Brealey, Myers. Principles of Corporate Finance. 7th edition

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Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 2. Present Value and the

Opportunity Cost of Capital

Companies, 2003

26 PART I Value

increase stockholders’ wealth. The way to do so is to invest in assets having pos-

itive net present values.

There are several institutional arrangements which help to ensure that man-

agers pay close attention to the value of the firm:

• Managers’ actions are subject to the scrutiny of the board of directors.

• Shirkers are likely to find that they are ousted by more energetic managers. This

competition may arise within the firm, but poorly performing companies are

also more likely to be taken over. That sort of takeover typically brings in a fresh

management team.

• Managers are spurred on by incentive schemes, such as stock options, which

pay off big if shareholders gain but are valueless if they do not.

Managers who focus on shareholder value need not neglect their wider obliga-

tions to the community. Managers play fair by employees, customers, and suppli-

ers partly because they know that it is for the common good, but partly because

they know that their firm’s most valuable asset is its reputation. Of course, ethical

issues do arise in financial management and, whenever unscrupulous managers

abuse their position, we all trust each other a little less.

FURTHER

READING

QUIZ

The pioneering works on the net present value rule are:

I. Fisher: The Theory of Interest, Augustus M. Kelley, Publishers. New York, 1965. Reprinted

from the 1930 edition.

J. Hirshleifer: “On the Theory of Optimal Investment Decision,” Journal of Political Economy,

66:329–352 (August 1958).

For a more rigorous textbook treatment of the subject, we suggest:

E. F. Fama and M. H. Miller: The Theory of Finance, Holt, Rinehart and Winston. New

York, 1972.

If you would like to dig deeper into the question of how managers can be motivated to maximize share-

holder wealth, we suggest:

M. C. Jensen and W. H. Meckling: “Theory of the Firm: Managerial Behavior, Agency Costs,

and Ownership Structure,” Journal of Financial Economics, 3:305–360 (October 1976).

E. F. Fama: “Agency Problems and the Theory of the Firm,” Journal of Political Economy,

88:288–307 (April 1980).

1. C

is the initial cash flow on an investment, and C

is the cash flow at the end of one

year. The symbol r is the discount rate.

a. Is C

usually positive or negative?

b. What is the formula for the present value of the investment?

c. What is the formula for the net present value?

d. The symbol r is often termed the opportunity cost of capital. Why?

e. If the investment is risk-free, what is the appropriate measure of r?

2. If the present value of $150 paid at the end of one year is $130, what is the one-year dis-

count factor? What is the discount rate?

3. Calculate the one-year discount factor DF

for discount rates of (a) 10 percent, (b) 20 per-

cent, and (c) 30 percent.

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Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 2. Present Value and the

Opportunity Cost of Capital

Companies, 2003

CHAPTER 2 Present Value and the Opportunity Cost of Capital 27

4. A merchant pays $100,000 for a load of grain and is certain that it can be resold at the

end of one year for $132,000.

a. What is the return on this investment?

b. If this return is lower than the rate of interest, does the investment have a positive

or a negative NPV?

c. If the rate of interest is 10 percent, what is the PV of the investment?

d. What is the NPV?

5. What is the net present value rule? What is the rate of return rule? Do the two rules give

the same answer?

6. Define the opportunity cost of capital. How in principle would you find the opportu-

nity cost of capital for a risk-free asset? For a risky asset?

7. Look back to the numerical example graphed in Figure 2.1. Suppose the interest rate is

20 percent. What would the ant (A) and grasshopper (G) do? Would they invest in the

office building? Would they borrow or lend? Suppose each starts with $100. How much

and when would each consume?

8. We can imagine the financial manager doing several things on behalf of the firm’s stock-

holders. For example, the manager might:

a. Make shareholders as wealthy as possible by investing in real assets with

positive NPVs.

b. Modify the firm’s investment plan to help shareholders achieve a particular time

pattern of consumption.

c. Choose high- or low-risk assets to match shareholders’ risk preferences.

d. Help balance shareholders’ checkbooks.

But in well-functioning capital markets, shareholders will vote for only one of these

goals. Which one? Why?

