A B C D E F G H

1 4/11/2010
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3 Chapter 7 Tool Kit for Stock Valuation
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5 VALUING COMMON STOCKS (Section 7.5)
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7 Stocks can be evaluated in two ways: (1) find the stock price directly by calculating the present value of the
8 expected future dividends, or (2) find the stock price indirectly by first calculating the value of the entire
9 corporation, which is the the present value of the firm's expected future free cash flows, and then subtracting
10 the value of the debt and preferred stock to find the total value of the common equity. Only the first approach
11 (the dividend model) is discussed in this chapter. The second approach (the corporate valuation model) is
described in Chapter 13.
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13
14 THE DISCOUNTED DIVIDEND APPROACH
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16
17 The value of any financial asset is the present value of the future cash flows provided by the asset. When an
18 investor buys a share of stock, he or she typically expects to receive cash in the form of dividends and then,
eventually, to sell the stock and to receive cash from the sale. However, the price the first investor receives is
19
dependent upon the dividends the next investor expects to earn, and so on for different generations of
20 investors. Thus, the stock's value ultimately depends on the cash dividends the company is expected to
21 provide and the discount rate used to find the present value of those dividends.
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23
24 Here is the basic dividend valuation equation:
25
26 D1 + D2 + . . . . DN
P0 =
27 ( 1 + rs ) ( 1 + rs ) 2 ( 1 + rs ) N
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29
30 The dividend stream theoretically extends on out forever, i.e., to n = infinity. Obviously, it would not be feasible
to deal with an infinite stream of dividends, but fortunately, a relatively simple equation has been developed
31 that can be used to find the PV of the dividend stream, provided it is growing at a constant rate.
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33
34 VALUING A CONSTANT GROWTH STOCK (Section 7.6)
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36
In the constant growth model, we assume that the dividend and stock will grow forever at a constant growth
37 rate. Naturally, assuming a constant growth rate for the rest of eternity is a rather bold assumption. However,
38 considering the implications of imperfect information, information asymmetry, and general uncertainty, the
39 assumption of constant growth is often reasonable. It is reasonable to guess that a given stock will experience
40 ups and downs throughout its life. By assuming constant growth, we are trying to find the average of the good
41 times and the bad times, and we assume that we will see both scenarios over the firm's life. In addition to a
42 constant growth rate, we also need the estimated long-term required return for the stock, and it too must be
43 constant. If these variables are constant, our price equation for common stock simplifies to the following
expression:
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 A B C D E F G H
45
46 D1
P0 =
47 ( rs – g )
48
49 In this equation, the long-run growth rate (g) can be approximated by multiplying the firm's return on assets by
50 the retention ratio. Generally speaking, the long-run growth rate of a firm is likely to fall between 5% and 8% a
51 year.
52
53 EXAMPLE: CONSTANT GROWTH
54 A firm just paid a $1.15 dividend and its dividend is expected to grow at a constant rate of 8%. What is its stock
55 price, assuming it has a required return of 13.4%?
56
57 D0 = $1.15
58 g= 8%
59 rs = 13.4%
60
61 D1 D0 (1 + g) $1.2420
P0 = = =
62 ( rs – g ) ( rs – g ) 0.0540
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64 P0 = $23.00
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66
67 How sensitive is the stock's price to changes in the dividend, the growth rate, and rs? We can construct a
68 series of data tables and a graph to examine this question.
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70
71 Resulting
72 % Change Last Price Changes in Dividends, Req's Return, and Growth: Effect on Stock Price
73 in D0 Dividend, D0 $23.00 Stock Price
74 -30% $0.81 $12
75 -15% $0.98
76 0% $1.15 $10
77 15% $1.32
78 30% $1.50
79 $8

80 % Change Req'd Return $23.00

81 -30% 9.38% $6
82 -15% 11.39% Growth Rate
83 0% 13.40%
84 15% 15.41% $4

85 30% 17.42% Dividend

86 $2
87 % Change Growth Rate $23.00 Required Return
88 -30% 5.60%
$0
89 -15% 6.80%
-30% -20% -10% 0% 10% 20% 30%
90 0% 8.00%
91 15% 9.20% Percent Change from Base

