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18. P/E Ratios. Web Cites Research projects a rate of return of 20 percent on new projects.
Management plans to plow back 30 percent of all earnings into the firm. Earnings this year
will be $2 per share, and investors expect a 12 percent rate of return on the stock.

a. What is the sustainable growth rate?
b. What is the stock price?
c. What is the present value of growth opportunities?
d. What is the P/E ratio?
e. What would the price and P/E ratio be if the firm paid out all earnings as dividends?
f. What do you conclude about the relationship between growth opportunities and P/E

19. Constant-Growth Model. Fincorp will pay a year-end dividend of $4.80 per share, which
is expected to grow at a 4 percent rate for the indefinite future. The discount rate is 12
Valuing Stocks 305

a. What is the stock selling for?
b. If earnings are $6.20 a share, what is the implied value of the firm™s growth opportuni-

20. P/E Ratios. No-Growth Industries pays out all of its earnings as dividends. It will pay its
next $4 per share dividend in a year. The discount rate is 12 percent.

a. What is the price-earnings ratio of the company?
b. What would the P/E ratio be if the discount rate were 10 percent?

21. Growth Opportunities. Stormy Weather has no attractive investment opportunities. Its re-
turn on equity equals the discount rate, which is 10 percent. Its expected earnings this year
are $3 per share. Find the stock price, P/E ratio, and growth rate of dividends for plowback
ratios of

a. zero
b. .40
c. .80

22. Growth Opportunities. Trend-line Inc. has been growing at a rate of 6 percent per year and
is expected to continue to do so indefinitely. The next dividend is expected to be $5 per

a. If the market expects a 10 percent rate of return on Trend-line, at what price must it be
b. If Trend-line™s earnings per share will be $8, what part of Trend-line™s value is due to as-
sets in place, and what part to growth opportunities?

23. P/E Ratios. Castles in the Sand generates a rate of return of 20 percent on its investments
and maintains a plowback ratio of .30. Its earnings this year will be $2 per share. Investors
expect a 12 percent rate of return on the stock.
a. Find the price and P/E ratio of the firm.
b. What happens to the P/E ratio if the plowback ratio is reduced to .20? Why?
c. Show that if plowback equals zero, the earnings-price ratio E/P falls to the expected rate
of return on the stock.

24. Dividend Growth. Grandiose Growth has a dividend growth rate of 20 percent. The dis-
count rate is 10 percent. The end-of-year dividend will be $2 per share.
a. What is the present value of the dividend to be paid in Year 1? Year 2? Year 3?
b. Could anyone rationally expect this growth rate to continue indefinitely?

25. Stock Valuation. Start-up Industries is a new firm which has raised $100 million by selling
shares of stock. Management plans to earn a 24 percent rate of return on equity, which is
more than the 15 percent rate of return available on comparable-risk investments. Half of all
earnings will be reinvested in the firm.

a. What will be Start-up™s ratio of market value to book value?
b. How would that ratio change if the firm can earn only a 10 percent rate of return on its

26. Nonconstant Growth. Planned Obsolescence has a product that will be in vogue for 3 years,
at which point the firm will close up shop and liquidate the assets. As a result, forecasted
dividends are DIV1 = $2, DIV2 = $2.50, and DIV3 = $18. What is the stock price if the dis-
count rate is 12 percent?

27. Nonconstant Growth. Tattletale News Corp. has been growing at a rate of 20 percent per
year, and you expect this growth rate in earnings and dividends to continue for another 3
a. If the last dividend paid was $2, what will the next dividend be?
b. If the discount rate is 15 percent and the steady growth rate after 3 years is 4 percent, what
should the stock price be today?

28. Nonconstant Growth. Reconsider Tattletale News from the previous problem.
a. What is your prediction for the stock price in 1 year?
b. Show that the expected rate of return equals the discount rate.

