Stocks Solutions_Manual Ch09

7/16/2019 Stocks Solutions_Manual Ch09

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Chapter 9

Stock Valuation

Before You Go On Questions and Answers

Section 9.1

1. What is NASDAQ?

The National Association of Securities Automatic Quotation (NASDAQ) system is one of the

largest electronic stock markets in the world, listing over three thousand companies. It was

created in 1971 by the National Assoc. of Securities Dealers

2. How do dealers differ from brokers?

A dealer differs from a broker in that a dealer takes ownership of assets and is exposed to

inventory risk, while a broker only facilitates a transaction on behalf of a client. Unlike

brokers, dealers are subject to capital risk, because they must finance their inventories of

securities.

3. List the major stock market indexes, and explain what they tell us.

The major U.S. Stock market indexes are the Dow Jones Industrial Average, New York

Stock Exchange, Standard & Poor’s 500 Index, and NASDAQ Composite Index. A stock

market index is essentially a listing of stocks that bear some commonality, such as being

traded on the same market exchange. They are used to measure the stock market

performance and many are used to benchmark the performance of portfolios, such as mutual

funds.



4. What does the price-earnings ratio tell us?

The price-earnings ratio is the firm’s current price divided by the current earnings. When

valuing a company’s stock, it is useful to look at a company’s ratio and compare it to that of

similar firms. In general, a high P/E ratio means high projected earnings in the future. In

addition, it also tells us how much investors are willing to pay per dollar of earnings.

5. Why do some people view preferred stock as a special type of a bond rather than a stock?

Just like debt, preferred stock is often callable and may be convertible into common stock.

Also, like debt, it has ―credit‖ ratings that are similar to those issued to bonds. But mostimportant, preferred stock has no voting rights and just like bonds pays fixed dividends. For

these reasons, many analysts treat preferred stock as a special kind of debt rather than as an

equity.

Section 9.2

1. What is the general formula used to calculate the price of a share of a stock? What does it

mean?

The general formula developed to value a share of stock is as follows:

It says that the price of a share of stock is the present value of all expected future dividends,

or: Stock price = PV (all future cash dividends).

2. What are growth stocks, and why do they typically pay little on dividends?

Growth stocks are defined as equity in any company whose earnings are growing faster than

the average firm and the higher growth rate is expected to continue for some time. Instead of

5

5

4

4

3

3

2

210

)R 1(

D

)R 1(

D

)R 1(

D

)R 1(

D

)R 1(

DP



paying dividends, these firms reinvest the earnings back into the firm to pursue other high-

return investment opportunities.

Section 9.3

1. What three different models are used to value stocks based on different dividend patterns?

Based on dividend patterns, we can use the following three models to value stock: (1) zero-

growth dividend model, (2) constant-growth dividend model, or (3) supernormal dividend

growth model.

2. Explain why the growth rate g must always be less than the rate of return R.

For the valuation equation to have any meaning, the firm’s dividend growth rate g must be

less than the rate of return R. It also makes intuitive sense, since no firm can in the long run

grow faster than the rate of economy. If that was the case, this one firm would become 100

percent of the economy and that makes no sense.

Section 9.4

1. Why can skipping payment of a preferred dividend be a bad signal?

A preferred dividend is treated like an interest payment on debt by financial markets and

investors. Although the firm will not be in default, any delay or failure to pay the dividends

by the firm will be treated seriously by the financial markets and make them think that the

firm is in financial difficulty.

2. How is a preferred stock with a fixed maturity valued?

Any preferred stock with a defined maturity date is similar to a bond with a fixed maturity

date. This similarity allows for the preferred stock to be valued in a similar fashion after

making adjustments for differences in the preferred stock relative to a bond. Using Equation

8.2, developed to value bonds, the coupons are replaced by the preferred dividends and the



number of payments is adjusted to four a year to accommodate the fact that firms pay the

dividends every quarter and not semiannually. This results in the model represented by

Equation 9.7.

Self Study Problems

9.1 Ted McKay has just bought the common stock of Ryland Corp. The company expects to

grow at the following rates for the next three years: 30 percent, 25 percent, and 15

percent. Last year the company paid a dividend of $2.50. Assume a required rate of return

of 10 percent. Compute the expected dividends for the next three years and also the

present value of these dividends.

Solution:

Expected dividends for Ryland Corp and their present value:

0 10% 1 2 3

├─────────┼─────────┼─────────┼─────────┼──────────› D0 = $2.50 D1 D2 D3

g 1 = 30% g 2 = 25% g 3 = 15%

D1 = D0 × (1 + g 1) = $2.50(1 + 0.30) = $3.25

D2 = D1 × (1 + g 2) = $3.25(1 + 0.25) = $4.06

D3 = D2 × (1 + g 3) = $4.06(1 + 0.15) = $4.67

Present value of the dividends = PV(D1) + PV(D2) + PV(D3)

= $2.96 + $3.36 + $3.51

= $9.83

9.2 Merriweather Manufacturing Company has been growing at a rate of 6 percent for the

past two years, and the CEO expects the company to continue to grow at this rate for the

next several years. The company paid a dividend of $1.20 last year. If your required rate



of return is 14 percent, what is the maximum price that you would be willing to pay for

this company’s stock?

