Chapter 04 - The Value of Common Stocks
CHAPTER 4
The Value of Common Stocks
Answers to Problem Sets
1. a.
True. This is a condition for equilibrium in well-functioning capital markets.
All stocks in a particular risk class must offer the same rate of return. If a certain stock,
for example, is priced above others in the equivalent risk class, investors would sell
their shares to buy cheaper shares from companies in the same risk class. This would
force the price of that higher priced stock down to the equilibrium price. The same
happens for a stock priced below equilibrium—investors would rush to buy the stock,
sending its price back up.
b.
True. When shareholders buy a particular stock, they receive cash from
the company in the form of future dividends. The rate of return that investors expect is
the expected dividend per share plus the expected price appreciation for the stock: r =
DIV + (P1 – P0) / P0.
Est. Time: 01-05
2.
Investors who buy stocks may get their return from capital gains as well as
dividends. But the future stock price always depends on subsequent dividends.
There is no inconsistency.
Est. Time: 01-05
3.
P0 = (5 + 110)/1.08 = $106.48.
Est. Time: 01-05
4.
r = 5/40 = .125, or 12.5%.
Est. Time: 01-05
5.
P0 = 10/(.08 − .05) = $333.33.
Est. Time: 01-05
6.
First we must determine the price based on dividends per share for years 1–4.
Then, we must account for the growth in earnings per share. With next year’s EPS
at $15 and EPS growing at 5% per year, the forecasted EPS at year 4 is $15 x
(1.05)4 = $18.23. Therefore, the forecasted price per share at year 4 is $18.23/.08
= $227.91. Therefore, the current price is:
4-1
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Chapter 04 - The Value of Common Stocks
Est. Time: 01-05
7.
Price = EPS1/r + PVGO. Recall that the price = DIV1/(r – g). Therefore, price =
$10/(.08 - .05) = $333.333. Therefore, 15/.08 + PVGO = 333.33; therefore PVGO =
$145.83.
Est. Time: 01-05
8.
With next year’s dividend at $10/share and next year’s price at $350/share
(calculated by taking the current year’s price of $333.33 x a 5% growth rate), Z’s
forecasted dividends and prices grow as follows:
Calculate the expected rates of return:
From year 0 to 1:
10  (350  333.33)
 .08
333.33
From year 1 to 2:
10.50  (367.50  350)
 .08
350
From year 2 to 3:
11.03  (385.88  367.50)
 .08
367.50
As shown, all three investors expect an 8% return over their respective one-, two.-,
and three-year investments.
4-2
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Chapter 04 - The Value of Common Stocks
Est. Time: 06-10
9.
a.
False. The value of a share equals the present value of the expected
future dividends per share. Earnings per share are not used to calculate share
price because part of earnings are used to reinvest in plant, equipment, and
working capital.
b.
True. The expected return is equal to the yearly dividend divided by the
share price. If the firm does not grow and all earnings are paid out as dividends,
then the expected return is also equal to the EPS/share price. Therefore, P0 =
DIV1/r = EPS1/r. We must still account for the present value of the growth
opportunities, however, so P0 = EPS1/r + PVGO.
Est. Time: 01-05
10.
PVGO = 0, and EPS1 equals the average future earnings the firm could generate
under no-growth policy.
Est. Time: 01-05
11.
Free cash flow is the amount of cash left over and available to pay out to investors
after all investments necessary for growth. In our simple examples, free cash flow
equals operating cash flow minus capital expenditure. Free cash flow can be
negative if investments are large.
Est. Time: 01-05
12.
The value at the end of a forecast period. Horizon value can be estimated
using the constant-growth DCF formula or by using price–earnings or market–
book ratios for similar companies.
Est. Time: 01-05
13.
If PVGO = 0 at the horizon date, H, horizon value = earnings forecasted for H +
1 divided by r.
Est. Time: 06-10
14.
Internet exercise; answers will vary.
Est. Time: 01-05
15.
Internet exercise; answers will vary.
Est. Time: 01-05
4-3
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Chapter 04 - The Value of Common Stocks
16. Internet exercise; answers will vary.
Est. Time: 06-10
17. Internet exercise; answers will vary.
Est. Time: 06-10
18.
PA 
DIV1
$10