29. Sustainable Growth. Computer Corp. reinvests 60 percent of its earnings in the firm. The
Challenge stock sells for $50, and the next dividend will be $2.50 per share. The discount rate is 15
Problems percent. What is the rate of return on the company™s reinvested funds?
30. Nonconstant Growth. A company will pay a $1 per share dividend in 1 year. The dividend
in 2 years will be $2 per share, and it is expected that dividends will grow at 5 percent per
year thereafter. The expected rate of return on the stock is 12 percent.

a. What is the current price of the stock?
b. What is the expected price of the stock in a year?
c. Show that the expected return, 12 percent, equals dividend yield plus capital apprecia-

31. Nonconstant Growth. Phoenix Industries has pulled off a miraculous recovery. Four years
ago it was near bankruptcy. Today, it announced a $1 per share dividend to be paid a year
from now, the first dividend since the crisis. Analysts expect dividends to increase by $1 a
year for another 2 years. After the third year (in which dividends are $3 per share) dividend
growth is expected to settle down to a more moderate long-term growth rate of 6 percent. If
the firm™s investors expect to earn a return of 14 percent on this stock, what must be its
32. Nonconstant Growth. 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 remain in business, the Metropolitan District Commission (MDC) has
agreed to pay whatever amount is necessary to yield CSI a 10 percent return on investment.
At the end of the year, CSI is expected to pay a $4 dividend. It has been reinvesting 40 per-
cent of earnings and growing at 4 percent a year.

a. Suppose CSI continues on this growth trend. What is the expected rate of return from
purchasing the stock at $100?
b. What part of the $100 price is attributable to the present value of growth opportunities?
c. Now the MDC announces a plan for CSI to treat Cambridge sewage. CSI™s plant will
therefore be expanded gradually over 5 years. This means that CSI will have to reinvest
80 percent of its earnings for 5 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?
33. Nonconstant Growth. Better Mousetraps has come out with an improved product, and the
world is beating a path to its door. As a result, the firm projects growth of 20 percent per
year for 4 years. By then, other firms will have copycat technology, competition will drive
Valuing Stocks 307

down profit margins, and the sustainable growth rate will fall to 5 percent. The most recent
annual dividend was DIV0 = $1.00 per share.

a. What are the expected values of DIV1, DIV2, DIV3, and DIV4?
b. What is the expected stock price 4 years from now? The discount rate is 10 percent.
c. What is the stock price today?
d. Find the dividend yield, DIV1/P0.
e. What will next year™s stock price, P1, be?
f. What is the expected rate of return to an investor who buys the stock now and sells it in
1 year?

1 People™s Energy™s high and low prices over the past 52 weeks have been 3915„16 and 281„2 per
Solutions to share. Its annual dividend was $2.00 per share and its dividend yield (annual dividend as a
Self-Test percentage of stock price) 7.0 percent. The ratio of stock price to earnings per share, the P/E
ratio, is 10. Trading volume was 68,100 shares. The highest price at which the shares traded
Questions during the day was $291„4, the lowest price was $281„8, and the closing price was $283„8,
which was $5/8 lower than the previous day™s closing price.
2 IBM™s forecast future profitability has fallen. Thus the value of future investment opportu-
nities has fallen relative to the value of assets in place. This happens in all growth industries
sooner or later, as competition increases and profitable new investment opportunities shrink.

DIV1 + P1 $5 + $105
3 P0 = = = $100
1+r 1.10

4 Since dividends and share price grow at 5 percent,
DIV2 = $5 — 1.05 = $5.25, DIV3 = $5 — 1.052 = $5.51
P3 = $100 — 1.053 = $115.76
P0 = + 2+
1+r (1 + r) (1 + r)3
$5.00 $5.25 $5.51 + $115.76
= + + = $100
1.10 1.102 1.103

DIV $25
5 P0 =
r = .20 = $125

6 The two firms have equal risk, so we can use the data for Androscoggin to find the expected
return on either stock:
DIV1 $5
r= +g= + .05 = .10, or 10%
P0 $100

7 We™ve already calculated the present value of dividends through Year 3 as $2.98. We can also
forecast stock price in Year 4 as
$1.73 — 1.05
P4 = = $36.33
.10 “ .05
P0 = PV (dividends through Year 3) + PV(DIV4) + PV(P4)
$1.73 $36.33
= $2.98 + +
1.104 1.104
= $2.98 + $1.18 + $24.81 = $28.97