Solution:

Present value of Merriweather stock:

0 14% 1 2 3

├─────────┼─────────┼─────────┼─────────┼──────────›

D0 = $1.20 D1 D2 D3

g = 6%

D1 = D0 × (1 + g )

= $1.20(1 + 0.06)

= $1.27

$15.88

)06.014.0(

27.1$

)gR (

DP 1

0

The maximum price you should be willing to pay for this stock is $15.88.

9.3 Clarion Corp. has been selling electrical supplies for the past 20 years. The company’s

product line has changed very little in the past five years, and the company does notexpect to add any new items for the foreseeable future. Last year, the company paid a

dividend of $4.45 to its common stockholders. The company is not expected to increase

its dividends for the next several years. If your required rate of return for such firms is 13

percent, what is the current value of this company’s stock?

Solution:

Present value of Clarion Corp. stock:

0 13% 1 2 3 Year

├─────────┼─────────┼─────────┼─────────┼──────────›

D0 = $4.45 D1 D2 D3

g = 0%



Since the company’s dividends are not expected to grow,

D0 = D1 =D2=……..D∞ = $4.45 = D

Current value of the stock = D/R

= $4.45/0.13

= $34.23

9.4 Barrymore Infotech is a fast -growing communications company. The company did not

pay a dividend last year and is not expected to do so for the next two years. Last year the

company’s growth accelerated, and management expects to grow the business at a rate of

35 percent for the next five years before growth slows to a more stable growth rate of 7

percent for the next several years. In the third year, the company has forecasted a

dividend payment of $1.10. Calculate the value of the company’s stock at the end of itsrapid growth period (i.e., at the end of five years). The required rate of return for such

stocks is 17 percent. What is the current value of this stock?

Solution:

Present value of Barrymore Infotech stock:

0 17% 1 2 3 4 5 6 7 8

├────┼────┼────┼────┼────┼────┼────┼────┤

D0 D1 D2 D3 D4 D5 D6

g 1 – g5 = 35% g 6 and beyond = 7%

D0 = D1 =D2 = 0

D3 = $1.10

D4 = D3 × (1 + g 4) = $1.10(1 + 0.35) = $1.485

D5 = D4 × (1 + g 5) = $1.485(1 + 0.35) = $2.005

D6 = D5 × (1 + g 6) = $2.005(1 + 0.07) = $2.145

Value of stock at t = 5 P5:

$21.45

)07.017.0(

145.2$

)gR (

DP 6

5



Present value of the dividends in years 1 to 5 = PV(D1) + PV(D2) + PV(D3) + PV(D4) + PV(D5)

39.2$

91.0$79.0$69.0$0$0$

)17.1(

005.2$

)17.1(

485.1$

)17.1(

10.1$0$0$

543

Current value of stock = PV(Dividends) + PV(P5)

= $2.39 + [$21.45/(1.17)5]

= $2.39 + $9.78

= $ 12.17

9.5 You are interested in buying the preferred stock of a bank that pays a dividend of $1.80

every quarter. If you discount such cash flows at 8 percent, what is the value of this

stock?

Solution:

Present value of preferred bank stock:

Quarterly dividend on preferred stock = D = $1.80

Required rate of return = 8%

Current value of stock: =

o

DP

R

$1.80 4

0.08

$90.00

Critical Thinking Questions

9.1 Why does the market price of a security vary from the true equilibrium price?



Let us start by first defining the market equilibrium price of a security as the price that

equates the demand for a security with the supply of the security. The role of the security

markets is to bring buyers and sellers together in the most efficient way such that

securities are bought and sold at the true equilibrium price. In reality, however, barriers

of various kinds including the geographic separation of the two parties make the market

price of a security slightly different than the true equilibrium price. The more efficient the

market place, the smaller the deviation between the two.

9.2 Why are investors and managers concerned about stock market efficiency?

The role of secondary markets is to bring buyers and sellers together. Ideally, we would

like security markets to be as efficient as possible. Markets are efficient when current

market prices of securities traded reflect all available information relevant to the security.

If this is the case, security prices will be near or at their equilibrium price. The more

efficient the market, the more likely this is to happen. This makes it easier for managers

to price the securities close to the equilibrium price.

What investors are most concerned about is having complete information

regarding a security’s current price and where that price information can be obtained.

Efficient markets allow them to trade at prices that are closer to the true equilibrium price

than otherwise possible.

Thus, both investors who provide funds and managers (firms) who raise money

are concerned when high transaction costs lead to inefficient markets.

9.3 Why are common stockholders considered to be more at risk than the holders of other

types of securities?

In the hierarchy of lenders of funds to a firm, common stockholders have the most to

lose. In the event of a firm becoming bankrupt, the law requires that creditors of different

types, including bondholders, be paid off first. Next, preferred stockholders are paid off.

Finally, common stockholders receive their investment if any funds are still available.



Thus, common stockholders receive their money back last and are placed at most risk.

This feature of common equity is referred to as residual claim.

9.4 Under what conditions does it make sense to use the constant-growth dividend model to

value a stock?

It only makes sense to use the constant-growth dividend model to value the stock of a

company that is already paying a dividend and when that dividend can reasonably be

expected to grow at a constant rate forever in other words, the stock of a mature

company.. As discussed in the chapter, the growth rate of such a company will be less

than the long term rate of inflation and the real growth rate of the economy. It must also

be less than the required rate of return on the stock, R.

