Futures and option

Hedging Strategies
Using Futures

Many of the participants in futures markets are hedgers. Their aim is to use futures
markets to reduce a particular risk that they face. This risk might relate to fluctuations
in the price of oil, a foreign exchange rate, the level of the stock market, or some other
variable. A perfect hedge is one that completely eliminates the risk. Perfect hedges are
rare. For the most part, therefore, a study of hedging using futures contracts is a study
of the ways in which hedges can be constructed so that they perform as close to perfect
as possible.

In this chapter we consider a number of general issues associated with the way hedges
are set up. When is a short futures position appropriate? When is a long futures
position appropriate? Which futures 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-forget strategies. We assume that no attempt is made to
adjust the hedge once it has been put in place. The hedger simply takes a futures
position at the beginning of the life of the hedge and closes out the position at the end
of the life of the hedge. In Chapter 18 we will examine dynamic hedging strategies in
which the hedge is monitored closely and frequent adjustments are made.

The chapter initially treats futures contracts as forward contracts (that is, it ignores
daily settlement). Later it explains an adjustment known as ‘‘tailing’’ that takes account
of the difference between futures and forwards.

3.1 BASIC PRINCIPLES

When an individual or company chooses to use futures markets to hedge a risk, the
objective is usually to take a position that neutralizes the risk as far as possible.
Consider a company that knows it will gain $10,000 for each 1 cent increase in the
price of a commodity over the next 3 months and lose $10,000 for each 1 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. The futures position should
lead to a loss of $10,000 for each 1 cent increase in the price of the commodity over
the 3 months and a gain of $10,000 for each 1 cent decrease in the price during this
period. If the price of the commodity goes down, the gain on the futures position
offsets the loss on the rest of the company’s business. If the price of the commodity

47

3C H A P T E R

goes up, the loss on the futures position is offset by the gain on the rest of the
company’s business.

Short Hedges

A short hedge is a hedge, such as the one just described, that involves a short position in
futures contracts. A short hedge is appropriate when the hedger already owns an asset
and expects to sell it at some time in the future. For example, a short hedge could be
used by a farmer who owns some hogs and knows that they will be ready for sale at the
local market in two months. A short hedge can also be used when an asset is not owned
right now but will be owned at some time in the future. Consider, for example, a US
exporter who knows that he or she will receive euros in 3 months. The exporter will
realize a gain if the euro increases in value relative to the US dollar and will sustain a
loss if the euro decreases in value relative to the US dollar. A short futures position
leads to a loss if the euro increases in value and a gain if it decreases in value. It has the
effect of offsetting the exporter’s risk.

To provide a more detailed illustration of the operation of a short hedge in a specific
situation, we assume that it is May 15 today and that an oil producer has just negotiated
a contract to sell 1 million barrels of crude oil. It has been agreed that the price that will
apply in the contract is the market price on August 15. The oil producer is therefore in
the position where it will gain $10,000 for each 1 cent increase in the price of oil over the
next 3 months and lose $10,000 for each 1 cent decrease in the price during this period.
Suppose that on May 15 the spot price is $80 per barrel and the crude oil futures price
for August delivery is $79 per barrel. Because each futures contract is for the delivery of
1,000 barrels, the company can hedge its exposure by shorting (i.e., selling) 1,000
futures contracts. If the oil producer closes out its position on August 15, the effect
of the strategy should be to lock in a price close to $79 per barrel.

To illustrate what might happen, suppose that the spot price on August 15 proves to
be $75 per barrel. The company realizes $75 million for the oil under its sales contract.
Because August is the delivery month for the futures contract, the futures price on
August 15 should be very close to the spot price of $75 on that date. The company
therefore gains approximately

$79�$75 ¼ $4

per barrel, or $4 million in total from the short futures position. The total amount
realized from both the futures position and the sales contract is therefore approximately
$79 per barrel, or $79 million in total.

For an alternative outcome, suppose that the price of oil on August 15 proves to be
$85 per barrel. The company realizes $85 per barrel for the oil and loses approximately

$85�$79 ¼ $6

per barrel on the short futures position. Again, the total amount realized is approxi-
mately $79 million. It is easy to see that in all cases the company ends up with
approximately $79 million.

Long Hedges

Hedges that involve taking a long position in a futures contract are known as long
hedges. A long hedge is appropriate when a company knows it will have to purchase a
certain asset in the future and wants to lock in a price now.