9. Why would one expect managers to act in shareholders’ interests? Give some reasons.

10. After the Salomon Brothers bidding scandal, the aggregate value of the company’s

stock dropped by far more than it paid in fines and settlements of lawsuits. Why?

PRACTICE

QUESTIONS

1. Write down the formulas for an investment’s NPV and rate of return. Prove that NPV

is positive only if the rate of return exceeds the opportunity cost of capital.

2. What is the net present value of a firm’s investment in a U.S. Treasury security yielding

5 percent and maturing in one year? Hint: What is the opportunity cost of capital? Ig-

nore taxes.

3. A parcel of land costs $500,000. For an additional $800,000 you can build a motel on the

property. The land and motel should be worth $1,500,000 next year. Suppose that com-

mon stocks with the same risk as this investment offer a 10 percent expected return.

Would you construct the motel? Why or why not?

4. Calculate the NPV and rate of return for each of the following investments. The oppor-

tunity cost of capital is 20 percent for all four investments.

Initial Cash Cash Flow

Investment Flow, C

in Year 1, C

1 10,000 18,000

2 5,000 9,000

3 5,000 5,700

4 2,000 4,000

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Brealey−Meyers:

Principles of Corporate

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I. Value 2. Present Value and the

Opportunity Cost of Capital

Companies, 2003

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28 PART I Value

a. Which investment is most valuable?

b. Suppose each investment would require use of the same parcel of land. Therefore

you can take only one. Which one? Hint: What is the firm’s objective: to earn a high

rate of return or to increase firm value?

5. In Section 2.1, we analyzed the possible construction of an office building on a plot of

land appraised at $50,000. We concluded that this investment had a positive NPV of

$7,143 at a discount rate of 12 percent.

Suppose E. Coli Associates, a firm of genetic engineers, offers to purchase the land

for $60,000, $30,000 paid immediately and $30,000 after one year. United States gov-

ernment securities maturing in one year yield 7 percent.

a. Assume E. Coli is sure to pay the second $30,000 installment. Should you take its

offer or start on the office building? Explain.

b. Suppose you are not sure E. Coli will pay. You observe that other investors demand

a 10 percent return on their loans to E. Coli. Assume that the other investors have

correctly assessed the risks that E. Coli will not be able to pay. Should you accept

E. Coli’s offer?

6. Explain why the discount rate equals the opportunity cost of capital.

7. Norman Gerrymander has just received a $2 million bequest. How should he invest it?

There are four immediate alternatives.

a. Investment in one-year U.S. government securities yielding 5 percent.

b. A loan to Norman’s nephew Gerald, who has for years aspired to open a big

Cajun restaurant in Duluth. Gerald had arranged a one-year bank loan for

$900,000, at 10 percent, but asks for a loan from Norman at 7 percent.

c. Investment in the stock market. The expected rate of return is 12 percent.

d. Investment in local real estate, which Norman judges is about as risky as the stock

market. The opportunity at hand would cost $1 million and is forecasted to be

worth $1.1 million after one year.

Which of these investments have positive NPVs? Which would you advise Norman

to take?

8. Show that your answers to Practice Question 7 are consistent with the rate of return rule

for investment decisions.

9. Take another look at investment opportunity (d) in Practice Question 7. Suppose a bank

offers Norman a $600,000 personal loan at 8 percent. (Norman is a long-time customer

of the bank and has an excellent credit history.) Suppose Norman borrows the money,

invests $1 million in real estate opportunity (d) and puts the rest of his money in op-

portunity (c), the stock market. Is this a smart move? Explain.

10. Respond to the following comments.

a. “My company’s cost of capital is the rate we pay to the bank when we borrow

money.”

b. “Net present value is just theory. It has no practical relevance. We maximize

profits. That’s what shareholders really want.”

c. “It’s no good just telling me to maximize my stock price. I can easily take a short

view and maximize today’s price. What I would prefer is to keep it on a gently

rising trend.”

11. Ms. Smith is retired and depends on her investments for retirement income. Mr. Jones

is a young executive who wants to save for the future. They are both stockholders in

Airbus, which is investing over $12 billion to develop the A380, a new super-jumbo

airliner. This investment’s payoff is many years in the future. Assume the investment

is positive-NPV for Mr. Jones. Explain why it should also be positive-NPV for Ms.

