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J. B. Williams™s original work remains very readable. See particularly Chapter V of:
J. B. Williams: The Theory of Investment Value, Harvard University Press, Cambridge,
Mass., 1938.
The following articles provide important developments of Williams™s early work. We suggest, how-
ever, that you leave the third article until you have read Chapter 16:
D. Durand: “Growth Stocks and the Petersburg Paradox,” Journal of Finance, 12:348“363
(September 1957).
M. J. Gordon and E. Shapiro: “Capital Equipment Analysis: The Required Rate of Profit,”
Management Science, 3:102“110 (October 1956).
M. H. Miller and F. Modigliani: “Dividend Policy, Growth and the Valuation of Shares,”
Journal of Business, 34:411“433 (October 1961).
Leibowitz and Kogelman call PVGO the “franchise factor.” They analyze it in detail in:
M. L. Leibowitz and S. Kogelman: “Inside the P/E Ratio: The Franchise Factor,” Financial
Analysts Journal, 46:17“35 (November“December 1990).
Myers and Borucki cover the practical problems encountered in estimating DCF costs of equity for
regulated companies; Harris and Marston report DCF estimates of rates of return for the stock
market as a whole:
S. C. Myers and L. S. Borucki: “Discounted Cash Flow Estimates of the Cost of
Equity Capital”A Case Study,” Financial Markets, Institutions and Instruments, 3:9“45
(August 1994).
Brealey’Meyers: I. Value 4. The Value of Common © The McGraw’Hill
Principles of Corporate Stocks Companies, 2003
Finance, Seventh Edition

CHAPTER 4 The Value of Common Stocks 83

R. S. Harris and F. C. Marston: “Estimating Shareholder Risk Premia Using Analysts™
Growth Forecasts,” Financial Management, 21:63“70 (Summer 1992).
The following book covers valuation of businesses in great detail:
T. Copeland, T. Koller, and J. Murrin: Valuation: Measuring and Managing the Value of Compa-
nies, John Wiley & Sons, Inc., New York, 1994.

1. True or false?
a. All stocks in an equivalent-risk class are priced to offer the same expected rate of
b. The value of a share equals the PV of future dividends per share.
2. Respond briefly to the following statement.
“You say stock price equals the present value of future dividends? That™s crazy! All the
investors I know are looking for capital gains.“
3. Company X is expected to pay an end-of-year dividend of $10 a share. After the divi-
dend its stock is expected to sell at $110. If the market capitalization rate is 10 percent,
what is the current stock price?
4. Company Y does not plow back any earnings and is expected to produce a level divi-

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dend stream of $5 a share. If the current stock price is $40, what is the market capital-
ization rate?
5. Company Z™s earnings and dividends per share are expected to grow indefinitely by 5
percent a year. If next year™s dividend is $10 and the market capitalization rate is 8 per-
cent, what is the current stock price?
6. Company Z-prime is like Z in all respects save one: Its growth will stop after year 4. In
year 5 and afterward, it will pay out all earnings as dividends. What is Z-prime™s stock
price? Assume next year™s EPS is $15.
7. If company Z (see question 5) were to distribute all its earnings, it could maintain a level
dividend stream of $15 a share. How much is the market actually paying per share for
growth opportunities?
8. Consider three investors:
a. Mr. Single invests for one year.
b. Ms. Double invests for two years.
c. Mrs. Triple invests for three years.
Assume each invests in company Z (see question 5). Show that each expects to earn an
expected rate of return of 8 percent per year.
9. True or false?
a. The value of a share equals the discounted stream of future earnings per share.
b. The value of a share equals the PV of earnings per share assuming the firm does
not grow, plus the NPV of future growth opportunities.
10. Under what conditions does r, a stock™s market capitalization rate, equal its earn-
ings“price ratio EPS1/P0?
11. What do financial managers mean by “free cash flow“? How is free cash flow related to
dividends paid out? Briefly explain.
12. What is meant by a two-stage DCF valuation model? Briefly describe two cases where
such a model could be used.
13. What is meant by the horizon value of a business? How is it estimated?
14. Suppose the horizon date is set at a time when the firm will run out of positive-NPV in-
vestment opportunities. How would you calculate the horizon value?
Brealey’Meyers: I. Value 4. The Value of Common © The McGraw’Hill
Principles of Corporate Stocks Companies, 2003
Finance, Seventh Edition

