is hedged by a company using a forwardcontract, there is no foreign exchange risk. When it is hedged using futures contracts, themarking-to-market process docs leave the company exposed to some risk. Explain the nature ofthis risk. In particular, consider whether the company is better off using a futures contract or aforward contract when
a. The value of the foreign currency falls rapidly during the life of the contract
b. The value of the foreign currency rises rapidly during the life of the contract
c. The value of the foreign currency first rises and then falls back to its initial value
d. The value of the foreign currency first falls and then rises back to its initial value
Assume that the forward price equals the futures price.
It is sometimes argued that a forward exchange rate is an unbiased predictor of future exchangerates. Under what circumstances is this so?
Show that the growth rate in an index futures price equals the excess return of the index over therisk-free rate. Assume that the risk-free interest rate and the dividend yield are constant.
Show that equation (3.7) is true by considering an investment in the asset combined with a shortposition in a futures contract. Assume that all income from the asset is reinvested in the asset. Usean argument similar to that in footnotes 2 and 4 and explain in detail what an arbitrageur woulddo if equation (3.7) did not hold.
The Value Line Index is designed to reflect changes in the value of a portfolio of over 1,600equally weighted stocks. Prior to March 9, 1988, the change in the index from one day to the nextwas calculated as the geometric average of the changes in the prices of the stocks underlying theindex. In these circumstances, does equation (3.12) correctly relate the futures price of the index toils cash price? If not, does the equation overstate or understate the futures price?Determination of Forward and Futures Prices 67
 
ASSIGNMENT QUESTIONS
 
3.24,
3.25,
3.26.
3.27.
3.28.
 
A stock is expected to pay a dividend of $1 per share in two months and in five months. The stockprice is $50, and the risk-free rate of interest is 8% per annum with continuous compounding forall maturities. An investor has just taken a short position in a six-month forward contract on thestock.a. What are the forward price and the initial value of the forward contract?b. Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8%per annum. What are the forward price and the value of the short position in the forwardcontract?
A bank offers a corporate client a choice between borrowing cash at 11% per annum andborrowing gold at 2% per annum. (If gold is borrowed, interest must be repaid in gold. Thus,100 ounces borrowed today would require 102 ounces to be repaid in one year.) The risk-freeinterest rate is 9.25% per annum, and storage costs are 0.5% per annum. Discuss whether the rateof interest on the gold loan is too high or too low in relation to the rate of interest on the cashloan. The interest rates on the two loans are expressed with annual compounding. The risk-freeinterest rate and storage costs are expressed with continuous compounding. Assume that noincome is earned on gold. Repeat your calculations for the situation where income of 1.5% perannum can be earned on gold.
A company that is uncertain about the exact date when it will pay or receive a foreign currencymay try to negotiate with its bank a forward contract that specifies a period during which deliverycan be made. The company wants to reserve the right to choose the exact delivery date to fit inwith its own cash flows. Put yourself in the position of the bank. How would you price theproduct that the company wants?
A forcign exchange trader working for a bank enters into a long forward contract to buy oncmillion pounds sterling at an exchange rate of 1.6000 in three months. At the same time, anothertrader on the next desk takes a long position in 16 three-month futures contracts on sterling. Thefutures price is 1.6000, and each contract is on 62,500 pounds. Within minutes of the trades beingexecuted the forward and the futures prices both increase to 1.6040. Both traders immediatelyclaim a profit of $4,000. The bank’s systems show that the futures trader has made a $4,000 profit,but the forward trader has made a profit of only $3,900. The forward trader immediately picks upthe phone to complain to the systems department. Explain what is going on here. Why are theprofits different?
A company enters into a forward contract with a bank to sell a foreign currency for K, at time 7}.The exchange rate at time 7, proves to be S; (> K,). The company asks the bank if it can roll the
contract forward until time T, (> 7,) rather than settle at time T,. The bank agrees to a newd
  
