Chapter 7 Stocks and Stock Valuation
Prepping for Exams
1. b.
2. b.
3. d.
4. a.
5. d.
6. a.
7. a.
8. d.
9. c.
10. c.
Problems
1. Anderson Motors, Inc. has just set the company dividend policy at $0.50 per year.
The company plans on being in business forever. What is the price of this stock if
a.
b.
c.
d.
e.
an investor wants a 5% return?
an investor wants an 8% return?
an investor wants a 10% return?
an investor wants a 13% return?
an investor wants a 20% return?
ANSWER
Use the constant dividend infinite dividend stream model:
Price = Dividend / r
a.
b.
c.
d.
e.
Price = $0.50 / 0.05 = $10.00
Price = $0.50 / 0.08 = $6.25
Price = $0.50 / 0.10 = $5.00
Price = $0.50 / 0.13 = $3.85
Price = $0.50 / 0.20 = $2.50
3. Singing Fish Fine Foods has a current annual cash dividend policy of $2.25. The price
of the stock is set to yield a 12% return. What is the price of this stock if the dividend
will be paid
a. for 10 years?
b. for 15 years?
183
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184
c.
d.
e.
f.
for 40 years?
for 60 years?
for 100 years?
forever?
ANSWER
Use the finite constant dividend model except with f (use infinite constant dividend
model)
Price = Dividend × (1 – 1/(1+r)n) / r
a.
b.
c.
d.
e.
f.
Price = $2.25 × (1 – 1/(1.12)10 / 0.12 = $2.25 × 5.6502 = $12.71
Price = $2.25 × (1 – 1/(1.12)15 / 0.12 = $2.25 × 6.8109 = $15.32
Price = $2.25 × (1 – 1/(1.12)40 / 0.12 = $2.25 × 8.2438 = $18.54
Price = $2.25 × (1 – 1/(1.12)60 / 0.12 = $2.25 × 8.3240 = $18.73
Price = $2.25 × (1 – 1/(1.12)100 / 0.12 = $2.25 × 8.3332 = $18.75
Price = $2.25 / 0.12 = $18.75
5. King Waterbeds has an annual cash dividend policy that raises the dividend each year
by 4%. Last year’s dividend was $0.40 per share. What is the price of this stock if
a.
b.
c.
d.
e.
an investor wants a 5% return?
an investor wants an 8% return?
an investor wants a 10% return?
an investor wants a 13% return?
an investor wants a 20% return?
ANSWER
Use the constant growth dividend model with an infinite dividend stream:
Price = Last Dividend × (1 + g) / (r – g)
a.
b.
c.
d.
e.
Price = $0.40 × (1.04) / (0.05 – 0.04) = $0.4160 / 0.01 = $41.60
Price = $0.40 × (1.04) / (0.08 – 0.04) = $0.4160 / 0.04 = $10.40
Price = $0.40 × (1.04) / (0.10 – 0.04) = $0.4160 / 0.06 = $6.93
Price = $0.40 × (1.04) / (0.13 – 0.04) = $0.4160 / 0.09 = $4.62
Price = $0.40 × (1.04) / (0.20 – 0.04) = $0.4160 / 0.16 = $2.60
7. Miles Hardware has an annual cash dividend policy that raises the dividend each year
by 3%. Last year’s dividend was $1.00 per share. Investors want a 15% return on this
stock. What is the stock’s price if
a.
b.
c.
d.
the company will be in business for 5 years and not have a liquidating dividend?
the company will be in business for 15 years and not have a liquidating dividend?
the company will be in business for 25 years and not have a liquidating dividend?
the company will be in business for 35 years and not have a liquidating dividend?
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Chapter 7  Stocks and Stock Valuation
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e. the company will be in business for 75 years and not have a liquidating dividend?
f. forever?
ANSWER
Use the constant growth dividend model with a finite dividend stream:
Price = Last Dividend × (1 + g) / (r – g) × [1 – ((1+g) / (1+r))n]
a. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))5]
= $1.03 / 0.12 × [1 - 0.5764] = $8.58 × [0.4236] = $3.64
b. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))15]
= $1.03 / 0.12 × [1 - 0.1915] = $8.58 × [0.8085] = $6.94
c. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))25]
= $1.03 / 0.12 × [1 - 0.0636] = $8.58 × [0.9364] = $8.03
d. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))35]
= $1.03 / 0.12 × [1 - 0.0211] = $8.58 × [0.9789] = $8.40
e. Price = $1.00 × (1.03) / (0.15 – 0.03) × [1 – ((1.03) / (1.15))75]
= $1.03 / 0.12 × [1 - 0.0003] = $8.58 × [0.9997] = $8.58
f. Price = $1.00 × (1.03) / (0.15 – 0.03) = $1.03 / 0.12 = $8.58
9. Fey Fashions expects the following dividend pattern over the next seven years:
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Year 7
$1.00
$1.10
$1.21
$1.33
$1.46
$1.61
$1.77
Then the company will have a constant dividend of $2.00 forever. What is the price
of this stock today if an investor wants to earn
a. 15%?
b. 20%?
ANSWER
There are two dividend patterns here; the first is a constant growth pattern for the next
seven years and then a constant dividend forever. Solve each part separately and then add
the two parts.
Part one: Constant growth for seven years, first find growth rate and note that there are
six changes in the dividend stream over the seven years.
g = ($1.77 / $1.00)1/6 – 1 = 1.771/6 – 1 = 10%
Next, use the finite dividend growth model
Price = Dividend × (1 + g) / (r – g) × [1 – ((1+g) / (1+r))n]
Part two: Use the constant dividend (infinite period) model and then discount the price at
period 7 back to the present
Price7 = Dividend8 / r
Price0 = Price7 / (1 + r)7
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186
a. Part One Price = $1.00 (1.10) / (0.15 – 0.10) × [ 1 – (1.10/1.15)7]
Part One Price = $22.00 × 0.2674 = $5.88
Part Two Price = ($2.00 / 0.15) / (1.15)7 = $13.33 / 2.66 = $5.01
Price = $5.88 + $5.01 = $10.89
b. Part One Price = $1.00 (1.10) / (0.20 – 0.10) × [ 1 – (1.10/1.20)7]
Part One Price = $11.00 × 0.4561 = $5.02
Part Two Price = ($2.00 / 0.20) / (1.20)7 = $10.00 / 3.5832 = $2.79
Price = $5.02 + $2.79 = $7.81
11. Fenway Athletic Club plans to offer its members preferred stock with a par value of
$100 and a 6% annual dividend rate. If a member wants the following returns, what
price should he or she be willing to pay?
a.
b.
c.
d.
Theo wants a 10% return.
Jonathan wants a 12% return
Josh wants a 15% return
Terry wants an 18% return
ANSWER
Use the constant dividend model with infinite horizon
Price = Dividend / r
a.
b.
c.
d.
Theo’s Price = $100 × 0.06 / 0.10 = $60.00
Jonathan’s Price = $100 × 0.06 / 0.12 = $50.00
Josh’s Price = $100 × 0.06 / 0.15 = $40.00
Terry’s Price = $100 × 0.06 / 0.18 = $33.33
13. Villalpondo Winery wants to raise $10 million from the sale of preferred stock. If the
winery wants to sell 1 million shares of preferred stock, what annual dividend will
they have to promise if investors demand
a.
b.
c.
d.
e.
f.
a 12% return?
an 18% return?
an 8% return?
a 6% return?
a 9% return?
a 7% return?
ANSWER
Use the constant dividend formula rearranged for the dividend,
Dividend = r × Price
And the price of the shares is $10,000,000 / 1,000,000 = $10.00 per share
a. A 12% return means: Dividend = 12% × $10.00 = $1.20 per year
b. A 18% return means: Dividend = 18% × $10.00 = $1.80 per year
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Chapter 7  Stocks and Stock Valuation
c.
d.
e.
f.
187
A 8% return means: Dividend = 8% × $10.00 = $0.80 per year
A 6% return means: Dividend = 6% × $10.00 = $0.60 per year
A 9% return means: Dividend = 9% × $10.00 = $0.90 per year
A 7% return means: Dividend = 7% × $10.00 = $0.70 per year
15. Using Yahoo! Finance (http://finance.yahoo.com/) and ticker symbol PEP, find