 $100.00.
r
0.10
PB 
DIV1
$5

 $83.33.
r g
0.10  0 .04
PC 
DIV1 DIV 2 DIV3 DIV 4 DIV5 DIV 6  DIV7
1 







.
1
2
3
4
5
6
1.10 1.10 1.10 1.10 1.10 1.10  0.10 1.10 6 
PC 
5.00
6.00 7.20 8.64 10.37 12.44  12.44
1 







  $104.50.
1
2
3
4
5
6
1.10
1.10 1.10 1.10
1.10
1.10  0.10 1.10 6 
At a capitalization rate of 10%, Stock C is the most valuable.
For a capitalization rate of 7%, The results are:
PA = $142.86
PB = $166.67
PC = $156.48
Therefore, Stock B is the most valuable at a 7% capitalization rate.
Est. Time: 06-10
19.
DIV1
$1.35  1.0275
 $1.35 
 $21.90.
r g
0.095  0 .0275
a.
P0  DIV0 
b.
First, compute the real discount rate as follows:
(1 + rnominal) = (1 + rreal)  (1 + inflation rate)
1.095 = (1 + rreal)  1.0275
(1 + rreal) = (1.095/1.0275) – 1 = .0657 = 6.57%
In real terms, g = 0. Therefore:
P0  DIV0 
DIV1
$1.35
 $1.35 
 $21.90
rg
0.0657
Est. Time: 06-10
4-4
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Chapter 04 - The Value of Common Stocks
20.
a.
Plowback ratio = 1 – payout ratio = 1.0 – 0.5 = 0.5.
Dividend growth rate = g = plowback ratio × ROE = 0.5 × 0.14 = 0.07.
Next, compute EPS0 as follows:
ROE = EPS0 /Book equity per share
0.14 = EPS0 /$50  EPS0 = $7.00
Therefore: DIV0 = payout ratio × EPS0 = 0.5 × $7.00 = $3.50.
EPS and dividends for subsequent years are:
Year
0
1
2
3
4
5
EPS
$7.00
$7.00 × 1.07 = $7.4900
$7.00 × 1.072 = $8.0143
$7.00 × 1.073 = $8.5753
$7.00 × 1.074 = $9.1756
$7.00 × 1.074 × 1.023 = $9.3866
DIV
$7.00 × 0.5 = $3.50
$7.4900 × 0.5 = $3.50 × 1.07 = $3.7450
$8.0143 × 0.5 = $3.50 × 1.072 = $4.0072
$8.5753 × 0.5 = $3.50 × 1.073 = $4.2877
$9.1756 × 0.5 = $3.50 × 1.074 = $4.5878
$9.3866 × 0.5 = $3.50 × 1.074 × 1.023 = $4.6933
EPS and dividends for year 5 and subsequent years grow at 2.3% per
year, as indicated by the following calculation:
Dividend growth rate = g = plowback ratio × ROE = (1 – 0.8) × 0.115 = 0.023.
b.
P0 