9.5 What does it mean when a company has a very high P/E ratio? Give examples of

industries in which you believe high P/E ratios are justified.

A high P/E ratio implies that investors believe that the firm has good prospects for

earnings growth in the future. In fact, they believe that the firm will have higher growth

potential than firms with lower P/E ratios. Companies in industries that are fast growinglike biotech or any hi-tech industry have high P/E ratios. In the past, firms like Cisco and

Intel had very high P/E ratios. As these firms matured and settled to annual growth rates

of 15 percent or less, their P/E ratios have declined.

9.6 Preferred stock is considered to be nonparticipating because

a. investors do not participate in the election of the firm’s directors.

b. investors do not participate in the determination of the dividend payout policy.

c. investors do not participate in the firm’s earnings growth.

d. none of the above.

c. Nonparticipating implies that the preferred dividend remains constant regardless

of any increase in the firm’s earnings. Thus, investors in a firm’s preferred stock



will not see higher dividends when the firm’s earnings increase. Nor will they see

a decrease if the firm’s earnings decrease.

9.7 Explain why preferred stock is considered to be a hybrid of equity and debt securities.

The law considers preferred stock as equity. Thus, holders are treated as the firm’s

owners. Also, like common stockholders, preferred stockholders have to pay taxes on

their dividend income. However, preferred stockholders do not have any voting rights. In

addition, they receive only a fixed dividend just like bondholders. If a firm is liquidated,

then they receive a stated value (par value) similar to bondholders. Preferred stock is

rated by credit rating agencies just like bonds. Some preferred issues are convertible to

the firm’s common stock just as convertible bonds. Some preferred issues are not perpetual and have a fixed maturity just like bonds.

Thus preferred stock is a hybrid security — like equity in some ways and like debt

security in others.

9.8 Why is stock valuation more difficult than bond valuation?

Despite the availability of mathematical models to value stocks, it is more difficult to

apply valuation techniques to stocks than to bonds. First, unlike bonds, firms are not in

default if dividends are not declared. This makes it difficult to determine the size and

timing of the cash flows. Second, common stock, unlike bonds, does not have a fixed

maturity, and hence, it is difficult to determine a terminal value unlike bonds, which have

a maturity value. Next, it is easier to calculate the present value of a bond because the

required rate of return is observable. In the case of stocks, it is rather difficult to estimate

a required rate of return for many stocks and classify them into different risk groups.

9.9 You are currently thinking about investing in a stock valued at $25.00 per share. The

stock recently paid a dividend of $2.25 and is expected to grow at a rate of 5 percent for

the foreseeable future. You normally require a return of 14 percent on stocks of similar

risk. Is the stock overpriced, underpriced, or correctly priced?



This stock is underpriced at $25. Using the constant-growth model, we can arrive at a

price of $26.25 for this stock. This makes the stock underpriced, and it should be

considered a good buy.

9.10 Stock A and Stock B are both priced at $50 per share. Stock A has a P/E ratio of 17,

while Stock B has a P/E ratio of 24. Which is the more attractive investment, considering

everything else to be the same, and why?

Stock A is the more attractive investment because it has a lower P/E ratio. The lower the

P/E ratio, the larger the amount of earnings supporting the stock price. This makes Stock

A a more attractive investment than Stock B.

Questions and Problems BASIC

9.1 Stock Market Index: What is a stock market index?

LO 1

Solution:

A stock market index is used to measure the performance of the stock market. These

indexes reflect the value of the stocks in a particular market, such as the NYSE or the

NASDAQ, or across markets and increase and decrease as the values of the stocks go up

and down. Examples of stock market indexes include the Dow Jones Industrial Average,

the New York Stock Exchange Index, the Standard & Poor’s Index, and the NASDAQComposite Index.

9.2 Stock Market Index: What is the Dow Jones Industrial Average?

LO 1



Solution:

The Dow is the most widely published stock index and reflects the value of the stocks of

30 large companies. The value of the shares of these companies represents about 20

percent of the market value of all U.S. stocks.

9.3 Stock Market Index: What does NASDAQ stand for? What is NASDAQ?

Solution:

National Association of Securities Automatic Quotation system. NASDAQ is one of the

largest electronic stock markets in the world, listing over three thousand companies.

LO 1

9.4 Dividend yield: What is the dividend yield? What does it tell us?

LO1

Solution:

A dividend yield is the ratio of the annual dividend payout divided by the current market

price of a stock. It tells us what percentage return we would earn from dividends alone if

we purchased the stock at its current price. Alternatively, it tells us what percentage of

the firm’s stock value is being distributed to stockholders each year.

9.5 Present value of dividends: Fresno Corp is a fast growing company that expects to grow

at a rate of 30 percent over the next two years and then slow down to a growth rate of 18

percent for the following three years. If the last dividend paid by the company was $2.15,

estimate the dividends for the next five years. Compute the present value of these

dividends if the required rate of return is 14 percent.

