Corporate Finance

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CHAPTER

20Corporate Risk ManagementIn this Chapter:

Why Companies Manage Corporate Risks

Managing Operational, Business and Financial Risks

Forwards and Futures

Swaps

Financial Options

Option Valuation

Real Options

Agency Costs

LEARNING OBJECTIVES

1. Explain the factors that make it desirable for firms to manage their risks.

2. Describe the risks faced by firms and how they are managed.

3. Define forward and futures contracts and be able to determine their prices.

4. Define interest rate and cross-currency swaps and know how they are valued.

5. Define a call option and a put option and describe the payoff function for each of these

options.

6. List and describe the factors that affect the value of an option.

7. Name some of the real options that occur in business and explain why traditional NPV

analysis does not accurately incorporate their values.

8. Describe how the agency costs of debt and equity are related to options.

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Gold conjures up visions of treasure hoardsand beautiful works of art. More prosai-cally, it is used today in dentistry and

electronics because of its resistance to corrosionand its excellent characteristics for conducting elec-tricity. While gold has largely lost its functions as acurrency and store of wealth, it is nevertheless indemand and a considerable number of miningcompanies are involved in its extraction – two ofthe largest are Barrick Gold and AngloGoldAshanti. For gold producers such as Barrick, it isthe reason for their huge investment in deep mineswith their expensive extraction equipment. Theyinvest in order to sell the gold they mine.Unfortunately, the value of gold fluctuates a lot.The collapse of the Bretton Woods agreement,when gold was worth $35/oz, considerably raisedits price such that it peaked at over $850/oz in1980. In the following years, the gold price drifteddownwards with occasional recoveries, until ittouched a low of $252.80/oz in 2000. Duringthis period, gold miners faced a dilemma. Mininggold was expensive and much of the cost had to be paid upfront in drilling the deep shafts to the oredeposits; the cost of extraction was also high. A declining price for their output was bad news.

Source: www.RealTerm.de, used by permission.

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Because of the falling gold price following the peak in the 1980s, Barrick Gold and AngloGoldAshanti, and other mining companies, faced the prospect of having to meet their large productioncosts out of a declining revenue stream as the gold price trended downwards. Their solution was tosell forward part or all of their production at a fixed price. This ensured a minimum price for theirproduction, gave them a stable cash flow and allowed them to raise finance, since investors werereassured they would be paid back with a high degree of certainty. By 1999, these forward salestransactions represented more than a year of total output by the entire industry.1

CHAPTER PREVIEW

The gold mining industry illustrates a key challenge for companies. What set of risks should

companies take and what risks can the company lay off elsewhere? A key factor in determining

corporate risk management policies is the nature and extent of these risks. Generally, companies will

accept risks in their core business areas where they have a degree of competitive advantage. On the

other hand, they will seek to eliminate or minimise other risks that have the capacity to derail the

company from its objective of creating value for shareholders. Generally, firms will seek to manage

macroeconomic risks, such as interest rate, commodity price, currency and credit risks.

Corporate risk management can take a number of forms, which boil down to either the way the

company organises itsmeans of production or the use of financial instruments, principally derivatives,

tomodify the underlying risks in acceptableways – ormore typically a combination of both processes.

We begin with a discussion of the rationale for corporate risk management before briefly

looking at corporate risk management processes. We then examine the way derivatives are valued.

Derivatives fall into two broad categories: those where the buyer or seller is locked into the contract

and those that allow the buyer to walk away, if they should choose to do so. There are a number of

different derivative instruments, variously known as forwards, futures, swaps and options. We will

show that the first three are all similar. Options are different in nature.

A forward contract, such as those used by goldmining companies, allows the buyer or seller to set

the price atwhich theywill purchase or sell a commodity or financial asset at a given date in the future.

Futures contracts do the same thing, but are tradedonanexchange, just likea company’s shares. Swaps

are slightly more complicated in that there are multiple cash flows, but serve essentially the same

purpose of setting prices now for a predeterminednumber of exchanges in future periods. The buyer or

seller of a forward, futures or swaphas specificobligations that last until the contract is completed and,

in particular, both parties must complete the transactionwhatever the circumstances in the future. On

the other hand, the buyer or owner of an option has the right, but not the obligation, to purchase or sell

a commodity or financial asset, such as a share, at a pre-specified price on or before a given date. This

means that option buyers only have to complete the transaction if they choose to do so.

We then turn to options on real assets, known as real options. Real options often arise in

corporate investment decisions. Managers often have options to delay investing in a project, expand

a project, abandon a project, change the technology employed in a project, and so on. You will see

that the value of these options is not adequately reflected in an NPV analysis.

We next revisit the agency costs of debt that we discussed in Chapter 16. In particular, we show

how option-like payoffs contribute to the dividend payout, asset substitution and underinvestment

conflicts. We follow this discussion with a related discussion of how option-like payoffs contribute

to conflicts between shareholders and the managers who work for them.

780 PART 7 CORPORATE RISK MANAGEMENT AND INTERNATIONAL DECISIONS

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WHY COMPANIES MANAGECORPORATE RISKS

Learning Objective 1Explain the factors that make it desirable forfirms to manage their risks.

In Chapter 6, we explained the nature of discounted

cash flows and valuation. Value depends on the size,

timing and riskiness of future cash flows and the rate

of return required by investors. This suggests that

corporate risk management may add value if it can

positively affect expected cash flows and the

required rate of return that is appropriate to those

cash flows. A simple example will illustrate the

point. Let us assume that a gold producer will

have an annual cash flow of D500 million over

the next 30 years, after which mining operations

cease.2 The appropriate risk-adjusted discount rate

for the cash flows is 12%. Recall that the discount

rate reflects the riskiness of the underlying cash

flows. The present value of the business today is

therefore simplyD500million times a 30-year annu-

ity at 12% (8.0552), or D4028 million (D500 �8.0552).Now the company decides to engage in risk

management activity that costs it D50 million per

year over the life of the mine. At the same time, the

required rate of return, given that the cash flows

have less risk, is reduced to 10% (the annuity factor

will now be 9.4269). The new value becomes

(D500 � D50) � 9.4269 ¼ D4242 million. Using

risk management in this situation has raised the

value of the firm by D214 million (D4242 �D4028¼ D214 million). Consequently, the owners

of the business are better off if the gold producer

undertakes risk management.

A number of different factors will influence the

extent to which firms manage their risks. These

include financial reporting, corporate taxation, the

costs of bankruptcy and contracting with providers

of capital, as well as issues such as agency costs and

employee compensation and retention. Further-

more, shareholders benefit when a company man-

ages its risks in ways they cannot reproduce

themselves. For instance, tax losses at the company

level are not directly transferable to shareholders,

so asymmetries in payoffs may lead firms to man-

age these risks.

If the company does not hedge, there will be

variability in the cash flows it generates from its

operations as economic conditions change and the

prices of its inputs and outputs change. Shortfalls,

as a result of adverse movements in output prices or

input costs, will either mean the company has to

raise money externally or reduce its future invest-

ments – and consequently may have to pass up on

attractive positive net present value projects. As we

have seen in Chapter 15, raising external capital is

costly and time-consuming. In addition, the issues

discussed in Chapter 16 about the effect of capital

structure and the problems of financial distress

apply. These affect the ability of firms to raise

external finance when distressed. A case in point

is the British company BAe, or British Aerospace as

it was then. In the 1980s, it had diversified away

from aerospace into property and automobiles (by

acquiring the Rover Group). In 1991, it suddenly

indicated that all was not well in its businesses and

announced a £432 million rights issue to repair its

balance sheet. Shareholders were angry at the un-

expected losses and having to subscribe more capi-

tal. For a while, therewas a real risk of the company

not getting the money it needed. The chairman and

other senior managers were forced to resign and a

new management team was recruited before share-

holders were willing to subscribe for the new shares

on offer.3

Taxes may also help explain why firms engage

in risk management. If the tax system operates in

such a way that the tax paid by the company rises

with the amount of profit or earnings, it becomes

attractive for firms to reduce the uncertainty of

future earnings. In this situation, a more volatile

earnings stream leads to higher expected taxes than

a less volatile earnings stream. The reason is that

firms have a number of potential tax offsets, such

as tax depreciation and allowances, which act to

reduce their taxable income. These are generally

fixed in size so that if pre-tax earnings increase,

they are likely to pay a higher tax rate overall. As


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we also saw in Chapter 3, many countries have a

low starter rate of company tax. For instance,

take the example of a company that has pre-tax

earnings of either D50 or D200 with equal proba-

bility and will pay an effective rate of tax of 25% if

its earnings are low or 35% if it has high earnings.

The expected profit before taxwill beD125 (D50�0.50þ D200� 0.50). The expected taxwill beD40

([D50 � 0.25] � 0.50 þ [D200 � 0.35] � 0.50).

The expected after-tax profit will then be D125 �D40 ¼ D85. Now consider the situation where the

company can use risk management to eliminate the

variability in future pre-tax earnings such that it

will have earnings before tax of D125 with com-

plete certainty. The corporate tax rate for this level

of income is 27%. The company will pay D34 in

tax (D125 � 0.27). The after-tax earnings are now

D125� D34¼ D91. The company has saved D6 in

taxes and increased after-tax profits from an

expected D85 to D91. In this situation, risk man-

agement creates value for shareholders.

As the opening vignette indicates, lenders are

concerned about repayment. For a given level of

debt, risk management can reduce the probability

that a company will find itself in the situation

where it is finding it hard or is unable to repay

the debt. As Chapter 16 indicates, in situations

where financial distress is costly, risk management

may increase the firm’s debt capacity. Higher debt

levels may also be desirable in reducing agency

problems and where this creates increased risk of

financial distress, risk management is likely to be

beneficial.

A key rationale for firms to engage in risk

management is that they are better able to address

problems of managerial motivation, capture the

benefits of tax management and reduce the costs of

financial distress in ways that shareholders cannot.

The ability of firms to manage their risks may also

allow them to exploit investment opportunities

that they would otherwise have to pass by because

it is costly or impossible to raise external finance.

There is less rationale for firms to manage

those risks that shareholders and lenders can easily

manage for themselves. For instance, it is not clear

that unrelated diversification at the company level

is beneficial since shareholders are able to create

well-diversified portfolios by holding shares in a

range of different companies at less cost and with

more flexibility.

The Risk Management ProcessCompanies need a risk management process. At its

simplest, it requires them to examine their

BUILDING INTUITI NRisk Management Can Help Firms Avoid Havingto Raise Capital When it is Difficult to Do SoA key question external providers of finance wish to resolve when firms come asking for new funds isthe company’s motivation. The added disclosure required when external finance is being soughtrapidly determines whether the company is in trouble – or not. If the finance is required to rescue thecompany, is this the management’s fault or just bad luck? It is difficult for outsiders to know whetherthe financially troubled condition of the company is a result of badmanagement or simply bad luck. Ifthe managers are at fault, lenders do not wish, as the saying goes, ‘to throw good money after bad’by providing more finance to a failing management. Better to wind up or sell the firm. Consequently,from the perspective of a company’s managers, seeking external finance when things have gonewrong is to be avoided as much as possible. Undertaking risk management that reduces thelikelihood of financial distress and the need for external finance makes sense when providers offinance find it difficult to understand what is happening in the business.


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operations in the broadest sense in order to recog-

nise the risks that can affect the firm’s future cash

flows. This involves identifying the risks, their

assessment or evaluation, the selection of the

risk management techniques, their implementation

and keeping the programme under review. For

instance, the pizza restaurant group would want

to look at where it sources its inputs and in what

way, what could gowrongwhen preparing, serving

and delivering pizzas. At the same time, it would

also consider how wider factors outside the com-

pany, such as the economy and social trends, might

affect the business’s future profitability.

The process can be broken down into a num-

ber of logical steps. These would typically include:

� Identification. This would involve the financial

manager in surveying the various business units

and determining the profile of the business risks

involved. Exposures can be simply classified

according to the way they could affect the firm’s

operations. For instance, the pizza restaurant

group may use an integrated accounting system –

failure here would have a major effect in that

the company may be unable to operate. Hence,

the risk that such a critical system could fail would

be classified as having a very significant impact.� Evaluation. Wherever possible, the impact of the

risk is quantified in monetary terms. This helps in

ranking the risks according to the severity of their

effect. When combined with estimates of their

frequency, this provides a way of scoring the

result. For instance, at individual pizza restau-

rants, itmaybe that there are often inconsistencies

in the till receipts against goods sold. However,

their monetary effect is likely to be very small.

Hence, while problems in this area are frequent,

their severity isminimal. A decisionwould need to

be reached as towhether this risk needsmanaging.

On the other hand, the IT system failure may be

very infrequent – but its impact on the business

could be seen as very severe.� Management. The final element is a clear frame-

work for managing the risks once they have been

identified and evaluated. Here a key criterion is

whether they have the capacity to derail the

firm’s strategy. The management of the risks is

therefore integrated into the company’s strategic

goals. At the operational level, the company will

establish procedures and assign responsibility to

oversee the management of these unacceptable

risks. Hence, it is often the function of the finan-

cial manager to use financial techniques or source

instruments to mitigate the risks. For instance, by

buying insurance cover against specific risks.� Review. The final step is to repeat the process

and keep the risks under review, since conditions

change and firms evolve over time.

Before You Go On

1. What two factors affect the value of a

business that can be modified when a

firm manages its risks?

2. What firm-specific reasons may prompt a

business to engage in risk management?

3. What are the adverse consequences to

companies from changes in input and out-

put prices?

BUILDING INTUITI NRisk Management Can Help Firms AddressCapital Market ImperfectionsCorporate risk management is desirable when capital market imperfections and asymmetries reducethe value of firms and make access to outside finance costly for firms that do not control risks. Unlessthis rationale is present, risk management should be left to capital providers.


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MANAGING OPERATIONAL,BUSINESS AND FINANCIALRISKS

Learning Objective 2Describe the risks faced by firms and how theyare managed.

Every transaction a firm undertakes includes mul-

tiple risks. For instance, Volkswagen sells its cars in

the Chinese market. In doing so, Volkswagen is

betting that its cars are competitive in that market.

However, it is also betting on the exchange rate

between the renminbi and the euro. In the past, the

renminbi has been linked to the US dollar and

hence has fluctuated against the euro as the US

dollar has risen and fallen over time.4 In order to

develop its market presence in China, Volkswagen

has to invest in this market by advertising the

attractions of its cars and developing a dealer

and repair network. These investments would be

lost if changes in the market made it unattractive to

the company. Volkswagen may be upbeat about

the opportunity to sell its cars in China, but be less

optimistic on the future of the exchange rate. A fall

in the renminbi would leave it selling cars at a loss

and hence the currency risk reduces the attractive-

ness of the fast-growing Chinese market. The solu-

tion is to split these risks, and for the company to

accept the risks in which it sees itself as having a

competitive advantage and removing those which

can derail its business strategy. The company

would therefore seek to manage the currency

risk in such a way as to eliminate the problem.

The risks that a company such as Volkswagen

faces are either operational risks or market risks.

Operational risks are either internal to the firm

or arise from the nature and extent of its activities.

The internal risks are largely under the control of

management in that decisions on how the firm

sets up and operates its production systems can

be organised so as to minimise the risks in-

volved. On the other hand, many of the external

risks are the result of changes in macroeconomic

conditions and relate to changes in interest rates,

commodity prices and exchange rates. These

are market risks, and companies seek to reduce

the effect of these on the firm’s operations and

profitability. In addition, transactions with third

parties create credit risk, which was discussed in

Chapter 14.

Operational risks

any risk arising from the execution of a

company’s business functions

Market risks

exposure to a change in the value of some

market factor, such as interest rates,

foreign exchange rates, equity or commod-

ity prices

Operational and BusinessRisksCompanies such as Volkswagen are involved in

complex activities and face a number of internal

and external risks. There are risks in its production

processes from potential factory fires, breakdown

in critical equipment and the development of new

technologies that render existing ones redundant or

uncompetitive. Some of these production risks are

insurable. Firms also have input and output risks.

The gold mining companies described at the start

of this chapter have significant output risk from

unexpected changes in the market price of gold

over time. These price risks affect both the costs of

a firm’s inputs and its outputs and hence its future

profitability. Typically, these risks include com-

modities, raw materials, finished products, interest

rates, energy, currencies and the prices of other

market-determined inputs and outputs.


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Production risk

all the elements of the production process

that can go wrong: for instance, fires and

equipment failures

Price risks

changes in the prices of a firm’s inputs and

outputs over time due to changes in demand

and supply

To the extent that a company has changes in

input prices (from unanticipated supply effects) and

outputs (from unanticipated demand effects), it will

experience variability in its cash flows. At a basic

level, it will want to ensure that revenues cover all its

costs. Since it incurs costs before revenues, this is to

some extent a timing problem. However, firms may

have trouble in raising prices. In the case of Volks-

wagen, if the demand for its carswas independent of

their price, a fall in the value of the renminbi against

the euro would be compensated for by increasing

the sales price in China to maintain the value in

euros. For most firms, a number of factors may

prevent this happening: (1) local competitors will

be largely unaffected by the movement in the cur-

rency; (2) demand may be significantly conditional

on price; and (3) other foreign suppliers may be

willing to cut their local currency prices tomaintain

or increase their market share.

Risk Management MethodsCompanies have a range of techniques that they can

use to reduce the risks they face. Some of these relate

to the way the firm operates. The solution for the

company is to anticipate that prices will change and

to position itself accordingly. Continuing ourVolks-

wagenexample, the companycouldorganise itself so

as to site inChina that part of its production facilities

that supplies the market. Then, costs and revenues

wouldbothbe inthe samecurrency.Whencashflows

are matched in this way, it is known as hedging.

Hedging

any technique designed to reduce or elimi-

nate risk; for example, taking two positions

that will offset each other if prices change

As indicated above, Volkswagen can set up

manufacturing facilities in the markets in which it

sells its cars. This works to an extent, but can lead

to a dispersion of production and higher costs than

concentrating facilities in units that can benefit

from economies of scale and scope. Typically, firms

will organise themselves to be efficient producers

and seek to address the remaining risks by using the

capital markets to hedge – a process known as

financial risk management. This involves the com-

pany in dealing in financial instruments that are

designed to transfer or modify risks. This can

involve the firm using insurance or derivatives. A

great advantage of these instruments is that they

are low cost and can be added and removed as

required as circumstances change. Contrast this to

the time and expense involved if the company

decides to change the location of its production

or switches the markets in which it sells.5

Financial risk management

the practice of protecting and creating eco-

nomic value in a firm by using financial

instruments to manage exposure to risks

Insurance

a contract that protects against the financial

losses (in whole or in part) of specified

unexpected events

Derivatives

financial instruments or securities whose val-

ue varies with the value of an underlying asset


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There are three generic ways in which a firm

can manage its various risks that involve hedging,

insurance and diversification. The choice of me-

thod will depend on a number of factors. When a

firm hedges, it reduces its exposure to the possibil-

ity of a loss but this also leads to the firm giving up

the possibility of a gain. Insurance means paying a

premium, the cost of the insurance, to avoid losses.