PepsiCo’s historical dividend payment and current price. Historical dividends are
available in the historical price section. Use these payments to find the annual
dividend growth rate. (If you have a quarterly pattern be sure to annualize this
quarterly growth rate.) Now, find the required rate of return for this stock, assuming
that the future dividend growth rate will remain the same and the company has an
infinite horizon. Does this return seem reasonable for PepsiCo?
ANSWER: (Note that we used 1998-2010 annual dividends for PepsiCo)
For this problem a thirteen-year dividend history is used and the following are the prior
thirteen year dividends adjusted for stock splits with a price on January 3, 2011 of
$65.75:
1998 $0.515
1999 $0.535
2000 $0.555
2001 $0.575
2002 $0.595
2003 $0.63
2004 $0.85
2005 $1.01
2006 $1.16
2007 $1.425
2008 $1.65
2009 $1.775
2010 $1.89
To find the average growth rate for the thirteen years with twelve changes we have:
($1.89 / $0.515)1/12 -1 = 0.1144 or 11.44%
Now by using the constant growth formula and the current price of $65.75 we solve for r:
Price = Last Dividend × (1 + g) / (r – g)
$65.75 = $1.89 × (1 + 0.1144) / (r – 0.1144)
$65.75 = $2.11 / (r – 0.1144)
r – 0.1144 = $2.11 / $65.75
r = 0.03209 + 0.1144 = 0.1464 or 14.64%
This looks like a reasonable required return for a stock.
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Brooks  Financial Management: Core Concepts, 2e
17. Using Yahoo! Finance, update the dividends for Coca-Cola for the last ten years. Find
both the arithmetic growth rate and the geometric growth rate of the dividends.
ANSWER
Year
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
Dividend
$1.76
$1.64
$1.52
$1.36
$1.24
$1.12
$1.00
$0.92
$0.80
$0.72
Change (percent)
$1.76-$1.64 = $0.12 (.12/1.64 = 7.32%
$1.64 – $1.52= $0.12 ($0.12/1.52 = 7.89%)
$1.52 – $1.36 = $0.16 ($0.16/$1.36 = 11.76%)
$1.36 – $1.24 = $0.12 ($0.12 / $1.24 = 9.68%)
$1.24 – $1.12 = $0.12 ($0.12 / $1.12 = 10.71%)
$1.12 – $1.00 = $0.12 ($0.12 / $1.00 = 12.00%)
$1.00 – $0.92 = $0.08 ($0.12 / $0.92 = 8.70%)
$0.92 – $0.80 = $0.12 ($0.12 / $0.80 = 15.00%)
$0.80 – $0.72 = $0.08 ($0.12 / $0.72 = 11.11%)
Average Change 10.46%
Geometric growth rate = ($1.76 / $0.72)1/9 – 1 = 10.44%
P/Y = 1, C/Y = 1
INPUT 9
?
KEYS N
I/Y
COMPUTE 10.44
-0.72
PV
0
PMT FV
1.76
19. Using Yahoo! Finance, update the dividends of Wal-Mart for the last ten years. Find
the arithmetic growth rate and the geometric growth rate of the dividends.
ANSWER
Year
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
Dividend
$1.212
$1.092
$0.952
$0.88
$0.672
$0.60
$0.52
$0.36
$0.30
$0.28
Change (percent)
$1.212 – $1.092 = $0.12 ($0.12/$1.092=10.99%)
$1.092 – $0.952=$0.14 ($0.14/$0.952= 14.71%)
$0.952 – $0.88 = $0.072 ($0.072/$0.88 = 8.18%)
$0.88 – $0.672 = $0.208 ($0.208 / $0.672 = 30.95%)
$0.672 – $0.60 = $0.072 ($0.072 / $0.60 = 12.00%)
$0.60 – $0.52 = $0.08 ($0.08 / $0.52 = 15.38%)
$0.52 – $0.36 = $0.16 ($0.16 / $0.36 = 44.44%)
$0.36 – $0.30 = $0.06 ($0.06 / $0.30 = 20.00%)
$0.30 – $0.28 = $0.02 ($0.02 / $0.28 = 7.14%)
Average Change 19%
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189
Geometric growth rate = ($1.212 / $0.28)1/9 – 1 = 17.68%
P/Y = 1, C/Y = 1
INPUT
9
KEYS
N
COMPUTE
?
I/Y
17.68
-0.28
PV
0
PMT
1.212
FV
21. Using the answer to Problem 17 on the Coca-Cola growth rates and the current
trading price, determine the current required rate of return for the company.
ANSWER
Coca-Cola price was at $65.22 as of January 3, 2011
 $1.76 1  0.1044 
Coca-Cola’s r  
  0.1044 = 0.1342 or 13.42%
$65.22