ery orice. K>, Exnlain how XK.ery price, Az, xplain how #
 68 CHAPTER 3
 
APPENDIX 3A
Proof That Forward and Futures Prices Are Equal When interest RatesAre Constant
This appendix demonstrates that forward and futures prices are equal when interest rates areconstant. Suppose that a futures contract lasts for n days and that F; is the futures price at the endof day i(0<i<n). Define 5 as the risk-free rate per day (assumed constant). Consider thefollowing strategy.’
1. Take a long futures position of ¢* at the end of day 0 (i.e., at the beginning of the contract).
2. Increase long position to e” at the end of day |.
3. Increase long position to e* at the end of day 2.
And so on.
This strategy is summarized in Table 3.5. By the beginning of day i, the investor has a long positionof e. The profit (possibly negative) from the position on day i is
(F; — Fe
Assume that the profit is compounded at the risk-free rate until the end of day n. Its value at theend of day n is(F; — Fee = (F; — Fae”
SOU = Fae
i=lThis is[Fy — Fy) + (Fant — Fy2) to + (Fi — Foe”? = (Fn — Foe™
Because F,, is the same as the terminal asset spot price, Sy, the terminal value of the investmentstrategy can be written(Sp — Fje”®
Table 3.5 The investment strategy to show that futures and forward prices are equal
Day 0 1 2 se ona] nFutures price Fo Fi Fy oo Fa-1 F,Futures position é es es se er) 0Gain/loss 0 (F.-Foje (Fyp-Fije® +: see (Fy — Fy_e®Gain/loss compounded todayn 0 (Fi-—Fyle® (Fo—Fije® =: ses (Fy — Fy-pe®
7 This strategy was proposed by J. C. Cox, J. E. Ingersoll, and S. A. Ross, “The Relation between Forward Pricesand Futures Prices,” Journal of Financial Economics, 9 (December 1981), 321-46.Determination of Forward and Futures Prices 69
 
An investment of Fo in a risk-free bond combined with the strategy just given yieldsFoe” + (Sp — Foe” = Spe?
at time 7. No investment is required for all the long futures positions described. It follows that anamount Fy can be invested to give an amount S;e”’ at time T.
Suppose next that the forward price at the end of day 0 is Gg. Investing Go in a riskless bond andtaking a long forward position of e” forward contracts also guarantees an amount Sye™ at time T.Thus, there are two investment strategies—one requiring an initial outlay of Fy and the otherrequiring an initial outlay of Go—both of which yield Se” at time T. It follows that, in theabsence of arbitrage opportunities,
In other words, the futures price and the forward price are identical. Note that in this nroof there isnothing special about the time period of one day. The futures price based on a contract withweekly settlements is also the same as the forward price when corresponding assumptions are
made. 
HEDGING STRATEGIES USINGFUTURES
Many of the participants in futures markets are hedgers. Their aim is to use futures markets toreduce a particular risk that they face. This risk might relate to the price of oil, a foreign exchangerate, the level of the stock market, or some other variable. A perfect hedge is one that completelyeliminates the risk. In practice, perfect hedges are rare. To quote one trader: ‘‘The only perfecthedge is in a Japanese garden.” For the most part, therefore, a study of hedging using futurescontracts is a study of the ways in which hedges can be constructed so that they perform as close toperfect as possible.
In this chapter we consider a number of general issues associated with the way hedges are set up.Whcen is a short futures position appropriate? When is a long futures position appropriatc? Whichfutures contract should be used? What is the optimal size of the futures position for reducing risk?At this stage, we restrict our attention to what might be termed hedge-and-forgert strategies. Weassume that no attempt is made to adjust the hedge once it has been put in place. The hedgersimply takes a futures-position at the beginning of the life of the hedge and closes out the positionat the end of the life of the hedge. In Chapter 14 we will examine dynamic hedging strategies inwhich the hedge is monitored closely and frequent adjustments are made.
Throughout this chapter we will treat futures contracts as forward contracts, that is, we willignore daily settlement. This means that we can ignore the time value of money in most situationsbecause all cash flows occur at the time the hedge is closed out.
4.1__BASIC PRINCIPLES
 
