500.
The wool must be within the tolerance described above. If the wool is of better quality than is specified in the contract, a premium must be paid. Conversely wool of lower quality involves a discount. It is the seller of the futures who must make delivery of the wool and he has the option to choose what quality he will deliver, subject to the schedule of discounts and premiums.
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
|Example 4.15: Margin
Suppose a contract requires initial margin of $7,000 and maintenance margin of $5,000. The following table illustrates the margining procedure and the cash flows required for the buyer of a futures contract.
Note that when the margin balance falls below the maintenance margin, it must be restored to the initial level. Note also that when the futures moves favorably (as at time 3) the marking to market cash inflow can be immediately withdrawn - it need not remain in the margin account.
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.
Cash Flows of Forward and Futures Contracts
|Forward Cash Flow||0||0||0||...||ST-FO0|
|Futures Cash Flow||0||FU1-FU0||FU2-FU1||...||FUT - FUT-1|
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 er 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 er(FU1-FU0), which he invests (or borrows) until maturity. At maturity, this has grown to er(T-1)er[FU1-FU0)] = erT(FU1-FU0).
At time 1 he increases his holding to e2r contracts. At time 2 the profit (possibly negative) on this position is e2r(FU2-FU1), which he invests (or borrows) until maturity. At maturity, this has grown to er(T-2)e2r(FU2-FU1) = erT(FU2-FU1). At time 2 he increases his holding to e3r and so on. At maturity, the total payoff on the futures position is:
e rT [(FU1 - FU0) + (FU2 - FU1) + ... + (FUT - FUT-1)] = e rT (ST - FU0)
where we note that FUT = ST. 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 S0erT.
Now consider the strategy of buying erT 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:
erT(ST - FO0) + FO0 erT = S0 erT
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.
|Time||Forward Position||Futures Position|
|Net Cash Flow||-FO0||-FU0|
Number of Contracts Purchased
Cash Flow from Contract
Investment in Bonds
e2r - er
|Net Cash Flow||0||0|
Number of Contracts Purchased
Cash Flow from Contract
Investment in Bonds
ert - er(t-1)
|Net Cash Flow||0||0|
Cash Flow from Contract
Payoff from Bonds
erT(ST - FO0)
erT(ST - FU0)
erT[FU0 + (FU1 - FU0) + ... + (FUT-1 - FUT-2)]
|Net Cash Flow||STerT||STerT|
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
F = S0 e (r-d)T
|F||The Futures price|
|S0||The current value of the S&P 500 stock index|
|r||The interest rate (annual continuously compounded T-Bill rate)|
|d||The dividend yield on the index (continuously compounded annual rate)|
|T||The time to maturity of the contract|
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.
Arbitrage Relationship Between Spot and Futures Contract
|Position||Time 0||Time T|
|Buy e-dT units of index||-e-dTS0||ST|
|Sell one Futures Contract||0||F - ST|
|Net Position||0||F - S0e(r-dT)|
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.
|Example 4.18: Futures arbitrage: Buy index - Sell
Suppose the S&P 500 stock index is at $295 and the six-month futures contract on that index is at $300. If the prevailing T-Bill rate is 7% and the dividend rate is 5%, an arbitrage opportunity exists because F=300 > S e(r-d)T = 297.96. The arbitrage can be executed by buying low and selling high. In this case, the futures contract is relatively overvalued, so we sell the futures and buy the index.
In particular, the strategy is to
This generates the following cash flows:
Hence this strategy generates an arbitrage profit of $2.04 six months from now.
|Example 4.19: Futures arbitrage: Sell index - Buy futures
Suppose the S&P 500 stock index is at $300 and the six-month futures contract on that index is at $300. If the prevailing T-Bill rate is 7% and the dividend rate is 5%, an arbitrage opportunity exists because F = 300 < S e(r-d)T = 303.02. The arbitrage can be executed by buying low and selling high. In this case, the futures contract is relatively undervalued, so we buy the futures and sell the index.
