Dr. Sudhakar Raju
FN 3000
ANSWERS TO CHAPTER 7 (EQUITY MARKETS & STOCK VALUATION)
1.
P0
=
D1
rg
=
D0 (1  g )
rg
1
=
$2.50(1  .05 )
.11  .05
1
=
$2.625
= $43.75
.06
Price in three years?
P3 =
D4
rg
=
D0 (1  g )
rg
4
=
$2.50(1  .05)
.11  .05
4
= $50.65
Note that you can also solve this as:
P3 = P0 (1 + g)3
= $43.75 (1 + .05)3
= $50.65
Price in 15 years?
P15 = P0 (1 + g)15
= $43.75 (1 + .05)15
= $90.95
2.
We need to find the required return of the stock. Using the constant growth model,
solve for r. Thus:
r = (D1 / P0) + g
r = ($1.80 / $47.00) + .065
r = .1033 or 10.33%
3.
The dividend yield is the dividend next year divided by the current price, so the
dividend yield is:
Dividend yield = D1 / P0
Dividend yield = $1.80 / $47.00
Dividend yield = .0383 or 3.83%
The capital gains yield, or percentage increase in the stock price, is the same as the
dividend growth rate (g). Thus:
Capital gains yield = 6.5%
1
4.
Using the constant growth model, we find the price of the stock today is:
P0 = D1 / (r – g)
P0 = $4.50 / (.12 – .04)
P0 = $56.25
5.
The required return of a stock is made up of two parts – the dividend yield and the
capital gains yield. Thus, the required return of this stock is:
r = Dividend yield + Capital gains yield
r = .0410 + .06
r = .1010 or 10.10%
6.
We know the stock has a required return of 13%, and the dividend and capital gains
yield are equal, so:
Dividend yield = (½) x 13% = 6.50%
Dividend yield = 6.50% = Capital gains yield
The current stock price is $60. Given that the dividend yield is 6.50% this implies
that D1 is given by:
.0650 = [D1 / P0]
.0650 = [D1 / $60]
D1 = .0650($60)
D1 = $3.90
This is the dividend next year. The dividend this year is then given by:
D1 = D0(1 + g)
$3.90 = D0 (1 + .0650)
D0 = $3.90 / 1.065
D0 = $3.66
7.
The price of any financial instrument is the present value of the future cash flows.
The future dividends of this stock are an annuity for eight years, so the price of the
stock is the present value of an annuity, which will be:
1 Shift P/ Yr
15 PMT
8N
11 I/Yr
PV = $77.19. Thus, current share price is $77.19.
2
8.
The price of a share of preferred stock is the fixed dividend payment divided by the
required return or discount rate (r). Thus:
FixedPMT
r
$7
$90.21 =
r
r = 7.76%
PPERP =
9.
Use the constant growth model to solve the equation for g. Thus:
P0 = [D1/ (r-g)]
g = r – (D1 / P0)
g = .12 – ($4.25 / $70)
g = .0593 or 5.93%
10. The price of a preferred stock is given by:
PPS
= [DPS]
/ [r]
= [$20 / .09]
= $222.22
Note that $222.22 is the price of the preferred stock in Year 19. To compute the current
stock price, discount $222.22 to the present thus:
1 Shift P/Yr
222.22 FV
19 N
9 I/Yr
PV = $43.22
11. Using the constant growth model and a required return of 15%, the stock price today
is:
P0 = D1 / (r – g)
P0 = $3.75 / (.15 – .05)
P0 = $37.50
The stock price today with a 10% return will be:
P0 = D1 / (r – g)
P0 = $3.75 / (.10 – .05)
3
P0 = $75.00
All else held constant, a higher required return implies that the stock will have a
lower price.
12. This stock pays no dividends for seven years. Once the stock begins paying
dividends, it will have a constant growth rate of dividends. The price of the stock in
Year 6 is given by:
P6 =
D7
rg
=
$7
.13  .05
= $87.50
The price of the stock today is simply the PV of the stock price in the future. Thus,
discount $87.50 back to the present as follows:
1 Shift P/Yr
87.50 FV
6N
13 I/Yr
PV = $42.03
13. The pattern of cash flows is as follows:
Yr 0 = $12
Yr 1 = $12 + $5 = $17
Yr 2 = $22
Yr 3 = $27
Yr 4 = $32
The stock price is the PV of future cash flows. Thus, ignore the cash flow of $12 in Year
0. The PV of the cash flows from Yr 1- 4 is given by:
1 Shift P/Yr
0 CFj
17 CFj
22 CFj
27 CFj
32 CFj
12 I/Yr
Shift NPV = $72.27
4
14. The pattern of CF’s is as follows:
YR
1
2
3
4
5
6
7
CFs
$9
$15
$17
$3
$3 x 1.05 = $3.15
$3.15 x 1.05 = $3.31
….