48 CHAPTER 3

Suppose that it is now January 15. A copper fabricator knows it will require 100,000
pounds of copper on May 15 to meet a certain contract. The spot price of copper is
340 cents per pound, and the futures price for May delivery is 320 cents per pound. The
fabricator can hedge its position by taking a long position in four futures contracts
offered by the COMEX division of the CME Group and closing its position on May 15.
Each contract is for the delivery of 25,000 pounds of copper. The strategy has the effect
of locking in the price of the required copper at close to 320 cents per pound.

Suppose that the spot price of copper on May 15 proves to be 325 cents per pound.
Because May is the delivery month for the futures contract, this should be very close to
the futures price. The fabricator therefore gains approximately

100,000�ð$3:25�$3:20Þ ¼ $5,000

on the futures contracts. It pays 100,000�$3:25 ¼ $325,000 for the copper, making the
net cost approximately $325,000�$5,000 ¼ $320,000. For an alternative outcome,
suppose that the spot price is 305 cents per pound on May 15. The fabricator then
loses approximately

100,000�ð$3:20�$3:05Þ ¼ $15,000

on the futures contract and pays 100,000�$3:05 ¼ $305,000 for the copper. Again, the
net cost is approximately $320,000, or 320 cents per pound.

Note that, in this case, it is clearly better for the company to use futures contracts
than to buy the copper on January 15 in the spot market. If it does the latter, it will pay
340 cents per pound instead of 320 cents per pound and will incur both interest costs
and storage costs. For a company using copper on a regular basis, this disadvantage
would be offset by the convenience of having the copper on hand.

1
However, for a

company that knows it will not require the copper until May 15, the futures contract
alternative is likely to be preferred.

The examples we have looked at assume that the futures position is closed out in the
delivery month. The hedge has the same basic effect if delivery is allowed to happen.
However, making or taking delivery can be costly and inconvenient. For this reason,
delivery is not usually made even when the hedger keeps the futures contract until the
delivery month. As will be discussed later, hedgers with long positions usually avoid
any possibility of having to take delivery by closing out their positions before the
delivery period.

We have also assumed in the two examples that there is no daily settlement. In
practice, daily settlement 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 day throughout the life of the hedge rather than all at the end.

3.2 ARGUMENTS FOR AND AGAINST HEDGING

The arguments in favor of hedging are so obvious that they hardly need to be stated.
Most companies 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, exchange rates, and commodity prices. It makes sense for them to

1
See Section 5.11 for a discussion of convenience yields.

Hedging Strategies Using Futures 49

hedge the risks associated with these variables as they become aware of them. The

companies can then focus on their main activities—for which presumably they do have

particular skills and expertise. By hedging, they avoid unpleasant surprises such as

sharp rises in the price of a commodity that is being purchased.

In practice, many risks are left unhedged. In the rest of this section we will explore

some of the reasons for this.

Hedging and Shareholders

One argument sometimes put forward is that the shareholders can, if they wish, do the

hedging themselves. They do not need the company to do it for them. This argument is,

however, open to question. It assumes that shareholders have as much information as the

company’s management about the risks faced by a company. In most instances, this is

not the case. The argument also ignores commissions and other transactions costs. These

are less expensive per dollar of hedging for large transactions than for small transactions.

Hedging is therefore likely to be less expensive when carried out by the company than

when it is carried out by individual shareholders. Indeed, the size of futures contracts

makes hedging by individual shareholders impossible in many situations.

One thing that shareholders can do far more easily than a corporation is diversify

risk. A shareholder with a well-diversified portfolio may be immune to many of the

risks faced by a corporation. 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 overall exposure to the price of copper. If companies are acting

in the best interests of well-diversified shareholders, it can be argued that hedging is

unnecessary in many situations. However, the extent to which managers are in practice

influenced by this type of argument is open to question.

Hedging and Competitors

If hedging is not the norm in a certain industry, it may not make sense for one

particular company to choose to be different from all others. Competitive pressures

within the industry may be such that 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 expect its profit margins to be roughly

constant. However, a company that does hedge can expect its profit margins to

fluctuate!

To illustrate this point, consider two manufacturers of gold jewelry, SafeandSure

Company and TakeaChance Company. We assume that most companies in the industry

do not hedge against movements 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 purchase of gold over the next

18 months. If the price of gold goes up, economic pressures will tend to lead to a

corresponding increase in the wholesale price of jewelry, so that TakeaChance Company’s

gross profit margin is unaffected. By contrast, SafeandSure Company’s profit margin will

increase after the effects of the hedge have been taken into account. If the price of gold

goes down, economic pressures will tend to lead to a corresponding decrease in the

wholesale price of jewelry. Again, TakeaChance Company’s profit margin is unaffected.