In this case, the company retains the possibility of

gain, but eliminates the exposure to potential loss.

Note the difference between hedging and insur-

ance: with hedging, the risk of loss is eliminated by

giving up the potential for gain; with insurance,

you pay a premium to eliminate the risk of loss and

retain the potential for gain.

Companies also use diversification to reduce

their risks. We know from the way portfolios work

that the aggregated portfolio risk will be less than

the sum of the individual risks as long as the

components of the portfolio are less than perfectly

correlated.6 As discussed earlier, while Volkswa-

gen will not aim to exactly match its production

facilities to its markets, nevertheless it does operate

a number of different production facilities spread

around the globe. This diversification makes sense

in that it does reduce Volkswagen’s overall risks.

But diversification of this kind is only advisable to

the extent that it does not adversely affect the firm’s

operational efficiency. As mentioned earlier, firms

will seek to be efficient producers and use financial

instruments to manage the remaining risks.

As we will see later, when we look at how

derivatives are valued, the cost of risk management

will depend on the future uncertainty. The higher

this uncertainty, the greater the costs involved.

While risk management is costly, there are real

benefits to companies from being able to manage

their risks by hedging, through insurance or via

BUILDING INTUITI NFinancial Risk Management Allows Firms toExploit their Comparative AdvantagesA key reason firms use financial hedging is that they want to optimise the way they go about theirbusiness but also only accept those risks in which they have a competitive advantage. By using thefinancial markets to lay off those risks that the firm is unwilling or unable to accept, it both ensures thatthese risks do not derail its strategy and allows it to concentrate resources in areas where it has thebest prospects of earning good returns.

BUILDING INTUITI NCorporate Risk Management Decisions are Basedon Cost--Benefit Trade-offsThere are a number of different methods that, taken together, companies use to manage their risks.Companies will weigh the costs of using the method against the benefit of risk reduction. Companiescan use diversification, insurance and hedging. These have different costs and benefits. Hence,there is no single solution that is appropriate in all circumstances. The benefits and costs of eachapproach have to be worked out for each method for all the different risks.


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diversification, and to reduce or eliminate the risks

that would otherwise lead firms to underinvest in

productive projects.

Over time, to cater to the needs of firms,

various organisations and contractual arrange-

ments have emerged to expand the scope of diver-

sification and by providing greater specialisation in

risk management. For instance, insurance compa-

nies cover a wide range of production and other

risks while derivatives markets in forward con-

tracts, futures, swaps and options have a promi-

nent place in financial markets.

Before You Go On

1. In what areas does a company face risks

from its business?

2. What are the different ways in which a

company can manage its risks?

3. What determines the balance between

operational hedging and financial hedging?

FORWARDS AND FUTURES

Learning Objective 3Define forward and futures contracts and beable to determine their prices.

A forward contract involves a delayed sale and

purchase by the two parties to the contract. Con-

sider the situation where Airbus is selling one of

its commercial jets to a customer. These are

usually manufactured to order and delivery may

take place several years later. What are the risks

for both sides if the terms and conditions are not

set at the outset? Exhibit 20.1 shows a payoff

diagram that graphically illustrates the way the

buyer and seller are exposed to the future

uncertain changes in the price of the airliner. First

and foremost, both parties have price risk in that –

when the delivery date finally arrives – the price

for Airbus jets has changed. Airbus will do well if

demand is high and aircraft prices have risen. The

buyer will lose out by having to pay more.

Equally, if demand is low and jetliner prices

have fallen, the buyer wins as they pay less –

and Airbus receives less. Given the uncertainties

about the future price, both parties stand to lose if

future prices are not the same as the current price

for the airliner. They both have an incentive to

ensure the contract for the aircraft specifies a price

for the aircraft today but payable upon delivery.

Commercial arrangements where prices and

quantities are agreed today for future delivery

are forward contracts. These contrast to spot

contracts, where the buyer and seller make an

immediate exchange.

Forward contract

agreement between two parties to buy or sell

an asset at a specified point of time in the

future at a price agreed today

Because forward contracts address theprice risk

facing buyers and sellers, they are very common in

business. In addition to commercial contracts, such

as that between Airbus and its client, there are

numerous financial forward contracts that can be

negotiated in the financial markets and these cover a

very wide range of business risks. There are con-

tracts on currencies, commodities, interest rates,

stock market indices and individual shares, credit

risks, energy and even the weather – to list the most

common types.

Valuing a Forward ContractAs the Airbus example illustrates, these contracts

work by ‘locking-in’ the prices at which firms buy

and sell in the future. What should determine the

price for the forward contract? In Chapter 5, we

looked at investment and future values and so we


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already know how to value future cash flows. A

forward contract works in exactly the same way,

except we want to start with the current price of

what will be delivered in the future and work out

its future value. At its simplest, a forward contract

therefore will be determined by Equation (5.1):

FVn ¼ PV� 1þ ið Þn

where:

FVn¼ future value of investment at the end

of period n

PV¼ original principal or present value

i¼ rate of interest per period, which is

often a year

n¼ number of periods (typically a year,

but it can be a quarter, a month, a

day or some other period of time)

(1 þ i)n¼ future value factor

Let us assume that Airbus is selling the A310

model to the customer as described earlier. These

are listed for immediate delivery at a price of

D50 million each. However, the agreed delivery

date is 2 years away and the annual rate of interest

for euros is 4%. The forward price will therefore

be D50 � (1 þ 0.04)2 ¼ D54.08 million. If

there are no other factors that affect the forward

price, this is a fair deal to both sides. The client

could immediately buy at D50 million. On the

€50 €60€40

Profit

Loss

A310price atmaturity

€10

(€10)

€50 €60€40

Profit

Loss

A310price atmaturity

€10

(€10)

Panel A: Airbus IndustriesPayoff Profile

Panel B: Customer’sPayoff Profile

Airbus gains ifmarket price for A310

million afteris above €502 years

Airbus loses ifmarket price for A310

is below €50 million after2 years

Client gains ifmarket price for A310

is below €50 million after2 years

Client loses ifmarket price for A310is above €50 million after2 years

Exhibit 20.1: Payoff Diagrams for the Price Risks Facing the A310 Buyer and Airbus Industries A payoffdiagram shows the profit and loss from the deferred purchase of the A310 jet. If, when the contract is agreed, thecontracted price is D50 million and it does not change, neither buyer nor seller gains. If the price rises, the buyer

loses and the seller gains, the profit and loss being the difference between the original price and the new price. So ifthe price rises to D60 million, the seller has a profit of D10 million and the buyer loses D10 million. The gains and

losses for both sides are the same and hence the payoffs to both parties are symmetrical.

Panel A shows the payoff profile for Airbus Industries. The company will profit if the market price of the A310 airlineris above D50 million in 2 years’ time. It will lose if the market price is below D50 million. How much it will lose will

depend on the future price uncertainty for the A310.

Panel B shows the customer’s payoff profile. You will notice it has exactly the reverse set of gains and losses.The customer’s risk is exactly the opposite of that of Airbus Industries in that it loses if the price rises and gainsif the price falls. Both Airbus and the customer have an incentive to manage the risk that the price will change.

They will do so by entering into a forward contract that establishes the price now that will be paid upondelivery in 2 years’ time.


C20 02/08/2011 19:31:5 Page 789

other hand, if they do not want the aircraft

immediately, they have use of the money for

the two intervening years and – if they invest it

at the 4% interest rate – they will earn D4.08

million, so will be no better or worse off from

buying immediately or waiting. What of Airbus?

If they sell the A310 for forward delivery, they

receive D54.08 million. The present value of this

is D50 million. By agreeing to sell at the higher

forward price, they are compensated for the delay

in receiving the money. Airbus has to wait two

years to get the price of the aircraft and – notion-

ally at least – may have to borrow the money

while it waits to deliver the A310 to the customer.

It can borrow the present value of the forward

price – which is, of course, D50 million. The

forward price is such that it is fair to both sides

and compensates them for the delayed delivery.

The pricing of forward contracts is known as

the cost of carry and equates the gains and losses

of both sides such that neither wins or loses. As

such, it is a zero net present value transaction in

that – as shown above – neither the buyer nor the

seller loses out from the delay. The cost of

carry may take account of more than just interest

rates as it includes all those elements that

change the value between the present and the

future and is the net cost to the seller in the

transaction. For instance, Airbus may have to

store the aircraft for the two years and will incur

costs from doing so. This would raise the future

price of the aircraft. On the other hand, it is

possible that Airbus could lease out the aircraft

and earn income over the two years, something

the client could also do. Airbus gains from this,

but the buyer loses the opportunity to gain the

rental income until the time for delivery.7 This

will reduce the forward price.

Cost of carry

the net cost of ‘carrying’ or holding an asset

for future delivery

The difference between the cash market price

and the forward price (PV – FV) will be the cost of

carry and will include the costs and benefits from

the delay. The elements that go into the cost of

carry and their effects on the forward price are as

follows:

� The interest rate (i) will work to increase the cost

of carry and hence the forward price.� Storage costs and wastage (u) will increase the

cost of carry and the forward price. Some assets

are subject to wastage when stored, such as

agricultural commodities which tend to deterio-

rate over time, and this will mean the amount

that can be delivered eventually will be less than

the amount stored initially.� Any income received prior to delivery will

decrease the cost of carry and the forward price

because it is a benefit to the seller. This is often

expressed as a yield (q). Expressed this way,

income on the asset can be viewed as a negative

interest rate.� A quantification of the benefits of immediate

ownership or availability. This is known as the

convenience yield (y) and can be thought of as

negative storage costs! For instance, companies

that use commodities which are vital to their

business operations – and where there is

restricted supply – may stockpile needed supplies

and forego income in order to have a guaranteed

availability. The convenience yield only applies

to consumption assets and will be zero for for-

wards on financial assets.

The effects of the factors that influence the cost

of carry are illustrated in Exhibit 20.2.

Convenience yield

a non-monetary return derived from

the physical ownership of an asset or

commodity


C20 02/08/2011 19:31:5 Page 790

Let us continue the A310 example and see how

the factors influence the cost of carry. We have

already worked out the case where the only factor

is interest rates. In this case, the forward price is

D54.08 million. If Airbus has to store the airliner

for the two years, it will incur costs. Assume that

storage costs are 1% of the aircraft’s value per

year. This is like adding on 1% to the interest rate,

so the future value will be D55.13 million [D50 �(1 þ 0.04 þ 0.01)2]. On the other hand, if Airbus

can lease out the aircraft for two years and earn 3%

of the value of the aircraft in lease payments, this

will have the effect of reducing the future value.

Without leasing, the future value isD54.08million.

The lease payments act like a negative interest rate

and reduce the future value, so we need to discount

the FV by the leasing rate (1 þ 0.03)2, such that

D54.08/(1.03)2 ¼ D50.98 million. If there were a

convenience yield attached to having physical own-

ership of the A310, this would also serve to reduce

the forward price.

The full cost-of-carry formula is therefore:

PV� 1þ iþ uð Þm1þ qþ yð Þm ¼ FVm ð20:1Þ

where:

PV¼ current price or present value

i¼ rate of interest per period (which is

often a year)

u¼ storage cost per period, expressed as an

interest rate

q¼ income from the asset per period,

expressed as an interest rate

y¼ convenience yield per period

FVm¼ future value at the maturity of the for-

ward contract

m¼ number of periods to maturity of the

forward contract; a period is typically a

year, but can be some other period,

such as a quarter, month or some other

unit of time

T=mT=0

Current valueof asset

(PV)

Future value of asset at time m

FVmCost of carry

Factors that raise the forward price:interest rates (i)•storage and wastage cost (u)•

Factors that lower the forward price:income from the asset (q)•convenience yield (y)•

Maturity of forward contract

Exhibit 20.2: Factors that Affect the Cost of Carry in a Forward Contract The price difference between thecurrent market, the present value (PV) and the forward price (FV) will be determined by the interplay of thosefactors that contribute to the cost of carry: (1) interest rates and (2) storage and wastage costs, which act to

increase the forward price; (3) income from the asset and (4) the convenience yield, which reducethe forward price. Note that, because of this interaction, it is quite possible that the forward price

is lower than the current market price.


C20 02/08/2011 19:31:6 Page 791

Learning byDoing

Application20.1

Problem: You are the purchasing manager atthe pizza restaurant chain and are becomingincreasingly alarmed by the way the price ofwheat is increasing and the effect it is having onthe ability to set prices, plan expenditure and itseffects on profit margins. Therefore, in order tofacilitate planning within the company and to fixthe cost of a major ingredient, you decide youwould like to hedge and ‘lock in’ the price of flourfor the coming year. You estimate you will need50 tonnes and decide that a 1-year forwardcontract is the appropriate hedging instrument.The current price for flour is D175 per tonne, theinterest rate is 4% and your contacts in theindustry tell you that storage costs are 2% peryear. At what price will you be able to execute aforward contract?

Approach: We need to apply Equation (20.1)to determine the future price at which the for-ward contract will be agreed. To get the correctvalue, we need to include both the currentinterest rate and the storage costs in the cost-of-carry formula.

Solution:Applying Equation (20.1) gives:

PV� 1þ i þ uð Þm1þ qþ yð Þm ¼ FVm

D175� 1þ 0:04þ 0:02ð Þ1þ 0þ 0ð Þ

¼ D185:50 per tonne

We should point out that the cost-of-carry

model is quite adaptable. For instance, we may not

be able to work out the storage cost as an interest

rate. Nevertheless, if we know what the storage

costs will be in money, we can still use the model.

Going back to our Airbus example, let us assume

that Airbus contracts with a maintenance company

to store the aircraft and the company says it will

need to be paid D0.60 million at the end of year 1

and D0.75 million at the end of year 2 to store,

maintain and service the aircraft. We can simply

apply our understanding of the way the cost-of-

carry model works and that these are costs that

need to be added to the agreed forward sale price.

When only interest rates affect the cost of carry, we

have a future value of D54.08 million. We can-

simply add to this the future value of the storage

costs. The timeline for the transaction will be as

follows:

1 2 Year 0 4%

€0.75 €0.60 €50 FV = ?

We need to work out the value at year 2, which

involves the following calculations:

PVAircraft � 1þ ið Þ2 ¼ FVAircraft; Year 2

þ FVStorage; Year 1 � 1þ ið Þ ¼ FVYear 1 storage; Year 2

þ FVStorage; Year 2

¼ Forward price

D50� 1:04ð Þ2 ¼ D54:08

þ D0:60� 1:04ð Þ ¼ D0:624

þ D0:75

¼ D55:454million

The forward price that Airbus requires so that it

is no better orworse off from selling theA310 today

is D55.454 million.8 This price includes the fore-

gone use of the sales proceeds and storage costs.


C20 02/08/2011 19:31:6 Page 792

Learning byDoing

Application20.2

Problem: You are the majority owner of thepizza restaurant group. The company is doingvery well at the moment and the shares arecurrently worth D80. However, you need tohave a considerable sum of money in two years’time to provide for your daughter’s universityeducation. You are aware that the value ofyour shares can fall over this period and if so,as a result, you may have to sell more shares thanyou would like. As your company intends to paya dividend of D5.20 at the end of the currentyear and you anticipate a dividend of D5.60 atthe end of year two, you want to receive thesedividends and not sell the shares until you actu-ally need the money. The current two-year risk-free interest rate is 4% per year.What will be thefair price for the forward sale of your shares?

Approach: We apply the cost-of-carry modeland adapt Equation (20.1) to take account ofthe specific dividends that will be paid on theshares over the life forward contract.

Application: We present-value the future divi-dend payments and find the price of the shares

excluding the two dividend payments and thenfuture-value this ex-dividend share price for twoyears:

Present value of dividends ¼ D5:201:04

þ D5:60

1:04ð Þ2¼ D5:00þ D5:18¼ D10:18

Ex-dividend share price ¼ D80:00� D10:18

¼ D69:82

Forward price ¼ D69:82� 1:04ð Þ2¼ D75:517

The forward price for the shares is D75.517each. You can now determine, based on themoney you need for your daughter’s education,how many shares you need to sell in the forwardcontract. Note that, as discussed in the text, dueto the value leakage from the dividends, the two-year forward price of D75.52 is below thecurrent market price of D80.

The Value of a ForwardContract Prior to MaturityAs Exhibit 20.1 indicates, the payoff for both

parties to a forward contract is symmetrical. The

buyer and seller’s gains and losses are the same, but

arise due to changes in themarket price of the A310

airliner. Once the terms of the forward contract are

agreed (we will take D54.08 million as the contract

price), the value of the contract to either party will

change as the market price of the asset to be

delivered changes. For instance, let us assume

that one year has elapsed. Airbus has raised the

price of its A310model to D52million. At the same

time, interest rates have also changed and are now

5% per year. What is the contract worth? The

payoff at maturity for Airbus, the seller, will be

the difference between the market price and the

agreed price, namely D54.08 � D52 ¼ D2.08

million. They will not receive this for another

year, so the present value will be D1.98 million

(D2.08/1.05). If Airbus is making a profit from the

transaction, then the customer must be losing an

equal amount.

Note the effect that greater price changes

have on the gains and losses from the contract

before maturity. If the A310 price had risen to


C20 02/08/2011 19:31:6 Page 793

D60million, the payoff from the contract would be

D5.92 million (D60 � D54.08). The present value

is D5.638 million (D5.92/1.05). What we find is

that the greater the changes in price over the life of

the forward contract, the greater the value of the

forward contract prior to maturity. This shows

that the greater the price uncertainty for a firm’s

inputs and outputs, the greater is the incentive to

hedge out these risks and the more valuable the

forward contract becomes.

Futures ContractsYou may have realised there is a problem with

forward contracts. Think of the situation facing

Airbus, if in two years’ time the market price of the

A310 is now D40 million. The customer has every

incentive not to honour the agreement, and buy the

same aircraft elsewhere and saveD14.08million by

doing so (D54.08 – D40). To ensure it is not left

nursing a loss, Airbus will only enter into the

forward transaction at the outset if it thinks the

customer will honour the forward contract re-

gardless of what happens to the future price at

maturity – and, of course, the customer has the

same worries. A major problem therefore is that

forward contracts are subject to what is called

counterparty credit risk and this materialises

when the other party fails to fulfil its obligations.

This will always happen to the party that stands to

gain from adhering to the contract. If the price after

two years was D60 million, the customer will not

renege on the contract even if they do not want the

airliner. This is because they can immediately resell

it at a profit! Problems with the creditworthiness of

counterparties in forward contracts limit the pos-

sible parties a company can deal with using for-

ward contracts to those that it knows will pay even

if it means they are losing out as a result.