92 30% 10.40%

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94
95 This chart shows that the stock price has a positive relationship with the dividend and the growth rate, and a
96 negative relationship with the required return. Furthermore, we see that the dividend has a linear relationship
97 with price, while the growth rate seems to have a quadratic relationship. The relationship between required
98 return and stock price is not only negative, but it is a quadratic relationship with greater convexity than the
growth rate. This indicates that the required return is the factor that more directly influences the stock price. In
99 other words, required return is the value driver in this valuation technique. However, the final effects also
100 depend on the amount of change in each of the three variables. If the required return and dividend are
101 expected to be stable, but the dividend growth rate is expected to change significantly, then the growth rate will
102 be the primary determinant of the stock price.
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105 DO STOCK PRICES REFLECT LONG-TERM OR SHORT-TERM CASH FLOWS?
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107 Managers often claim that stock prices are "short-term" in nature in the sense that they reflect what is
108 happening in the near-term and ignore the long-term. We can use the results for the constant growth model to
109 shed light on this claim.
110
111
The first step is to forecast the dividends for the next 5 years. Then we find the present value of these
112
dividends and compare that PV with the current stock price, which reflects the PV of all future dividends.
113
114
115 P0 = $23.00
116 D0 = $1.15
117 g= 8%
118 rs = 13.4%
119
120 Year 0 1 2 3 4 5
121 Dividend $1.15 $1.24 $1.34 $1.45 $1.56 $1.69
122
123 PV of dividends in Years 1 through 5 = $4.98
124 Current stock price = $23.00
125 Percent of current stock price due to
126 dividends in Years 1 through 5 = 21.6%
127 Percent of current stock price due to
128 dividends beyond Year 5 = 78.4%
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130 For most stock, the percentage of the current price that is due to long-term cash flows is over 80%.
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132 EXPECTED RATE OF RETURN ON A CONSTANT GROWTH STOCK (Section 7.7)
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134 Using the constant growth equation introduced earlier, we can re-work the equation to solve for rs. In doing so,
135 we are now solving for an expected return. The expression we are left is:
136
137 D1
rs = + g
138 P0
139
140 This expression tells us that the expected return on a stock comprises two components. First, it consists of the
141 expected dividend yield, which is simply the next expected dividend divided by the current price. The second
142 component of the expected return is the expected capital gains yield. The expected capital gains yield is the
143 expected annual price appreciation of the stock, and is given by g. This shows us the dual role of g in the
144 constant growth rate model. Not only does g indicate expected dividend growth, but it is also the expected
145 stock price growth rate.
146
147 EXAMPLE: EXPECTED RATE OF RETURN ON A CONSTANT GROWTH STOCK
148 You buy a stock for $23, and you expect the next annual dividend to be $1.242. Furthermore, you expect the
149 dividend to grow at a constant rate of 8%. What is the expected rate of return on the stock, and what is the
150 dividend yield of the stock?
151
152 P0 $23.00
153 D1 $1.242
154 g 8%
155
156 rs 13.40%
157 Dividend Yield + Capital Gains Yield = Expected Rate of Return
158 Dividend yield 5.40% 5.40% + 8% = 13.40%
159
160 EXAMPLE: EXPECTED PRICE IN THE FUTURE
161 What is the expected price of this stock in five years?
162
163 N = 5
164 Using the growth rate we find that:
165
166 P5 = $33.79 =B152*(1+B154)^B163
167
168
169 VALUING NONCONSTANT GROWTH STOCKS (Section 7.8)
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171
172 For many companies, it is unreasonable to assume that they grow at a constant growth rate. Hence, valuation
173 for these companies proves a little more complicated. The valuation process, in this case, requires us to
estimate the short-run nonconstant growth rate and predict future dividends. Then, we must estimate a
174
constant long-term growth rate at which the firm is expected to grow. Generally, we assume that after a certain
175 point of time, all firms begin to grow at a rather constant rate. Of course, the difficulty in this framework is
176 estimating the short-term growth rate, how long the short-term growth will hold, and the long-term growth rate.
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179 Specifically, we will predict as many future dividends as we can and discount them back to the present. Then
180 we will treat all dividends to be received after the convention of constant growth rate with the Gordon constant
181 growth model described above. The point in time when the dividend begins to grow constantly is called the
182 horizon date. When we calculate the constant growth dividends, we solve for a terminal value (or a continuing
183 value) as of the horizon date. The terminal value can be summarized as:
184
185 DN+1 DN (1 + g)
TVN = PN = =
186 ( rs – g ) ( rs – g )
187
188 This condition holds true, where N is the terminal date. The terminal value can be described as the expected
189 value of the firm in the time period corresponding to the horizon date.
190
191 EXAMPLE: NONCONSTANT GROWTH
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193 A company's stock just paid a $1.15 dividend, which is expected to grow at 30% for the next three years. After
194 three years the dividend is expected to grow constantly at 8% forever. The stock's required return is 13.4%,
195 what is the price of the stock today?
196
197 D0 = $1.15
198 rs = 13.4%
199 gs = 30% Short-run g; for Years 1-3 only.
200 gL = 8% Long-run gL; for all years after Year 3.
201
202 Growth rate 30% 30% 30% 8% 8%
203 Year 0 1 2 3 4
204 Dividends $1.15 $1.4950 $1.9435 $2.5266 $2.7287
205
206 PV of dividends discounted at rs
207 $1.3183
208 1.5113
209 1.7326 $2.7287
210 $4.5622 = PV of nonconstant dividends $50.5310 = Horizon value = ──────
211 $34.6512 = PV of horizon value 5.4%
212 $39.2135 = P0 rs – gL
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214
215 PREFERRED STOCK (Section 7.11)
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217 Consider an issue of preferred stock that pays a $10 dividend and has a required return of 10%. What is the
218 price of this preferred stock?
219
220 Vps = Dps ÷ rps
221 = $10.00 ÷ 10.00%
222 = $100.00
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225 Some preferred stock has a maturity date. Consider a firm whose preferred stock matures in 50 years, pays a
$10 annual dividend, has a par value of $100, and has a required return of 8%. What is the price of this
226 preferred stock?
227
228 Years to Maturity (N): 50
229 Annual Dividend (PMT): $10
230 Par value (FV): $100
231 Required return, rd (I/YR): 8%
232
233 Vps = $124.47
234
235
236 STOCK MARKET EQUILIBRIUM (Section 7.12)
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238 Changes in Equilibrium Stock Prices
239 Small changes in the market's expectations can cause large changes in stock price!
240
241 Original New
242 Risk-free rate, rRF 8% 7%
243 Market risk premium, rM – rRF 4% 3%
244 Stock i’s beta coefficient, b i 2 1
245 ri 16.00% 10.00%
246 Stock i’s expected growth rate, gi 5% 6%
247 D 0 $2.8571 $2.8571
248 Price of Stock i $27.27 $75.71
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nt value of7 the
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urn on assets by
en 5% and 50 8% a
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What is54 its stock
55
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62
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64
65
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construct 67 a
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Effect on Stock
72 Price
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Dividend 85
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93
94
wth rate,95
and a
ear relationship
96
ween required
97
exity than98the
he stock price. In
99
effects also
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he growth101rate will
102
103
104
105
106
what is107
growth 108
model to
109
110
111
of these
112
dividends.
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
80%. 130
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133
or rs. In 134
doing so,
135
136
137
138
139
140 of the
, it consists
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the
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144
145
146
147
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150
151
152
153
154
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156
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158
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167
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171
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172
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174
hat after a certain
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176 rate.
erm growth
177
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179
the present. Then
e Gordon 180
constant
tantly is181
called the
ue (or a continuing
182
be summarized
183 as:
184
185
186
187
188
s the expected
189
190
191
192
193After
hree years.
194
turn is 13.4%,
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%. What 217
is the
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50 years, pays a
ce of this
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SECTION 7.5
SOLUTIONS TO SELF-TEST