When an individual or company chooses to use futures markets to hedge a risk, the objective isusually to take a position that neutralizes the risk as far as possible. Consider a company thatknows it will gain $10,000 for each 1 cent increase in the price of a commodity over the next threemonths and lose $10,000 for each | cent decrease in the price during the same period. To hedge,the company’s treasurer should take a short futures position that is designed to offset this risk. Thefutures position should lead to a loss of $10,000 for each I cent increase in the price of thecommodity over the three months and a gain of $10,000 for each 1 cent decrease in the price duringthis period. If the price of the commodity goes down, the gain on the futures position offsets theloss on the rest of the company’s business. If the price of the commodity goes up, the loss on thefutures position is offset by the gain on the rest of the company’s business.
70Hedging Strategies Using Futures 71
Short Hedges
A short hedge is a hedge, such as the one just described, that involves a short position in futurescontracts. A short hedge is appropriate when the hedger already owns an asset and expects to sell itat some time in the future. For example, a short hedge could be used by a farmer who owns somehogs and knows that they will be ready for sale at the local market in two months. A short hedgecan also be used when an asset is not owned right now but will be owned at some time in thefuture. Consider, for example, a U.S. exporter who knows that he or she will receive euros in threemonths. The exporter will realize a gain if the euro increases in value relative to the U.S. dollar andwill sustain a loss if the euro decreases in value relative to the U.S. dollar. A short futures positionleads to a loss if the euro increases in value and a gain if it decreases in value. It has the effect ofoffsetting the exporter’s risk.
To provide a more detailed illustration of the operation of a short hedge in a specific situation,we assume thai it is May 15 today and that an oil producer has just negotiated a contract to seil| million barrels of crude oil. It has been agreed that the price that will apply in the contract is themarket price on August 15. The oil producer is therefore in the pasition considered above where itwill gain $10,000 for each 1 cent increase in the price of oil over the next three months and lose$10,000 for each | cent decrease in the price during this period. Suppose that the spot price onMay 15 is $19 per barrel and the August crude oil futures price on the New York MercantileExchange (NYMEX) is $18.75 per barrel. Because each futures contract on NYMEX is for thedelivery of 1,000 barrels, the company can hedge its exposure by shorting 1,000 August futurcscontracts. If the oil producer closes out its position on August 15, the effect of the strategy shouldbe to lock in a price close to $18.75 per barrel.
As an example of what might happen, suppose that the spot price on August 15 proves to be$17.50 per barrel. The company realizes $17.5 million for the oil under its sales contract. BecauseAugust is the delivery month for the futures contract, the futures price on August 15 should be veryclose to the spot price of $17.50 on that date. The company therefore gains approximately
$18.75 — $17.50 = $1.25
per barrel, or $1.25 million in total from the short futures position. The total amount realized fromboth the futures position and the sales contract is therefore approximately $18.75 per barrel, or$18.75 million in total.
For an alternative outcome, suppose that the price of oil on August 15 proves to be $19.50 perbarrel. The company realizes $19.50 for the oil and loses approximately
$19.50 — $18.75 = $0.75
per barrel on the short futures position. Again, the total amount realized is approximately $18.75million. It is casy to see that in all cases the company ends up with approximately $18.75 million.
Long Hedges
Hedges that involve taking a long position in a futures contract are known as /ong hedges. A longhedge is appropriate when a company knows it will have to purchase a certain asset in the futureand wants to lock in a price now.
Suppose that it is now January 15. A copper fabricator knows it will require 100,000 pounds ofcopper on May [5 to meet a certain contract. The spot price of copper is 140 cents per pound, and
the May futures price is 120 cents per pound. The fabricator can hedge its position by taking a72 CHAPTER 4
long position in four May futures contracts on the COMEX division of NYMEX and closing itsposition on May 15. Each contract is for the delivery of 25,000 pounds of copper. The strategy hasthe effect of locking in the price of the required copper at close to 120 cents per pound.
Suppose that the price of copper on May [5 proves to be 125 cents per pound. Because May isthe delivery month for the futures contract, this should be very close to the futures price. Thefabricator therefore gains approximatcly
100,000 x ($1.25 — $1.20) = $5,000
on the futures contracts. It pays 100,000 x $1.25 = $125,000 for the copper, making the total costapproximately $125,000 — $5,000 = $120,000. For an alternative outcome, suppose that thefutures price is 105 cents per pound on May 15. The fabricator then loses approximately
100,000 x ($1.20 — $1.05) = $15,000
on the futures contract and pays 100,000 x $1.05 = $105,000 for the copper. Again, the total costis approximately $120,000, or 120 cents per pound.
Note that it is better for the company to use futures contracts than to buy the copper onJanuary [5 in the spot market. If it does the latter, it will pay 140 cents per pound instead of120 cents per pound and will incur both interest costs and storage costs. For a company usingcopper on a regular basis, this disadvantage would be offset by the convenience yield associatedwith having the copper on hand (see Section 3.12 for a discussion of convenience yields).However, for a company that knows it will not require the copper until May 15, the convenienceyield has no value.
Long hedges can also be used to partially offset an existing short position. Consider an investorwho has shorted a certain stock. Part of the risk faced by the investor is related to the performanceof the stock market as a whole. The investor can neutralize this risk by taking a long position inindex futures contracts. This type of hedging strategy is discussed further later in the chapter.
The examples we have looked at assume that the futures position is closed out in the deliverymonth. The hedge has the same basic effect if delivery is allowed to happen. However, making ortaking delivery can be a costly business. For this reason, delivery is not usually made even whenthe hedger keeps the futures contract until the delivery month. As will be discussed later, hedgerswith long positions usually avoid any possibility of having to take delivery by clusing out theirpositions before the delivery period.
We have also assumed in the two examples that a futures contract is the same as a forwardcontract. In practice, marking to market does have a small effect on the performance of a hedge.As explained in Chapter 2, it means that the payoff from the futures contract is realized day by daythroughout the life of the hedge rather than all at the end.
4.2 ARGUMENTS FOR AND AGAINST HEDGING
 