The constant growth phase begins after year 4. Using the constant growth model, one
can figure out the price at year 4 thus:
P4 = D5 / r-g = [$3.15] / [.11 - .05] = 3.15/.06 = $52.50
Imagine that you sell the stock at the end of year 4 after receiving the dividend of $3.
The price that you would receive for the stock is $52.50. The total cash flows from
holding the stock is then given by:
YR
1
2
3
4
CFs
$9
$15
$17
$3 + $52.50 = $55.50
The current stock price can be computed thus:
1 Shift P/Yr
0 CFj
9 CFj
15 CFj
17 CFj
55.50 CFj
11 I/Yr
Shift NPV = $69.27
Thus, the intrinsic price of the stock today should be about $69.27.
15. The pattern of CF’s is as follows:
YR
0
1
2
3
4
CFs
$2.90
$2.90 x 1.20 = $3.48
$3.48 x 1.20 = $4.18
$4.18 x 1.20 = $5.01
$5.01 x 1.06 = $5.31
5
5
6
7
$5.31 x 1.06 = $5.63
…….
….
The constant growth phase begins after year 3. Using the constant growth model, one
can figure out the price at year 3 thus:
P3 = D4 / r-g = [$5.31] / [.14 - .06] = $5.31 /.08 = $66.38
Imagine that you sell the stock at the end of year 3 after receiving the dividend of
$5.01. The price that you would receive for the stock is $66.38. The total cash flows
from holding the stock are then given by:
YR
1
2
3
CFs
$3.48
$4.18
$5.01 + $66.38 = $71.39
The current stock price can be computed thus:
1 Shift P/Yr
0 CFj
3.48 CFj
4.18 CFj
71.39 CFj
14 I/Yr
Shift NPV = $54.46
Thus, the intrinsic price of the stock today should be about $54.46.
16. The constant growth model can be applied even if the dividends are declining by a
constant percentage (negative growth). The price of the stock today will be:
P0 = [D0 (1 + g)] / [(r – g)]
P0 = [$9(1 – .07)] / [(.10 – (–.07)]
P0 = $49.24
17.
P0
$84 =
=
D0 (1  g )
rg
D0 (1  .06)
.13  .06
1
1
D0 = [($84) (.07)] / [1.06] = $5.55
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18 a.The total return (total yield / required return) on a stock is given by:
D1
+g
P0
where r = total yield, (D1/P0) is the dividend yield, g is the capital gains yield.
r=
The total yield on a stock is made up of two components – dividend yield and capital
gains yield. All stocks – W, X, Y and Z – have a total yield of 18%. To find the
components of the total return, find the stock price for each stock. Using this stock price
and the dividend, calculate the dividend yield thus:
Stock W: P0 = [D0 (1 + g) / (r – g)]
P0 = [$3.25 (1.10) / (.18 – .10)]
P0 = $44.69
Dividend yield = D1/P0
Dividend yield = [$3.25(1.10)] / [$44.69]
Dividend yield = .08 or 8%
Capital gains yield = Total Yield – Dividend yield
Capital gains yield = .18 – .08
Capital gains yield = .10 or 10%
Stock X: P0 = D0 (1 + g) / (r – g)
P0 = $3.25/(.18 – .00)
P0 = $18.06
Dividend yield = D1/P0
Dividend yield = $3.25/$18.06
Dividend yield = .18 or 18%
Capital gains yield = Total return – Dividend yield
Capital gains yield = .18 – .18
Capital gains yield = .00 or 0%
Stock Y: P0 = D0 (1 + g) / (r – g)
P0 = [$3.25(1 – .05)] / [.18 – (–.05)]
P0 = $13.42
Dividend yield = D1/P0
Dividend yield = $3.25(.95)/$13.42
Dividend yield = .23 or 23%
Capital gains yield = Total return – Dividend yield
Capital gains yield = .18 – .23
Capital gains yield = –.05 or –5%
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Stock Z: The cash flows for Stock Z are as follows:
YR
0
1
2
3
4
5
6
7
CFs
$3.25
$3.25 x 1.20 = $3.90
$3.90 x 1.20 = $4.68
$4.68 x 1.12 = $5.24
$5.24 x 1.12 = $5.87
…….
…….
…….
The constant growth phase begins after year 2. Using the constant growth model, one
can figure out the price at year 2 thus:
P2 = D3 / r-g = [$5.24] / [.18 - .12] = $5.24 /.06 = $87.33
Imagine that you sell the stock at the end of year 2 after receiving the dividend of
$4.58. The price that you would receive for the stock is $87.33. The total cash flows
from holding the stock are then given by:
YR
1
2
CFs
$3.90
$4.68 + $87.33 = $92.01
The current stock price can be computed thus:
1 Shift P/Yr
0 CFj
3.90 CFj
92.01 CFj
18 I/Yr
Shift NPV = $69.39
Thus, the intrinsic price of the stock today should be about $69.39. The dividend
yield is then given by:
Dividend yield = D1/P0
Dividend yield = [$3.90 / $69.39]
Dividend yield = .0562 or 5.62%
Capital gains yield = Total return – Dividend yield
Capital gains yield = .18 – .0562
Capital gains yield = .1238 or 12.38%
18 b. In all cases, the total yield (required return) is 18%, but the yield is distributed
differently between current income and capital gains. High-growth stocks have an
appreciable capital gains component but a relatively small current income yield;
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conversely, mature, negative-growth stocks provide a high current income but also
price depreciation over time.