DIV 3
DIV1
DIV 2
DIV 4
1 
 DIV 5






1
2
3
4
4
1.115
1.115 1.115 1.115
 0.115 1.115 
3.745
4.007 4.288 4.588 
4.693
1 





 $45.65
1
2
3
4
4 
1.115
1.115 1.115 1.115
 0.115 - 0.023 1.115 
The last term in the above calculation is dependent on the payout ratio
and the growth rate after year 4.
Est. Time: 11-15
21.
a.
An Incorrect Application. Hotshot Semiconductor’s earnings and
dividends have grown by 30% per year since the firm’s founding 10 years
ago. Current stock price is $100, and next year’s dividend is projected at
$1.25. Thus:
r
DIV1
1.25
g
 0 .30  0 .3125  31.25%
P0
100
This is wrong because the formula assumes perpetual growth; it is not
possible for Hotshot to grow at 30% per year forever.
4-5
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Chapter 04 - The Value of Common Stocks
A Correct Application. The formula might be correctly applied to the Old
Faithful Railroad, which has been growing at a steady 5% rate for
decades. Its EPS1 = $10, DIV1 = $5, and P0 = $100. Thus:
r
DIV1
5
g
 0 .05  0 .10  10.0%
P0
100
Even here, you should be careful not to blindly project past growth into the
future. If Old Faithful hauls coal, an energy crisis could turn it into a
growth stock.
b.
An Incorrect Application. Hotshot has current earnings of $5.00 per share.
Thus:
r
EPS1
5

 0 .05  5.0%
P0
100
This is too low to be realistic. The reason P0 is so high relative to earnings
is not that r is low, but rather that Hotshot is endowed with valuable growth
opportunities. Suppose PVGO = $60:
P0 
EPS1
 PVGO
r
5
100   60
r
Therefore, r = 12.5%.
A Correct Application. Unfortunately, Old Faithful has run out of valuable
growth opportunities. Since PVGO = 0:
P0 
EPS1
 PVGO
r
100 
10
0
r
Therefore, r = 10.0%.
Est. Time: 06-10
22.
Share price 
EPS1 NPV

r
rg
Therefore:
Ρα 
EPSα 1
rα

NPV
(rα  0.15)
4-6
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Chapter 04 - The Value of Common Stocks
Ρβ 
EPSβ1
rβ

NPVβ
(rβ  0.08)
The statement in the question implies the following:
NPVβ
(rβ  0.08)
 EPSβ1
NPVβ 

  NPVα

 r
(rβ  0.08)  (rα  0.15)
 β
 EPSα1
NPVα 



(rα  0.15) 
 rα
Rearranging, we have:
NPVβ
rβ
NPVα
r
 α 

(rα  0.15) EPS α1 (rβ  0.08) EPSβ1
a.
NPV < NPV; everything else equal.
b.
(r - 0.15) > (r - 0.08); everything else equal.
c.
NPVβ
NPVα

; everything else equal.
(rα  0.15) (rβ  0.08)
d.
rβ
rα

; everything else equal.
EPSα1 EPSβ1
Est. Time: 06-10
23.
a.
Growth-Tech’s stock price should be:
P
b.

$0.50 $0.60
$1.15  1
$1.24
  $23.81


 

2
3
3
(1.12) (1.12) (1.12)  (1.12) (0.12  0 .08) 
The horizon value P3 contributes:
PV(PH ) 
c.
1
1.24

 $22.07
3
(1.12) (0.12  0 .08)
Without PVGO, P3 would equal earnings for year 4 capitalized at 12%:
P3 = EPS3/r =
$2.49
 $20.75
0.12
(PH ) 
$1.24
 $31
(0.12  0 .08)
Therefore: PVGO = $31.00 – $20.75 = $10.25
4-7
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Chapter 04 - The Value of Common Stocks
d.
The PVGO of $10.25 is lost at year 3. Therefore, the current stock price
of $23.81 will decrease by:
$10.25
 $7.30
(1.12) 3
The new stock price will be: $23.81 – $7.30 = $16.51.
Est. Time: 11-15
24.
a.
Here we can apply the standard growing perpetuity formula with
DIV1 = $4, g = 0.04 and P0 = $100:
r
DIV1
$4
g
 0 .04  0 .08  8.0%
P0
$100
The $4 dividend is 60% of earnings. Thus:
EPS1 = 4/0.6 = $6.67
Also: P0 
EPS1
 PVGO
r
$100 
$6.67
 PVGO
0.08
PVGO = $16.63
b.
DIV1 will decrease to: 0.20  6.67 = $1.33.
However, by plowing back 80% of earnings, CSI will grow by 8% per year
for five years. Thus:
Year
DIVt
1
1.33
2
1.44
3
1.56
4
1.68
5
1.81
6
5.88
EPSt
6.67
7.20
7.78
8.40
9.07
9.80
7, 8 . . .
Continued
growth at
4%
Note that DIV6 increases sharply as the firm switches back to a 60%
payout policy in year 6, from a $1.81 dividend to a $5.88 dividend.
Forecasted stock price in year 5 is:
P5 
DIV6
5.88

