CHAPTER 5: Practice questions
Exercise 16: 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% a year forever.
c. Stock C is expected to pay a dividend of $5 next year. Thereafter, dividend growth is
expected to be 20% a year for 5 years (until year 6) and zero thereafter.
If the market capitalization rate for each stock is 10%, which stock is the most valuable?
a.
=
PA
b. =
PB
c. PC =
DIV1
$10
= = $100.00
0.10
r
DIV1
$5
=
= $83.33
r−g
0.10 − 0.04
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 percent, Stock C is the most valuable.
Exercise 21. See BMA for the question text.
Extremely high P/EPS ratios can be misleading for a number of reasons. In the case of
Textron, the extremely high P/EPS of 63 resulted from a large one-time loss which reduced
EPS below what it would otherwise have been, and (perhaps) below what investors
expected Textron’s EPS to be in the future.
A somewhat more common scenario resulting in an extremely high P/EPS ratio is a
growth stock which investors expect will experience significant increases in earnings in
the near term future. Mathematically, this is a result similar to Textron’s, but the cause
of the expected increase in future earnings and dividends is different.
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Exercise 24. See BMA for the question text.
Share price =
EPS1 NPV
+
r
r−g
Therefore:
Ρα =
EPS α 1
Ρβ =
EPSβ1
rα
rβ
+
NPVα
(rα − 0.15)
+
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
 EPS α1
NPVα 


+
(rα − 0.15) 
 rα
2
Rearranging, we have:
NPVβ
rβ
NPVα
r
× α <
×
(rα − 0.15) EPS α1 (rβ − 0.08) EPS β1
a.
NPVα < NPVβ, everything else equal.
b.
(rα - 0.15) > (rβ - 0.08), everything else equal.
c.
NPVβ
NPVα
, everything else equal.
<
(rα − 0.15) (rβ − 0.08)
d.
rβ
rα
, everything else equal.
<
EPS α1 EPSβ1
Exercise 26. Compost Science, Inc. (CSI), is in the business of converting Boston’s sewage sludge
into fertilizer. The business is not 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% book return on equity. At the end of the year CSI is expected to pay
$4 dividend. It has been reinvesting 40% of earnings and growing 4% 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% of its earnings for five years. Starting in year 6, however, it will again be able to pay out
60% of earnings. What will be CSI’s stock price once this announcement is made and its
consequences for CSI are known?
(a) To calculate the expected long-run rate of return we can apply the standard growing perpetuity
formula with DIV1 = $4, g = 0.04 and P0 = $100:
P0 =
⇔=
r
DIV1
(r − g)
DIV1
$4
+=
+ 0.04
= 0.08
= 8.0%
g
P0
$100
From the text we know that the dividends ($4) are 60% of earnings. So we can calculate the
earnings per share as:
3
DIV
EPS
DIV
4
⇔ EPS =
=
=$6,67
Payout ratio 0,6
Payout ratio =
An from here it is now easy to proceed with the estimation of the PVGOs:
EPS1
+ PVGO
r
6,67
100 =
+ PVGO
0,08
PVGO = $16,63
=
P0
(b) With this new project the dividends of the firm will be affected because the payout ratio
is now only 20%. DIV1 will thus decrease to: 0.20 × 6.67 = $1.33
However, by plowing back 80 percent of earnings, CSI will grow by 8 percent per year for
five years (because we hold constant the relationship between reinvestment and growth).
Thus:
Year
EPSt
DIVt
1
6.67
1.33
2
7.20
1.44
3
7.78
1.55
4
8.40
1.68
5
9.07
1.81
6
9.80
5.88
7, 8 . . .
Continued
growth at
4 percent
***Note that DIV6 increases sharply as the firm switches back to a 60% payout policy.
To forecast the stock price in year 5 we proceed as follows:
P5 =
DIV6
5.88
=
= $147.00
r − g 0.08 − 0 .04
Therefore, CSI’s stock price will increase to:
P0 =
1.33 1.44
1.55
1.68 1.81 + 147
+
+
+
+
= $106.21
2
3
1.08 1.08
1.08
1.08 4
1.08 5
Exercise 28. Permian Partners (PP) produces from aging oil fields in west Texas. Production is
1,8 million barrels per year in 2006, but production is declining at 7% per year for the foreseeable
future. Costs of production, transportation, and administration add up to $25 per barrel. The
average oil price was $65 per barrel in 2006.
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PP has 7 million shares outstanding. The cost of capital is 9%. All of PP’s net income is
distributed as dividends. For simplicity assume that the company will stay in business forever and
that costs per barrel are constant at $25. Also, ignore taxes.
(a) What is the PV of a PP share? Assume that oil prices are expected to fall to $60 per barrel in
2007, $55 per barrel in 2008, and $50 per barrel in 2009. After 2009, assume a long-term
trend of oil-price increase at 5% per year.
(b) What is PP’s EPS/P ratio and why is not equal to the 9% cost of capital?
a. First, we use the following Excel spreadsheet to compute net income (or dividends) for 2006
through 2010:
2006
1.8000
65
25
Production (million barrels)
Price of oil/barrel
Costs/barrel
Revenue
Expenses
Net Income (= Dividends)
117,000,000
45,000,000
72,000,000
2007
1.6740
60
25
100,440,000
41,850,000
58,590,000
2008
1.5568
55
25
2009
1.4478
50
25
2010
1.3465
52.5
25
85,625,100
38,920,500
46,704,600
72,392,130
36,196,065
36,196,065
70,690,915
33,662,340
37,028,574
Next, we compute the present value of the dividends to be paid in 2007, 2008 and 2009:
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 2010 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 2009) of revenues beginning in 2010 is:
PV2009 =
70,690,915
= $622,827,4 45
0.09 - (-0.0235)
Similarly, the present value (in 2009) of expenses beginning in 2010 is:
PV2009 =
33,662,340
= $210,389,6 25
0.09 - (-0.07)
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Subtracting these present values gives the present value (in 2009) of net income, and then
discounting back three years to 2006, we find that the present value of dividends paid
in 2010 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
b.
EPS2006 = $72,000,000/7,000,000 = $10.29
EPS/P = $10.29/$62.78 = 0.164
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