However, SafeandSure Company’s profit margin goes down. In extreme conditions,

50 CHAPTER 3

SafeandSure Company’s profit margin could become negative as a result of the ‘‘hedging’’

carried out! The situation is summarized in Table 3.1.

This example emphasizes the importance of looking at the big picture when hedging.

All the implications of price changes on a company’s profitability should be taken into

account in the design of a hedging strategy to protect against the price changes.

Hedging Can Lead to a Worse Outcome

It is important to realize that a hedge using futures contracts can result in a decrease or

an increase in a company’s profits relative to the position it would be in with no

hedging. In the example involving the oil producer considered earlier, if the price of oil

goes down, the company loses money on its sale of 1 million barrels of oil, and the

futures position leads to an offsetting gain. The treasurer can be congratulated for

having had the foresight to put the hedge in place. Clearly, the company is better off

than it would be with no hedging. Other executives in the organization, it is hoped, will

appreciate the contribution made by the treasurer. If the price of oil goes up, the

company gains from its sale of the oil, and the futures position leads to an offsetting

loss. The company is in a worse position than it would be with no hedging. Although

the hedging decision was perfectly logical, the treasurer may in practice have a difficult

time justifying it. Suppose that the price of oil at the end of the hedge is $89, so that the

company loses $10 per barrel on the futures contract. We can imagine a conversation

such as the following between the treasurer and the president:

President: This is terrible. We’ve lost $10 million in the futures market in the space
of three months. How could it happen? I want a full explanation.

Treasurer: The purpose of the futures contracts was to hedge our exposure to the

price of oil, not to make a profit. Don’t forget we made $10 million
from the favorable 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 worry when 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. The fact is that it went up. I really do not know what you were
doing playing the futures markets like this. Our shareholders will

expect us to have done particularly well this quarter. I’m going to have
to explain to them that your actions reduced profits by $10 million. I’m
afraid this is going to mean no bonus for you this year.

Table 3.1 Danger in hedging when competitors do not hedge.

Change in
gold price

Effect on price of
gold jewelry

Effect on profits of
TakeaChance Co.

Effect on profits of
SafeandSure Co.

Increase Increase None Increase

Decrease Decrease None Decrease

Hedging Strategies Using Futures 51

Treasurer: That’s unfair. I was only. . .

President: Unfair! You are lucky not to be fired. You lost $10 million.

Treasurer: It all depends on how you look at it . . .

It is easy to see why many treasurers are reluctant to hedge! Hedging reduces risk for the

company. However, it may increase risk for the treasurer if others do not fully under-

stand what is being done. The only real solution to this problem involves ensuring that

all senior executives within the organization 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 communicated to both the company’s

management and the shareholders. (See Snapshot 3.1 for a discussion of

hedging by gold mining companies.)

3.3 BASIS RISK

The hedges in the examples considered so far have been almost too good to be true. The

hedger was able to identify the precise date in the future when an asset would be bought

or sold. The hedger 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 as this. 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 the futures contract.

Snapshot 3.1 Hedging by Gold Mining Companies

It is natural for a gold mining company to consider hedging against changes in the
price of gold. Typically it takes several years to extract all the gold from a mine.
Once a gold mining company decides to go ahead with production at a particular
mine, it has a big exposure to the price of gold. Indeed a mine that looks profitable at
the outset could become unprofitable if the price of gold plunges.

Gold mining companies are careful to explain their hedging strategies to potential
shareholders. Some gold mining companies do not hedge. They tend to attract
shareholders who buy gold stocks because they want to benefit when the price of
gold increases and are prepared to accept the risk of a loss from a decrease in the
price of gold. Other companies choose to hedge. They estimate the number of ounces
of gold they will produce each month for the next few years and enter into short
futures or forward contracts to lock in the price for all or part of this.

Suppose you are Goldman Sachs and are approached by a gold mining company
that wants to sell you a large amount of gold in 1 year at a fixed price. How do you
set the price and then hedge your risk? The answer is that you can hedge by
borrowing the gold from a central bank, selling it immediately in the spot market,
and investing the proceeds at the risk-free rate. At the end of the year, you buy the
gold from the gold mining company and use it to repay the central bank. The fixed
forward price you set for the gold reflects the risk-free rate you can earn and the lease
rate you pay the central bank for borrowing the gold.

52 CHAPTER 3

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 before its delivery
month.