Counterparty credit risk

the risk that the other party to a transaction

will be unable or unwilling to honour their

commitments

Futures contracts were developed specifically

to deal with the counterparty problem. They do

this in a number of ways:

� All contracts are made with a clearing house and

not directly between buyers and sellers. This

means if one of the parties defaults, the contract

is still good for the other party since they have a

contract with the clearing house.� To protect the clearing house from losses due to

defaults, both buyers and sellers have to post a

goodwill deposit when buying or selling a futures

contract, known as margin, to cover possible

losses. The amount that is posted is enough to

cover anticipated daily price changes in the con-

tract plus an additional safety margin.� The values of futures contracts to the buyer

and seller are updated daily and the amounts

debited and credited to the goodwill deposit. If

the amount in the goodwill deposit account

(margin account) falls below some predeter-

mined level, further margin is required by the

party incurring the losses. If this fails to materi-

alise, the contract is terminated and the

margin account used to cover any losses by

the clearing house.� Contracts are standardised to facilitate the mar-

ket and trading is carried out through an organ-

ised exchange.

Futures contract

a standardised, transferable, exchange-

traded contract that requires the delivery

of a specified asset at a predetermined price

on a specified future date

Margin

collateral that the holder of a futures

contract has to deposit to cover the credit

risk


C20 02/08/2011 19:31:6 Page 794

These institutional and functional changes

made to the way forward contracts work create

exchange-traded futures contracts. Because these

contracts are standardised and a central counter-

party acts as the buyer and seller for market users,

there is a liquid market in futures. Buyers can enter

the market very rapidly and find sellers through the

exchange. Unlike a forward contract that has to be

unwound with the other party, when the buyer

comes to sell through the exchange they can easily

find another market user who wants to take on

their position. Furthermore, transaction costs are

very low and this adds to the attraction of the

instruments for short-term risk management pur-

poses. As a result, there are large volumes of futures

contracts being traded on numerous exchanges.

The most important ones in Europe are the

NYSE/Euronext/Liffe group and EUREX. The

largest exchange in the world is the Chicago-based

CME Group.

WEB

The major exchanges have information

about their contracts and how they can be

used. The two major ones in Europe are

the NYSE Euronext group http://www

.euronext.com and Eurex http://www

.eurexchange.com. The largest exchange in

the world is the CME Group in Chicago

http://www.cmegroup.com.

Typically, a futures contract will be based on a

representative asset for the particular asset class or

a recognised benchmark asset. For instance, the

copper futures contract traded on the London

Metal Exchange, a commodities futures exchange,

specifies that it must be Grade A copper bars

conforming to a defined standard of purity. A

number of asset types have no representative asset.

An example is corporate bonds, and there is no

corporate bond futures contract since there is no

such thing as a ‘representative company’. In this

case, market participants have to use government

bonds, for which there are futures contracts. This

means that hedging will be less than perfect. For

companies, using futures or forwards involves a

trade-off between the advantages of having a ready

market and low transaction costs and using a

standardised contract, in futures; and being able

to agree to buy and sell a specific asset and the

problems of credit risk and illiquidity, in forward

contracts.

Apart from the institutional arrangements

and the fact that using futures requires both

buyer and seller to post margin, as far as compa-

nies are concerned, forwards and futures serve

very much the same purpose: both types of con-

tract allow firms to set the prices at which they

enter into a specific purchase or sale transaction in

the future and to manage the price risk for inputs

or outputs.

Before You Go On

1. What are the elements that go to deter-

mine the price at which a forward contract

is agreed? Which elements will increase

the forward price and which elements will

reduce the forward price?

2. How does a forward contract create coun-

terparty credit risk?

3. What are the main differences between a

forward contract and a futures contract?

SWAPS

Learning Objective 4Define interest rate and cross-currency swapsand know how they are valued.

Companies often enter into long-term agreements

that have predetermined cash flows. For instance, a

companymay borrow via issuing a fixed-rate bond.

Other companies, in particular small ones that do


C20 02/08/2011 19:31:6 Page 795

not have access to the bondmarket, have to borrow

at a variable rate from a bank or other financial

institution. These fixed or variable-rate loans may

create undesirable risks. Managers like to be able

to plan ahead and know the costs of the various

factors of production. For a company to borrow at

a variable rate creates the risk that interest rates

increase over the life of the loan. This is likely to

happen just when there is also pressure on the

firm’s profit margins and sales. It is therefore

desirable to manage the interest rate risk. It is

possible to use forward and futures contracts to

do this. However, these have some disadvantages:

there may not be suitable contracts for the longer

maturities or they are expensive and the prices will

change with the maturity of the contracts. Think

back to the Airbus example: if the contract had

been for three years, the forward price would have

been more than the two-year price, given the way

the cost-of-carry formula works. Furthermore, for-

wards and futures only cover a single purchase or

sale transaction.

Interest rate swap

exchange agreement where one party

exchanges a stream of interest payments

for another party’s stream of cash flows

Swaps get around the problems with forwards

and futures by using the same price for all the

exchanges in cash flows. Take the situation where

a company borrows money at a variable rate, but

would rathermake fixed payments on its loan – i.e.,

just like a bond’s payments. What it would like to

do is exchange the variable-rate liability for a fixed

set of payments. This is precisely what interest rate

swaps do. They are agreements where one party

agrees to make a set of fixed interest payments to

another party conditional upon the other party

making variable payments in exchange. To deter-

mine the payment amounts, the contract specifies a

notional amount of principal to calculate what

each party is due. The variable rate is determined

using an index of interest rates, such as the euro

interbank offered rate (Euribor). Take the example

of SEBA AG, a German machinery manufacturer

that has borrowed D20 million at a floating rate

from Commerzbank. The company wants to lock

in the interest it will pay on this loan and enters into

a five-year interest swap that exactly matches the

amount and maturity of the loan. The fixed rate is

preset at 5% and hence the fixed side (also called

the coupon) on the swap will be D1 million

(D20 million � 0.05) and that for the floating

side will be D20 million � Euribort. This is

‘reset’ at each period, which for simplicity we

will assume is 1 year, although in most cases it

is more frequent – typically, every six months, to

match the interest due on the loan. The swapwould

therefore have the following cash flows:

In this transaction SEBA has borrowed via a

loan, where the interest rate is set by reference to

Euribor. By using the interest rate swap, the com-

pany has transformed the payment flows such that

5 Years 4 3 2 1 0

–€1 –€1 –€1 –€1 –€1 [A] Fixed rate payment(millions)

[B] Indexed (floating rate)payment

þ Euribor1� D20 million

þ Euribor2 �D20 million




Combining A þ B

Net cash flow �D1 þ Euribor1� D20 million

�D1 þ Euribor2� D20 million





C20 02/08/2011 19:31:6 Page 796

its loan now has a fixed interest payment of 5% per

year with a known future interest payment at the

end of each year of D1 million. Exhibit 20.3 illus-

trates the way the loan’s variable interest rate is

transformed into a fixed rate by adding the interest

rate swap.

An interest rate swap allows a company such

as SEBA to make either fixed-rate payments or

floating-rate payments. Hence, the swap would

work equally well if SEBA had borrowed at a fixed

rate and wanted to make a floating-rate payment.

Because companies and financial institutions often

have offsetting needs, a market in swaps brokered

via major banks has evolved and a major bank will

usually be the counterparty to any corporate swap

transaction.

Learning byDoing

Application20.3

Problem: As the financial manager at the pizzarestaurant group, you note there is an inverserelationship between the revenues of the restau-rants and interest rates. This means that if interestrates rise, the group’s cash flow suffers

disproportionately since revenues go down justwhen interest costs go up. The pizza restaurantbusiness has borrowings of D5 million thatmature in four years’ time. These consist of aloan that has an interest rate indexed to Euribor

SEBA

Euriborinterestpaymenton loan

SEBA

SEBA AG enters into an interest rate swap transaction with Ribo Bank

Ribo Bank(swap counterparty)

5%

Euribor

Effect of the interest rate swap on SEBA AG’s payment flows on its loan

Ribo Bank(swap counterparty)

5%

EuriborEuriborinterestpaymenton loan

SEBA

Exhibit 20.3: How the Interest Rate Swap Transforms SEBA AG’s Floating-Rate Liability into aFixed-Rate Liability The interest rate swap transforms SEBA’s floating-rate loan payments into fixed-ratepayments when SEBA contracts to pay the fixed rate, or coupon, on the swap and receive the floating-rate

payments. As a result, the floating rate it receives matches the payments it makes on the loan and itsobligation is now to make the fixed payment of 5% on the notional amount of the swap. As the loan and the

swap are both for D20 million, the company has a fixed-rate payment each year of D1 million fromentering into the swap.


C20 02/08/2011 19:31:7 Page 797

plus 2%. The four-year interest rate swaps rate is3.25%. What can you do to reduce the effect ofhigher interest rates on the group’s cash flow?What will be the fixed rate if you use an interestrate swap?

Approach: You need to enter into an interestrate swap to ‘lock in’ the current swaps rate plusthe margin over Euribor – the interest rate index –that the pizza restaurant pays on its loan for thenext four years.

Application: The pizza restaurant group agreesto pay on the fixed (coupon) side of the four-yearswap, which is 3.25% per year. In exchange, it

will receive Euribor from the swaps counterparty.This will result in the following:

LoanInterestRate Swap Net

Payments in � 3.25% � 3.25%Payments

out� (Euriborþ 2.0%)

þ Euribor � 2.0%

� 5.25%

By entering into the swap, the pizza restau-rant group can obtain a fixed rate of 5.25% onits borrowings. Even if interest rates rise over thefour years, the group’s total interest expense isnow fixed at D262 500 per year (D5 million �0.0525).

Valuing Interest Rate SwapsThe value of an interest rate swap will be the

difference between the payments that the com-

pany makes and those it receives. We can apply

our understanding of how cash flows are valued

by noting that a swap is created if we borrow at a

floating rate and agree to pay the indexed rate and

use the proceeds to invest in a fixed-rate par bond.

The cash flows from these transactions will look

as follows:

The net payments will be simply the interest

differential between the fixed side payment and the

then prevailing floating-rate payment. Since the

swap is simply the product of a package made up

of a floating-rate borrowing and a fixed-rate

lending, we can use our understanding of how to

value thefixed side and thefloating side todetermine

the swap’s value. The swap’s value will simply be:

Value of interest rate swap

¼ Value of bond with swap coupon rate

� value of loan with swap floating rate

ð20:2Þ

In Chapter 8, we learned that the way to

price bonds is to discount their cash flows using

Equation (8.1):

PB ¼ C1

1þ iþ C2

1þ ið Þ2 þ � � � þ Cn þ Fn

1þ ið Þn

n Years 4 3 2 1 0

[A] Borrow amountP at a floating rate (i %)

þP � P � i1% �P � i1% �P � i1% �P � i1% �P � i1%� P

[B] Invest proceedsP in a fixed-rate parbond paying k%

�P þ P � k% þP � k% þP � k% þP � k% þP � k%þ P

Combining A þ B

Net cash flow — �P � i1%þP � k%

�P � i1%þP � k%

�P � i1%þP � k%

�P � i1%þP � k%

�P � i1%þP � k%


C20 02/08/2011 19:31:7 Page 798

where:

PB¼ price of the bond or present value of the

stream of cash payments

Ct¼ coupon payment in period t, where t¼ 1,

2, 3, . . . , n

Fn¼ par value or face value (principal

amount) to be paid at maturity

i¼market interest rate (discount rate or

market yield)

n¼ number of periods to maturity

This will work well for the fixed side. But what

of the floating side?We knowwhat the interest rate

is for the first period since we will know the value

of the index, but we do not know what the interest

rates will be at t ¼ 2 and thereafter. This makes it

seemingly impossible to value the floating-rate

loan. However, this is to ignore the fact that at

the start of period 2, the loan’s interest rate will be

set by the index at the then current prevailing

interest rate. This means that the loan value will

be its par value or principal amount. This means

that the value of the loan will be its principal

amount at the reset date.

Let us check this out by assuming that we have

perfect foresight and know the interest rates that

will prevail on the fixed and floating sides of the

following D100 million five-year interest rate swap

that has a fixed-rate payment of 4.383%:

The five-year interest rate is 4.383% per year.

We first present-value the cash flows, treating the

fixed-rate side as a bond and the floating-rate side

as a loan (where, exceptionally, we know what

these floating-rate payments will be):

PB ¼ D4:383

1:04383þ D4:383

1:04383ð Þ2 þD4:383

1:04383ð Þ3

þ D4:383

1:04383ð Þ4 þD4:383

1:04383ð Þ5 þD100

1:04383ð Þ5

PB ¼ D100million

PL ¼ D4:01:04383

þ D4:21:04383ð Þ2 þ

D4:4

1:04383ð Þ3

þ D4:6

1:04383ð Þ4 þD4:8

1:04383ð Þ5 þD100

1:04383ð Þ5

PL ¼ D100million

The value of this swap is zero since both sides

have equal value (PB ¼ PL). We would call this an

‘at-market’ swap since it has a zero net present

value. Just as with forwards and futures, the price

at which we can enter swaps that are being offered

in the market is their fair value. In the above swap,

neither side stands to win or lose. Of course, in

practice, the payer and receiver of the floating

payment do not know in advance what these

payments will be. However, since we know that

the floating side remains at or very close to the

notional principal on the swap, all the value change

will occur as the present value of the fixed pay-

ments rise and fall with changes in interest rates.9

An ‘off-market’ interest rate swap is valued in

the same way as an at-market interest rate swap by

noting that the floating-rate side is unaffected by

changes in interest rates.What will be the value of a

five-year swap when the fixed side or coupon

payment is not 4.383%, but 3.90% and 4.70%,

respectively?

5 Years 4 3 2 1 0

[A] Fixed-rate payment (millions) �D4.383 �D4.383 �D4.383 �D4.383 �D4.383[B] Indexed (floating-rate)

payments (millions)þD4.0 þD4.2 þD4.4 þD4.6 þD4.8

Combining A þ BNet cash flow �D0.383 �D0.183 þD0.017 þD0.217 þD0.417


C20 02/08/2011 19:31:8 Page 799

To answer this question, we simply need to

recalculate the fixed side of the swap with the new

coupon rates:

PB;3:9% ¼ D3:91:04383

þ D3:91:04383ð Þ2 þ

D3:9

1:04383ð Þ3

þ D3:9

1:04383ð Þ4 þD3:9

1:04383ð Þ5 þD100

1:04383ð Þ5

PB ¼ D97:874million

PB;4:7% ¼ D4:71:04383

þ D4:71:04383ð Þ2 þ

D4:7

1:04383ð Þ3

þ D4:7

1:04383ð Þ4 þD4:7

1:04383ð Þ5 þD100

1:04383ð Þ5

PB ¼ D101:397million

So in the case where the coupon rate is less than

the market interest rate or at-market swaps coupon

rate (3.9% < 4.383%), the value of the swap will

be þD2.126 million (�D97.874 þ D100). In the

case where the coupon rate on the swap is greater

than themarket interest rate (4.7%> 4.383%), the

value of the swap will be �D1.397 million. Of

course, there are two sides to a swap, just as there

are in forwards and futures, and hence the gains

and losses here will depend on whether one is

paying or receiving the fixed rate. The situation

will therefore be:

Swaps are direct obligations between the two

parties, like forwards, and have the same problem

with counterparty credit risk. Credit risk will arise

if the present value of the future receipts is greater

than the present value of future payments.

Learning byDoing

Application20.4

Problem: The current four-year ‘at-market’ inter-est rate swaps rate is 4.00%. You have a swapwith exactly four years to maturity with a notionalprincipal amount of D50 million and you arereceiving a fixed rate of 3.75% on the swap.What is the swap’s value?

Approach: We apply the swap valuationapproach where we treat the value of theswap as the difference between a fixed-ratebond (PB) and a floating-rate loan (PL) in orderto work out the net present value of the swap,taking the floating-side value to be the notionalprincipal amount.

Application: The amount of interest (or the cou-pon payment) on the fixed side of the swap willbe D1 875 000 (D50 000 000 � 0.0375). Thepresent value of the bond element (PB) will there-fore be:

PB ¼ D1 8750001:04

þ D1 875000

1:04ð Þ2þ D1 875000

1:04ð Þ3

þ D1 875000

1:04ð Þ4þ D50 000 000

1:04ð Þ4¼ D1 802 885þ D1 733 543þ D1 666 868

þ D44 342 967

¼ D49 546 263

Coupon Rate> At-market

Swaps Rate

Coupon Rate< At-market

Swaps Rate

Receive thefixed rate(Pay thefloating

rate)

Swap will

have anegativevalue

Swap will

have apositivevalue

Pay the fixedrate(Receive the

floatingrate)

Swap will

have apositivevalue

Swap will

have anegativevalue


C20 02/08/2011 19:31:8 Page 800

The value of the swap will be PB – PL, thatis þD49 546 263 � D50 000 000, or�D453 737. From your perspective, the swap

is a liability since the present value of the pay-ments out exceeds the present value of thepayments tobereceived.That is,3.75%<4.00%.

Cross-Currency SwapsA cross-currency swap is like an interest rate swap

except that instead of being in one currency, it in-

volves the exchange of cash flows between two

different currencies. So, for instance, one side of a

cross-currency swapmay be denominated inUS dol-

lars and the other side in euros. In this case, for the

swap to work, both parties must exchange both the

interestpayments.Forexample,ifAirbusentersintoa

cross-currency swap for D100 million at an agreed

exchange rate of US D1.3000 ¼ D1, with fixed

interest payments of 3.5% in euros and 4.1% in US

dollars for five years –where it pays inUSdollars and

receives in euros – the cashflowswill be as follows:10

To receive the US dollars, Airbus provides

D100 million at the start of the transaction.

The euro-side interest payments are D3.5 million

(D100� 0.035). On the dollar side, Airbus initially

receives D130 million (that is, D100m � 1.3000)

based on the agreed exchange rate. The interest is

$5.33 million ($130 � 0.041). At the maturity of

the swap, both parties re-exchange the principal. A

key feature of the cross-currency swap is that the

exchange rate is fixed throughout. In the next

chapter we discuss how exchange rates work and

why managing exchange rate risk is important.

What we need to understand at this point is that

by using a cross-currency swap, Airbus has effec-

tively made a fixed-rate loan in euros against a

fixed-rate borrowing in US dollars.

Cross-currency swap

the exchange of principal and interest in one

currency for the principal and interest in

another currency

There are a good many reasons why Airbus, or

any other company, might want to enter into a

cross-currency swap. In Airbus’s case, its costs and

borrowings will be largely in euros, but its airliner

sales will be largely in US dollars. Therefore, the

motivation may be to reduce the effect of currency

movements on its costs, which are largely in euros.

Other motivations include using the company’s

borrowing cost advantage in euros to fund US

dollar-denominated investments, such as a North

American subsidiary. The motivations for corpo-

rate risk management discussed at the start of the

chapter stimulate the corporate use of cross-

currency swaps.

Learning byDoing

Application20.5

Problem: The pizza restaurant group is consid-ering expanding its operations into Sweden. Youhave been tasked with providing the necessary

finance to support this move, which is estimatedto need D2 million. You have been in contactwith a Swedish bank and they say the company

5 Years 4 3 2 1 0

Cash flows in euros (millions)

+€3.5 +€3.5 +€3.5 +€3.5 +€3.5 –€100 +€100

Cash flows in US dollars (millions)

–$5.33 –$5.33 –$5.33 –$5.33 –$5.33 +$130 –$130


C20 02/08/2011 19:31:8 Page 801

can borrow Swedish krona (SKr) 20 million at afixed rate of 6.20% per year for five years. Inyour research, you discover you can do a cross-currency swap between the euro and the Swed-ish krona for five years at 6.10% per yearagainst a fixed rate in euros of 5.10% peryear. The company currently has the ability toraise fixed rate using an interest rate swap as perLearning by Doing 20.3 at 5.25% per year.Which represents the better financing deal?