84 PART I Value

PRACTICE 1. Look in a recent issue of The Wall Street Journal at “NYSE-Composite Transactions.“

a. What is the latest price of IBM stock?
b. What are the annual dividend payment and the dividend yield on IBM stock?
c. What would the yield be if IBM changed its yearly dividend to $1.50?
d. What is the P/E on IBM stock?
e. Use the P/E to calculate IBM™s earnings per share.
f. Is IBM™s P/E higher or lower than that of Exxon Mobil?
g. What are the possible reasons for the difference in P/E?
2. The present value of investing in a stock should not depend on how long the investor
plans to hold it. Explain why.
3. Define the market capitalization rate for a stock. Does it equal the opportunity cost of
capital of investing in the stock?
4. Rework Table 4.1 under the assumption that the dividend on Fledgling Electronics is
$10 next year and that it is expected to grow by 5 percent a year. The capitalization rate
is 15 percent.
5. In March 2001, Fly Paper™s stock sold for about $73. Security analysts were forecasting
a long-term earnings growth rate of 8.5 percent. The company was paying dividends of
$1.68 per share.
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a. Assume dividends are expected to grow along with earnings at g 8.5 percent per
year in perpetuity. What rate of return r were investors expecting?
b. Fly Paper was expected to earn about 12 percent on book equity and to pay out
about 50 percent of earnings as dividends. What do these forecasts imply for g?
For r? Use the perpetual-growth DCF formula.
6. You believe that next year the Superannuation Company will pay a dividend of $2 on
its common stock. Thereafter you expect dividends to grow at a rate of 4 percent a year
in perpetuity. If you require a return of 12 percent on your investment, how much
should you be prepared to pay for the stock?
7. Consider the following three stocks:
a. Stock A is expected to provide a dividend of $10 a share forever.
b. Stock B is expected to pay a dividend of $5 next year. Thereafter, dividend growth
is expected to be 4 percent a year forever.
c. Stock C is expected to pay a dividend of $5 next year. Thereafter, dividend growth
is expected to be 20 percent a year for 5 years (i.e., until year 6) and zero thereafter.
If the market capitalization rate for each stock is 10 percent, which stock is the most
valuable? What if the capitalization rate is 7 percent?
8. Crecimiento S.A. currently plows back 40 percent of its earnings and earns a return of
20 percent on this investment. The dividend yield on the stock is 4 percent.
a. Assuming that Crecimiento can continue to plow back this proportion of earnings
and earn a 20 percent return on the investment, how rapidly will earnings and
dividends grow? What is the expected return on Crecimiento stock?
b. Suppose that management suddenly announces that future investment
opportunities have dried up. Now Crecimiento intends to pay out all its earnings.
How will the stock price change?
c. Suppose that management simply announces that the expected return on new
investment would in the future be the same as the market capitalization rate. Now
what is Crecimiento™s stock price?
9. Look up General Mills, Inc., and Kellogg Co. on the Standard & Poor™s Market Insight
website (www.mhhe.com/edumarketinsight). The companies™ ticker symbols are
GIS and K.
a. What are the current dividend yield and price“earnings ratio (P/E) for each
company? How do the yields and P/Es compare to the average for the food
Brealey’Meyers: I. Value 4. The Value of Common © The McGraw’Hill
Principles of Corporate Stocks Companies, 2003
Finance, Seventh Edition

CHAPTER 4 The Value of Common Stocks 85

industry and for the stock market as a whole? (The stock market is represented by
the S & P 500 index.)
b. What are the growth rates of earnings per share (EPS) and dividends for each
company over the last five years? Do these growth rates appear to reflect a steady
trend that could be projected for the long-run future?
c. Would you be confident in applying the constant-growth DCF valuation model to
these companies™ stocks? Why or why not?
10. Look up the following companies on the Standard & Poor™s Market Insight website
(www.mhhe.com/edumarketinsight): Citigroup (C), Dell Computer (DELL), Dow
Chemical (DOW), Harley Davidson (HDI), and Pfizer, Inc. (PFE). Look at “Financial
Highlights” and “Company Profile” for each company. You will note wide differences
in these companies™ price“earnings ratios. What are the possible explanations for these
differences? Which would you classify as growth (high-PVGO) stocks and which as in-
come stocks?
11. Vega Motor Corporation has pulled off a miraculous recovery. Four years ago, it was near
bankruptcy. Now its charismatic leader, a corporate folk hero, may run for president.
Vega has just announced a $1 per share dividend, the first since the crisis hit. Ana-
lysts expect an increase to a “normal” $3 as the company completes its recovery over
the next three years. After that, dividend growth is expected to settle down to a mod-