In particular, the strategy is to
This generates the following cash flows:
Hence this strategy generates an arbitrage profit of $3.02 six months from now.
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.
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.
|Example 4.22: LIBOR Conventions.
If 3-month (90 actual days) LIBOR is quoted as 8%, the interest payable on a $1 million loan at the end of the 3-month borrowing period is
(.08)(90/360) $1,000,000 = ((.08)/4) $1,000,000 = $20,000
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 FT = 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 4.23: Settlement of Eurodollar
Suppose you purchased 1 December Eurodollar futures contract on November 15 when the price was 94.86. If interest rates fall 100 basis points between November 15 and expiration of the futures contract in December, what is your total gain or loss on the contract at settlement?
First note that no money changes hands at the time you buy the contract. This is the nature of all futures contracts. The November 15 price of 94.86 implies that the LIBOR rate of interest was 100-94.86=5.14% at that time. If LIBOR falls 100 basis points by the time the December contract expires, LIBOR will then be 4.14%. Therefore, the expiration futures price will be 100-4.14=95.86. The total gain is therefore:
0.25 (1,000,000)(Ft-F0) = 0.25 (1,000,000)(0.9586-0.9486) = $2,500
That is, to settle the contract, your counterparty will give you $2,500.
Example 4.24 contains a detailed illustration of how the Eurodollar futures contract can be used to hedge interest rate risk.
|Example 4.24: Hedging with the
Eurodollar Futures Contract.
It is currently November 15 and your company is aware that it needs to borrow $1 million on December 16 to pay a liability which falls due on that day. The loan can be repaid on March 16 when an account receivable will be collected. The current LIBOR rate is 5.14%. Your company is concerned that interest rates will rise between now and December 16, in which case you will pay a higher rate of interest on your loan. How can your company lock in the current rate of 5.14%?
Your company stands to lose if interest rates increase. Therefore, you want enter a futures position that increases in value if interest rates rise. Then, if interest rates rise, your company loses by paying higher interest charges on the loan, but your company gains by profiting on the futures position. Conversely, if interest rates fall, your company gains by paying lower interest charges on the loan, but your company loses on the futures position. Ideally, the loss and the gain would exactly cancel, whether interest rates rise or fall.
From the construction of the Eurodollar futures contract, we know that if the interest rate rises, the futures price will fall. Therefore, you will sell 1 December Eurodollar futures contract at 94.86. Underlying this contract is a notional 3-month $1 million dollar loan to be entered into on December 16 (the day the contract expires).
If we could lock in the rate of 5.14%, the total interest on the loan would be 0.0514($1 million)/4 = $12,850.
First, suppose that on December, 16 LIBOR is 6.14%. Interest on the
loan will be
Now suppose that on December, 16 LIBOR is 4.14%. Interest on the loan
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 4.26: Settlement of the
S&P 500 Futures Contract.
It is currently November 15 and the S&P 500 index is at 382.62. The December S&P 500 futures price is 383.50. If you buy 1 December S&P 500 futures contract, how much will you gain if the futures price at expiration is $393.50?
The gain on your futures position is $500(Ft-F0) = $500(393.50-383.50)=$5,000. That is, to settle the contract, your counterparty will give you $5,000.
Example 4.27 contains a detailed illustration of how the S&P 500 futures contract can be used to hedge stock market risk.
|Example 4.27: Hedging with the
S&P 500 Futures Contract.
A portfolio manager holds a portfolio that mimics the S&P 500 index. The S&P 500 index started the year at 306.8 and is currently at 382.62. The December S&P 500 futures price is currently 383.50. The manager's fund was valued at $76.7 million at the beginning of the year. Since the fund has already generated a handsome return for the year, the manager wishes to lock in its current value. That is, he is willing to give up potential increases in order to ensure that the value of the fund does not decrease. How does he lock in the current value of the fund?