Approach: We need to compare the two alter-natives, which are (1) borrowing directly fromthe Swedish bank or (2) borrowing in euros andusing the cross-currency swap to obtain theSwedish krona for the new venture.

Solution: The two alternatives provide the fol-lowing cost of borrowing:

(1) Direct borrowing from the Swedish bank is6.20%.

(2) The swaps rate is euros 5.10% and 6.10% inSwedish krona. The company pays 0.15%onits euro borrowing (5.25% – 5.10%) but pays0.20% less on its Swedish krona via the swap(6.30%–6.10%).Netting the two differencesmeans that it is saving a modest 0.05% peryear (0.15% – 0.20%) by borrowing in eurosandswapping intoSwedishkrona.Borrowingin euros and swapping to fixed rate in eurosgivesanall-in costof6.15%inSwedishkrona.

Valuing Cross-Currency SwapsThe valuation of cross-currency swaps is the same

as that for interest rate swaps. We simply present-

value the cash flows of the two sides and convert

one of the present values into the other currency

using the prevailing exchange rate. Let us value the

Airbus swap given above and, to do so, we will

assume that one year has passed and interest rates

and the exchange rate have both changed. The

exchange rate has now moved to $1.3500 ¼ D1,that is, the US dollar has fallen against the euro.

The interest rate in US dollars has risen slightly to

4.5%, as has that in euros, which is now 3.75%.

The original market conditions at the initiation

of the cross-currency swap and the new market

conditions and changes are given below:

Using the current market conditions, we now

revalue the swap using the same approach that we

used for the interest rate swap. We therefore pres-

ent-value the remaining cash flows as follows:

PB;euro ¼ D3:5

1:0375þ D3:5

1:0375ð Þ2 þD3:5

1:0375ð Þ3

þ D103:51:0375ð Þ4

¼ D99:087million

PB;US dollars ¼$5:33

1:045þ $5:33

1:045ð Þ2 þ$5:33

1:045ð Þ3

þ $135:33

1:045ð Þ2 ¼ $128:134million

The last step is to convert one of the currencies

into the other at the current exchange rate for the

US dollar and the euro ($1.3500/D ). The value of

the swap in euros is therefore D4.173 million

(D99.087 – $128.134/$1.3500). Of course, this

value will depend on whether one is receiving

the euro cash flows or the US dollar ones. For

one side, it is a gain and for the other side, a

loss. To better understand where the gains and

losses are coming from, we can break the swap

Original MarketConditions

Market ConditionsAfter One Year

Change From OriginalMarket Conditions

Euro interest rate 3.50% 3.75% þ0.25%US dollar interest rate 4.10% 4.50% þ0.40%Foreign exchange rate $1.300/D $1.3500/D þ$0.0500Maturity 5 years 4 years �1 year


C20 02/08/2011 19:31:8 Page 802

into its constituent value change components: (1)

change in value of euro component, (2) change in

value of US dollar component and (3) changes in

the exchange rate. We therefore have:

The table shows that the components of value

change have led to the cross-currency swap either

being an asset (if the party is paying US dollars

and receiving euros) or a liability (if receiving US

dollars and paying euros).

We will look at the cross-currency swap from

Airbus’s perspective, but this is simply a mirror

image to that of the counterparty on the other

side of the swap. From the table we can see that

there is a change in value of �D0.913 million

(D99.087 � D100) from the increase in the rate of

interest on the euro side of the swap. The original

interest rate was 3.50% and it has increased to

3.75%. This leads to a reduction in the present

value of the cash flows denominated in euros

that Airbus will receive. The same has happened

for the US dollar side, where interest rates have

risen from 4.10% to 4.50% and the value has

fallen from $130 million to $128.134 million

(�$1.866). Whether this is good or bad news

depends on whether one is paying or receiving

US dollars on the cross-currency swap. As the

dollar side is a liability to Airbus since it is paying,

a reduction in value is good news. This is because

Airbus can now terminate the swap at the current

market conditions and pay back less than

originally borrowed. To put it another way, to

replace the dollar-denominated cash flows, Air-

bus can provide $128.134 million rather than the

original $130 million, thus saving $1.866 million.

If Airbus had been receiving the US dollar cash

flows, it would have lost money from the change

in interest rates – as it has done from the increase

in the rate of interest in the euro. The same logic

applies for the change in the value of exchange

rate between the euro and the US dollar. The

original swap required $130 million to equate

to D100 million; with the fall in value of the

US dollar, $135 million is needed. This means

an additional $5 million is needed, depending on

whether one is due to repay or receive dollars.

Since Airbus is due to repay $130 million, it now

needs fewer euros to repay the originally con-

tracted amount of $130 million. At the start,

D100 million bought $130 million, now D100

million buys $135 million, so Airbus would only

need to provide $130/$135 � D100 million

(D96.296), saving D3.704 million. Adding all

these effects together, and converting the dollars

to euros, gives a net change in value of D4.173

million. This is the sum that the euro payer (US

dollar receiver) needs to pay to the euro receiver

(US dollar payer) to terminate the swap. Since

Airbus has contracted to pay US dollars and

receive euros, it will receive D4.173 million if

the swap is terminated by mutual agreement.11

(millions)OriginalValue

NewValue

Value if Paying theEuros and Receivingthe US Dollars

Value if Paying theUS Dollars andReceiving the Euros

(1) Euro-side value change D100 D99.087 þD0.913 �D0.913(2) US dollar-side valuechange

$130 $128.134 �$1.866or �D1.382(�$1.866/$1.3500)

þ$1.866or þD1.382($1.866/$1.3500)

(3) Change frommovement in the currency

$130 $135 �$5or �D3.704(�$5/$1.3500)

þ$5or þD3.704($5/$1.3500)

(1 þ 2 þ 3) Net effect ofchanges in value

�D4.173 þD4.173


C20 02/08/2011 19:31:9 Page 803

Learning byDoing

Application20.6

Problem: The pizza restaurant decided to usethe cross-currency swap discussed in Learning byDoing 20.5 and entered into a 5-year agreementto pay Swedish krona (SKr) and receive euros.The fixed rate on the swap is 6.30% in krona peryear and 5.10% in euros. The amount of theswap is D2 million and SKr 19.6 million,respectively.

It is now the pizza restaurant’s year end andthe auditors want to know what is the swap’scurrent value as, under the IFRS rules, derivativetransactions need to be marked-to-market andreported on the company’s balance sheet. Thatis, they need to be revalued to their fair value forfinancial reporting purposes. Since the swapwas initiated, one year has passed and theexchange rate of the euro to the Swedish kronais now at SKr 9.9/D , the Swedish krona interestrate for four-year swaps is 6.25% and that foreuros is 5.05%.What is the swap’s fair value forreporting purposes?

Approach: We need to apply the valuationapproach for cross-currency swaps where wepresent-value the two sets of remaining cashflows for the four years at the now-prevailinginterest rates and convert the two sides to acommon currency before determining the netvalue. The pizza restaurant group is payingSwedish krona (which is the liability side) andreceiving euros (the asset side). Since the report-ing currency is the euro, it is necessary to convertthe value of the cross-currency swap to thiscurrency.

Solution: We first need to calculate the remain-ing cash flows on the two sides of the swap. Theeuro side is worth D2 million and the interest rateis 5.10%, so the fixed payments are D102 000per year (D2 million � 0.051%). On the kronaside, the fixed payment is SKr 1 196 000(SKr 19.6 � 6.10).

PB;euro ¼ D102 000

1:0505þ D102 000

1:0505ð Þ2 þD102 0001:0505ð Þ3 þ

D2 102 0001:0505ð Þ4

¼ D2 003 542

PB;SKr ¼ SKr 1 196 000

1:0625þ SKr 1 196 000

1:0625ð Þ2

þ SKr 1 196 000

1:0625ð Þ3 þ SKr 21 196 000

1:0625ð Þ4 ¼ SKr 19 498 706


C20 02/08/2011 19:31:9 Page 804

The final stage is to convert the krona valueinto euros (we do this since the pizza restau-rant reports its results in euros) at the currentexchange rate, which gives a value in euros ofD1 969 566 (SKr 19 498 706/SKr 9.9/D ).The company receives the euros, so this is a

cash inflow and pays the krona, so the netvalue of the swap is D33 976 (D2 003 542� D1 969 566). This is a positive value, so thisis the amount that will be reported as a long-term financial asset on the balance sheet at theyear-end.

Before You Go On

1. How can we characterise the cash flows

from an interest rate swap and a cross-

currency swap?

2. Why does a swap only have credit risk

when it has a positive value?

3. In what ways do swaps transform the risk

of firms’ assets and liabilities?

FINANCIAL OPTIONS

Learning Objective 5Define a call option and a put option anddescribe the payoff function for each of theseoptions.

A financial option is a derivative in that, like for-

wards, futures and swaps, its value is derived from

the value of another asset. The owner of a financial

optionhas the right, but not theobligation, tobuyor

sell an asset on or before a specified date for a

specified price. The asset that the owner has a right

to buy or sell is known as the underlying asset. The

last date on which an option can be exercised is

called the exercise date, or expiration date, and the

price at which the option holder can buy or sell the

asset is called the strike price, or exercise price.

Financial option

the right to buy or sell a financial security,

such as a share of stock, on or before a

specified date for a specified price

Underlying asset

the asset fromwhich the value of an option is

derived

Exercise (expiration) date

the last date on which an option can be

exercised

Strike (exercise) price

the price at which the owner of an option has

the right to buy or sell the underlying asset

Call OptionsLet us consider how the value of an option is

derived from the value of an underlying asset.

Suppose you own an option to buy one share of

Siemens AG, the German engineering company,

for D50 and today is the exercise date – if you do

not exercise the option today, it will expire and

become worthless. If the price of Siemens shares is

less than D50, it does not make sense to exercise

your option, because if you did, you would be

paying D50 for something you could buy for less

than D50 in the open market. Similarly, if the share

price is D50, there is no benefit to be had from

exercising your option. If, however, the price is

above D50, then you will benefit from exercising

the option. Even if you do not want to own

the Siemens share, you can buy it for D50 and


C20 02/08/2011 19:31:9 Page 805

immediately turn around and sell it for a profit. The

value of the option to you is the difference between

the market price of Siemens shares and the strike

price of the option. For example, if the Siemens

shares are trading for D60 in the market, then the

option is worth D10 (D60 share price � D50strike price) to you. If the shares are trading at

D70, then the value of the option is D20 (D70 �D50), and so on.

The relation between the value of an option

and the price (value) of the underlying asset – such

as the Siemens shares – is known as the option

payoff function. Part A in Exhibit 20.4 illustrates

the payoff function at expiration (actually, the

instant before the option expires) for the owner

of an option that is like the option on the Siemens

shares we just discussed. This option is known as a

call option because it gives the owner the right to

buy, or ‘call’, the underlying asset.

Option payoff function

the function that shows how the value of an

option varies with the value of the under-

lying asset

Call option

an option to buy the underlying asset

Strike price0

Val

ue o

f Cal

lO

ptio

n at

Exp

iratio

n

Value (price) of Underlying Asset

A. Owner (buyer) of a call option

Strike price

0

Val

ue o

f Sel

ler’s

Pos

ition

at E

xpira

tion

of C

all O

ptio

n

Value (price) of Underlying Asset

B. Seller of call option

The value of a call option increasesone for one with an increase in thevalue of the underlying asset when thevalue of that asset is above the strikeprice.

The value of the seller’s positiondecreases one for one with an increasein the value of the underlying asset when the value of that asset is abovethe strike price.

Exhibit 20.4: Payoff Functions for a Call Option at Expiration At the instant before it expires, the value of acall option to the owner equals either: (1) zero, if the value of the underlying asset is less than or equal to the strike

price, or (2) the value of the underlying asset less the value of the strike price, if the value of the underlyingasset is greater than the strike price.

The value of the seller’s position equals either: (1) zero, if the value of the underlying asset is less than or equal tothe strike price, or (2) the strike price less the value of the underlying asset if the value of the underlying

asset is greater than the strike price.


C20 02/08/2011 19:31:9 Page 806

With an exercise price of D50, the value of theSiemens call option equals D0 if the price of the

underlying shares is D50 or less. As we noted

earlier, it would not make sense to exercise the

option if the price of the shares is not greater

than D50. Since an option is the right to buy or

sell an underlying asset, rather than an obligation

to buy or sell, the owner of the option can simply

let it expire if it does not make sense to exercise it.

This limits the downside for the owner of the

option to D0. In this way, options are very differ-

ent to the forwards, futures and swaps discussed

earlier.

If the underlying asset price is above the strike

price, the value of the call option at exercise

increases unit for unit with the price of the under-

lying asset. You can see this relation in part A of the

exhibit. For every euro that the asset price exceeds

the strike price, the value of the call option

increases by one euro. In other words, the slope

of the payoff function equals one when the under-

lying asset price is above the exercise price.

Part B of Exhibit 20.4 illustrates the payoff

function for a person who sells a call option (also

known as writing the option). Notice that the

payoff function for the seller (or writer) is the

mirror image of that for the owner (buyer) of

the call option. This makes sense, since any gain

for the owner is a loss for the seller. To see why this

is true, let us return to the Siemens option example.

Recall that if the shares are trading at D60 when

the option expires, the call option is worth D10 to

the owner, who can purchase the shares for

D50 and then immediately sell them on the market

for D60. The seller of the call option, though, must

sell shares that are worth D60 forD50 – resulting in

a D10 loss.

Part B of Exhibit 20.4 shows that the payoff to

the seller of the call option is never positive. It is

negative when the price of the underlying asset is

greater than the strike price, and it equals zero

when the price of the underlying asset is equal to or

less than the strike price. You may be wondering

why anyone would ever sell a call option if the

return were never positive. The reason is simply

that the buyer pays the seller a fee to purchase the

option. This fee, known as the call premium, makes

the total return to the seller positive when the

price of the underlying asset is near or below the

strike price.

Call premium

the price that the buyer of a call option pays

the seller for that option

A call premium is just like the premium you

pay when you purchase insurance for your car. In

return for the insurance premium, the insurance

company agrees to pay you if certain events occur,

such as if you collide with another car or if a

hailstorm damages the car. The seller of a call

option is simply selling insurance to the buyer

which pays the buyer when the value of the under-

lying asset is above the strike price.

Put OptionsWhile the owner of a call option has the right to

buy the underlying asset at a pre-specified price on

or before the expiration date, the owner of a put

option has the right to sell the underlying asset at a

pre-specified price. The payoff function for the

owner of a put option is similar to that for a call

option but it is the reverse in the sense that the

owner of a put option profits if the price of the

underlying asset is below the strike price. This is

illustrated in Exhibit 20.5.

Put option

an option to sell the underlying asset

Part A of the exhibit shows that the owner of a

put option will not want to exercise that option if

the price of the underlying asset is above the strike

price. Obviously, it does not make sense to sell an

asset for less than you can get on the open market.


C20 02/08/2011 19:31:9 Page 807

When the value of the underlying asset is below the

strike price, however, the owner of the put option

will find it profitable to exercise the option. For

example, suppose that you own a put option that is

expiring today and that entitles you to sell shares in

Siemens for D50. If the current price of Siemens

shares in the market is D45, the put option is worth

D5, because exercising the option will enable you

to buy the shares for D45 and then turn around and

sell them for D50. Similarly, if the current price of

Siemens shares is D30, the put option is worth D20,

because you can buy the shares for D30 and sell

them for D50.

Part B of Exhibit 20.5 shows that the payoff for

the seller of the put option is negativewhen the price

of the underlying asset is below the strike price. This

is because the seller of the put option is obliged to

purchase the asset at a price that is higher than its

market price. For instance, in the Siemensput option

example, if the exercise price is D50 and the current

market price isD30, the seller of the put optionmust

buy the shares for D50 but can only sell them for

D30. This results in a D20 loss.

As with a call option, the payoff for the seller of

a put option, which is illustrated in part B of Exhibit

20.5, is never positive. The seller of a put option

hopes to profit from the fee, or put premium, that he

or she receives from the buyer of the put option.

Put premium

the price that the buyer of a put option pays

the seller of that option

Strike priceValue (price) of Underlying Asset

0

Val

ue o

f Put

Opt

ion

at E

xpira

tion

A. Owner (buyer) of a put option

Strike priceValue (price) of Underlying Asset

0

Val

ue o

f Sel

ler’s

Pos

ition

at E

xpira

tion

of P

ut O

ptio

n

B. Seller of put option

The value of a put option increasesone for one with a decrease in thevalue of the underlying asset when thevalue of that asset is below the strikeprice.

The value of the seller’s positiondecreases one for one with an decreasein the value of the underlying asset when the value of that asset is belowthe strike price.

Exhibit 20.5: Payoff Functions for Put Option at Expiration At the instant before it expires, the value of a putoption to the owner equals either: (1) zero, if the value of the underlying asset is greater than or equal to the

strike price, or (2) the strike price less the value of the underlying asset, if the value of the underlyingasset is less than the strike price.

The value to the seller of a put option equals either: (1) zero, if the value of the underlying asset is greater than orequal to the strike price, or (2) the value of the underlying asset less the strike price,

if the value of the underlying asset is less than the strike price.


C20 02/08/2011 19:31:9 Page 808

American, European andBermudan OptionsAt the beginning of this section, we said that the

owner of a financial option has the right to buy or

sell a specific asset on or before a specified date for

a specified price. In the real world, there are actu-

ally several different arrangements concerning

when an option can be exercised. Some options

can only be exercised on the expiration date. These

are known as European options. Other options,

known as American options, can be exercised at

any point in time on or before the expiration date.

There are also exotic options, such as so-called

Bermudan options, which can be exercised only on

specific dates during the life of the option. Most

exchange-traded options are American options.

More on the Shapes of OptionPayoff FunctionsIt is important to note that the payoff functions in

Exhibits 20.4 and 20.5 illustrate the values of

options to owners and sellers at the instant before

they expire.These payoff functions have similar, but

somewhat different, shapes at earlier points in time.

We discuss why this is the case in the next section.

It is also important to recognise that the

payoff functions in Exhibits 20.4 and 20.5 are

not straight lines for all possible values of the

underlying asset. Each payoff function has a

‘kink’ at the strike price. This kink exists because

the owner of the option has a right, not an

obligation, to buy or sell the underlying asset.

If it is not in the owner’s interest to exercise the

option, he or she can simply let it lapse. Later, we

will discuss how this feature of options causes

agency problems and how it can be useful in

managing the risks faced by a firm.

WEB

You can learn more about call options and

put options on the Options.Net website at:

http://www.theoptions.net/option-trading-

strategies/pay-off-diagrams-for-option/.