19. The highest dividend yield will occur when the stock price is the lowest. So, using the 52week low stock price, the highest dividend yield was:
Dividend yield = Dividend / PLow
Dividend yield = $1.12 /$54.64
Dividend yield = .0205 or 2.05%
The lowest dividend yield occurred when the stock price was the highest, so:
Dividend yield = D/PHigh
Dividend yield = $1.12/$80.25
Dividend yield = .0140 or 1.40%
20. Note that the current dividend for IBM from the information provided in the stock
tables is $.72. The pattern of CF’s is then as follows:
YR
0
1
2
3
4
5
6
7
CFs
$.72
$.72 x 1.1350 = $.8172
$.8172 x 1.1350 = $.9275
$.9275 x 1.1350 = $1.0527
$1.0527 x 1.1350 = $1.1949
$1.1949 x 1.1350 = $1.3562
$1.3562 x 1.05= $1.4240
(Note that g in year 6 is 5%)
….
The constant growth phase begins after year 5. Using the constant growth model, one
can figure out the price at year 5 thus:
P5 = D6 / r-g = [$1.4240] / [.11 - .05] = $23.73
Imagine that you sell the stock at the end of year 5 after receiving the dividend of
$1.3562. The price that you would receive for the stock is $23.73. The total cash
flows from holding the stock are then given by:
YR
1
2
3
4
5
CFs
$.8172
$.9275
$1.0527
$1.1949
$1.3562 + $23.73 = $25.09
The current stock price can be computed thus:
9
1 Shift P/Yr
0 CFj
.8172 CFj
.9275 CFj
1.0527 CFj
1.1949 CFj
25.09 CFj
11 I/Yr
Shift NPV = $17.94
Thus, the intrinsic price of the stock today should be about $17.94. Given that the
actual closing price of the stock (from the stock market table) is $93.10, the stock
seems to be substantially overvalued. In fact, the stock is trading at a price five times
as large as the intrinsic price.
The factors that would affect the stock price are the dividend growth rate, both the
supernormal growth rate of 13.50% and the long-term growth rate of 5%, the length
of the supernormal growth, and the required return.
21. Using the constant growth model, solve for r. Note that the current dividend for
Duke Energy from the stock market table is $1.10. Thus D1 = D0 (1 + g) = $1.10
(1.0250) = $1.13. Duke Energy’s closing stock price from the stock market table is
$25.33.
r = (D1 / P0) + g
r = [$1.13/ $25.33] + .0250
r = 0.0695 or 6.95%
The required return depends on the company and the industry. Since Duke Energy is a
regulated utility company, there is little room for growth. This is the reason for the relatively
high dividend yield. Since the company has little reason to keep retained earnings for new
projects, a majority of net income is paid to shareholders in the form of dividends.
22. Using the constant growth model, solve for r. Note that D1 = D0 (1 + g) = $.50 (1 .0950) = $.4525. J.C. Penney’s closing stock price from the stock market table is $42.63.
Thus:
r = (D1 / P0) + g
r = [$.4525 / $42.63] + (–.095)
r = –0.844 or –8.44%
This number CANNOT be correct since it implies that investor’s require a return of about
negative 8%. Since the required return can never be negative, JC Penney investors must
believe that the dividend growth rate over the past 10 years is not indicative of future growth
in dividends.
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For JC Penney, store sales had fallen in recent years, while at the same time industry store
sales had increased. Additionally, JC Penney previously owned its own credit subsidiary that
had lost money in recent years. The company also experienced increased competition from
Wal-Mart, among others.
23. The annual dividend paid to stockholders is $0.28, and the dividend yield is .90%.
(Note that .90% is not even 1%. In fact, .90% is 90% of 1% or .0090). Thus:
Dividend yield = [Dividend / Stock Price]
.0090 = $0.28 / P0
P0 = $0.28/.009
P0 = $31.11
The dividend yield quoted in the stock market table is rounded. This means the price
calculated using the dividend will be slightly different from the actual price. Note
that D1 = D0 (1 + g) = $.28 (1 + .02) = $.2856. The required return is then given by:
r = (D1 / P0) + g
r = [$0.2856 / $32.05] + .02
r = 0.0289 or 2.89%
It should be noted that the return of 2.89% is extraordinarily low given that Tootsie Roll is a
risky stock. A total yield of 2.89% is lower than the yield on risk -free, 3 month T-bills. The
possible reason is that the constant dividend growth model is inappropriate for Tootsie Roll.
Tootsie Roll’s dividends may be following some other pattern not captured by the constant
growth model.
11