First note that at the December futures price of 383.50, the return on the index, since the beginning of the year, is 383.5/306.80-1 = 25%. If the manager is able to lock in this return on his fund, the value of the fund will be 1.25($76.7 million) = $95.875 million. Since the notional amount underlying an S&P 500 futures contract is 500(383.50) = $191,750, the manager can lock in the 25% return by selling 95,875,000/191,750=500 contracts. To illustrate that this position does indeed form a perfect hedge, we examine the net value of the hedged position under two scenarios.
First, suppose the value of the S&P 500 index is 303.50 at the end of December. In this case, the value of the fund will be (303.50/383.50)95.875 million = 75.875 million. The gain on the futures position will be -500(500)(303.50-383.50) = 20 million. Hence the total value of the hedged position is 75.875+ 20 = 95.875 million, locking in a 25% return for the year.
Now suppose that the value of the S&P 500 index is 403.50 at the end of December. In this case, the value of the fund will be 403.50/383.50(95.875 million) = 100.875 million. The gain on the futures position will be -$500(500)(403.50-383.50) = -5 million. Hence the total value of the hedged position is 100.875-5=95.875 million, again locking in a 25% return for the year.
Another source of risk that an individual or organization may wish to hedge is foreign exchange risk. For example, a person who will be travelling overseas in the coming months may wish to hedge the value of the amount of money he intends to spend abroad against a devaluation of his domestic currency relative to the foreign currency. An exporter who sells goods overseas on credit may wish to hedge against a devaluation of the foreign currency in which payment occurs.
A number of foreign currency futures contracts trade on the International Monetary Market division of the Chicago Mercantile Exchange. The currencies on which contracts are based, and the underlying notional amount are listed in the following Table. Delivery months for all contracts are March, June, September, and December. Prices are quoted as US dollars per unit of foreign currency. For example, if one Swiss franc buys 69.15 US cents, the price will be quoted as 0.6915.
Denomination of Foreign Currency Futures Contracts
|British Pound||62,500 L|
|Canadian Dollar||100,000 C$|
|German Mark||125,000 DM|
|Japanese Yen||12,500,000 Y|
|Swiss Franc||125,000 SF|
|French Franc||250,000 FF|
|Australian Dollar||125,000 A$|
Example 4.29 contains a detailed illustration of hedging exchange risk.
|Example 4.29: Hedging with the
Swiss Franc Futures Contract.
Your company sells 10 machines to a Swiss company. The sale price is 100,000 Swiss Francs each and payment is to be made at the end of the calendar year. The December futures price for Swiss Francs is 0.6915. You are worried that the Swiss Franc will depreciate against the US Dollar between now and the end of the year. How can you hedge this exchange rate risk?
Note that since (1) the total exposure is one million Swiss Francs and (2) each futures contract is for 125, 000 Francs, eight contracts are required to hedge the exposure. Further, since (1) the company stands to lose if the Swiss Franc depreciates (each Swiss Franc can be converted back into a smaller number of Dollars) and (2) the futures contracts decrease in value if the Swiss Franc depreciates (since the basis of the contract is Swiss Francs per Dollar), the contracts should be sold.
To illustrate that selling eight futures contracts provides an adequate hedge, first suppose that the value of the Swiss Franc is 0.30 at the end of December. In this case, the US Dollar value of the payment for the machines will be 0.30(10)(100,000) = $300,000. The gain on the futures position will be -8(125,000)(0.30-0.6915) = $391,500. Hence the total income is $691,500, which equals the unhedged income in dollars if the exchange rate does not fluctuate.
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 BT 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 FT. Prior to the closeout, both BT and FT are uncertain, although F0 is known. The usual computation of the funds that the person will have at closeout is:
Net Revenue(bond sale plus futures) = Q[BT + (FT - F0)] =QF0 + Q[BT - FT]
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 (BT-FT). 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.
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.
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.
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
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
F = S0 e(r+q)T
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.
F = S0 e(r-d)T
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.