Decision-Making Example 20.1

When it Makes Sense toExercise an OptionSituation: Youowna call optionandaput optionon Fiat shares. The strike price for both of theseoptions isD8and both options expire today. If thecurrent price of Fiat shares is D7, would youexercise either of these options? If so, which one?

Decision: You should exercise the put option.It allows you to sell Fiat shares for D8 that wouldcost you only D7 to buy. It does not make senseto exercise the call option because the strikeprice is greater than the market price of Fiatshares.

BUILDING INTUITI NPayoff Functions for Options are Not LinearPayoff functions for options are not straight lines. This is because the owners of options have the right,rather than the obligation, to buy or sell the underlying assets. If it is not in the owner’s best interest toexercise an option, he or she can simply let it expire without exercising it. This limits the owner’spotential loss to the value of the premium he or she paid for the option. This makes options fromforwards, futures and swaps where the gains and losses are symmetrical.


C20 02/08/2011 19:31:9 Page 809

Before You Go On

1. What is a call option and what do the

payoff functions for the owner and seller

of a call option look like?

2. What is a put option, and what do the

payoff functions for the owner and seller

of a put option look like?

3. Why does the payoff function for an

option have a kink in it?

OPTION VALUATION

Learning Objective 6List and describe the factors that affect the valueof an option.

We saw in the last section that determining the

value of a call or a put option at the instant before

it expires is relatively simple. For a call option, if

the value of the underlying asset is less than or

equal to the strike price, the value of the option to

the owner is zero. If the value of the underlying

asset is greater than the strike price, the value to

the owner is simply the value of the underlying

asset minus the strike price. For a put option, if

the value of the underlying asset is greater than or

equal to the strike price, the value of the option is

zero to the owner. If the value of the underlying

asset is less than the strike price, the value to the

owner is the strike price minus the value of the

underlying asset.

It is more complicated to determine the value

of an option at a point in time before its expiration

date. We do not know exactly how the value of the

underlying asset will change over time and there-

fore we do not know what value we will ultimately

receive from the option. In this section, we discuss

the key variables that affect the value of an option

prior to expiration and describe one method that is

commonly used to value options. Our objective is

not to make you an expert in option valuation but

rather to help you develop some intuition about

what makes an option more or less valuable. This

intuition will help you better understand how

options affect corporate finance decisions.

Limits on Option ValuesWe will begin by using some common sense to put

limits on what the value of a call option can

possibly be prior to its expiration date. We focus

on call options here because, as you will see, there

is a simple relation that enables us to calculate the

value of a put option once we know the value of a

call option with the same strike price and expira-

tion date.

We already know that the value of a call

option can never be less than zero, since the

owner of the option can always decide not to

exercise it, if doing so is not beneficial. A second

limit on the value of a call option is that it can

never be greater than the value of the underlying

asset. It would not make sense to pay more for the

right to buy an asset than you would pay for the

asset itself. These two limits suggest that the value

of a call option prior to expiration must be in the

shaded area in part A of Exhibit 20.6. The shaded

area is bounded below by the horizontal axis,

because the value of the option must be greater

than zero, and it is bounded above by the line that

slopes upward at a 45-degree angle, because an

option value greater than this would exceed the

value of the underlying asset.

There are two other limits on the value of a call

option prior to expiration, and these limits are

somewhat more subtle. First, the value of a call

option prior to the expiration date will never be less

than the value of that option if it were exercised

immediately. This is true because there is always a

possibility that the value of the underlying asset

will be greater than it is today at some time before

the option expires. Of course, it is possible that the

value will be lower but, since the value of the

option cannot be less than zero and there is no

limit on how high it can go, the expected effect of

an increase in the value of the underlying asset on

the value of the option is greater than the expected


C20 02/08/2011 19:31:9 Page 810

effect of a decrease. The bottom line is that, prior to

expiration, the value of a call option will be greater

than the value represented by the solid line in part

A of Exhibit 20.4.12

The final limit arises because of the time value

of money. When we consider the value of a call

option at some time prior to expiration, we must

compare the current value of the underlying asset

with the present value of the strike price, dis-

counted at the risk-free interest rate. We would

be comparing apples and oranges if we did not do

this. The present value of the strike price is the

amount that an investor would have to invest in

risk-free securities at any point prior to the expira-

tion date to ensure that he or she would have

enough money to exercise the option when it

expired. Thus, when we compare the value of a

call option prior to expiration with the value at

expiration, represented by the solid line in part A of

Exhibit 20.4, we must use the present value of the

strike price to draw the line. The shaded area in

part B of Exhibit 20.6 illustrates the possible values

for a call option prior to expiration under all four

of the limits we have discussed.

In practice, we find that, prior to expiration,

call options have a shape that is very similar to the

one illustrated by the dotted line in part C of

Exhibit 20.6. Notice that this dotted line appro-

aches zero as the value of the underlying asset gets

very small relative to the strike price. This makes

sense because, with a very low asset value, it

becomes highly unlikely that the owner of the

option will ever choose to exercise it.

On the right side of the dotted line, you can see

that the value of a call option prior to expiration

approaches the value of the call option at expiration.

A. Possible values with first two limits

0

Val

ue o

f Cal

l Opt

ion

The first two limits tell us that the valueof a call option prior to expiration mustfall within this shaded area.

Current Value of Underlying Asset

B. Possible values with all four limits

0

Val

ue o

f Cal

l Opt

ion


C. Typical payoff function for call option prior to expiration

0

Val

ue o

f Cal

l Opt

ion


Present Value of Strike Price

The four limits tell us that the value ofa call option prior to expiration willactually fall within this shaded area.

Strike Price

Value of call optionprior to expiration

Exhibit 20.6: Possible Values of a Call Option Prior to Expiration The value of a call option: (1) must be greateror equal to zero (horizontal axis) and (2) cannot be greater than the value of the underlying asset (45-degree line).

In addition to the two limits illustrated in part A, the value of a call option prior to expiration: (3) will never be less thanthe value of the option if it were exercised immediately where (4) the value of the option is calculated using the

present value of the strike price, discounted from the expiration date at the risk-free interest rate. These conditions areboth illustrated by the lower 45-degree angle.

Part C shows the typical relation between the value of a call option prior to expiration and its value at expiration. Thevalue of the option prior to expiration is farthest from the value of the option at expiration when the price of the

underlying asset is near the strike price.


C20 02/08/2011 19:31:9 Page 811

This is because, when the current value of the under-

lying asset is far to the right of the kink in the

payoff function, the probability that this value

will fall below the strike price is very small. In other

words, the expected effect of an increase in the

value of the underlying asset on the value of the

option is no longer much greater than the expected

effect of adecrease. In this situation, the call option is

verymuch like a forward contract on the underlying

asset.

Finally, notice that the dotted line is furthest

above the value of the call option at expiration

when the price of the underlying asset is near the

strike price. At the strike price, the expected effect

of an increase in the value of the underlying asset

on the value of the option exceeds the expected

effect of a decrease by the greatest amount.

Variables that AffectOption ValuesFive variables affect the value of a call option prior

to expiration. Four of them are related to the

following questions:

1. How likely is it that the value of the underlying

asset will be higher than the strike price the

instant before the option expires?

2. How far above the strike price might it be?

The first two variables are relatively easy to

understand. They are the current value of the under-

lying asset and the strike price. The higher the

current value of the underlying asset, themore likely

it is that the value of the assetwill be above the strike

pricewhen the call optionnears expiration. Further-

more, the higher the current value of the asset, the

greater the likely difference between the value of the

asset and the strike price. This means that, holding

the strike price constant, investors will paymore for

a call option if the underlying asset value is higher,

because the expected value of the option as it nears

expiration is higher.13 For example, suppose that

you are considering purchasing a three-month

American call optiononSiemens shareswith a strike

price of D50. You should be willing to paymore for

this option if the current price of Siemens shares is

D55 than if it is D50.

The opposite relation applies to the strike price.

That is, the lower the strike price, the more likely

that the value of the underlying asset will be higher

than the strike price when the option nears expira-

tion. In addition, the lower the strike price, the grea-

ter the likely difference between these two amounts.

Thus, the lower the strike price, the more valuable

the option is likely to be at expiration. Of course, if

the option is expected to be more valuable at expi-

ration, it will also be more valuable at any point

prior to expiration. Returning to our Siemens exam-

ple, we see that a call option with a strike price of

D45 is worth more than a call option with a strike

price of D50.

We turn next to two variables that affect the

value of call options in somewhat more subtle

ways. These variables are the volatility of the value

of the underlying asset and the time until the

expiration of the option. To understand how these

factors affect the value of a call option, recall from

part C of Exhibit 20.6 that the payoffs function for

a call option prior to expiration is not symmetric. If

the value of the underlying asset is well above the

strike price, then the value of the option varies in

much the same way as the value of the underlying

asset. However, if the value of the underlying asset

is well below the strike price, then the value of the

option approaches zero but changes at a much

lower rate than the value of the underlying asset

changes. It does not matter if the underlying asset

value is just a little bit below the strike price or is

worthless – a call option cannot be worth less

than zero.

To show how the volatility of the underlying

asset value affects the value of an option, we will

consider a call option on an underlying asset that

has a value exactly equal to the strike price of the

option. The value of this option will increase more

when the value of the underlying asset goes up than

it will decrease when the value of the underlying

asset goes down. Let us suppose that the value of

the underlying asset is equally likely to go up or

down. In this case, the further the value of the asset

is likely to move (the greater its volatility), the


C20 02/08/2011 19:31:9 Page 812

higher will be the value of a call option on this

asset. In other words, the greater the volatility of

the underlying asset value, the higher the value of a

call option on the asset prior to expiration.

In our Siemens example, suppose the strike

price for a call option on Siemens shares is D50,

the current price of the shares is D50 and the optionexpires in one year. Further, suppose that the

standard deviation, s, of the return on the Siemens

shares is 30% per year. Recall from the discussion

in Chapter 7 that with a standard deviation of

30%, there is a 5% chance that the Siemens share

price will change by more than 58.8% (1.96 stan-

dard deviations � 30%) by the time the option

expires. In other words, there is a 5% chance

that the Siemens share price will be less than

D20.60 (D50� [1 – 0.588]) or greater than D79.40

(D50 � [1þ0.588]) in a year. If, instead of 30%,

the standard deviation of Siemens shares were 40%

per year, there would be a 5% chance that the price

would be below D10.80 or above D89.20. (You

should check these numbers to make sure you

know how they are calculated.) As you can see,

with the higher standard deviation the share price

is more volatile. Investors will pay more for an

option on a share that has a more volatile price,

because the potential change in the price is greater.

The time until the expiration affects the value

of a call option through its effect on the volatility

of the value of the underlying asset. The greater

the time to maturity, the more the value of the

underlying asset is likely to change by the time the

option expires. For example, we will return once

again to the Siemens example. Suppose that the

option expires in two years rather than in one

year. People who study statistics have found that

the standard deviation of the return on an asset

increases over time by the square root of n, where

n is the number of periods. Thus, if the standard

deviation of the return on Siemens shares is 30%

per year, the standard deviation over two years

will be:

s2 years ¼ s� nð Þ1=2 ¼ 30� 2ð Þ1=2 ¼ 30� 1:414

¼ 42:42%

Clearly, then, a two-year option will be worth

more than a one-year option if all the other char-

acteristics of the two options are the same.

We have now discussed four of the five vari-

ables that affect the value of an option. The fifth

variable is the risk-free rate of interest. The value of

a call option increases with the risk-free interest

rate. Exercising a call option involves paying cash

in the future for the underlying asset. The higher

the interest rate, the lower the present value of the

amount that the owner of a call option will have to

pay to exercise it.

WEB

You can read about what affects the values of

financial options and how they are traded at

the websites for the Chicago Board Options

Exchange (CBOE) at: http://www.cboe.com/

and the International Securities Exchange

(ISE) at: http://www.iseoptions.com/.

The Binomial OptionPricing ModelIn this section, we use a simple model to show how

we can calculate the value of a call option at some

point in time before the expiration date. This model

assumes that the underlying asset will have one of

only two possible values when the option expires.

The value of the underlying asset will either increase

to some value above the strike price or decrease to

some value below the strike price.

Arbitrage

buying and selling assets in a way that takes

advantage of price discrepancies and yields a

profit greater than that which would be

expected based solely on the risk of the

individual investments

To solve for the value of the call option using

this model, we must assume that investors have no


C20 02/08/2011 19:31:9 Page 813

arbitrage opportunities with regard to this option.

Arbitrage is the act of buying and selling assets in a

way that yields a return above that suggested by the

SecurityMarket Line (SML), whichwe discussed in

Chapter 7. In other words, the absence of arbitrage

opportunities means that investors cannot earn a

return that is greater than that justified by the

systematic risk associated with an investment. As

an example of an arbitrage opportunity, suppose

that the shares of a particular company are being

sold for a lower price in one country than in

another country. An investor could simultaneously

buy the shares in the country where they are less

expensive and sell them in the country where they

are more expensive. Assuming that the profit

exceeds any transaction costs, the investor would

earn an instantaneous risk-free profit. Since it is

instantaneous, this profit would be, by definition,

above the SML because the SML would predict

that the expected return on a risk-free investment is

zero if the holding period is zero.

To value the call option in our simplemodel,we

will first create a portfolio that consists of the asset

underlying the call option and a risk-free loan. The

relative investments in these two assets will be

selected so that the combination of the asset and

the loan has the same cash flows as the call option,

regardless of whether the value of the underlying

asset goes up or down. This is called a replicating

portfolio, since it replicates the cash flows of the

option.The replicatingportfoliomust have the same

value as the option today, since it has the same cash

flows as the call option in all possible future out-

comes. If the replicating portfolio did not have the

samevalue as theoption, an investor could construct

an arbitrage portfolio by buying the cheaper of the

two and selling the more expensive of the two. Such

trading would eventually drive the values of the

option and the replicating portfolio together.

To see how a replicating portfolio is construc-

ted, consider an example. Suppose that DRYAD

SA shares currently trade for D50 and that its price

will be either D70 or D40 in one year. We want to

determine the value of a call option to buy DRYAD

shares for D55 in one year. First, notice that the

value of this option is D15 if the share price goes up

to D70 (D70� D55¼ D15) and that it is zero if the

share price goes down to D40, since the option will

not be exercised. Suppose also that the risk-free

rate is 5%.

We can construct a portfolio consisting of x

DRYAD SA shares and a risk-free loan with a value

of y euros that produces a payoff of either D70 or

D40. As youwill see, this risk-free loanmay involve

either borrowing or lending. For each risk-free euro

lend, we know that wewill receive D1.05 regardlessof what happens to the price of the DRYAD shares.

In the same way, if we borrow D1, we will owe

D1.05 at the end of the year. The value of the shares,

the risk-free loan, and the option today and at

expiration can be illustrated as follows:


C20 02/08/2011 19:31:9 Page 814

The value of each asset when the share price goes

up to D70 is shown on the right arrow and the

value when the shares go down to D40 is shown on

the left arrow. Notice that we do not know the

value of the option today – that is what we are

trying to calculate.

We can write two equations that define the

replicating portfolio that we want to construct:

D15 ¼ D70� xð Þ þ 1:05� yð ÞD0 ¼ D40� xð Þ þ 1:05� yð Þ

The first equation represents the case in which the

share price increases to D70 and the second equa-

tion represents the case in which the share price

goes down to D40. The first equation says that we

want the portfolio to be worth D15 when the share

price increases to D70 and that the D15 value will

consist of x shares worth D70 and a risk-free loan

with a face value of y and a value in one year of

D1.05 per euro of face value. Similarly, the second

equation says that if the share price falls to D40,

we want the portfolio to be worth zero (D0). In

this case, the portfolio will consist of x shares

worth D40 and a risk-free loan with a face value

of y and a value in one year of D1.05 per euro of

face value.

Since we have two equations and there are two

unknowns, x and y, we can solve for the values of

the unknowns. Recall from your algebra class that

we can solve for x and y by first writing one

equation in terms of either x or y and then substi-

tuting the result into the second equation. For

example, the first equation can be written in terms

of x as follows:

x ¼ D15� 1:05� yð ÞD70

Now, substituting into the second equation

gives us:

D0 ¼ D40� D15� 1:05� yð ÞD70

� �þ 1:05� yð Þ

We can now solve this equation for y. For example,

we can write this relation as follows:

D0 ¼ D40� D15� 1:05� yð ÞD70

� �þ 1:05� yð Þ

D0 ¼ D8:5714� 0:6� yð Þ þ 1:05� yð ÞD0 ¼ D8:5714þ 0:45y

0:45y ¼ �D8:5714

Therefore:

y ¼ �D8:5714

0:45¼ �D19:05

Finally, substituting this value back into the first

equation gives us the value of x:

x ¼ D15� 1:05��D19:05ð ÞD70

x ¼ D15þ D20:00

D70

x ¼ 0:5

This tells us that the replicating portfolio con-

sists of half a DRYAD SA share (x ¼ 0.50) and a

D19.05 risk-free loan (y¼�19.05).14 The negative

value for y tells us that we would borrow, rather

than lend, D19.05 at the risk-free interest rate. If

we buy half a share and borrow D19.05, then in

one year our replicating portfolio will have exactly

the same payoff as the call option with a strike

price of D55.

If the value of the shares declined to D40, we

would own half a share worth D20 and we would

owe D19.05 � 1.05 ¼ D20 on the loan. Since the

value of the half-share would exactly equal the

amount owed on the loan, the portfolio would

have a total value of exactly zero. In contrast, if

the value of the shares increased to D70, the half a

sharewould beworthD35. Sincewewould still owe

only D20 in this case, the portfolio would have a

total value of D15. Since these payoffs are the same

as those for the option, this portfolio must have the

same value as the option.

At this point, we know what the replicating

portfolio is and we know that the replicating port-

folio must have the same value as the call option.

Now all we have to do to estimate the value of the

call option is figure out what is the present value of

the replicating portfolio. To do this, we simply


C20 02/08/2011 19:31:9 Page 815

determine how much of our own money we would

actually have to invest to construct the replicating

portfolio. In our example, we could use the D19.05

loan to help purchase the shares, so we would not

have to come upwith all themoney for the shares on

our own. In fact, since DRYAD SA shares are

currently worth D50, a half share would cost only

D25. Therefore, we would have to come up with

only D5.95 (D25.00 – D19.05) over and above the

amount received from the loan to buy the shares.

Since D5.95 is the amount of money that we would

actually have to invest to obtain the replicating

portfolio, it is the value of this portfolio and there-

fore the value of the option.

The equation for calculating the value of the

replicating portfolio, and therefore the value of the

call option, can be expressed as follows:

Value of the call option today

¼ C ¼ D50� yð Þ þ 1� yð Þ¼ D50� 0:5ð Þ þ 1��D19:05ð Þ¼ D5:95

Notice, too, that the strike price, the current

price of the underlying shares, the possible future

prices of the underlying shares and the risk-free

interest rate are all that entered into our calcula-

tions. We did not even mention the probabilities

that the share price would go up or down at any

point. That is because the volatility of the under-

lying shares value is accounted for by how far

apart the two possible future values are. Similarly,

the time to expiration is not directly considered.