LO 4

Solution:

0 1 2 3 4 5



├───────┼────────┼───────┼────────┼───────┤

D0 = $2.15 g 1-2 = 30%; g 3-5 = 18%; k CS = 14%

D1 = D0 × (1 + g 1) = $2.15(1.30) = $2.795

D2 = D1 × (1 + g 2) = $2.795(1.30) = $3.634

D3 = D2 × (1 + g 3) = $3.634(1.18) = $4.288

D4 = D3 × (1 + g 4) = $4.288(1.18) = $5.06

D5 = D4 × (1 + g 4) = $5.06(1.18) = $5.97

$14.24

10.3$00.3$89.2$80.2$45.2$

)14.1(

97.5$

)14.1(

06.5$

)14.1(

288.4$

)14.1(

634.3$

)14.1(

795.2$)Dividends(PV

54321

9.6 Zero growth: Nynet, Inc., paid a dividend of $4.18 last year. The company does not

expect to increase its dividend i the foreseeable future. If the required rate of return is

18.5 percent, what is the current price of the stock?

LO 4

Solution:

D0 = $4.18; g = 0; R = 18.5%

$22.60

185.0

18.4$

R

DP0

9.7 Zero growth: Knight Supply Corp. has not grown for the past several years and

management expects this lack of growth to continue. The firm last paid a dividend of

$3.56. If you require a rate of return of 13 percent, what is the current stock price?

LO 4

Solution:

D0 = $3.56; g = 0; R = 13%

$27.39

13.0

56.3$

R

DP0



9. 8 Zero growth: Ron Santana is interested in buying the stock of First National Bank.

While the bank expects no growth in the near future, Ron is attracted by the dividend

income. Last year, the bank paid a dividend of $5.65. If Ron Santana requires a return of

14 percent on such stocks, what is the maximum price he should be willing to pay for a

share of the bank’s stock?

LO 4

Solution:

D0 = $5.65; g = 0; R = 14%

$40.36

14.0

65.5$

R

DP

0

9.9 Zero growth: The current stock price of Largent, Inc., is $44.72. If the required rate of

return is 19 percent, what is the dividend paid by this firm, if this dividend is not

expected to grow in the future?

LO 4

Solution:P0 = $44.72; R = 19%; D = ?;

$8.50

19.072.44$D

19.0

D72.44$

R

DP0

9.10 Constant growth: Moriband Corp. just paid a dividend of $2.15 yesterday. The company

is expected to grow at a steady rate of 5 percent for the foreseeable future. If investors in

stocks of companies like Moriband require a rate of return of 15 percent, what should be

the market price of Moriband’s stock?

LO 4



Solution:

D0 = $2.15; g = 5%; R = 15%

1 00

D D (1 )P

R R

$2.15(1.05)

0.15 0.05

g

g g

$22.58

9.11 Constant growth: Nyeil, Inc., is a consumer products firm that is growing at a constant

rate of 6.5 percent. The firm’s last dividend was $3.36. If the required rate of return is 18

percent, what is the market value of this stock if the dividends grow at the same rate as

the firm?

LO 4

Solution:

D0 = $3.36; g = 6.5%; R = 18%

10

D $3.36(1.065)P

R 0.18 0.065 g

$31.12

9.12 Constant growth: Reco Corp. is expected to pay a dividend of $2.25 next year. The

forecast for the stock price a year from now is $37.50. If the required rate of return is 14

percent, what is the current stock price? Assume constant growth.

LO 4

Solution:

D1 = $2.25; P1 = $37.50; R = 14%



2 11

10

D D (1 )P

R R

$2.25(1 )$37.50

0.14

$37.50(0.14 ) $2.25 2.25

$5.25 37.5 2.25 2.25

$5.25 $2.25 37.5 2.25

$3.007.55%

$39.75

DP

R

$2.25

0.14 0.0755

g

g g

g

g

g g

g g

g g

g

g

$34.88

9.13 Constant growth: Proxicam, Inc., is expected to grow at a constant rate of 7 percent. If

the company’s next dividend, which will be paid in a year, is $1.15 and its current stock

price is $22.35, what is the required rate of return on this stock?

LO 4

Solution:

D1 = $1.15; P0 = $23.00; g = 7%

10

1

0

DP

R

DR

P

$1.150.07

$22.35

0.0515 0.07

g

g

12.15%



9.14 Preferred stock valuation: X-Centric Energy Company has issued perpetual preferred

stock with a stated (par) value of $100 and a dividend of 4.5 percent. If the required rate

of return is 8.25 percent, what is the stock’s current market price?

LO 6

Solution:

D = 4.5% ($100) = $4.50; R = 8.25%

$54.55

0825.0

50.4$

R

DP0

9.15 Preferred stock valuation: The First Bank of Ellicott City has issued perpetual preferred

stock with a $100 par value. The bank pays a quarterly dividend of $1.65 on this stock.

What is the current price of this preferred stock given a required rate of return of 11.6

percent?LO 6

Solution:

Quarterly dividend = $1.65

Required rate of return = R = 11.6%

$56.90

116.0

)465.1(P0

9.16 Preferred stock: The preferred stock of Axim Corp. is currently selling at $47.13. If

your required rate of return is 12.2 percent, what is the dividend paid by this stock?

LO 6



Solution:

P0 = $47.13; R = 12.2%

$5.75

122.013.47$D

122.0

D13.47$P0

9.17 Preferred stock: Each quarter, Sirkota, Inc., pays a dividend on its perpetual preferred

stock. Today, the stock is selling at $63.37. If the required rate of return for such stocks is

15.5 percent, what is the quarterly dividend paid by this firm?