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:
2

Basis ¼ 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

basis should be zero at the expiration of the futures contract. Prior to expiration, the
basis may be positive or negative. From Table 2.2, we see that, on May 26, 2010, the
basis was negative for gold and positive for short maturity contracts on soybeans.

As time passes, the spot price and the futures price for a particular month do not

necessarily change by the same amount. As a result, the basis changes. An increase in
the basis is referred to as a strengthening of the basis; a decrease in the basis is referred
to as a weakening of the basis. Figure 3.1 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:

S1: Spot price at time t1
S2: Spot price at time t2
F1: Futures price at time t1
F2: Futures price at time t2
b1: Basis at time t1

b2: Basis at time t2.

Time

t1 t2

Futures price

Spot price

Figure 3.1 Variation of basis over time.

2
This is the usual definition. However, the alternative definition Basis ¼ Futures price�Spot price is

sometimes used, particularly when the futures contract is on a financial asset.

Hedging Strategies Using Futures 53

We will assume that a hedge is put in place at time t1 and closed out at time t2. As an

example, we will consider the case where the spot and futures prices at the time the

hedge is initiated are $2.50 and $2.20, respectively, and that at the time the hedge is

closed out they are $2.00 and $1.90, respectively. This means that S1 ¼ 2:50, F1 ¼ 2:20,
S2 ¼ 2:00, and F2 ¼ 1:90.

From the definition of the basis, we have

b1 ¼ S1 � F1 and b2 ¼ S2 � F2

so that, in our example, b1 ¼ 0:30 and b2 ¼ 0:10.
Consider first the situation of a hedger who knows that the asset will be sold at time t2

and takes a short futures position at time t1. The price realized for the asset is S2 and the

profit on the futures position is F1 � F2. The effective price that is obtained for the asset

with hedging is therefore

S2 þ F1 � F2 ¼ F1 þ b2

In our example, this is $2.30. The value of F1 is known at time t1. If b2 were also known

at this time, a perfect hedge would result. The hedging risk is the uncertainty associated

with b2 and is known as basis risk. Consider next a situation where a company knows it

will buy the asset at time t2 and initiates a long hedge at time t1. The price paid for the

asset is S2 and the loss on the hedge is F1 � F2. The effective price that is paid with

hedging is therefore

S2 þ F1 � F2 ¼ F1 þ b2

This is the same expression as before and is $2.30 in the example. The value of F1 is

known at time t1, and the term b2 represents basis risk.

Note that basis risk can lead to an improvement or a worsening of a hedger’s

position. Consider a short hedge. If the basis strengthens (i.e., increases) unexpectedly,

the hedger’s position improves; if the basis weakens (i.e., decreases) unexpectedly, the

hedger’s position worsens. For a long hedge, the reverse holds. If the basis strengthens

unexpectedly, the hedger’s position worsens; if the basis weakens unexpectedly, the

hedger’s position improves.

The asset that gives rise to the hedger’s exposure is sometimes different from the

asset underlying the futures contract that is used for hedging. This is known as cross

hedging and is discussed in the next section. It leads to an increase in basis risk. Define

S
�
2 as the price of the asset underlying the futures contract at time t2. As before, S2 is

the price of the asset being hedged at time t2. By hedging, a company ensures that the

price that will be paid (or received) for the asset is

S2 þ F1 � F2

This can be written as

F1 þðS
�
2 � F2Þþ ðS2 � S

�
2Þ

The terms S
�
2 � F2 and S2 � S

�
2 represent the two components of the basis. The S

�
2 � F2

term is the basis that would exist if the asset being hedged were the same as the asset

underlying the futures contract. The S2 � S
�
2 term is the basis arising from the difference

between the two assets.

54 CHAPTER 3

Choice of Contract

One key factor affecting basis risk is the choice of the futures contract to be used for

hedging. This choice has two components:

1. The choice of the asset underlying the futures contract

2. The choice of the delivery month.

If the asset being hedged exactly matches an asset underlying a futures contract, the first

choice is generally fairly easy. In other circumstances, it is necessary to carry out a

careful analysis to determine which of the available futures contracts has futures prices

that are most closely correlated with the price of the asset being hedged.

The choice of the delivery month is likely to be influenced by several factors. In the

examples given earlier in this chapter, we assumed that, when the expiration of the

hedge corresponds to a delivery month, the contract with that delivery month is chosen.

In fact, a contract with a later delivery month is usually chosen in these circumstances.