Smith.

12. Answer this question by drawing graphs like Figure 2.1. Casper Milktoast has

$200,000 available to support consumption in periods 0 (now) and 1 (next year). He

EXCEL

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 2. Present Value and the

Opportunity Cost of Capital

Companies, 2003

CHAPTER 2 Present Value and the Opportunity Cost of Capital 29

wants to consume exactly the same amount in each period. The interest rate is 8 per-

cent. There is no risk.

a. How much should he invest, and how much can he consume in each period?

b. Suppose Casper is given an opportunity to invest up to $200,000 at 10 percent risk-

free. The interest rate stays at 8 percent. What should he do, and how much can he

consume in each period?

c. What is the NPV of the opportunity in (b)?

13. We said that maximizing value makes sense only if we assume well-functioning capi-

tal markets. What does “well-functioning” mean? Can you think of circumstances in

which maximizing value would not be in all shareholders’ interests?

14. Why is a reputation for honesty and fair business practice important to the financial

value of the corporation?

Dollars, year 0, millions

Owner's preferred

consumption pattern

3.75

1.6

Dollars, year 1, millions

42.6

FIGURE 2.2

See Challenge Question 2.

CHALLENGE

QUESTIONS

1. It is sometimes argued that the NPV criterion is appropriate for corporations but not for

governments. First, governments must consider the time preferences of the community

as a whole rather than those of a few wealthy investors. Second, governments must

have a longer horizon than individuals, for governments are the guardians of future

generations. What do you think?

2. In Figure 2.2, the sloping line represents the opportunities for investment in the capital

market and the solid curved line represents the opportunities for investment in plant

and machinery. The company’s only asset at present is $2.6 million in cash.

a. What is the interest rate?

b. How much should the company invest in plant and machinery?

c. How much will this investment be worth next year?

d. What is the average rate of return on the investment?

e. What is the marginal rate of return?

f. What is the PV of this investment?

g. What is the NPV of this investment?

h. What is the total PV of the company?

i. How much will the individual consume today?

j. How much will he or she consume tomorrow?

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Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 2. Present Value and the

Opportunity Cost of Capital

Companies, 2003

Visit us at www.mhhe.com/bm7e

30 PART I Value

3. Draw a figure like Figure 2.1 to represent the following situation.

a. A firm starts out with $10 million in cash.

b. The rate of interest r is 10 percent.

c. To maximize NPV the firm invests today $6 million in real assets. This leaves

$4 million which can be paid out to the shareholders.

d. The NPV of the investment is $2 million.

When you have finished, answer the following questions:

e. How much cash is the firm going to receive in year 1 from its investment?

f. What is the marginal return from the firm’s investment?

g. What is the PV of the shareholders’ investment after the firm has announced its

investment plan?

h. Suppose shareholders want to spend $6 million today. How can they do this?

i. How much will they then have to spend next year? Show this on your drawing.

4. For an outlay of $8 million you can purchase a tanker load of bucolic acid delivered in

Rotterdam one year hence. Unfortunately the net cash flow from selling the tanker load

will be very sensitive to the growth rate of the world economy:

Slump Normal Boom

$8 million $12 million $16 million

a. What is the expected cash flow? Assume the three outcomes for the economy are

equally likely.

b. What is the expected rate of return on the investment in the project?

c. One share of stock Z is selling for $10. The stock has the following payoffs after

one year:

Slump Normal Boom

$8 $12 $16

Calculate the expected rate of return offered by stock Z. Explain why this is the

opportunity cost of capital for your bucolic acid project.

d. Calculate the project’s NPV. Is the project a good investment? Explain why.

5. In real life the future health of the economy cannot be reduced to three equally proba-

ble states like slump, normal, and boom. But we’ll keep that simplification for one more

example.