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erate long-term growth rate of 6 percent.
Vega stock is selling at $50 per share. What is the expected long-run rate of return
from buying the stock at this price? Assume dividends of $1, $2, and $3 for years 1, 2,
3. A little trial and error will be necessary to find r.
12. P/E ratios reported in The Wall Street Journal use the latest closing prices and the last 12
months™ reported earnings per share. Explain why the corresponding earnings“price
ratios (the reciprocals of reported P/Es) are not accurate measures of the expected rates
of return demanded by investors.
13. Each of the following formulas for determining shareholders™ required rate of return
can be right or wrong depending on the circumstances:

a. r g
b. r

For each formula construct a simple numerical example showing that the formula can
give wrong answers and explain why the error occurs. Then construct another simple
numerical example for which the formula gives the right answer.
14. Alpha Corp™s earnings and dividends are growing at 15 percent per year. Beta Corp™s
earnings and dividends are growing at 8 percent per year. The companies™ assets, earn-
ings, and dividends per share are now (at date 0) exactly the same. Yet PVGO accounts
for a greater fraction of Beta Corp™s stock price. How is this possible? Hint: There is
more than one possible explanation.
15. Look again at the financial forecasts for Growth-Tech given in Table 4.3. This time
assume you know that the opportunity cost of capital is r .12 (discard the .099 figure
calculated in the text). Assume you do not know Growth-Tech™s stock value. Otherwise
follow the assumptions given in the text.
a. Calculate the value of Growth-Tech stock.
b. What part of that value reflects the discounted value of P3, the price forecasted for
year 3?
c. What part of P3 reflects the present value of growth opportunities (PVGO) after
year 3?
Brealey’Meyers: I. Value 4. The Value of Common © The McGraw’Hill
Principles of Corporate Stocks Companies, 2003
Finance, Seventh Edition

86 PART I Value

d. Suppose that competition will catch up with Growth-Tech by year 4, so that it can
earn only its cost of capital on any investments made in year 4 or subsequently.
What is Growth-Tech stock worth now under this assumption? (Make additional
assumptions if necessary.)
16. Look up Hawaiian Electric Co. (HI) on the Standard & Poor™s Market Insight website
(www.mhhe.com/edumarketinsight). Hawaiian Electric was one of the companies in
Table 4.2. That table was constructed in 2001.
a. What is the company™s dividend yield? How has it changed since 2001?
b. Table 4.2 projected growth of 2.6 percent. How fast have the company™s dividends
and EPS actually grown since 2001?
c. Calculate a sustainable growth rate for the company based on its five-year average
return on equity (ROE) and plowback ratio.
d. Given this updated information, would you modify the cost-of-equity estimate
given in Table 4.2? Explain.
17. Browse through the companies in the Standard & Poor™s Market Insight website
(www.mhhe.com/edumarketinsight). Find three or four companies for which the
earnings-price ratio reported on the website drastically understates the market capi-
talization rate r for the company. (Hint: you don™t have to estimate r to answer this
question. You know that r must be higher than current interest rates on U.S. govern-
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ment notes and bonds.)
18. The Standard & Poor™s Market Insight website (www.mhhe.com/edumarketinsight)
contains information all of the companies in Table 4.6 except for Chubb and Weyer-
haeuser. Update the calculations of PVGO as a percentage of stock price. For simplicity
use the costs of equity given in Table 4.6. You will need to track down an updated fore-
cast of EPS, for example from MSN money (www.moneycentral.msn.com) of Yahoo
19. Compost Science, Inc. (CSI), is in the business of converting Boston™s sewage sludge
into fertilizer. The business is not in itself very profitable. However, to induce CSI to re-
main in business, the Metropolitan District Commission (MDC) has agreed to pay
whatever amount is necessary to yield CSI a 10 percent book return on equity. At the
end of the year CSI is expected to pay a $4 dividend. It has been reinvesting 40 percent
of earnings and growing at 4 percent a year.
a. Suppose CSI continues on this growth trend. What is the expected long-run rate of
return from purchasing the stock at $100? What part of the $100 price is
attributable to the present value of growth opportunities?
b. Now the MDC announces a plan for CSI to treat Cambridge sewage. CSI™s plant will,
therefore, be expanded gradually over five years. This means that CSI will have to
reinvest 80 percent of its earnings for five years. Starting in year 6, however, it will
again be able to pay out 60 percent of earnings. What will be CSI™s stock price once
this announcement is made and its consequences for CSI are known?
20. List at least four different formulas for calculating PV(horizon value) in a two-stage
DCF valuation of a business. For each formula, describe a situation where that formula
would be the best choice.
21. Look again at Table 4.7.
a. How do free cash flow and present value change if asset growth rate is only 15
percent in years 1 to 5? If value declines, explain why.
b. Suppose the business is a publicly traded company with one million shares outstand-
ing. Then the company issues new stock to cover the present value of negative free cash
flow for years 1 to 6. How many shares will be issued and at what price?
c. Value the company™s one million existing shares by the two methods described in
Section 4.5.
22. Icarus Air has one million shares outstanding and expects to earn a constant $10 mil-
lion per year on its existing assets. All earnings will be paid out as dividends. Suppose
Brealey’Meyers: I. Value 4. The Value of Common © The McGraw’Hill
Principles of Corporate Stocks Companies, 2003
Finance, Seventh Edition