 $147
r  g 0.08  0 .04
Therefore, CSI’s stock price will increase to:
P0 
1.33 1.44
1.56
1.68 1.81  147




 $106.22
2
3
1.08 1.08
1.08
1.08 4
1.08 5
Est. Time: 11-15
4-8
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Chapter 04 - The Value of Common Stocks
25.
a.
First, we use the following Excel spreadsheet to compute net income (or
dividends) for 2013 through 2017:
Production (million barrels)
Price of oil/barrel ($)
Costs per barrel ($)
2013
1.8000
65
25
2014
1.6740
60
25
Revenue
Expenses
Net Income (= Dividends)
117,000,000
45,000,000
72,000,000
100,440,000
41,850,000
58,590,000
2015
1.5568
55
25
85,625,100
38,920,500
46,704,600
2016
1.4478
50
25
72,392,130
36,196,065
36,196,065
2017
1.3465
52.5
25
70,690,915
33,662,340
37,028,574
Next, we compute the present value of the dividends to be paid in 2014,
2015, and 2016:
P0 
58,590,000 46,704,600 36,196,065


 $121,012,624
1.09
1.09 2
1.09 3
The present value of dividends to be paid in 2017 and subsequent years
can be computed by recognizing that both revenues and expenses can
be treated as growing perpetuities. Since production will decrease 7%
per year while costs per barrel remain constant, the growth rate of
expenses is: –7.0%.
To compute the growth rate of revenues, we use the fact that production
decreases 7% per year while the price of oil increases 5% per year, so
that the growth rate of revenues is:
[1.05 × (1 – 0.07)] – 1 = –0.0235 = –2.35%
Therefore, the present value (in 2016) of revenues beginning in 2017 is:
PV2012 
70,690,915
 $622,827,4 45
0.09 - (-0.0235)
Similarly, the present value (in 2016) of expenses beginning in 2017 is:
PV2012 
33,662,340
 $210,389,6 25
0.09 - (-0.07)
Subtracting these present values gives the present value (in 2016) of
net income, and then discounting back three years to 2013, we find
that the present value of dividends paid in 2017 and subsequent years
is: $318,477,671
The total value of the company is:
$121,012,624 + $318,477,671 = $439,490,295
Since there are 7,000,000 shares outstanding, the present value per
share is:
$439,490,295 / 7,000,000 = $62.78
4-9
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Chapter 04 - The Value of Common Stocks
b.
EPS2013 = $72,000,000/7,000,000 = $10.29.
EPS/P = $10.29/$62.78 = 0.164, or 16.4%.
This is greater than the cost of capital because production and earnings
are declining.
Est. Time: 15-20
26.
[Note: In this problem, the long-term growth rate, in year 9 and all later years,
should be 8%.]
The free cash flow for years 1 through 10 is computed in the following table:
Year
1
2
3
4
5
6
7
8
9
10
Asset value
10
11.2
12.54
14.05
15.31
16.69
18.19
19.29
20.44
21.67
Earnings
1.2
1.34
1.51
1.69
1.84
1.92
2.00
2.03
1.64
1.73
Investment
1.20
1.34
1.51
1.26
1.38
1.50
1.09
1.16
1.23
1.30
Free Cash Flow
0.00
0.00
0.00
0.43
0.46
0.42
0.91
0.87
0.41
0.43
Computing the present value of the free cash flows, following the approach in the
text, we find that the present value of the free cash flows occurring in years 1
through 7 is:
PV 
0
0
0
0.43 0.46 0.42
0.91






 $1.28
1
2
3
4
5
6
1.10
1.10 1.10 1.10 1.10 1.10
1.10 7
The present value of the growing perpetuity that begins in year 8 is:
 1