LO 6

Solution:

P0 = $63.37; R = 15.5%

82.9$

155.037.63$D

155.0

D37.63$P0

Annual dividend = $9.82

Quarterly dividend = $9.82 /4 = $2.46

INTERMEDIATE

9.18 Constant growth: Kay Williams is interested in purchasing the common stock of

Reckers, Inc., which is currently priced at $37.45. The company is expected to pay a

dividend of $2.58 next year and to grow at a constant rate of 7 percent.

a. What should the market value of the stock be if the required rate of return is 14

percent?

b. Is this a good buy? Why or why not?

LO 4



Solution:

a. 10

D $2.58P

R 0.14 0.07 g

$36.86

b. The stock is overpriced and not a good buy.

9.19 Constant growth: The required rate of return is 23 percent. Ninex Corp. has just paid a

dividend of $3.12 and is expected to grow at a constant rate of 5 percent. What is the

expected price of the stock three years from now?

LO 4

Solution:

R = 23%; D0 = $3.12; g = 5%

4

4 03

4

D D (1 )P

R R

3.12(1.05)

0.23 0.05

g

g g

$21.07

9.20 Constant growth: Jenny Banks is interested in buying the stock of Fervan, Inc., which is

increasing at a constant rate of 6 percent. Last year, the firm paid a dividend of $2.65.

Her required rate of return is 16 percent. What is the current value of this stock? What

would be the price of the stock in year 5?

LO 4

Solution:

g = 6%, D0 = $2.65, R = 16%

1

0

6

65

D $2.65(1.06)

P R 0.16 0.06

D $2.65(1.06)P

R 0.16 0.06

g

g

$28.09

$37.59



9.21 Constant growth: You own shares of Old World DVD Company stock and are

interested in selling them. With so many people downloading these days, sales, profits,

and dividends at Old World have been declining 6 percent per year. The firm just paid a

dividend of $1.15 per share. The required rate of return for a stock this risky is 15

percent. If dividends are expected to decline at 6 percent per year, what is a share of the

stock worth today?

LO 4

Solution:

R = 15%; D0 = $1.15; g = -6%

1 0

0

D D (1 )P

R R

$1.15 (1 0.06)

0.15 ( 0.06)

g

g g

$5.15

9.22 Nonconstant growth: You own a company that competes with Old World DVD

Company (in the previous problem). Instead of selling DVDs, however, your company

sells music downloads from a web site. Things are going well now, but you know that it

is only a matter of time before someone comes up with a better way to distribute music.Your company just paid a $1.50 per share dividend and you expect to increase the

dividend 10 percent next year. However, you then expect your dividend growth rate to

begin going down – to 5 percent the following year, 2 percent the next year, and to -3

percent per year thereafter. Based upon these estimates, what is the value of a share of

your company’s stock? Assume that the required rate of return is 12 percent.

LO 5

Solution:

g 1 = 10%, g 2 = 5%, g 3 =2%, g = -3%, D0 = $1.50 R = 12%

D1 = $1.50 × 1.1 = $1.65, D2 = $1.65 × 1.05 = $1.7325, D3 = $1.7325 ×1.02 =

$1.7672, D4 = $1.7672 × 0.97 = $1.7142.



43

D $1.7142P $11.43

R 0.12 ( 0.03) g

1 2 3 30 2 3 3

0 2 3 3

0

0

D D D PP

1 R (1 R) (1 R) (1 R)

$1.65 $1.7325 $1.7672 $11.43P

1.12 (1.12) (1.12) (1.12)

P $1.47 $1.38 $1.26 $8.14

P

$12.25

9.23 Nonconstant growth: Tre-Bien, Inc., is a fast growing technology company.

Management projects rapid growth of 30 percent for the next two years, then a growthrate of 17 percent for the following two years. After that, a constant growth rate of 8

percent is expected. The firm expects to pay its first dividend of $2.45 a year from now.

If dividends will grow at the same rate as the firm and the required rate of return on

stocks with similar risk is 22 percent, what is the current price of the stock?

LO 5

Solution:

g 1 = g 2 = 30%, g 3 = g4 = 17%, g = 8%, D1 = $2.45, R = 22%

D1 = $2.45, D2 = $2.45 × 1.30 = $3.19, D3 = $3.19 × 1.17 = $3.73

D4 = $3.73 × 1.17 = $4.36, D5 = 4.36 × 1.08 = $4.71

54

D $4.71P $33.64

R 0.22 0.08 g

1 2 3 4 40 2 3 4 4

0 2 3 4

0

0

D D D D PP

1 R (1 R) (1 R) (1 R) (1 R)

$2.45 $3.19 $3.73 ($4.36 33.64)P

1.22 (1.22) (1.22) (1.22)

P $2.01 $2.14 $2.05 17.15

P

$23.35



9.24 Nonconstant growth: ProCor, a biotech firm, forecasted the following growth rates for

the next three years: 35 percent, 28 percent, and 22 percent. The company then expects to

grow at a constant rate of 9 percent forever. The company paid a dividend of $1.75 last

week. If the required rate of return is 20 percent, what is the value of this stock?