However, the time to expiration affects how high

and how low the share price can be when the

option expires.15

This model may seem surprisingly simple.

However, that is largely because we chose to illus-

trate a simple example. The model can be extended

in several ways. For example, we can incorporate

possible prices for the underlying asset between

now and the expiration date of the option. The

underlying asset price might take one of two values

one month (or day or hour) from now, and then for

each of those values there might be two possible

values in the following month (day or hour), and so

on. Solving a model such as this requires us to work

backwards from the expiration date to find the

value of the option at each intermediate date and

price until we finally arrive at the value of the

option today. Most modern option pricing models

are extensions of this type of model.

Learning byDoing

Application20.7

Valuing a Call OptionProblem: You are considering purchasing a calloption on Le Terrain Agricole SA shares. LeTerrain Agricole shares currently trade for D35and you predict that its price will be either D25or D50 in one year. The call option wouldenable you to buy Le Terrain Agricole sharesin one year for D30. What is this option worth ifthe risk-free interest rate is 4%?

Approach: The value of the option can bedetermined by computing the cost of constructing

a portfolio that replicates the payoffs from thatoption.

Solution: The option will be worth D20 if theshare price rises to D50 (D50 � D30 strikeprice) and will be worth D0 if the share pricedeclines to D25. Therefore, the replicating port-folio for this option can be determined from thefollowing two equations:

D20 ¼ D50� xð Þ þ 1:04� yð ÞD0 ¼ D25� xð Þ þ 1:04� yð Þ


C20 02/08/2011 19:31:10 Page 816

Solving for xand y, we find that x¼0.80andy ¼ D19.23. Therefore, the replicating portfolioconsists of 0.8 Le Terrain Agricole shares and aD19.23 loan. Since 0.8 of a share would costD28 (0.8 � D35) and D19.23 of this amount

would be covered by the loan, this replicatingportfolio would cost D8.77 (D28.00 � D19.23)to construct. Therefore, the call option isworth D8.77.

Put--Call Parity

Put--call parity

the relation between the value of a call

option on an asset and the value of a put

option on the same asset that has the same

exercise price

To this point, our discussion has focused on call

options. As mentioned earlier, this is possible

because there is a simple relation that enables us

to calculate the value of a put option once we know

the value of a call option with the same strike price

and expiration date. This relation is called put–call

parity. The formula for put–call parity is:

P ¼ CþXe�rt � V ð20:3Þwhere P is the value of the put option, C is the

value of the call option, X is the strike price, r is

the risk-free interest rate, t is the amount of time

before the option expires, and V is the current

value of the underlying asset. The term e�rt is the

exponential function that you can calculate using

the ‘ex’ key on your calculator; it is simply a

discount factor that assumes continuous com-

pounding. It is important to make sure that the

r and t are both stated in the same units of time

(for example, months or years).

To see how this formula works, we will con-

sider the option on the DRYAD SA shares that we

just valued. We know that C ¼ D5.95, X ¼ D55,

r ¼ 0.05, t ¼ 1 and V ¼ D50. Substituting these

values into the put–call parity formula and solving

for P, we get:

P ¼ D5:95þ D55e� 0:05ð Þ 1ð Þ � D50¼ D5:95þ D52:32� D50¼ D8:27

Notice that the variables used in this calculation

are the same variables that determine the value of a

call option. This means that the same factors that

affect the value of a call option also affect the value

of a put option. Notice, too, that the value of

the put option (D8.27) is greater than the value

of the call option (D5.95) in this example. This will

not always be true. However, it is true in our

example because the current share price of D50

is below the D55 strike price.

Learning byDoing

Application20.8

Valuing a Put OptionProblem: In Learning by Doing Application20.7, we found that a call option on Le TerrainAgricole SA shares is worth D8.77 when theshare price is D35, the strike price is D30, therisk-free interest rate is 4% and the time to

maturity is 1 year. What is the value of a putoption on the shares if the strike price and allother variables have the same values?

Approach: Use the put–call parity relation,Equation (20.3), to calculate the value of a putoption.


C20 02/08/2011 19:31:10 Page 817

Solution: The value of the put option is asfollows:

P ¼ C þ Xe�rt � VP ¼ D8:77þ D30e� 0:04ð Þ 1ð Þ � D35

¼ D8:77þ D28:82� D35¼ D2:59

Note that the value of the put option is lessthan the value of the call option in this example.This is because the current price of the shares isabove the strike price.

Options and RiskManagementWehave seenhowoptions havekinkedpayoffs. This

makes them very useful for corporate risk manage-

ment. To see how risks can be managed using

options, consider an oil company that is producing

and selling oil to refiners. Suppose that the price of

crude oil has recently risen above $130 per barrel

and the company wants to make sure that, even if

prices drop below $125 per barrel, it will receive at

least $125 per barrel for each barrel of oil that it sells

during the next three months. If the company plans

to sell 100000barrels ofoil in thenext threemonths,

the financial manager can hedge the price risk by

purchasing put options on 100 000 barrels of oil

with a strike price of $125 per barrel plus the cost of

the options. The maturity dates on the options must

be selected to match the timing of the company’s oil

output over the next three months. In addition, the

actual strike prices on the options must be slightly

greater than $125 to account for the premiums that

the company pays to purchase the options. This will

ensure that the company actually receives $125 per

barrel after paying for the options.

One interesting benefit of using options in this

way is that they provide downside protection but

do not limit the upside to the company if oil prices

continue to increase. Put options give the company

the right to sell its oil at the strike price if crude oil

prices fall but, because there is no obligation to sell,

the company can still benefit if oil prices increase.

As discussed earlier, this is just like buying insur-

ance. In fact, insurance contracts can be seen as

specialised put options.

In addition to using options and other deriva-

tive instruments to manage commodity price risks,

as in the oil company example, companies can use

these instruments to manage risks associated with

changing interest rates and exchange rates. Large

swings in interest rates can cause a great deal of

volatility in the net income of a highly financially

leveraged company whose managers rely on float-

ing-rate debt. As interest rates go up and down, the

company’s interest expense also goes up and down,

which can lead to cash flow problems.

Options can also be used to manage risks

associated with foreign exchange rates. For exam-

ple, as we discussed earlier, the revenues that a

company reports can be strongly affected by

changes in exchange rates if the company manu-

factures products in Europe and has significant

sales in foreign currencies. If the euro strengthens

against foreign currencies, the company will have

to increase the overseas prices of its products in

order to maintain the same euro price per unit.

This, in turn, can prompt consumers in overseas

markets to purchase fewer of the company’s prod-

ucts. By using options and other derivative instru-

ments to protect against exchange rate movements,

managers can limit declines in revenues that occur

because of such movements.

Finally, options can be used to manage risks as-

sociatedwith equity prices. This is especially impor-

tant to companies that have traditional defined-

benefit pension plans, which provide retirees with

guaranteed retirement payments. Companies are

required to put money aside to cover the costs of

these payments and this money is partially invested

in equities. When the stock market declines signifi-

cantly, these companies must replace any lost value

with new contributions, which must come from

earnings. As you might expect, companies are

very interested in managing the risk that they will

have to make such contributions.


C20 02/08/2011 19:31:10 Page 818

Before You Go On

1. What are the limits on the value of a call

option prior to its expiration date?

2. What variables affect the value of a call

option?

3. Why are the variables that affect the value

of a put option the same as those that

affect the value of a call option?

REAL OPTIONS

Learning Objective 7Name some of the real options that occur inbusiness and explain why traditional NPVanalysis does not accurately incorporate theirvalues.

Many investments in business involve real options –

options on real assets. NPV analysis does not ad-

equately reflect the value of these options. While it

is not always possible to directly estimate the value

of the real options associated with a project, it is

important to recognise that they exist when we

perform a project analysis. If we do not even

consider them, we are ignoring potentially impor-

tant sources of value. In this section, we provide an

overview of the types of real options commonly

associated with real investments.

WEB

You can find a list of websites with informa-

tion about real options at: http://www.real-

options.com/resources_links.htm.

Real option

an option for which the underlying asset is a

real asset

Options to Defer InvestmentCompanies often have considerable flexibility as to

the timing of their investments. For instance, con-

sider the case of an oil company that owns property

expected to contain oil deposits. The oil company

can choose to wait to see what happens to oil prices

before deciding whether to invest in developing the

deposits. This ability to wait and see involves what

is known as an option to defer investment. The

underlying asset in this option is the stream of cash

flows that the developed oil field would produce,

while the strike price is the amount of money that

the company would have to spend to develop it

(drill the well and build any necessary infrastruc-

ture). Just as the value of shares might go up or

down, the value of the cash flows produced by the

oil field might increase or decrease with the price

of oil.

Property developers often purchase options

on land. For example, a developer might pay a

landowner D100 000 for a one-year option to

purchase a property at a particular price. By ac-

cepting the payment, the landowner agrees not to

sell the property to anyone else for a year. Such an

option provides the developer with time to make a

final decision regarding whether or not to actually

purchase the land and proceed with a project.

Since the developer will still have to buy the land

if he or she decides to proceed with the project,

the cost of the option reflects a cost of being able

to collect more information before making a final

decision.

The value of an option to defer investment is

not reflected in an NPV analysis. Recall that the

NPV rule tells us to accept a project with a positive

NPV and to reject one with a negative NPV. NPV

analysis does not allow for the possibility of defer-

ring an investment decision. It assumes that we

invest either now or never. However, if we have the

option of deferring an investment decision, it may

make sense to do so. After all, a project that has a

negative NPV today might have a positive NPV at

some point in the future. The price of the product

may increase, production costs may decline or the

cost of capital may go down, making the project


C20 02/08/2011 19:31:10 Page 819

attractive. We need not assume that an investment

that is unattractive today will never be attractive.

Options to Make Follow-OnInvestmentsAnother very important type of real option is an

optiontomakefollow-oninvestments.Someprojects

open the door to future business opportunities that

wouldnototherwisebeavailable.Forexample,atthe

end of 2008, �Electricit�e de France (EDF) acquired a

controlling interest in the UK’s sole nuclear energy

utility, British Energy plc, for £12.5 billion. At first

glance,thisdidnotlooklikeaverygoodmovesincean

NPV analysis of the purchase carried out by outside

analysts indicated that the acquisition would be, at

best, only marginally positive. However, the move

created options for awide range of follow-on invest-

ments. TheNPV analysis did not take account of the

fact that the UK electricity market, which tradition-

ally relied largely on fossil fuels, was changing and

renewable and nuclear power generation were both

seen as theway forward.ByacquiringBritishEnergy

and agreeing to build two new power stations in the

UK,basedon its tried and testeddesigns, EDFwould

be able to rapidly add to its generating capacity if

market demandand the economics of nuclear power

made further investments attractive. In fact, the

acquisition provided EDF with several different

options to make follow-on investments, not just to

make additional investments in nuclear capacity.

Without these, EDF would probably not have been

willing toacquireBritishEnergy–orpay theamount

it did. In other words, acquiring British Energy pro-

vided EDF with options to enter other areas of the

UK’s energy market.

Another example of an option to make follow-

on investments concerns an investment in a new

technology that can be extended to other products.

For instance, in the early 1990s, Airbus invested in

a computer-aided aircraft design system as part of

the development of the A380 series aircraft. This

system allowed the company to complete much

more of the design work for a new aircraft on a

computer before building a prototype, thereby

lowering the cost of designing and building a

new aircraft. While the cost of the new system

and the associated facilities was high, the invest-

ment provided benefits that extended well beyond

the project. For example, the technologies could be

used in the design of other new aircraft, both

civilian and military. By reducing the cost of devel-

oping new aircraft, the design system had the

potential to make projects economically attractive

that would not have been attractive otherwise.

Options to make follow-on investments are

inherently difficult to value because, at the time

we are evaluating the original project, it may not be

obvious what the follow-on projects will be. Even if

we know what the projects will be, we are unlikely

to have enough information to estimate what they

are worth. Of course, this makes it impossible to

estimate directly the value of any option associated

with them. Nevertheless, it is important for man-

agers to consider options to make follow-on invest-

ments when evaluating projects. Doing so is a

central part of the process of evaluating projects

in the context of the overall strategy of the firm.

Projects that lead to investment opportunities that

are consistent with a company’s overall strategy

are more valuable than otherwise similar projects

that do not.

WEB

Real options are considered by NASA when

space systems and other investments are

evaluated. See the following page on the

NASA website for references to additional

readings in this area: http://ceh.nasa.gov/

webhelpfiles/Real_Option_Valuation.htm.

Options to Change OperationsIn addition to options to defer investment and

options to make follow-on investments, which

are real options related to the investment decisions

themselves, there are also real options that are

related to the flexibility managers have once an

investment decision has been made. These options,

which include the options to change operations


C20 02/08/2011 19:31:10 Page 820

and to abandon a project, affect the NPV of a

project and must be taken into account at the time

the investment decision is made.

In an NPV analysis, we discount the expected

cash flows from a project. We often consider sev-

eral alternative scenarios and use our estimates of

the probabilities associated with those scenarios to

compute the expected cash flows. While this sort of

analysis does consider alternative scenarios, it does

not fully account for the fact that once a project has

begun, the managers at a company have options to

change operations as business conditions change.

This means that there is a value associated with

being able to change operations that is not fully

reflected in a scenario analysis.

The changes that managers might make can

involve something as simple as reducing output if

prices decline or increasing output if prices

increase. Businesses do this all the time in response

to changing demand for their goods and services.

At the extreme, managers might temporarily sus-

pend operations entirely if business conditions are

weak. This is quite common in the auto industry,

where we often hear of plants being temporarily

shut down during periods of slow auto sales. Other

changes in operations can involve fundamentally

altering the way in which a product is produced, as

when a new production technology becomes avail-

able, making the old technology uncompetitive.

Having the flexibility to react to changing

business conditions can be very valuable. Since

we do not know how conditions are likely to

change, however, it can be difficult to estimate

just how valuable this flexibility will be. Never-

theless, we can see that managers do recognise the

importance of flexibility by observing how they

structure projects. For example, most modern

office buildings do not have permanent internal

walls. Not having permanent walls provides flexi-

bility in configuring the offices and workspaces in

the building. If more people must be put into a

building than originally anticipated, the workspa-

ces can be compressed to fit them. If the company

finds that it does not need all of the space, having a

flexible interior makes it easier to change things so

that the excess space can be leased. Similarly, when

a company plans to build a new manufacturing

facility, it often acquires more land than is imme-

diately needed and designs the facility to accom-

modate the addition of unexpected increases in

production capacity.

Building flexibility into a project costs money,

but this can be money well spent if things change

unexpectedly. The flexibility to expand, scale back

or temporarily shut down a project, or to change

the methods or technology employed in a project,

are all options that managers should consider when

evaluating projects. Projects with more flexibility

in these dimensions are inherently more valuable.

Options to Abandon ProjectsA project can also be terminated if things do not go

as well as anticipated.16 In other words, manage-

ment often has an option to abandon a project. The

ability to choose to terminate a project is a bit like a

put option. By shutting down the project, manage-

ment is saving money that would otherwise be lost

if the project kept going. The amount saved repre-

sents the gain from exercising this option.

As with flexibility, we can see that managers

recognise the importance of having an option to

abandon a project by observing the way they design

projects.Consider, for example, thatmost industrial

buildings are built like big boxes that can easily be

reconfigured as manufacturing spaces, warehouses

or even retail outlets, depending on which use is

most valuable. Suppose a company is building a

facility to use as a warehouse. If the building is only

able to accommodate a warehouse, it might end up

sitting empty for long periods of time – for example,

if the area has excesswarehouse space at some point

in the future. Designing the building so that it can be

reconfigured relatively inexpensively for some other

use increases the likelihood that the building will

remain fully utilised in the future.

Concluding Comments on NPVAnalysis and Real OptionsWe have stated that NPV analysis does not deal

well with real options. This is true because the

riskiness of a project that has real options


C20 02/08/2011 19:31:10 Page 821

associated with it varies with time and the appro-

priate discount rate varies with the risk. For exam-

ple, deciding to expand operations may be very

risky, but until the decision is actually made, the

option to expand is relatively risk free. In order to

use NPV analysis to value such an option, we

would not only have to estimate all the cash flows

associated with the expansion, but would also have

to estimate the probability that we would actually

undertake the expansion and determine the appro-

priate rate at which to discount the incremental

cash flows from the expansion back to the present.

The discount rate might even change with the value

of the underlying asset.

In some cases, we can incorporate the value of

a real option into an investment analysis by valuing

the option separately and then adding this value to

the NPV estimate. In these cases, we value the real

option using valuation methods similar to those

used to value financial options.

Decision-Making Example 20.2

The Value of Real OptionsSituation: You work for a company that manu-factures cardboard packaging for consumerproduct companies under long-term contracts.For example, your company manufactures theboxes for several popular cereal and pharma-ceutical products. You have just won a large five-year contract to produce packaging materialsfor a company that sells furniture on the Internet.Since this contract will require you to producemuch larger boxes than you currently can pro-duce, you must purchase some new equipment.You have narrowed your choices to two alter-natives. The first is a capital-intensive processthat will cost more up-front but will be lessexpensive to operate. This process requiresvery specialised equipment that can be usedonly for the type of packaging that your furniture

client needs. The second alternative is a labour-intensive process that will require a smaller up-front investment but will have higher unit costs.This process involves equipment that can be usedto produce a wide range of other packages. Ifthe expected life of both alternatives is 10 yearsand you estimate the NPV to be the same forboth, which should you choose?

Decision: You should choose the labour-intensivealternative. Your contract is only for five yearsand there is a chance that it will not be renewedbefore the equipment’s useful life is over. Ifthe contract is not renewed, it will be easierto convert the labour-intensive equipment toanother use. In other words, the labour-intensivealternative gives you the added value of havingthe option to abandon producing packagingfor furniture.

Before You Go On

1. What is a real option?

2. What are four different types of real

options commonly found in business?

3. Is it always possible to estimate the

value of a real option? Why or why not?

AGENCY COSTS

Learning Objective 8Describe how the agency costs of debt andequity are related to options.

Agency conflicts arise between shareholders

and debtholders and between shareholders and


C20 02/08/2011 19:31:10 Page 822

managers because the interests of shareholders,

lenders (creditors) and managers are not perfectly

aligned. In fact, their interests can greatly diverge.

One reason is that the claims they have against the

cash flows produced by the firm have payoff func-

tions that look like different types of options. We

now discuss how these payoff functions lead to

agency conflicts and their related costs.

Agency Costs of DebtIn Chapter 16, we discussed agency costs that arise

ina company that usesdebtfinancing.Wenoted that

these costs occur because the incentives of people

who lend to a company differ from those of the

shareholders. If you were to carefully reread those

discussions now, youmight notice that the problems

we discussed arise because the payoff functions for

shareholders and lenders (creditors) differ like those

for the different options we have been discussing.