CHAPTER 4 The Value of Common Stocks 87

that next year Icarus plans to double in size by issuing an additional one million shares
at $100 a share. Everything will be the same as before but twice as big. Thus from year
2 onward the company earns a constant $20 million, all of which is paid out as divi-
dends on the 20 million shares. What is the value of the company? What is the value of
each existing Icarus Air share?
23. Look one more time at Table 4.1, which applies the DCF stock valuation formula to
Fledgling Electronics. The CEO, having just learned that stock value is the present value
of future dividends, proposes that Fledgling pay a bumper dividend of $15 a share in
period 1. The extra cash would have to be raised by an issue of new shares. Recalculate
Table 4.1 assuming that profits and payout ratios in all subsequent years are un-
changed. You should find that the total present value of dividends per existing share is
unchanged at $100. Why?

1. Look again at Tables 4.3 (Growth-Tech) and 4.7 (Concatenator Manufacturing). Note

the discontinuous increases in dividends and free cash flow when asset growth slows
down. Now look at your answer to Practice Question 11: Dividends are expected to
grow smoothly, although at a lower rate after year 3. Is there an error or hidden incon-
sistency in Practice Question 11? Write down a general rule or procedure for deciding

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how to forecast dividends or free cash flow.
2. The constant-growth DCF formula

is sometimes written as
r bROE

where BVPS is book equity value per share, b is the plowback ratio, and ROE is the ra-
tio of earnings per share to BVPS. Use this equation to show how the price-to-book ra-
tio varies as ROE changes. What is price-to-book when ROE r?
3. Portfolio managers are frequently paid a proportion of the funds under manage-
ment. Suppose you manage a $100 million equity portfolio offering a dividend yield
(DIV1/P0) of 5 percent. Dividends and portfolio value are expected to grow at a con-
stant rate. Your annual fee for managing this portfolio is .5 percent of portfolio value
and is calculated at the end of each year. Assuming that you will continue to man-
age the portfolio from now to eternity, what is the present value of the management

Reeby Sports
Ten years ago, in 1993, George Reeby founded a small mail-order company selling high-
quality sports equipment. Reeby Sports has grown steadily and been consistently profitable
(see Table 4.8). The company has no debt and the equity is valued in the company™s books
at nearly $41 million (Table 4.9). It is still wholly owned by George Reeby.
George is now proposing to take the company public by the sale of 90,000 of his existing
shares. The issue would not raise any additional cash for the company, but it would allow
Brealey’Meyers: I. Value 4. The Value of Common © The McGraw’Hill
Principles of Corporate Stocks Companies, 2003
Finance, Seventh Edition

88 PART I Value


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