LO 5

Solution:

g 1 = 35%; g 2 = 28%; g 3 = 22%; g 4 = 9%; D0 = $1.75; R = 20%

1 0 1

2 1 2

3 2 3

4 3

43

1 2 3 30 2 3 3

0

D D (1 ) $1.75 1.35 $2.36

D D (1 ) $2.36 1.28 $3.02

D D (1 ) $3.02 1.22 $3.69

D D (1 ) $3.69 1.09 $4.02

D $4.02P $36.57

R 0.20 0.09

D D D PP

1 R (1 R) (1 R) (1 R)

$2.36 $3.02P

1.20 (1.20)

g

g

g

g

g

2 3 3

0

0

$3.69 $36.57

(1.20) (1.20)

P $1.97 $2.10 $2.14 $21.16

P

$27.37

9.25 Nonconstant growth: Revarop, Inc., is a fast- growth company that is expected to grow

at a rate of 23 percent for the next four years. It is then expected to grow at a constant rate

of 6 percent. Revarop’s first dividend, of $4.25, will be paid in year 3. If the required rate

of return is 17 percent, what is the current value of the stock if the dividends are expected

to grow at the same rate as the company?

LO 5



Solution:

g 1-4 = 23%; g = 6%; D3 = $4.25; R = 17%

D4 = D3 × 1.23 = $4.25 × 1.23 = $5.23

5

4

1 2 3 4 40 2 3 4

0 3 4

0

D $5.23 (1.06)

P $50.40R 0.17 0.06

D D D D PP

1 R (1 R) (1 R) (1 R)

$4.25 ($5.23 $50.48)P 0 0

(1.17) (1.17)

P $2.65 $29.69

g

$32.34

9.26 Nonconstant growth: Quansi, Inc., management expects to pay no dividends for the next

six years. It has projected a growth rate of 25 percent for the next seven years. After

seven years, the firm will grow at a constant rate of 5 percent. Its first dividend, to be

paid in year 7, will be $3.25. If the required rate of return is 24 percent, what is the stock

worth today?

LO 5

Solution:

g constant = 5%; R = 24%; D7 = $3.25; D1 – D6 = 0

87

7 70 7 7

D $3.25 (1.05)P $17.96

R 0.24 0.05

D P $3.25 $17.96P

(1 R) (1.24)

g

$4.71

9.27 Nonconstant growth: Staggert Corp. will pay dividends of $5.00, $6.25, $4.75, and

$3.00 for the next four years. Thereafter, the company expects its dividend growth rate to

be constant at 6 percent. If the required rate of return is 18.5 percent, what is the current

value of the stock?



LO 5

Solution:

D1 = $5; D2 = $6.25; D3 = 4.75; D4 = $3; g = 6%; R = 18.5%;

1 2 3 4 40 2 3 4 4

54

0 2 3 4

0

D D D D PP

1 R (1 R) (1 R) (1 R) (1 R)

D $3 (1.06)P $25.44

R 0.185 0.06

$5 $6.25 $4.75 ($3 $25.44)

P 1.185 (1.185) (1.185) (1.185)

P $4.22 $4.45 $2.86 $14.42

g

$25.95

9.28 Nonconstant growth: Diaz Corp. is expected to grow rapidly at a rate of 35 percent for

the next seven years. The first dividend, to be paid three years from now, will be worth

$5. After seven years, the company (and the dividends it pays) will settle to a constant

growth rate of 8.5 percent. What is the value of this stock with a required rate of return of

14 percent?

LO 5

Solution:

g 1-7 = 35%; D3 = $5.00; g = 8.5%; R = 14%

D1 = D2 = 0; D3 = $5

D4 = $5 × 1.35 = $6.75

D5 = $6.75 × 1.35 = $9.11

D6 =$9.11 × 1.35 = $12.30

D7 = $12.30 × 1.35 = $16.61

D8 = $16.61 × 1.085 = $18.02



87

1 2 3 4 5 6 7 70 2 3 4 5 6 7

3 4 5 6 7

D $18.02P $327.64

R 0.14 0.085

D D D D D D D PP 1 R (1 R) (1 R) (1 R) (1 R) (1 R) (1 R)

$5 $6.75 $9.11 $12.30 ($16.61 $327.64)0 0

(1.14) (1.14) (1.14) (1.14) (1.14)

$3.38 $4.00 $4.73 $5.60

g

$137.58

$155.29

9.29 Nonconstant growth: Tin-Tin Waste Management, Inc., is growing rapidly. Dividends

are expected to grow at rates of 30 percent, 35 percent, 25 percent, and 18 percent over

the next four years. Thereafter, management expects dividends to grow at a constant rate

of 7 percent. The stock is currently selling at $47.85 and the required rate of return is 16

percent. Compute the dividend for the current year (D0).

LO 5

Solution:

g 1 = 30%; g 2 = 35%; g 3 = 25%; g 4 = 18%; g = 7%; R = 16%; P0 = $47.85

1 2 3 4 40 2 3 4 4

1 2 3 4 4

0 0 0

2 3

0

0

4 4

D D D D PP

(1 ) (1 ) (1 ) (1 ) (1 )

D (1.30) D (1.30) (1.35) D (1.30) (1.35) (1.25)$47.85

1.16 (1.16) (1.16)

D (2.5886) (1.07)D (1.30) (1.35) (1.25) (1.18) 0.16 0.07

(1.16) (1.16)

g g g g g

0

0

D [1.12 1.30 1.41 1.43 17.00]

$47.85D

$22.26

$2.15



ADVANCED

9.30 Equation 9.4 shows the relation between a stock’s price and the dividend that is expected

next year if dividends are expected to grow at a constant rate forever. If a firm pays all of

its earnings as dividends, show how Equation 9.4 can be rearranged to calculate thatfirm’s P/E ratio. What does this tell us about the factors that determine a firm’s P/E ratio?