The arguments in favor of hedging are so obvious that they hardly need to be stated. Mostcompanies are in the business of manufacturing or retailing or wholesaling or providing a service.They have no particular skills or expertise in predicting variables such as interest rates, exchangerates, and commodity prices. It makes sense for them to hedge the risks associated with thesevariables as they arise. The companies can then focus on their main activities—in whichHedging Strategies Using Futures 73
 
presumably they do have particular skills and expertise. By hedging, they avoid unpleasantsurprises such as sharp rises in the price of a commodity.
In practice, many risks are left unhedged. In the rest of this section we will explore some of thereasons.
eee Osneaging ang onarenoiaers
One argument sometimes put forward is that the shareholders can, if they wish, do the hedgingthemselves. They do not need the company to do it for them. Ihis argument is, however, open toquestion. It assumes that shareholders have as much information about the risks faced by acompany as does the company’s management. In most instances, this is not the case. Theargument also ignores commissions and other transactions costs. These are less expensive perdollar of hedging for large transactions than for small transactions. Hedging is therefore likely tobe less expensive when carried out by the company than by individual shareholders. Indeed, thesize of futures contracts makes hedging by individual sharcholders impossible in many situations.
One thing that shareholders can do far more easily than a corporation is diversify risk. Asharcholder with a well-diversified portfolio may be immune to many of the risks faced by acorporation. For example, in addition to holding shares in a company that uses copper, a well-diversified shareholder may hold shares in a copper producer, so that there is very little overallexposure to the price of copper. If companies are acting in the best interests of well-diversifiedshareholders, it can be argued that hedging is unnecessary in many situations. However, theextent to which managements are in practice influenced by this type of argument is open toquestion.
Hedging and Competitors
If hedging is not the norm in a certain industry, it may not make sense for one particular companyto choose to be different from all the others. Competitive pressures within the industry may be suchthat the prices of the goods and services produced by the industry fluctuate to reflect raw material
costs. interest rates. exchange rates. and so on. A company that does not hedge can exnect its profitcosts, interest rates, exchange rates, and so on. A company that does not hedge can expect its profit
margins to be roughly constant. However, a company that does hedge can expect its profit marginsto fluctuate!
To illustrate this point, consider two manufacturers of gold jewelry, SafeandSure Company andTakeaChance Company. We assume that most jewelry manufacturers do not hedge againstmovements in the price of gold and that TakeaChance Company is no exception. However,SafeandSure Company has decided to be different from its competitors and to use futures contracts
to hedge its purchases of gold over the next 18 m
Has ‘SafeandSure Company reduced its risks? If the price of gold goes up, economic pressureswill tend to lead to a corresponding increase in the wholesale price of the jewelry, so thatTakeaChance Company’s profit margin is unaffected. By contrast, SafeandSure Company’s profitmargin will increase after the effects of the hedge have been taken into account. If the price of goldgoes down, economic pressures will tend to lead to a corresponding decrease in the wholesale priceof the jewelry. Again, TakeaChance Company’s profit margin is unaffected. However, SafeandSureCompany’s profit margin goes down. In extreme conditions, SafeandSure Company’s profitmargin could become negative as a result of the “hedging” carried out! The situation issummarized in Table 4.1.
The foregoing example emphasizes the importance of looking at the big picture when hedging.
 74 CHAPTER 4
Table 4.1 Danger in hedging when competitors do not
 