Your company has identified two more projects, B and C. Each will require a $5 mil-

lion outlay immediately. The possible payoffs at year 1 are, in millions:

Slump Normal Boom

B4 6 8

C 5 5.5 6

You have identified the possible payoffs to investors in three stocks, X, Y, and Z:

Current Price

Payoff at Year 1

per Share Slump Normal Boom

X 95.65 80 110 140

Y40 40 4448

Z 10 8 12 16

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Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 2. Present Value and the

Opportunity Cost of Capital

Companies, 2003

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CHAPTER 2 Present Value and the Opportunity Cost of Capital 31

a. What are the expected cash inflows of projects B and C?

b. What are the expected rates of return offered by stocks X, Y, and Z?

c. What are the opportunity costs of capital for projects B and C? Hint: Calculate the

percentage differences, slump versus normal and boom versus normal, for stocks

X, Y, and Z. Match up to the percentage differences in B’s and C’s payoffs.

d. What are the NPVs of projects B and C?

e. Suppose B and C are launched and $5 million is invested in each. How much will

they add to the total market value of your company’s shares?

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 3. How to Calculate

Present Values

Companies, 2003

CHAPTER THREE

H O W T O

C A L C U L A T E

PRESENT VALUES

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 3. How to Calculate

Present Values

Companies, 2003

IN CHAPTER 2 we learned how to work out the value of an asset that produces cash exactly one year

from now. But we did not explain how to value assets that produce cash two years from now or in

several future years. That is the first task for this chapter. We will then have a look at some shortcut

methods for calculating present values and at some specialized present value formulas. In particular

we will show how to value an investment that makes a steady stream of payments forever (a perpe-

tuity) and one that produces a steady stream for a limited period (an annuity). We will also look at in-

vestments that produce a steadily growing stream of payments.

The term interest rate sounds straightforward enough, but we will see that it can be defined in var-

ious ways. We will first explain the distinction between compound interest and simple interest. Then

we will discuss the difference between the nominal interest rate and the real interest rate. This dif-

ference arises because the purchasing power of interest income is reduced by inflation.

By then you will deserve some payoff for the mental investment you have made in learning about

present values. Therefore, we will try out the concept on bonds. In Chapter 4 we will look at the val-

uation of common stocks, and after that we will tackle the firm’s capital investment decisions at a

practical level of detail.

Do you remember how to calculate the present value (PV) of an asset that produces

a cash flow (C

) one year from now?

The discount factor for the year-1 cash flow is DF

, and r

is the opportunity cost

of investing your money for one year. Suppose you will receive a certain cash in-

flow of $100 next year (C

⫽ 100) and the rate of interest on one-year U.S. Treasury

notes is 7 percent (r

⫽ .07). Then present value equals

The present value of a cash flow two years hence can be written in a similar

way as

is the year-2 cash flow, DF

is the discount factor for the year-2 cash flow, and r

is the annual rate of interest on money invested for two years. Suppose you get an-

other cash flow of $100 in year 2 (C

⫽ 100). The rate of interest on two-year Trea-

sury notes is 7.7 percent per year (r

⫽ .077); this means that a dollar invested in

two-year notes will grow to 1.077

⫽ $1.16 by the end of two years. The present

value of your year-2 cash flow equals

PV ⫽

11 ⫹ r

⫽

100

11.0772

⫽ $86.21

PV ⫽ DF

⫻ C

⫽

11 ⫹ r

PV ⫽

1 ⫹ r

⫽

100

1.07

⫽ $93.46

PV ⫽ DF

⫻ C

⫽

1 ⫹ r

3.1 VALUING LONG-LIVED ASSETS

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 3. How to Calculate

Present Values

Companies, 2003

Valuing Cash Flows in Several Periods

One of the nice things about present values is that they are all expressed in current

dollars—so that you can add them up. In other words, the present value of cash

flow A ⫹ B is equal to the present value of cash flow A plus the present value of

cash flow B. This happy result has important implications for investments that

produce cash flows in several periods.

We calculated above the value of an asset that produces a cash flow of C

in year 1,

and we calculated the value of another asset that produces a cash flow of C

in year 2.