0.87
  $22.20
PV  

7
 (1.10) (0.10  0 .08) 
Therefore, the present value of the business is:
$1.28 + $22.20 = $23.48 million
Est. Time: 15-20
27. a. The constant growth formula P= DIV/(r − g) can be used here:
$200 = $8.5/(r-.075)
Solving for r, r = 0.1175. That is, investors expect an 11.75% rate of return.
4-10
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 04 - The Value of Common Stocks
b. Given the Mexican Motors is plowing back 50% of earnings after earning 12%
on equity, the firm can expect to grow at an annual rate of only 6% (ROE x
plowback = 0.12 x 0.50 = 0.06).
Applying the constant growth formula again and solving for r we get:
$200 = $8.5/(r-.06)
r = 0.1025. That is, investors should expect only a 10.25% rate of return.
Est. Time: 15-20
28.
a. With no further growth in the fifth year and beyond, the present value of the
$2.6 million in free cash flow will be worth $28.89 million (PV = $2.6/0.09) for
2022.
Computing the present value of the free cash flows in 2016 we find:
PV2016 
0
1
2
2.3
2.6
28.89





 $22.93 million
1
2
3
4
5
1.09
1.09 1.09 1.09 1.09
1.09 6
b. With 12 million shares outstanding Phoenix with sell for $1.91 per share
($22.93/12).
c. With net income of $1 million, or 0.0833 per share, and a price per share of
$1.91, Phoenix is selling at a PE ratio of 22.93.
d. To show the return is equal to 9% each, we calculate the present value of all
future cash flows for each ending period, including the reinvestment of free
cash flow.
At the end of 2017 Phoenix will be worth:
PV 
1
2
2.3
2.6
28.89




 $24.99 million
1
2
3
4
1.09
1.09 1.09 1.09
1.09 5
equal to a return of 9%, (24.99 – 22.93)/22.93 = 0.09.
At the end of 2018 Phoenix will be worth:
PV 1 
2
2.3
2.6
28.89



 $27.24 million
1
2
3
1.09
1.09 1.09
1.09 4
equal to a return of 9%, (27.24 – 24.99)/24.99 = 0.09.
4-11
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Chapter 04 - The Value of Common Stocks
At the end of 2019 Phoenix will be worth:
PV 1x(1.09)  2 
2.3
2.6
28.89


 $29.70 million
1
2
1.09
1.09
1.09 3
equal to a return of 9%, (29.70 – 27.24)/ 27.24 = 0.09.
At the end of 2020 Phoenix will be worth:
PV  1 x 1.09 2  2 x (1.09)  2.3 
2.6
28.89

 $32.37 million
1
1.09
1.09 2
equal to a return of 9%, (32.37 – 29.70)/ 29.70 = 0.09.
At the end of 2021 Phoenix will be worth:
PV  1 x 1.09 3  2 x 1.09 2  2.3 x 1.09  2.6 
28.89
 $35.28 million
1.09 1
equal to a return of 9%, (35.28 – 32.37)/ 32.37 = 0.09.
Est. Time: 15-20
29.
This question requires individual student self study and there is no set solution.
30.
a.
b. To work with real rates, the nominal cost of capital and growth rates given
need to be adjusted for inflation. For example, if the inflation rate was 1%,
the real rates are as follows:
Growth rate =
Cost of capital =
With these rates:
4-12
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 04 - The Value of Common Stocks
31.
From the equation given in the problem, it follows that:
P0
ROE  (1 b)
1 b