Change in Effect on price _ Effect on profits of Effect on profitsgold price of gold jewelry TakeaChance Co. of SafeandSure Co.
Increase Increase None IncreaseDecrease Decrease None Decrease
All the implications of price changes on a company’s profitability should be taken into account inthe design of a hedging strategy to protect against the price changes.
Other Considerations
It is important to realize that a hedge using futures contracts can result in a decrease or an increasein a company’s profits relative to the position it would be in with no hedging. In the exampleinvolving the oil producer considered earlier, if the price of oil goes down, the company losesmoney on its sale of 1 million barrels of oil and the futures position leads to an offsetting gain. Thetreasurer can be congratulated for having had the foresight to put the hedge in place. Clearly, thecompany is better off than it would be with no hedging. Other executives in the organization, it ishoped, will appreciate the contribution made by the treasurer. If the price of oil goes up, thecompany gains from its sale of the oil, and the futures position leads to an offsetting loss. Thecompany is in a worse position than it would be with no hedging. Although the hedging decisionwas perfectly logical, the treasurer may in practice have a difficult time justifying it. Suppose thatthe price of oil is $21.75 at the end of the hedge, so that the company loses $3 per barrel on thefutures contract. We can imagine a conversation such as the following between the treasurer andthe president.
PRESIDENT: This is terribie. We've iost $3 miilion in the futures market in the space of threemonths. How could it happen? I want a full explanation.
TREASURER: The purpose of the futures contracts was to hedge our exposure to the price ofoil—not to make a profit. Don’t forget that we made about $3 million from thefavorable effect of the oil price increases on our business.
PRESIDENT: What’s that got to do with it? That’s like saying that we do not need to worrywhen our sales are down in California because they are up in New York.
TREASURER: If the price of oil had gone down...
PRESIDENT: I don’t care what would have happened if the price of oil had gone down. Thefact is that it went up. I really do not know what you were doing playing thefutures markets like this. Our shareholders will expect us to have doneparticularly well this quarter. I’m going to have to explain to them that youractions reduced profits by $3 million. I’m afraid this is going to mean no bonusfor you this year.
TREASURER: That’s unfair. I was only...
PRESIDENT: Unfair! You are lucky not to be fired. You lost $3 million.
TREASURER: It all depends how you look at it...
It is easy to See why many treasurers are reluctant to hedge! Hedging reduces risk for the company.Hedging Strategies Using Futures 75
However, it may increase risks for the treasurer if others do not fully understand what is beingdone. The only real solution to this problem involves ensuring that all senior executives within theorganization fully understand the nature of hedging before a hedging program is put in place.Ideally, hedging strategies are set by a company’s board of directors and are clearly communicatedto both the company’s management and its shareholders.
4.3 BASIS RISK
 