Following our additivity rule, we can write down the value of an asset that produces

cash flows in each year. It is simply

We can obviously continue in this way to find the present value of an extended

stream of cash flows:

This is called the discounted cash flow (or DCF) formula. A shorthand way to

write it is

where ⌺ refers to the sum of the series. To find the net present value (NPV) we add

the (usually negative) initial cash flow, just as in Chapter 2:

Why the Discount Factor Declines as Futurity Increases—

And a Digression on Money Machines

If a dollar tomorrow is worth less than a dollar today, one might suspect that a dol-

lar the day after tomorrow should be worth even less. In other words, the discount

factor DF

should be less than the discount factor DF

. But is this necessarily so,

when there is a different interest rate r

for each period?

Suppose r

is 20 percent and r

is 7 percent. Then

Apparently the dollar received the day after tomorrow is not necessarily worth less

than the dollar received tomorrow.

But there is something wrong with this example. Anyone who could borrow

and lend at these interest rates could become a millionaire overnight. Let us see

how such a “money machine” would work. Suppose the first person to spot the

opportunity is Hermione Kraft. Ms. Kraft first lends $1,000 for one year at 20 per-

cent. That is an attractive enough return, but she notices that there is a way to earn

⫽

11.072

⫽ .87

⫽

1.20

⫽ .83

NPV ⫽ C

⫹ PV ⫽ C

⫹

11 ⫹ r

PV ⫽

11 ⫹ r

PV ⫽

1 ⫹ r

⫹

11 ⫹ r

⫹

11 ⫹ r

⫹

…

PV ⫽

1 ⫹ r

⫹

11 ⫹ r

34 PART I Value

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 3. How to Calculate

Present Values

Companies, 2003

an immediate profit on her investment and be ready to play the game again. She

reasons as follows. Next year she will have $1,200 which can be reinvested for a

further year. Although she does not know what interest rates will be at that time,

she does know that she can always put the money in a checking account and be

sure of having $1,200 at the end of year 2. Her next step, therefore, is to go to her

bank and borrow the present value of this $1,200. At 7 percent interest this pres-

ent value is

Thus Ms. Kraft invests $1,000, borrows back $1,048, and walks away with a profit

of $48. If that does not sound like very much, remember that the game can be

played again immediately, this time with $1,048. In fact it would take Ms. Kraft

only 147 plays to become a millionaire (before taxes).

Of course this story is completely fanciful. Such an opportunity would not last

long in capital markets like ours. Any bank that would allow you to lend for one

year at 20 percent and borrow for two years at 7 percent would soon be wiped out

by a rush of small investors hoping to become millionaires and a rush of million-

aires hoping to become billionaires. There are, however, two lessons to our story.

The first is that a dollar tomorrow cannot be worth less than a dollar the day after

tomorrow. In other words, the value of a dollar received at the end of one year

(DF

) must be greater than the value of a dollar received at the end of two years

(DF

). There must be some extra gain

from lending for two periods rather than

one: (1 ⫹ r

)

must be greater than 1 ⫹ r

Our second lesson is a more general one and can be summed up by the precept

“There is no such thing as a money machine.”

In well-functioning capital markets,

any potential money machine will be eliminated almost instantaneously by in-

vestors who try to take advantage of it. Therefore, beware of self-styled experts

who offer you a chance to participate in a sure thing.

Later in the book we will invoke the absence of money machines to prove several

useful properties about security prices. That is, we will make statements like “The

prices of securities X and Y must be in the following relationship—otherwise there

would be a money machine and capital markets would not be in equilibrium.”

Ruling out money machines does not require that interest rates be the same for

each future period. This relationship between the interest rate and the maturity of

the cash flow is called the term structure of interest rates. We are going to look at

term structure in Chapter 24, but for now we will finesse the issue by assuming that

the term structure is “flat”—in other words, the interest rate is the same regardless

of the date of the cash flow. This means that we can replace the series of interest

rates r

, r

, ..., r

, etc., with a single rate r and that we can write the present value

formula as

PV ⫽

1 ⫹ r

⫹

11 ⫹ r2

⫹

…

PV ⫽

1200

11.072

⫽ $1,048

CHAPTER 3 How to Calculate Present Values 35

That is, 1,000 ⫻ (1.04813)

147

⫽ $1,002,000.

The extra return for lending two years rather than one is often referred to as a forward rate of return. Our

rule says that the forward rate cannot be negative.

The technical term for money machine is arbitrage. There are no opportunities for arbitrage in well-

functioning capital markets.