BVPS r  (b  ROE) (r / ROE)  b
Consider three cases:
ROE < r  (P0/BVPS) < 1
ROE = r  (P0/BVPS) = 1
ROE > r  (P0/BVPS) > 1
Thus, as ROE increases, the price-to-book ratio also increases, and, when
ROE = r, price-to-book equals one.
Est. Time: 06-10
32.
Assume the portfolio value given, $100 million is the value as of the end of the
first year. Then, assuming constant growth, the value of the contract is given by
the first payment (0.5% of portfolio value) divided by (r – g). Also:
r = dividend yield + growth rate
Hence:
r – growth rate = dividend yield = 0.05 = 5.0%
Thus, the value of the contract, V, is:
V
0.005  $100 million
 $10 million
0.05
For stocks with a 4% yield:
r – growth rate = dividend yield = 0.04 = 4.0%
Thus, the value of the contract, V, is:
V
0.005  $100 million
 $12.5 million
0.04
Est. Time: 11-15
33.
a. Under the new faster growth assumption, Table 4.7 is reproduced here.
($ millions)
1
2
Year
3
4
5
6
7
8
9
10
4-13
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 04 - The Value of Common Stocks
Asset value
Earnings
Net investment
Free cash flow
10.00 12.00 14.40 17.28 19.35 21.68 24.28 25.73 27.28 28.91
1.20 1.44 1.73 2.07 2.32 2.60 2.91 3.09 3.27 3.47
2.00 2.40 2.88 2.07 2.32 2.60 1.46 1.54 1.64 1.73
-0.80 -0.96 -1.15 0.00 0.00 0.00 1.46 1.54 1.64 1.73
Return on equity
Asset growth rate
Earnings growth
rate
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.12
0.12
0.12
0.11
0.12
0.11
0.06
0.08
0.06
0.08
0.06
0.20
0.20
0.20
0.12
0.12
0.12
0.06
0.06
0.06
The present value of the near-term cash flows is
PV 
- 0.80 - 0.96 - 1.15
0
0
0
1.46






 $2.39million
1
2
3
4
5
6
1.10
1.10
1.10 1.10 1.10 1.10
1.10 7
Horizon value (PV looking forward from period 6) =
PVH = 1.46/(0.10-0.06) = $36.42 million
Horizon value (discounted back to PV in period 0) =
36.42/1.16 = $20.56 million
PV(business) = PV(free cash flow) + PV(horizon value)
= -$2.39 + $20.56 = $18.17 million
The revised present value says the division’s PVGO have risen with the higher growth
rate in the near term.
b. Under the revised growth rates the company will need to issue new shares in the
amount of $2.39 million. The need to issue shares does not, however, change the
value of the company. The value is dependent only on the free cash flows. However, if
new shares are issued to fund the negative present value of the new cash flows, the
value of existing shares would be diluted.
Est. Time: 15-20
4-14
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 04 - The Value of Common Stocks
34. a.
1
10
2
1.5
0.5
0.20
0.15
Asset value start
Earnings
Net Investment
FCF
ROE
Asset growth rate
Earnings growth rate
2
11.5
2.3
1.725
0.575
0.20
0.15
0.15
3
13.225
2.645
1.984
0.661
0.20
0.15
0.15
4
15.209
3.042
2.281
0.760
0.20
0.15
0.15
5
17.490
3.498
2.624
0.875
0.20
0.15
0.15
6
20.114
2.816
2.011
0.805
0.14
0.10
-0.195
7
22.125
3.097
2.212
0.885
0.14
0.10
0.10
For the perpetuity that starts in year 6:
= 20.114
= R12.68 billion
PVGO =
b.
= R70 million
In a competitive market it is to be expected that the return on equity would
decline to the cost of capital as competitors come into the market and drive out
excess returns.
c.
1
2
3
10.00
11.50
13.23
15.21
17.49
19.24
21.16
Earnings
2.00
2.28
2.60
2.96
3.38
3.85
Net
investment
1.50
1.73
1.98
2.28
1.75
Free cash
flow
0.50
0.56
0.62
0.68
Earnings
growth
from
previous
period (%)
14
14
14
14
Asset
value
4
5
6
7
8
9
10
23.28
25.61
28.17
4.39
5.00
5.71
6.50
1.92
2.12
2.33
2.56
2.82
1.63
1.93
2.27
2.68
3.14
3.69
14
14
14
14
14
14
4-15
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 04 - The Value of Common Stocks
=
$27.60
PVGO =
4-16
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.