8 a. The sustainable growth rate is
g = return on equity — plowback ratio
= 10% — .40 = 4%

b. First value the company. At a 60 percent payout ratio, DIV1 = $3.00 as before. Using the
constant-growth model,
P0 = = $37.50
.12 “ .04

which is $4.17 per share less than the company™s no-growth value of $41.67. In this ex-
ample Blue Skies is throwing away $4.17 of potential value by investing in projects with
unattractive rates of return.
c. Sure. A raider could take over the company and generate a profit of $4.17 per share just
by halting all investments offering less than the 12 percent rate of return demanded by
investors. This assumes the raider could buy the shares for $37.50.

MINICASE mon stock was distributed among 15 grandchildren and nephews
Terence Breezeway, the CEO of Prairie Home Stores, wondered
of Jacob Breezeway, most of whom had come to depend on gener-
what retirement would be like. It was almost 20 years to the day
ous regular dividends. The commitment to high dividend payout1
since his uncle Jacob Breezeway, Prairie Home™s founder, had
asked him to take responsibility for managing the company. had reduced the earnings available for reinvestment and thereby
Now it was time to spend more time riding and fishing on the constrained growth.
old Lazy Beta Ranch. Mr. Breezeway believed the time had come to take Prairie
Under Mr. Breezeway™s leadership Prairie Home had grown Home public. Once its shares were traded in the public market, the
slowly but steadily and was solidly profitable. (Table 3.7 shows Breezeway descendants who needed (or just wanted) more cash to
earnings, dividends, and book asset values for the last 5 years.) spend could sell off part of their holdings. Others with more inter-
Most of the company™s supermarkets had been modernized and est in the business could hold on to their shares and be rewarded
its brand name was well-known. by higher future earnings and stock prices.
Mr. Breezeway was proud of this record, although he wished But if Prairie Home did go public, what should its shares sell
that Prairie Home could have grown more rapidly. He had for? Mr. Breezeway worried that shares would be sold, either by
passed up several opportunities to build new stores in adjacent Breezeway family members or by the company itself, at too low a
counties. Prairie Home was still just a family company. Its com- price. One relative was about to accept a private offer for $200, the

2000 2001 2002 2003 2004
Financial data for Prairie
Book value, start of year $62.7 66.1 69.0 73.9 76.5
Home Stores, 2000“2004
Earnings $9.7 9.5 11.8 11.0 11.2
(figures in millions)
Dividends $6.3 6.6 6.9 7.4 7.7
Retained earnings $3.4 2.9 4.9 2.6 3.5
Book value, end of year $66.1 69.0 73.9 76.5 80.0

1. Prairie Home Stores has 400,000 common shares.
2. The company™s policy is to pay cash dividends equal to 10 percent of start-of-year book value.

1The company traditionally paid out cash dividends equal to 10 percent of start-of-period book value. See
Table 5.6.
Valuing Stocks 309

2005 2006 2007 2008 2009 2010
Financial projections for
Rapid-Growth Scenario
Prairie Home Stores,
2005“2010 (figures in Book value, start of year 80 92 105.8 121.7 139.9 146.9
millions) Earnings 12 13.8 15.9 18.3 21.0 22.0
Dividends 0 0 0 0 14 14.7
Retained earnings 12 13.8 15.9 18.3 7.0 7.4
Book value, end of year 92 105.8 121.7 140.0 146.9 154.3
Constant-Growth Scenario
Book value, start of year 80 84 88.2 92.6 97.2 102.1
Earnings 12 12.6 13.2 13.9 14.6 15.3
Dividends 8 8.4 8.8 9.3 9.7 10.2
Retained earnings 4 4.2 4.4 4.6 4.9 5.1
Book value, end of year 84 88.2 92.6 97.2 102.1 107.2

1. Both panels assume earnings equal to 15 percent of start-of-year book value. This profitability rate is


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