 
The hedges in the examples considered so far have been almost too good to be true. The hedgerwas able to identify the precise date in the future when an asset would be bought or sold. Thehedger was then able to use futures contracts to remove almost all the risk arising from the price of
the asset on that date. In practice, hedging is often not quite as straightforward. Some of the
reasons are as follows:
1. The asset whose price is to be hedged may not be exactly the same as the asset underlying thefutures contract.
2. The hedger may be uncertain as to the exact date when the asset will be bought or sold.3. The hedge may require the futures contract to be closed out well before its expiration date.
These problems give rise to what is termed basis risk. This concept will now be explained.
The Basis
The basis in a hedging situation is as follows:'Basis = Spot price of asset to be hedged — Futures price of contract used
If the asset to be hedged and the asset underlying the futures contract are the same, the basisshould be zero at the expiration of the futures contract. Prior to expiration, the basis may bepositive or negative. From the analysis in Chapter 3, when the underlying asset is a low-interest-rate currency or gold or silver, the futures price is greater than the spot price. This means that thebasis is negative. For high-interest-rate currencies and many commodities, the reverse is true, andthe basis is positive.
When the spot price increases by more than the futures price, the basis increases. This is referredto as a strengthening of the basis. When the futures price increases by more than the spot price, thebasis declines. This is referred to as a weakening of the basis. Figure 4.1 illustrates how a basis might
change over time in a situation where the basis is positive prior to expiration of the futures
contract.To examine the nature of basis risk, we will use the following notation:S,: Spot price at time t,Sy: Spot price at time tF,: Futures price at time ¢,
' This is the usual definition. However, the alternative definitionBasis = Futures price — Spot price
is sometimes used, particularly when the futures contract is on a financial asset.76 CHAPTER 4
Spot price
Futures price
 
th Ly
Figure 4.1. Variation of basis over time
F,: Futures price at time t)b,: Basis at time ¢;b,: Basis at time ty
We will assume that a hedge is put in place at time ¢, and closed out at time ft). As an example, wewill consider the case where the spot and futures prices at the time the hedge is initiated are $2.50and $2.20, respectively, and that at the time the hedge is closed out they are $2.00 and $1.90,respectively. This means that S; =2.50, F; =2.20, S. =2.00, and F,=1.90.
From the definition of the basis, we have
bh =S,-Fiby = S) — Fy
and hence, in our example, 6; = 0.30 and 5, =0.10.
Consider first the situation of a hedger who knows that the asset will be sold at time t, and takesa short futures position at time t). The price realized for the asset is S, and the profit on the futuresposition is F, — Fy. The effective price that is obtained for the asset with hedging is therefore
StF -Fp=F +h,
In our example, this is $2.30. The value of F, is known at time ¢,. If 6) were also known at thistime, a perfect hedge would result. The hedging risk is the uncertainty associated with b, and isknown as basis risk. Consider next a situation where a company knows it will buy the asset at timet and initiates a long hedge at time ¢;. The price paid for the asset is S; and the loss on the hedgeis F, — Fy. The effective price that is paid with hedging is therefore
S+F, -F, =F, +b,
This is the same expression as before and is $2.30 in the example. The value of F, is known at timet,, and the term 4, represents basis risk.
For investment assets such as currencies, stock indices, gold, and silver, the basis risk tends to bemuch less than for consumption commodities. The reason, as shown in Chapter 3, is that arbitrageHedging Strategies Using Futures 77
 
arguments lead to a well-defined relationship between the futures price and the spot price of aninvestment asset. The basis risk for an investment asset arises mainly from uncertainty as to thelevel of the risk-free interest rate in the future. In the case of a consumption commodity,imbalances between supply and demand and the difficulties sometimes associated with storingthe commodity can lead to large variations in the convenience yield. This in turn leads to a bigincrease in the basis risk.
The asset that gives rise to the hedger’s exposure is sometimes different from the asset underlyingthe hedge. The basis risk is then usually greater. Define Sj as the price of the asset underlying thefutures contract at time t,. As before, $) is the price of the asset being hedged at time t). Byhedging, a company ensures that the price that will be paid (or received) for the asset is
This can be written asFy + (Sz — Fy) + (8) — 53)
‘The terms Sy — F, and S) — Sj represent the two components of the basis. The S¥ — F, term is thebasis that would exist if the asset being hedged were the same as the asset underlying the futurescontract. The S$; — S; term is the basis arising from the difference between the two assets.
Note that basis risk can jead to an improvement or a worsening of a hedger’s position. Consider ashort hedge. If the basis strengthens unexpectedly, the hedger’s position improves; if the basisweakens unexpectedly, the hedger’s position worsens. For a long hedge, the reverse holds. If thebasis strengthens unexpectedly, the hedger’s position worsens; if the basis weakens unexpectedly, thehedger’s position improves.
Choice of Contract
One key factor affecting basis risk is the choice of the futures contract to be used for hedging. Thischoice has two components:
1, The choice of the