If D1 = $3.00, P0 = $50, and the expected P at t=1 is equal to $52, what are the stock’s expected dividend yield,
capital gains yield, and total return for the coming year?

D1 $3.00
P0 $50.00
Expected P1 $52.00

Exp. dividend yield 6.0% =B6/B7

Exp. capital gains yield 4.0% =(B8-B7)/B7
Exp. total return 10.0% =C10+C11
d dividend yield,
SECTION 7.6
SOLUTIONS TO SELF-TEST

A stock is expected to pay a dividend of $2 at the end of the year. The required rate of return is r s = 12%. What would
stock’s price be if the growth rate were 4%?

D1 $2.00
g 4%
rs 12%

Stock price $25.00

A stock is expected to pay a dividend of $2 at the end of the year. The required rate of return is r s = 12%. What would
stock’s price be if the growth rate were 0%?

D1 $2.00
g 0%
rs 12%

Stock price $16.67

n is r s = 12%. What would the

n is r s = 12%. What would the

SECTION 7.7
SOLUTIONS TO SELF-TEST

If D0 = $4.00, rs = 9%, and g = 5% for a constant growth stock, what are the stock’s expected
dividend yield and capital gains yield for the coming year?

D0 $4.00
g 5%
rs 9%

Expected D1 $4.20

Stock price $105.00

Expected dividend yield 4.00%

Expected capital gains yield 5.00%

Alternatively, you know that the capital gains yield is equal to the growth rate.

Expected capital gains yield = growth rate = 5.00%

Because the total return is rs, the dividend yield is rs minus the capital gains yield:

Expected dividend yield 4.00%

SECTION 7.8
SOLUTIONS TO SELF-TEST

Suppose D0 = $5.00 and rs = 10%. The expected growth rate from Year 0 to Year 1
(g0 to 1) = 20%, the expected growth rate from Year 1 to Year 2 (g1 to 2) = 10%, and the constant rate
beyond Year 2 is gn = 5%. What are the expected dividends for Year 1 and Year 2? What is the
expected horizon value price at Year 2? What is the expected P0?

D0 $5.00
g0 to 1 20%
g1 to 2 10%
gn 5%
rs 10%
Year
1 2
D1 D2
Expected dividends $6.00 $6.60

Expected P2 $138.60

PV of expected dividends $10.91

PV of expected P2 $114.55

Expected P0 $125.45
SECTION 7.11
SOLUTIONS TO SELF-TEST

A preferred stock has an annual dividend of $5. The required return is 8%. What is the V ps?

Dps $5.00
rps 8%

Vps $62.50
?