23. Using the answer to Problem 19 on the Wal-Mart growth rates and the current trading
price, determine the current required rate of return for the company.
ANSWER
Wal-Mart’s price was at $54.56
 $01.212 1 0.1768 
Wal-Mart’s r  
  0.1768 = .2029 or 20.29%
$54.56


25. Given the growth rates for Coca-Cola, Johnson & Johnson, Wal-Mart, and Intel from
the dividend history in Problems 21 through 24, what price would you predict for
each stock if they all had a required return of 18%? Why are Wal-Mart and Intel
prices troublesome?
ANSWER
 $1.76 1  0.1044 
Coca-Cola, P  
  $25.71
0.18

0.1044


 $2.11 1  0.1304 
Johnson and Johnson, P  
  $48.09
 0.18  0.1304 
 $1.212 1 0.1768 
Walmart, P  
  $445.71
0.18  0.1768 

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Brooks  Financial Management: Core Concepts, 2e
 $0.632 1  0.1458 
Intel, P  
  $21.18
0.18  0.1458 

Wal-Mart’s predicted price is unrealistically high since the required rate is so close to the
growth rate. Intel’s price on the other hand seems okay.
27. Peterson Packaging Incorporated does not currently pay dividends. The company will
start with a $0.50 dividend at the end of year three and grow it by 10% for each of the
next six years until it nearly reaches $1.00. After six years of growth, it will fix its
dividend at $1.00 forever. If you want a 15% return on this stock, what should you
pay today, given this future dividend stream?
ANSWER
The first step is to look at the timing and amount of the cash flow that you will receive as
a shareholder.
Expected Dividend Stream of Peterson Packaging
T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 … T ∞
--- $0.00 $0.00 $0.50 $0.55 $0.61 $0.67 $0.73 $0.81 $0.89 $1.00 …$1.00
We notice that there are three distinct dividend patterns -- a period of no dividends, T1 to
T2; a period of constant growth dividends, T3 to T9; and a period of constant dividends,
T10 to T∞. To price this stock we need to determine the present value of each pattern.
The first pattern, no dividends, is rather easy to value. It is zero.
The second pattern is a constant growth dividend stream with a finite horizon.
But we need to realize that this pattern does not start until the end of year three.
Therefore, when we apply the dividend growth model we will be getting a price for the
end of year two and we will receive seven dividends:
Price2 
n
Div3   1  g  
 1 

r  g   1  r  
Price2 
  1  0.10 7 
$0.50
 1 

 0.15  0.10   1  0.15  
Price2 = $10.00 × (0.2674) = $2.674
The price at the end of period 2 is a future value. Now we must discount this future value
at 15% for its present value:
PV 
FV
1 r 
n
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Chapter 7  Stocks and Stock Valuation
Price0 = PV 
$2.674
1  0.15
2

191
$2.674
 $2.02
1.3225
The final dividend pattern is a perpetuity and we can use equation 7.1 here:
Price9 
Dividend10
r
Price9 
$1.00
 $6.67
0.15
And again, we must discount this future value at 15% for its present value:
Price0 
$6.67
1  0.15
9
 $1.90
Finally, adding the three pieces we get:
Price of Stock = $0.00 + $2.02 + $1.90 = $3.92
Although we now have a way of pricing stocks through discounting dividends, we must
realize that dividends are not a promised future cash flow. In fact, firms can increase,
reduce, or even suspend cash dividends. And even though past dividends may be a good
predictor of future dividends, the timing and amount of dividends can and do vary across
time for a company. The dividend models are really expected dividend models, and if we
were to write these models correctly we would use an expectations operator with the
dividends, E(Div0) and E(Div1) instead of Div0 and Div1.
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