LO 4

Solution:

1 10

D EP

R R g g

, where E1 is the earnings pershare next year.

Rearranging the formula:

0

1

P 1

E R g

This formula tells us that the P/E ratio is determined by both the risk of the dividend cash

flows and the rate at which they are expected to grow.

9.31 Riker Departmental Stores management has forecasted a growth rate of 40 percent for the

next two years, followed by growth rates of 25 percent and 20 percent for the following

two years. It then expects growth to stabilize at a constant rate of 7.5 percent forever. The

firm paid a dividend of $3.50 recently. If the required rate of return is 18 percent, what is

the current value of Riker’s stock?

LO 5

Solution:

g 1-2 = 40%; g 3 = 25%; g 4 = 20%; g = 7.5%; D0 = $3.50; R = 18%



1 0 1

2 1 2

3 2 3

4 3 4

5 4

54

1 20 2

D D (1 g ) $3.50 1.40 $4.90

D D (1 g ) $4.90 1.40 $6.86

D D (1 g ) $6.86 1.25 $8.575

D D (1 g ) $8.575 1.20 $10.29

D D (1 g) $10.29 1.075 $11.06

D $11.06P $105.35

R 0.18 0.075

D DP

1 R (1 R)

g

3 4 4

3 4 4

0 2 3 4 4

0

0

D D P

(1 R ) (1 R) (1 R)

$4.90 $6.86 $8.575 $10.29 $105.35P1.18 (1.18) (1.18) (1.18) (1.18)

P $4.15 $4.93 $5.22 $5.31 $54.34

P

$73.94

9.32 Courtesy Bancorp issued perpetual preferred stock a few years ago. The bank pays an

annual dividend of $4.27, and your required rate of return is 12.2 percent.

a. What is the value of the stock given your required rate of return?b. Should you buy this stock if its current market price is $34.41? Explain.

LO 4

Solution:

a. D = $4.27; R = 12.2%

$35.00

122.0

27.4$

R

DP0

b. Since the stock is worth $35.00 but can be purchased for $34.41, you should buy

this stock.

9.33 Rhea Kirby owns shares in Ryoko Corp. Currently, the market price of the stock is

$36.34. Management expects dividends to grow at a constant rate of 6 percent for the



foreseeable future. Its last dividend was $3.25. Rhea’s required rate of return for such

stocks is 16 percent. She wants to find out whether she should sell his shares or add to her

holdings.

a. What is the value of this stock?

b. Based on your answer above, should Rhea buy additional shares in Ryoko Corp?

Why or why not?

LO 4

Solution:

a. 10

D $3.25 (1.06)P

R 0.16 0.06 g

$34.45

b. No, she should not buy more shares. This stock is overpriced with the stock

selling at a higher price than what it is worth. She should sell her shares.

9.34 Perry, Inc., paid a dividend of $2.50 yesterday. You are interested in investing in this

company, which has forecasted a constant-growth rate of 7 percent forever. Your

required rate of return is 18 percent.

a. Compute the expected dividends D1, D2, D3, and D4.

b. Find the present value of these four dividends.c. What is the expected value of the stock four years from now (i.e., P4)?

d. What is the value of the stock today based on the answers to parts b. and c.?

e. Use the equation for constant growth (Equation 9.4) and compute the price of the

stock today.

LO 4

Solution:

a. D0 = $2.50 g = 7% R = 18%



1

2

2

3

3

44

D $2.50 (1.07) $2.675

D $2.50 (1.07) $2.86

D $2.50 (1.07) $3.063

D $2.50 (1.07) $3.277

b.

$7.87

69.1$86.1$05.227.2$

)18.1(

277.3$

)18.1(

063.3$

)18.1(

86.2$

)18.1(

675.2$)Dividends(PV

4321

c.

5 4

54

D D (1 ) $3.277 (1.07) $3.506

D $3.506P

R 0.18 0.07

g

g

$31.88

d.

$24.31

44.16$87.7$)P(PV)Dividends(PVP

44.16$)18.1(

88.31$)P(PV

40

44

e. For a constant-growth stock:

10

D $2.675P

R 0.18 0.07 g

$24.32

9.35 Zweite Pharma is a fast -growing drug company. Management forecasts that in the next

three years, its growth rates will be 30 percent, 28 percent, and 24 percent, respectively.

Last week it paid a dividend of $1.67. After three years management expects growth to

stabilize at a rate of 8 percent.. The required rate of return is 14 percent.



a. Compute the dividends for each of the next three years and calculate their present

value.

b. Calculate the price of the stock at the end of year 3 when the firm settles to a

constant-growth rate.

c. What is the current price of the stock?

LO 5

Solution:

g 1 = 30%; g 2 = 28%; g 3 = 24%; g = 8%; D0 = $1.67; R = 14%

a.

1 0 1

2 1 2

3 2 3

1 2 3

2 3

2 3

D D (1 ) $1.67 (1.30) $2.171

D D (1 ) $2.171 (1.28) $2.779

D D (1 ) $2.779 (1.24) $3.446

D D DPV(Dividends)

1 R (1 R) (1 R)

$2.171 $2.779 $3.446

1.14 (1.14) (1.14)

$1.90 $2.14 $2.33

g

g

g

$6.37

b.