To understand why this is the case, consider a

company that has a single loan outstanding. This

loan will mature next year and all of the interest

and principalwill be due at that time.Now, consider

what happens when the debt matures. On the one

hand, if the value of the company is less than the

amount owed on the debt, the shareholders will

simply default and the lenders will take control of

theassetsofthecompany.Theshareholderclaimswill

be worth zero in this case. If, on the other hand, the

valueofthecompanyisgreaterthantheamountowed

onthe loan, the shareholderswillpayoff the loanand

retain control of the assets. In this case, the share-

holderclaimswillbeworththedifferencebetweenthe

valueof thefirmand theamountowed to the lenders.

In other words, the payoff function for the

shareholders looks exactly like that for the owner

of a call option, where the strike price is the

amount owed on the loan and the underlying asset

is the firm itself. If the value of the firm exceeds the

strike price, the shareholders will choose to exer-

cise their option; and if it does not exceed the strike

price, they will let their option expire unexercised.

Part A of Exhibit 20.7 illustrates the payoff func-

tion for the shareholders in this simple example.

The payoff function for the lenders in our

example is illustrated in part B of Exhibit 20.7.

If the value of the firm is less than the amount

owed, the lenders receive only the assets of the firm;

and if the value of the firm is greater than the

amount owed, the lenders receive only the amount

owed. One way to think about the payoff function

for the lenders is that when they lend money to the

firm, they are essentially selling a put option to the

shareholders.17 This option gives the shareholders

the right to ‘put’ the assets to the lenders with a

strike price that equals the amount they owe.When

the value of the firm is less than the strike price, the

shareholders will exercise their option by default-

ing. Of course, the shareholders are able to default

and walk away only because our bankruptcy laws

limit their liability to the amount that they have

invested in the company.

The Dividend Payout ProblemKnowing that debt and equity claims are like

options in which the underlying asset is the firm,

we can use the intuition gained from the discussion

of the determinants of option value to better under-

stand the agency costs of debt. The incentives that

shareholders of a leveraged firm have to pay them-

selves dividends arise because of their option to

default. If a company faces some realistic risk of

going bankrupt, the shareholders might decide that

they are better off taking money out of the firm by

paying themselves dividends. This situation can

arise because the shareholders know that the bank-

ruptcy laws limit their possible losses. If the firm

goes bankrupt and the lenders end up receiving, for

example, 50% rather than 80% of what they are

owed, it will make no difference to the sharehold-

ers, who will get nothing from the liquidation of

the company’s assets in either case.

The Asset Substitution ProblemIn Chapter 16, we saw that when bankruptcy is

possible, shareholders have an incentive to invest in

very risky projects, some of which might even have

negative NPVs. Shareholders have this incentive

because they receive all of the benefits if things turn

out well but do not bear all of the costs if things

turn out poorly. Since equity claims are like call

options on the assets of the firm, this asset


C20 02/08/2011 19:31:10 Page 823

substitution problem should not be surprising. We

pointed out earlier in this chapter that the more

volatile the value of the underlying asset, the more

valuable a call option on that asset will be. Share-

holders of leveraged firms know this and therefore

have an incentive to invest in risky projects that

increase the overall volatility of the value of their

companies’ assets.

Lenders, in contrast, do not want the firm to

invest in high-risk projects. As you can see from

their payoff function in Exhibit 20.7, the lenders

bear costs as the value of the firm drops below the

amount they are owed but do not benefit at all as

the value of the firm’s assets increases above the

amount that they are owed. Lenders to companies

that are worth more than they are owed can only

expect to lose when a project increases the overall

riskiness of a company’s assets.

The Underinvestment ProblemChapter 16 also explained that shareholders have

incentives to turn down positive NPV projects

when all of the benefits are likely to go to the

lenders. You can see how this underinvestment

problem arises from the differences in the payoff

functions in Exhibit 20.7. Suppose that the com-

pany will owe D10 million when the loan matures,

that the company is currently worth D5million and

that the loan matures next week. This company is

financially distressed because its assets are not even

worth as much as its outstanding debt – so it is

unlikely to have enough money to finance new

investments. Now suppose that management iden-

tifies a positive NPV project that would require a

D3 million investment and that has a positive NPV

of D1 million which will be realised before the debt

payment must be made. Management would have

Firm Value

Val

ue o

f Equ

ity

0Face Value of Loan

When the firm value is below the face valueof the debt, the shareholders default andthe equity is worth zero.

Firm Value

Val

ue o

f Loa

n

0Face Value of Loan

When the firm value is below the face valueof the debt, the shareholders default andthe lenders receive the value of the firm.

When the value of the firm is above theface value of the debt and the equity isworth the difference between the firmvalue and the face value of the debt.

When the value of the firm is above theface value of the debt, the shareholdersrepay the debt and the lenders receivethe face value of the debt.

Exhibit 20.7: Payoff Functions for Shareholders and Lenders The equity in a leveraged company is like acall option on the underlying assets of the firm. The shareholders exercise their option by paying off the debt ifthe firm is worth more than the face value of the debt when the debt matures. If the value of the firm is lowerthan the face value of the debt, the shareholders can default (let their option expire) without incurring losses

beyond their investment in the firm.

The lenders’ payoff function is like that for the seller of a put option. They have effectively agreed topurchase the firm for an amount that equals the face value of the firm’s debt, at the discretion

of the shareholders.


C20 02/08/2011 19:31:10 Page 824

a hard time convincing the shareholders to invest

an additionalD3million in the firm, because even if

the investment turns out to be worth D4million, all

of the money will go to the lenders. The share-

holders have a strong incentive to turn down this

positive NPV project.

Agency Costs of EquitySo far, we have assumed that managers act in the

best interests of the shareholders. Since managers

are hired to manage the firm on behalf of the

shareholders, this might appear to be a reasonable

assumption. However, as you already know, man-

agers do not always act in the shareholders’ best

interests. This is because the payoff function for a

manager can be quite different from that for share-

holders. In fact, a manager’s payoff function can

look a lot like a lender’s payoff function.

To see how this is possible, consider the con-

nection between managers’ personal wealth and

the performance of the companies for which they

work. The present value of managers’ future earn-

ings is a large part of their overall wealth. If a

company gets into financial difficulty and a man-

ager is viewed as responsible, that manager could

lose his or her job and find it difficult to obtain a

similar job at another company. Of course, the

most obvious way for a company to get into

financial difficulty is to default on its debt. There-

fore, as long as a company is able to avoid default-

ing on its debt, a manager has a reasonable chance

of retaining his or her job. Once the firm defaults,

the chances of job loss increase dramatically. In

addition, researchers have found that senior man-

agers of financially distressed large public compa-

nies who lose their jobs find it difficult to obtain

similar jobs afterwards.18 We might also expect

that the worse the company’s financial distress, the

worse the manager’s future employment prospects

and the lower the present value of the compensa-

tion that he or she can expect to receive in the

future. If this is so, when the value of a firm is less

than the amount it owes, the payoff function for a

manager will look something like that for the

lender in part B of Exhibit 20.7 – it will slope

downwards as the value of the firm decreases.

On the positive side, we would expect the

present value of a manager’s future earnings to

increase with the value of the firm when this value

is above the amount that the company owes to its

lenders. Managers will receive larger bonuses and

larger pay raises and any shares or options that

they receive will be more valuable. However, these

increases will not be nearly as large as those for

shareholders. The shareholders are not likely to

give the managers a large proportion of any

increase in firm value. The net result is that the

payoff function for managers can look something

like the one in Exhibit 20.8.

The fact that the payoff function for a man-

ager resembles that for a lender means that man-

agers, like lenders, have incentives to invest in less

risky assets and to distribute less value through

dividends and share repurchases than the share-

holders would like them to. These tendencies are

reinforced by the fact that managers are individ-

uals who do not hold diversified portfolios, since

most of their wealth is tied to the performance of

their firms. Managers tend to make conservative

investment, financing and dividend decisions

because the personal cost to them of failure can

be very great.

Boards of directors understand how the

incentives of managers differ from those of share-

holders. Consequently, boards put a great deal of

effort into designing compensation plans that

make the payoff functions for managers look as

much as possible like those of shareholders. Ulti-

mately, this is a key to minimising agency conflicts

between shareholders and the managers that rep-

resent them.

Before You Go On

1. What do the payoff functions for share-

holders and lenders look like?

2. What does the payoff function for a typi-

cal manager look like?


C20 02/08/2011 19:31:10 Page 825

SUMMARY OF LEARNING OBJECTIVES

1. Explain the factors that make it desirable for firms to manage their risks.

Companies have a number of risks from inputs, production and outputs that increase the

variability of their cash flows. Factors that will influence the decision to manage these risks

include: financial reporting, corporate taxation, bankruptcy costs, the cost of capital, agency costs

and employee compensation and retention. Firms start by first identifying the risks they face,

evaluating them and managing these in appropriate ways and keeping their risks under review. A

keymotivation for firms tomanage certain risks is that they can add value by so doing and are able

to manage some risks that shareholders cannot, such as tax losses.

2. Describe the risks faced by firms and how they are managed.

Risks from a firm’s operations and unanticipated changes to market prices or rates lead to

undesirable cash flow volatility. Companies can use insurance against production risks. Firms can

hedge their risks by taking positions that offset each other if prices change. For market risks, they

can adjust their exposures to risks associated with commodity prices, interest rates, foreign

exchange rates and equity prices by using financial risk management. Derivatives, such as forward

contracts, futures, swaps and options, are frequently used since they are flexible and are low cost.

The cost of risk management depends on future uncertainty and this has to be weighed against the

benefits of risk reduction.

3. Define forward and futures contracts and be able to determine their prices.

A forward contract is an agreement to buy and sell an asset at a predetermined price at a future

date. The key elements for the delayed sale determined in advance are (1) the forward date, (2) the

0

Val

ue o

f Man

ager

’sF

utur

e C

ompe

nsat

ion

Face Valueof Loan

Firm Value

Exhibit 20.8: Representative Payoff Function for a Manager The payoff function for a manager with a typicalcompensation arrangement is more similar in shape to the payoff function for a lender than for a shareholder. Whilea shareholder’s payoff function is flat to the left of the face value of the loan, the value of the manager’s compensationis downward sloping, much like the payoff of a lender. When the value of the firm is greater than the face value of theloan, the value of the manager’s compensation does not increase as much as the value of the firm’s shares (the line ofthe payoff function is not as steep). Because managers’ payoff functions differ from those for shareholders, managers

have incentives to take actions that are not in the best interests of shareholders.


C20 02/08/2011 19:31:10 Page 826

asset and (3) the agreed price. A futures contract is a standardised forward contract that is traded on

an organised exchange and is very similar to a forward contract. The pricing of forwards and futures

is known as the cost of carry and is based on time value of money principles such that the forward

price, when the transaction is agreed, is determined so that neither side is worse off as a result. The

risk-free interest rate to the forward date and any costs associated with storing the asset will raise

the future price. Income earned by the asset and the demand for physical ownership – known as

the convenience yield – will reduce the forward price. The balance between these factors will

determine whether the forward price is higher or lower than the current price. Over time, the

relationship between the contracted forward price and the current price will change and one party

will own an asset and the other will have a liability. This means that forward contracts have credit

risk. Futures contracts were developed to address this problem, as well as providing liquidity

since futures contracts are made with a centralised clearing house that facilitates buying and

selling on the exchange. Futures users are required to post margin that protects the clearing house

against default.

4. Define interest rate and cross-currency swaps and know how they are valued.

An interest rate swap is an agreement to exchange a set of fixed future cash flows against a set

of cash flows based on an index of interest rates using a notional principal amount that determines

the amounts to be paid. A cross-currency swap involves exchanging cash flows in one currency

against corresponding payments in another currency. As such, a cross-currency swap can be

considered a package of borrowing and lending where a term loan in one currency is financed by

lending in another currency.When initially transacted, swap terms are designed so that the present

value of the cash flows from both sides is equal. Changes in market conditions mean that, over

time, the present values of each side of the swap will diverge and the swap will become either a

liability or an asset. A swapwill have credit risk if the present value of the cash flows to be received

is greater than the present value of the cash flows to be paid out. Companies use swaps for asset-

liability risk management purposes and, in the case of cross-currency swaps, to fund or borrow in

different currencies without incurring exchange rate risk.

5. Define a call option and a put option, and describe the payoff function for each of these options.

Anoption is the right, but not the obligation, to buy or sell an asset for a given price on or before

a specific date. The price is called the strike or exercise price and the date is called the exercise date or

expiration date of the option. The right to buy the asset is known as a call option. The payoff from a

call option equals zero if the value of the underlying asset is less than the strike price at expiration. If

the value of the underlying asset is higher than the strike price at expiration, then the payoff from a

call option is equal to the value of the asset value minus the strike price. The right to sell the asset is

called a put option. The payoff from a put option is zero if the value of the underlying asset is greater

than the strike price at expiration. If the value is lower than the strike price, then the payoff from a

put option equals the strike price minus the value of the underlying asset.

6. List and describe the factors that affect the value of an option.

The value of an option is affected by five factors: (1) the current price of the underlying asset,

(2) the strike price of the option, (3) the volatility of the value of the underlying asset, (4) the time

left until the expiration of the option and (5) the risk-free rate.

7. Name some of the real options that occur in business and explain why traditional NPV analysis

does not accurately incorporate their values.

Real options that are associated with investments include options to defer investment, make

follow-on investments, change operations and abandon projects. Traditional NPV analysis is

designed to make a decision to accept or reject a project at a particular point in time. It is not


C20 02/08/2011 19:31:11 Page 827

designed to include the potential value associated with deferring the investment decision.

Incorporating the value of the other options into an NPV framework is technically possible

but would be very difficult to do because the rate used to discount the cash flows would change

over time with their riskiness. In addition, the information necessary to value real options using

the NPV approach is not always available.

8. Describe how the agency costs of debt and equity are related to options.

The chapter discusses two principal classes of agency conflicts. The first is between share-

holders and lenders. When there is a risk of bankruptcy, shareholders may have incentives to

increase the volatility of the firm’s assets, turn down positive NPV projects or pay out assets in the

form of dividends. Shareholders have these incentives because their payoff functions look like

those for the owner of a call option.

The other principal class of agency conflicts is between managers and owners. Managers tend

to prefer less risk than shareholders do and prefer to distribute fewer assets in the form of

dividends because their payoff functions are more like those of lenders than those of shareholders

are. These preferences are magnified by the fact that managers are risk-averse individuals whose

portfolios are not well diversified.

SUMMARY OF KEY EQUATIONS

Equation Description Formula

(20.1) Cost of carryPV� 1þ iþ uð Þm

1þ qþ yð Þm ¼ FVm

(20.2) Interest rate swapvalue

Value of interest rate swap ¼Value of bond with swap coupon rate – value of loan withfloating rate

(20.3) Put–call parity P ¼ CþXe�rT � V

SELF-STUDY PROBLEMS

20.1. What will determine whether a firm should –

or should not – manage particular risks in its

business?

20.2. You own property which has a value of

D5 million and will pay rental income of

D450 000 at the end of the first year and

D500 000 at the end of the second year. You

have been approached by a property com-

pany and they would like you to sell the

property to them at the end of the second

year but at a price agreed today. The interest

rate is 4.3% per year. What would be a fair

price for the property, if agreed now?


C20 02/08/2011 19:31:11 Page 828

20.3. Your company is considering opening a new

factory in theMiddle East to serve the grow-

ing demand for your product there. Your

home currency is the euro but you need US

dollars for the investment that will cost

US$25 million and will last for five years.

You decide that a cross-currency swap is the

best way of financing the investment.

The exchange rate between the US dollar

and the euro is $1.3475 ¼ D1 and the five-

year swap rates for the euro and the US

dollar are 3.5% and 4.2% per year, respec-

tively, paid annually. Lay out the cash flows

for the swap.

20.4. Deutsche Euroshop AG shares are currently

selling for D12. Over the next year, the

company is undertaking a new supermarket

project. If the project is successful, the com-

pany’s shares are expected to rise to D24. If

the project fails, the shares are expected to

fall to D8. The risk-free interest rate is 6%.

Calculate the value today of a one-year call

option on one Deutsche Euroshop share

with a strike price of D20.

20.5. Fiera Milano S.p.A. is an Italian company

that organises trade fairs and is listed on the

Milan Stock Exchange. The company’s

shares are currently trading at D50. Depend-

ing on the outlook for the economy and the

demand for trade conferences, the com-

pany’s share price is expected to be either

D65 or D30 in six months. The risk-free

interest rate is 8% per year. What is the

value of a put option on one Fiera Milano

share that has a D40 strike price?

SOLUTIONS TO SELF-STUDY PROBLEMS

20.1. The decision will be based on assessing the

costs versus the benefits. Firms will manage

those risks for which the benefits can only be

captured by the firm but not its owners.

These include, but are not limited to, corpo-

rate taxation, bankruptcy costs, the cost of

capital from outside providers, employee

compensation and retention and financial

reporting.

20.2. Wewant to apply the cost of carry model to

determine the forward price, knowing that

we have discrete value distributions at the

end of years 1 and 2. We start by present-

valuing these at the risk-free interest rate

and subtracting them from the value of the

property before then future-valuing the

property at the risk-free interest rate for

two years:

PVYear 1 income ¼ D450 000

1:043¼ D431 448

PVYear 1 income ¼ D500 000

1:043ð Þ2 ¼ D459 623

Forward price ¼ ðD5 000 000� D431 448

� D459 623Þ � 1:043ð Þ2¼ D4 469 895

The two-year forward price will be

D4469 895.

20.3. The first step is to determine the amount of

euros that are required in exchange for

receiving US25 million at the start of the

swap. With an exchange rate of $1.3475/Dthis requires $25 000 000/$1.3475 ¼D18 552 876 to be paid at the start. The

interest on the euro side will therefore be


C20 02/08/2011 19:31:11 Page 829

D18 552 876� 0.035 ¼ D649 351 per year.

For the US dollar side, it is $25 000 000 �0.042 ¼ $1 050 000 per year. From the per-

spective of the company, the cross-currency

swap cash flows will look as follows:

Cash Flows for the Cross-Currency SwapTime(years) Euros US dollars

0 �18 552 876 25 000 0001 649 351 �1 050 0002 649 351 �1 050 0003 649 351 �1 050 0004 649 351 �1 050 0005a 649 351 �1 050 0005b 18 552 876 �25 000 000

20.4. First determine the payoffs for the shares, a

risk-free loan and the call option under the

two possible outcomes. In one year, the

share price is expected to be either D8 or

D24. The loan will be worth D1.06 regard-

less of whether the project is successful. If

the project fails, the share price will be less

than the strike price of the call option. The

option will not be exercised and will be

worth D0. If the project is successful,

the share price will be higher than the strike

price of the call option. The option will

be exercised and its value will be the differ-

ence between the share price and the strike

price, D4.