The reason is that futures prices are in some instances quite erratic during the delivery

month. Moreover, a long hedger runs the risk of having to take delivery of the physical

asset if the contract is held during the delivery month. Taking delivery can be expensive

and inconvenient. (Long hedgers normally prefer to close out the futures contract and

buy the asset from their usual suppliers.)

In general, basis risk increases as the time difference between the hedge expiration and

the delivery month increases. A good rule of thumb is therefore to choose a delivery

month that is as close as possible to, but later than, the expiration of the hedge. Suppose

delivery months are March, June, September, and December for a futures contract on a

particular asset. For hedge expirations in December, January, and February, the March

contract will be chosen; for hedge expirations in March, April, and May, the June

contract will be chosen; and so on. This rule of thumb assumes that there is sufficient

liquidity in all contracts to meet the hedger’s requirements. In practice, liquidity tends to

be greatest in short-maturity futures contracts. Therefore, in some situations, the hedger

may be inclined to use short-maturity contracts and roll them forward. This strategy is

discussed later in the chapter.

Example 3.1

It is March 1. A US company expects to receive 50 million Japanese yen at the end

of July. Yen futures contracts on the CME Group have delivery months of March,

June, September, and December. One contract is for the delivery of 12.5 million

yen. The company therefore shorts four September yen futures contracts on

March 1. When the yen are received at the end of July, the company closes out

its position. We suppose that the futures price on March 1 in cents per yen is

0.7800 and that the spot and futures prices when the contract is closed out are

0.7200 and 0.7250, respectively.

The gain on the futures contract is 0:7800�0:7250 ¼ 0:0550 cents per yen. The
basis is 0:7200�0:7250 ¼�0:0050 cents per yen when the contract is closed out.
The effective price obtained in cents per yen is the final spot price plus the gain on

the futures:

0:7200þ0:0550 ¼ 0:7750

Hedging Strategies Using Futures 55

This can also be written as the initial futures price plus the final basis:

0:7800þð�0:0050Þ ¼ 0:7750

The total amount received by the company for the 50 million yen is 50�0:00775
million dollars, or $387,500.

Example 3.2

It is June 8 and a company knows that it will need to purchase 20,000 barrels of

crude oil at some time in October or November. Oil futures contracts are currently

traded for delivery every month on the NYMEX division of the CME Group and

the contract size is 1,000 barrels. The company therefore decides to use the

December contract for hedging and takes a long position in 20 December con-

tracts. The futures price on June 8 is $68.00 per barrel. The company finds that it

is ready to purchase the crude oil on November 10. It therefore closes out its

futures contract on that date. The spot price and futures price on November 10

are $70.00 per barrel and $69.10 per barrel.

The gain on the futures contract is 69:10�68:00 ¼ $1:10 per barrel. The basis
when the contract is closed out is 70:00�69:10 ¼ $0:90 per barrel. The effective
price paid (in dollars per barrel) is the final spot price less the gain on the

futures, or

70:00�1:10 ¼ 68:90

This can also be calculated as the initial futures price plus the final basis,

68:00þ0:90 ¼ 68:90

The total price paid is 68:90�20,000 ¼ $1,378,000.

3.4 CROSS HEDGING

In Examples 3.1 and 3.2, the asset underlying the futures contract was the same as the

asset whose price is being hedged. Cross hedging occurs when the two assets are

different. Consider, for example, an airline that is concerned about the future price

of jet fuel. Because jet fuel futures are not actively traded, it might choose to use heating

oil futures contracts to hedge its exposure.

The hedge ratio is the ratio of the size of the position taken in futures contracts to the

size of the exposure. When the asset underlying the futures contract is the same as the

asset being hedged, it is natural to use a hedge ratio of 1.0. This is the hedge ratio we

have used in the examples considered so far. For instance, in Example 3.2, the hedger’s

exposure was on 20,000 barrels of oil, and futures contracts were entered into for the

delivery of exactly this amount of oil.

When cross hedging is used, setting the hedge ratio equal to 1.0 is not always

optimal. The hedger should choose a value for the hedge ratio that minimizes the

variance of the value of the hedged position. We now consider how the hedger can do

this.

56 CHAPTER 3

Calculating the Minimum Variance Hedge Ratio

The minimum variance hedge ratio depends on the relationship between changes in the
spot price and changes in the futures price. Define:

�S: Change in spot price, S, during a period of time equal to the life of the hedge

�F: Change in futures price, F, during a period of time equal to the life of the
hedge.

We will denote the minimum variance hedge ratio by h
�
. It can be shown that h

�
is the