4 3

43

D D (1 ) $3.446 (1.08) $3.722

D $3.722P

R 0.14 0.08

g

g

$62.03

c.

$48.24

87.41$37.6$

)P(PV)Dividends(PVP

87.41$)14.1(

03.62$)P(PV

30

33



9.36 Triton, Inc., expects to grow at a rate of 22 percent for the next five years and then settle

to a constant-growth rate of 6 percent. The company recently paid a dividend of $2.35.

The required rate of return is 15 percent.

a. Find the present value of the dividends during the rapid growth period if the

dividends grow at the same rate as the company.

b. What is the value of the stock at the end of year 5?

c. What is the value of the stock today?

LO 5

Solution:

g 1-5 = 22%; g = 6%; D0 = $2.35; R = 15%

a.

1 0 1

2 1 2

3 2 3

4 3 4

5 4

1 2 3 4

2 3

D D (1 ) $2.35 (1.22) $2.867

D D (1 ) $2.867 (1.22) $3.498

D D (1 ) $3.498 (1.22) $4.267

D D (1 ) $4.267 (1.22) $5.206

D D (1 ) $5.206 (1.22) $6.351

D D D DPV(Dividends)

1 R (1 R) (1 R) (1

g

g

g

g

g

5

4 5

2 3 4 5

D

R) (1 R)

$2.867 $3.498 $4.267 $5.206 $6.351

1.15 (1.15) (1.15) (1.15) (1.15)

$2.49 $2.65 $2.81 $2.98 $3.16

$14.09

b.

6 5

65

D D (1 ) $6.351 (1.06) $6.732

D $6.732P

R 0.15 0.06

g

g $74.80



c.

5 5

0 5

$74.80PV(P ) $37.19

(1.15)

P PV(Dividends) PV(P )

$14.09 $37.19

$51.28

9.37 Ceebros Builders is expanding very fast and expects to grow at a rate of 25 percent for

the next four years. The company recently paid a dividend of $3.60 but does not expect to

pay any dividends for the next three years. In year 4, management intends to pay a $5

dividend and thereafter to increase its dividend at a constant rate of 6 percent. The

required rate of return on such stocks is 20 percent.

a. Calculate the present value of the dividends during the fast-growth period.

b. What is the value of the stock at the end of the fast -growth period (P4)?

c. What is the value of the stock today?

d. Would today’s stock value be affected by the length of time you intend to hold the

stock?

LO 5

Solution:

a. g 1-4 = 25% g 5 = 6% D0 = $3.60 D4 = $5.00 R = 20%

$2.414)20.1(

5$000)Dividends(PV

b. 54

D $5.00 (1.06)P

R 0.20 0.06 g

$37.86

c. $20.67 40)20.1(

86.37$41.2$P

d. No, the length of the holding period has no bearing on today’s stock price.



Sample Test Problems

9.1 Mason Corp. is a manufacturer of consumer staples and has experienced no growth for

the past five years while paying a dividend of $3.50 every year. The CFO expects thefirm to have no growth for the foreseeable future. If the required rate of return is 10

percent, what is the price of this stock today?

Solution:

D0 = $3.50; g = 0; R = 10%

$35.00

10.0

50.3$

R

DP0

9.2 Bucknell, Inc., recently paid a dividend of $2.10. Management forecasts dividend growth

of 6 percent for the foreseeable future. What is the value of the stock today with a

discount rate of 13 percent?

Solution:

D0 = $2.10; g = 6%; R = 13%

1 00

D D (1 )P

R R

$2.10 (1.06)

0.13 0.06

g

g g

$31.80

9.3 Bradley Corp. is growing at a constant rate of 7.2 percent every year. Last week the

company paid a dividend of $1.85. If dividends will grow at the same rate as the firm and

the required rate of return is 15 percent, what should be the stock’s price four years from

now?

Solution:

D0 = $1.85; g = 7.2%; R = 15%



5 5

5 0

54

D D (1 ) $1.85 (1.072) $2.619

D $2.619P

R 0.15 0.072

g

g

$33.58

9.4 Wichita Technologies is expected to grow at a rate of 35 percent for the next three years

and then stabilize with annual growth of 7 percent. The company will pay no dividend for

the first two years and pay a dividend of $1.25 in year 3. What will be the value of the

company’s stock when the company’s supernormal growth ends? What is the value of the

stock today? Assume that dividends will grow at the same rate as the firm once Wichita

starts paying them. The firm’s required rate of return is 12 percent.

Solution:

g 1-3 = 35%; g = 7%; D1 = D2 = $0; D3 = $1.25; R = 12%

4 3

4

3

1 2 3 30 2 3 3

0 3 3

0

0

D D (1 ) $1.25 (1.07) $1.338

D $1.338P $26.76

R 0.12 0.07

D D D PP

1 R (1 R) (1 R) (1 R)

$1.25 $26.76P $0 $0

(1.12) (1.12)

P $0.89 $19.05

P

g

g

$19.94

9.5 UNC Bancorp has issued preferred stock with no maturity date. It has a par value of $100

and pays a quarterly dividend of $2.25. If the required rate of return is 8 percent, what is

the value of the stock today?



Solution:

Quarterly dividend = $2.25

Required rate of return = R = 8%

0

($2.25 4)P

0.08

$112.50

Documents

Stocks Solutions_Manual Ch09