The shares and loan can be used to

create a replicating portfolio which has

the same payoff as the call option:

ðD8� xÞ � ð1:06� yÞ ¼ D0ð$24� xÞ � ð1:06� yÞ ¼ D4

Solving the two equations yields: x ¼0.25, y ¼ �D1.887

The value of the call option is the same

as the current value of this portfolio:

ðD12� 0:25Þ � ðD1��D1:887Þ ¼ 1:11

20.5. Here we solve directly for the value of the

put option. First we determine the payoffs

for the shares, a risk-free bond and the

put option under the two possible outcomes.

To determine the payoff of the bond in

six months’ time, we must calculate the

six-month risk-free interest rate given the

one-year risk-free rate listed in the problem

statement:

Six-month risk-free rate

¼ ð1þ 0:08Þ1=2 � 1 ¼ 1:039; or 3:9%


C20 02/08/2011 19:31:11 Page 830

The payoffs are therefore:

Now we can use the shares and the

bond tocreatea replicatingportfolio,which

will give the same payoff as the put option:

ðD30� xÞ � ð1:039� yÞ ¼ D10ðD65� xÞ � ð1:039� yÞ ¼ D0

Solving the two equations, we deter-

mine x ¼ �0.286, y ¼ D17.87The value if the put option is the same

as the current value of this portfolio:

ðD50��0:286Þ � ðD1� D17:87Þ¼ D3:58

Alternatively, you could solve this

problem by calculating the value of a

call option with the same strike price of

D40 and then using the put–call parity

relation. The value of the call option is

D15.09 (you may like to check this by

calculating it yourself) and the value of

the associated put option calculated using

the put–call parity relation is D3.52. The

difference (D3.58 vs. D3.52) is due to

rounding and the compounding assump-

tion for the discount rate.

CRITICAL THINKING QUESTIONS

20.1. A manufacturer of consumer products

which is based in France is considering

entering a new market in a Latin American

country by exporting its products for sale

there. Detail the various risks it has from

expanding into this new market.

20.2. There are active markets in forward con-

tracts on financial securities, such as

exchange rates, equities and interest rates

and on commodities, the principal ones

being base and precious metals, agricul-

tural and energy commodities. Why will

there be a consumption yield for

commodity forward contracts and not

for financial securities? What are the impli-

cations for the forward price from this

difference?

20.3. For a company wanting to hedge its expo-

sures, what are the attractions and disad-

vantages of using futures markets rather

than forward markets for this purpose?

20.4. A swap contract is simply an exchange of

two sets of cash flows over an agreed period.

The interest rate swap exchanges a set of

predetermined andfixed payments based on

a notional principal for a floating set of


C20 02/08/2011 19:31:11 Page 831

payments based on an interest rate index.

What other possible types of swaps can be

created given the way such swaps work?

20.5. Which is likely to have more credit risk, an

interest rate swap or a cross-currency

swap – and why?

20.6. A writer of a call option may or may not

actually own the underlying asset. If he or

she owns the asset, and therefore will have

the asset available to deliver should the

option be exercised, he or she is said to be

writing a covered call. Otherwise, he or she

is writing a naked call and will have to buy

the underlying asset on the open market

should the option be exercised. Draw the

payoff diagram of a covered call (including

the valueof the ownedunderlyingasset) and

compare it with the payoff of other options.

20.7. What kinds of real options should be con-

sidered in the following situations?

a. Fiat S.p.A. is considering two sites for a

new car factory. One is just large

enough for the planned facility, while

the other is three times larger.

b. Hellenistic Cruises is purchasing three

new cruise ships to be built sequentially.

The first ship will commence construc-

tion todayandwill takeoneyear tobuild.

The second will then be started. Helle-

nistic Cruises can cancel the order for a

given cruise ship at any time before con-

struction begins for a small fee.

20.8. Zukunft Betrieb AG is considering a factory

that will include an option to expand

operations in three years. If things go

well, the anticipated expansion will have

a value of D10 million and will cost D2

million to undertake. Otherwise, the antici-

pated expansion will have a value of only

D1 million and will not take place. What

information would we need in order to

analyse this capital budgeting problem

using the traditional NPV approach that

we would not need using option valuation

techniques?

20.9. Companies frequently include employee

share options as part of the compensation

for their managers and sometimes for all

their employees. These options allow the

holder to buy the shares of the company for

a preset price like any other option, but

they usually have very long maturities, of

up to 10 years not being uncommon. The

goal of share option plans is to align the

incentives of employees and shareholders.

What are the implications of these plans for

current shareholders?

20.10. You are a bondholder of DRYAD SA.

Using option-pricing theory, explain

what agency concerns you would have if

DRYAD SA were in danger of bankruptcy.

20.11. A bond covenant is part of a bond contract

that restricts the behaviour of the firm,

barring it from taking certain actions.

Using the terminology of options, explain

why a bond contract might include a cove-

nant preventing the firm frommaking large

dividend payments to its shareholders.

QUESTIONS AND PROBLEMS

Basic20.1. Managing corporate risks: Why do com-

panies usually seek to hedge out the risks

from financial markets?

20.2. Managing corporate risks: Renault, the

French carmaker, sells its vehicles within

Europe and elsewhere. What effect has

the introduction of the euro in France,

Germany, Spain, Italy and other member

countries had on Renault’s sales in these

countries?


C20 02/08/2011 19:31:11 Page 832

20.3. Forward contracts: What are the three

elements that have to be defined in a for-

ward contract?

20.4. Forward contracts: What are the payoff

profiles for (a) a long forward and (b) a

short forward at maturity?

20.5. Forward contract valuation: What factors

raise the price of a forward contract and

what factors reduce the value of a forward

contract?

20.6. Swaps: There are four possible types of

cross-currency swaps based on the nature

of the cash flows to the two parties. What

are the four possible types?

20.7. Option characteristics: Explain how the

payoff functions differ for the owner

(buyer) and the seller of a call option. Of

a put option.

20.8. Option valuation: What is the value of an

option if the share price is zero?What if the

share price is extremely high (relative to the

strike price)?

20.9. Option valuation: Like owners of shares,

owners of options can lose nomore than the

amount they invested. They are far more

likely to lose that full amount but they

cannot lose more. Do sellers of options

have the same limitation on their losses?

20.10. Option valuation: What is the value at

expiration of a call option with a strike

price of D65 if the share price is D1? D50?

D65? D100? D1000?

20.11. Option valuation: Suppose you have an

option to buy NASDAL shares for D100.

The option expires tomorrow and the cur-

rent price of NASDAL shares is D95. How

much is your option worth?

20.12. Option valuation: You hold an American

option to sell one share of Cimbalom. The

option expires tomorrow. The strike price

of the option is D50 and the current share

price isD49.What is the value of exercising

the option today? If you wanted to sell the

option instead, about how much would

you expect to receive?

20.13. Realoptions:What is the differencebetween

a financial option and a real option?

20.14. Real options: List and describe four differ-

ent types of real options that are associated

with investment projects.

20.15. Agency costs: How are options related to

the agency costs of debt and equity?

Intermediate20.16. Managing corporate risks: Why do com-

panies prefer to use financial hedging, if

available, rather than operational hedging?

When might operational hedging be a bet-

ter choice?

20.17. Risk management methods: When might a

company prefer to use insurance rather

than hedging to protect itself against a

particular risk?

20.18. Forward contract valuation: If the current

asset price is D350 and the risk-free interest

rate is 3% per year, the asset provides a

continuous dividend yield of 5.2% per

year, what will be the forward price for

the asset in a forward contract if the agreed

delivery date is 18 months?

20.19. Forward contracts:Whatwillbethevalueof

the forward contract in 20.18 if the contract

nowhassixmonthstomaturity,thespotasset

price is now D355, the risk-free interest rate

is 3.6% per year and the dividend yield is

now 4.7% per year? If you had taken a long

position in the contract in 20.18, is the for-

ward contract now an asset or a liability?

20.20. Interest rate swap valuation: Valencia Fab-

ricaci�on SA has an interest rate swap where

it pays a fixed rate of 4.6% per year. The

notional amount of the swap isD30million

and the swap has currently exactly 3 years

tomaturity. The current 3-year swap rate is

3.9%. What is the value of the swap and,

from Valencia Fabricaci�on’s perspective, is

the swap an asset or a liability?

20.21. Option valuation: Shares ofMotores Socra-

tes SA are currently trading forD40 andwill

either rise to D50 or fall to D35 in one


C20 02/08/2011 19:31:11 Page 833

month. The risk-free interest rate for

one month is 1.5%. What is the value of a

one-month call option with a strike price

of D40?

20.22. Option valuation: Again assume that the

price of Motores Socrates SA shares will

either rise to D50 or fall to D35 in one

month and that the risk-free interest rate

for one month is 1.5%. How much is an

optionwith a strike priceofD40worth if the

current share price is D45 instead of D40?

20.23. Option valuation: You are considering a

three-month put on Budowlanych Kra-

kow. The company’s shares currently trade

at Zloty 10.0 and in three months will

either rise to Zl. 15.0 or fall to Zl. 7.0.

The risk-free interest rate for three months

is 2%. What is the appropriate price for a

put with a strike price of Zl. 9.0?

20.24. Option valuation: You hold a European

put option on Cannello S.p.A. with a strike

price of D100. Things have not been going

too well for Cannello. The current share

price is D2 and you think that it will either

rise to D3 or fall to D1.50 at the expiration

of your option. The appropriate risk-free

interest rate is 5%. What is the value of the

option? If this were an American option,

would it be worth more?

20.25. Other options: A golden parachute is part

of a manager’s compensation package that

makes a large lump-sum payment in the

event that the manager is fired (or loses his

or her job in a merger, for example). This

seems ill-advised to most people when first

hearing about it. Explain how a golden

parachute can help reduce agency costs

between shareholders and managers.

Advanced20.26. Consider the following two strategies for

investing in a company’s shares:

a. buy the shares immediately and hold

them for 6 months for D100 before sell-

ing these at the end of the six months;

b. takea longposition ina6-month forward

contract for D102 and immediately sell

the shares at thematurity of the contract.

The six-month rate of interest is 2%.

What will your payoff be in six months’

time fromboth strategies if the shareprice is

D110 and D95? What is the effect on the

payoffs if after you have decided, the com-

pany subsequently announces and pays a

dividend of D5 at the end of month five?

20.27. You want to enter into four sequential for-

ward contracts with maturities of 6, 12, 18

and 24 months, respectively. The risk-free

rate of interest for the four periods is 3.0,

3.5, 3.7 and 4.0% per year, respectively. If

the spotprice isD350 today,whatwill be the

forward prices at which you can transact, if

the asset has a dividend yield of 3.6% per

year?Whatdo the prices you calculate tell us

about the way forward markets work?

20.28. Two years ago, FabricaSc~ao Azulejos de

Lisboa SA (FALSA) entered into an interest

rate swap for D25 million with a maturity

of 7 years where the company makes a

fixed payment of 4.5% per year against

Euribor. Now the company wants to ter-

minate the swap. The five-year swap rate is

4.0%. Will FALSA pay or receive to termi-

nate the swap and how much is involved?

20.29. Dynamo Plastics plc entered into a five-year

cross-currency swap for £10million against

the eurowhen the exchange ratewasD1.25/

£ and the sterling fixed interest rate was

4.5% per year and that for the euro was

3.7% per year. Dynamo Plastics agreed to

pay pounds sterling and receive the euro.

Exactly two years have passed and the com-

panywants to terminate the swap. The euro

is now trading at D1.10, the sterling 3-yearfixed swap rate is 3.2% and the euro 3-year

swap rate is 2.8% per year. What is the

value change on the swap, will Dynamo

Plastics gain or lose from termination, and

what are the component value changes from

the changes in market conditions?


C20 02/08/2011 19:31:11 Page 834

20.30. Consider the following payoff diagram.

Find a combination of calls, puts, risk-

free bonds and shares that has this payoff.

(You need not use all of these instruments,

and there are many possible solutions.)

20.31. Consider the payoff structures of the fol-

lowing two portfolios:

a. Buying a call option on one share in one

month at a strike price of D50 and

saving the present value of D50 (so

that at expiration it will have grown

to D50 with interest).

b. Buying a put option on one share in one

month at a strike price of D50 and

buying one share.

What conclusion can you draw about

the relation between call prices and put

prices?

20.32. One way to extend the binomial pricing

model is by including multiple time peri-

ods. Suppose Splittime, Inc. shares are cur-

rently trading at $100. In one month, the

price will either increase by $10 (to $110)

or decrease by $10 (to $90). The following

month will be the same. The price will

either increase by $10 or decrease by

$10. Notice that in two months, the price

could be $120, $100 or $80. The risk-free

rate is 1% per month. Find the value today

of an option to buy one share of Splittime

in two months for a strike price of $105.

(Hint: To do this, first find the value of the

option at each of the two possible one-

month prices. Then use those values as

the payoffs at one month and find the value

today.)

20.33. Spin The Wheel Company has assets cur-

rently worth £10 million in the form of

one-year risk-free bonds that will return

10%. The company has debt with a face

value of £5.5 million due in one year. (No

interest payments will bemade.) The share-

holders decided to sell £8 million of the

risk-free bonds and to invest the money in a

very risky venture. This venture consists of

Mr William Kid’s taking the money now

and, in one year, flipping a coin. If it comes

up heads, Mr Kid will pay Spin The Wheel

£17.6 million. If it is tails, Spin The Wheel

gets nothing. (Notice that this is a zero

NPV investment.)

a. What is the value of the debt and equity

before the shareholders make this

‘investment’?

b. Using the binomial pricing model, with

the payoff to the equity holders repre-

senting the option and the assets of the

company representing the underlying

asset, estimate the value of the equity

after the shareholders make the

investment.

c. What is the new value of the debt after

the investment?

20.34. The payoff function for the holder of

straight debt looks like that for the seller

of a put option. Convertible debt is straight

debt plus a call option on a firm’s shares.

How does the addition of a call option to

straight debt affect the concern that lenders

have about the asset substitution problem,

and why?


C20 02/08/2011 19:31:11 Page 835

SAMPLE TEST PROBLEMS

20.1. Draw the payoff diagram representing the

payoff for a combination of buying a call

with a strike price of D40 and selling a call

with a strike price of D50. What would the

buyer of such an option hope would happen

to the share price?

20.2. Of the five variables identified as affecting the

value of an option, whichwill have the oppo-

site effects on the value of a put and the value

of a call? That is, for which variables will a

given change increase the value of a call and

decrease the value of a put (or vice versa)?

20.3. What kinds of real options are being

described?

a. Fred’s Cheap Cars buys the empty field

adjacent to its car lot.

b. Midway through construction, Maxival

AG stops construction of an office

building that it had planned to use as

a corporate headquarters.

c. Lidl, the German discount retailer,

opens its first new store in Morocco.

20.4. If you fail to account for the real options

available in a given project, what error

might you make in your capital budgeting

decision?

20.5. Suppose you are a wheat farmer. Assuming

that there is an active market in wheat

futures contracts, what trades might you

want to use to protect yourself against fall-

ing wheat prices? What would be the cost of

using them?

ENDNOTES

1. Since 2000, the price has risen spectacularly such that by the spring of 2010, the pricewas over $1100/oz.

Needless to say, gold producers have quickly moved to remove many of the forward contracts that

locked them in at lower prices. Notably, Barrick Gold announced in the autumn of 2009 that it would

spend $2.9 billion to repurchase forward and other derivative contracts.

2. For simplicity, we will assume there is no salvage or environmental costs at the end of the mining

operations.

3. In 1992, BAe made provisions and write-downs of £1 billion, at that time the largest corporate write-

down in UK history, to cover staff redundancies and losses in its regional aircraft division.

4. We discuss the foreign exchange market and currency management in the next chapter when looking

at international financial management.

5. Economists refer to investment in production facilities as irreversible. The costs involved are largely

upfront and will be hard to recover later if the company should change its mind. We will look at this

problem from a capital budgeting perspective later in the chapter when we discuss real options.

6. Portfolio theory is discussed in Chapter 7.

7. In this case, there is depreciation in the value to be considered since, if Airbus leases the aircraft, it will

no longer be ‘new’. The buyer will be receiving a less valuable airplane andwould not be willing to pay

the full price for a new aircraft. This will reduce the forward price. In financial contracts where

depreciation is not an issue, the income received by the seller prior to delivery acts as a negative

interest rate and reduces the forward price.


C20 02/08/2011 19:31:11 Page 836

8. We could also have present-valued these costs and then future-valued the total. It will give the same

result [D50 þ D0.60/(1.04) þ D0.75/(1.04)2] � (1.04)2 ¼ D55.454 million.

9. Revisit Chapter 8 to review the discussion on how interest rates affect the prices of bonds.

10. The way foreign exchange is quoted and how the currency market works is examined in detail in the

next chapter.

11. It can be shown that using this money, Airbus can replace the cross-currency swap with a swap using

the current market conditions and be no better or worse off as a result. The sum paid to terminate the

swap is used to subsidise the future payments on the new ‘at-market’ swap such that Airbus has

undisturbed cash flows that are exactly the same as those of the original swap.

12. Even if the value of the option ever fell below the line to the right of the exercise price in part A of

Exhibit 20.1, it would not stay there. This is because investors would be able to make an instant profit

by buying the option, exercising it to get the underlying asset and then selling the underlying asset.

Such trading by investors would drive the price of the option back above the line.

13. We are focusing in this discussion on what the value of the underlying asset is likely to be immediately

before the option expires because it does not generally make sense to exercise an option before then as

long as there is a chance that the value of the underlying asset could increase further. An exception is

when the value of the underlying asset is not expected to be higher as the expiration of the option nears

because value is being distributed to the owners of the underlying asset (for example, through

dividend payments). In a situation like this, it can be appropriate to exercise a call option immediately

before such a payment. There are also situations where it is advantageous to exercise a put option

early. Such situations can arise if it is very likely that the option will be exercised at expiration. When

this happens, the value received from exercising the option today can exceed the present value of the

amount that is expected to be received if the option is exercised immediately before expiration.

14. We can also compute the value of x and y by noting that the combined positions in the upper fork

and the lower fork are equal if we hold the replicating portfolio and sell the call option (otherwise,

the portfolio is not riskless). This means that D70� xð Þ þ 1:05� yð Þ � D15 ¼ D40� xð Þþ1:05� yð Þ � 0. Simplifying, we have D70� xð Þ � D40� xð Þ ¼ D15. Therefore, as before, x ¼0.5. Knowing x, we can now solve for y, since D40� 0:5ð Þ ¼ 1:05� yð Þ and therefore, y ¼D19.05 as before.

15. There are other ways to solve the binomial pricing problem than by actually finding an equivalent

portfolio. They differ only in the calculations, however. The underlying concepts are identical. See any

advanced investments textbook for details.

16. An exception exists where a contractual agreement prevents the project from being terminated

without payment of a penalty that is equivalent to the remaining value of the project.

17. This payoff function is actually like that from the combination of selling a put option and buying a

risk-free loan. Lenders receive the face value of the loan from the risk-free bond, but they might have

to pay some or all of that value in losses on the put option. Since the risk-free loan payout is unaffected

by changes in the value of the firm, it does not affect the discussion above.

18. S. C. Gilson, Management turnover and financial distress, Journal of Financial Economics 25 (1989)

241–262.


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