Bond Value and Required Return Chart - Excel by tfe14389

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1   09model                                                             1/7/2011 20:15                                               5/8/2000
2
3                                               Chapter 9. Model for Valuing Common Stock
4
5
6   This model is similar to the bond valuation models developed in Chapter 8 in that we employ discounted cash flow
7   analysis to find the value of a firm's stock.
8
9   Stocks can be evaluated in two ways: (1) by finding the present value of the expected future dividends, or (2) by finding
10   the present value of the firm's expected future operating income, subtracting the value of the debt and preferred stock
11   to find the total value of the common equity, and then dividing that total value by the number of shares outstanding to
12   find the value per share. Both approaches are examined in this spreadsheet.
13
14
15   THE DISCOUNTED DIVIDEND APPROACH
16
17   The value of any financial asset is equal to the present value of 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,
19   to sell the stock and to receive cash from the sale. Moreover, the price any investor receives is dependent upon the
20   dividends the next investor expects to earn, and so on for different generations of investors. Thus, the stock's value
21   ultimately depends on the cash dividends the company is expected to provide and the discount rate used to find the
22   present value of those dividends.
23
24   Here is the basic dividend valuation equation:
25
26                              D1               +             D2              +          . . . .       Dn
P0 =                                                      2
27                           (1+ks)                        (1+ks)                                   ( 1 + k s )n
28
29   The dividend stream theoretically extends on out forever, i.e., n = infinity. Obviously, it would not be feasible to deal
30   with an infinite stream of dividends, but fortunately, an equation has been developed that can be used to find the PV of
31   the dividend stream, provided it is growing at a constant rate.
32
33   Naturally, trying to estimate an infinite series of dividends and interest rates forever would be a tremendously difficult
34   task. Now, we are charged with the purpose of finding a valuation model that is easier to predict and construct. That
35   simplification comes in the form of valuing stocks on the premise that they have a constant growth rate.
36
37
38   VALUING STOCKS WITH A CONSTANT GROWTH RATE
39
40   In this stock valuation model, we first assume that the dividend and stock will grow forever at a constant growth rate.
41   Naturally, assuming a constant growth rate for the rest of eternity is a rather bold statement. However, considering the
42   implications of imperfect information, information asymmetry, and general uncertainty, perhaps our assumption of
43   constant growth is reasonable. It is reasonable to guess that a given will experience ups and downs throughout its life.
44   By assuming constant growth, we are trying to find the average of the good times and the bad times, and we assume that
45   we will see both scenarios over the firm's life. In addition to assuming a constant growth rate, we will be estimating a
46   long-term required return for the stock. By assuming these variables are constant, our price equation for common
47   stock simplifies to the following expression:
48
49                             D1
P0 =
50                           (ks-g)
51
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52    In this equation, the long-run growth rate (g) can be approximated by multiplying the firm's return on assets by the
53    retention ratio. Generally speaking, the long-run growth rate of a firm is likely to fall between 5 and 8 percent a year.
54
55    PROBLEM
56    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 price,
57    assuming it has a required return of 13.4%?
58
59                              D0             \$1.15
60                              g               8%
61                              ks             13.4%
62
63                                              D1                                   D 0 (1+g)                    \$1.24
P0 =                                      =                            =
64                                            (ks-g)                                 (ks-g)                        0.05
65
66                              P 0=           \$23.00
67
68
69    How sensitive is the stock price to changes in the dividend, the growth rate, and ks?
70    We can construct a series of data tables and a graph to examine this question.
71
72
73                                           Resulting
74       % Change              Last            Price
75         in D0           Dividend, D0       \$23.00
76        -30%                \$0.81           \$16.10
77        -15%                \$0.98           \$19.55
78          0%                \$1.15           \$23.00
79         15%                \$1.32           \$26.45
80         30%                \$1.50           \$29.90                                             Stock price sensitivity
\$90.00
81
82       % Change          Req'd Return        \$23.00                    \$80.00
Div
83        -30%                9.38%            \$90.00                    \$70.00
k
Stock Price

84        -15%               11.39%            \$36.64                    \$60.00
g
85         0%                13.40%            \$23.00                    \$50.00
86         15%               15.41%            \$16.76
\$40.00
87         30%               17.42%            \$13.18
88                                                                       \$30.00

89       % Change          Growth Rate         \$23.00                    \$20.00
90        -30%               5.60%             \$15.57                    \$10.00
91        -15%               6.80%             \$18.61                         -30%      -20%         -10%          0%          10%   20%       30%
92         0%                8.00%             \$23.00                                                      % change in input
93         15%               9.20%             \$29.90
94         30%               10.40%            \$42.32
95
96    From this chart, we see that the stock price has a positive relationship with the dividends and the growth rate, and a
97    negative relationship with the required return. Furthermore, we see that the dividend has a linear relationship with
98    price, while the growth rate seems to have a quadratic relationship. The required return stock price is not only
99    negative, but is a quadratic relationship with greater convexity than the growth rate. This indicates that the required
100   return is the factor that more directly influences the stock price. In other words, required return is the value driver in
101   this valuation technique.
102
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103   A special case of this constant growth model is a stock that has a constant growth rate of zero. This scenario of zero
104   growth is consistent with preferred stock, which pays a constant dividend in perpetuity. This kind of valuation was
105   outlined in Chapter 7, and is executed by dividing the dividend by the required return (because, the g in the constant
106   growth model drops out of the equation).
107
108   PROBLEM
109   Consider an issue of preferred stock that pays a \$1.15 dividend and has a required return of 13.4%. What is the price
110   of this preferred stock?
111
112          P =                D                /             k
113          P =              \$1.15              /          13.40%
114          P =              \$8.58
115
116   An important consideration to be made is that this kind of constant growth assumption only makes sense if you are
117   valuing a mature firm with somewhat stable growth rates. There are some special scenarios when the Gordon DCF
118   constant growth model will not make sense, which will be discussed later.
119
120
121   EXPECTED RATE OF RETURN ON A CONSTANT GROWTH STOCK
122
123   Using the constant growth equation introduced earlier, we can re-work the equation to solve for ks. In doing so, we are
124   now solving for an expected return. The expression we are left is:
125
126                             D1
ks =                                 +             g
127                             P0
128
129   This expression tells us that the expected return on a stock comprises two components. First, it consists of the
130   expected dividend yield, which is simply the next expected dividend divided by the current price. The second component
131   of the expected return is the expected capital gains yield. The expected capital gains yield is the expected annual price
132   appreciation of the stock, and is given by g. This shows us the dual role of g in the constant growth rate model. Not
133   only does g indicate expected dividend growth, but it is also the expected stock price growth rate.
134
135   PROBLEM
136   You buy a stock for \$23, and you expect the next annual dividend to be \$1.242. Furthermore, you expect the dividend to
137   grow at a constant rate of 8%. What is the expected rate of return on the stock, and what is the dividend yield of the
138   stock?
139
140          P0               \$23.00
141          D1               \$1.24
142          g                  8%
143
144          ks               13.40%
145
146    dividend yield         5.40%
147
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148   PROBLEM
149   What is the expected price of this stock in five years?
150
151          N =                 5
152   Using the growth rate we find that:
153
154          P5                \$33.79
155
156
157   VALUING STOCKS WITH NON-CONSTANT GROWTH
158
159   For many companies, it is unreasonable to assume that it grows at a constant growth rate. Hence, valuation for these
160   companies proves a little more complicated. The valuation process, in this case, requires us to estimate the short-run
161   non-constant growth rate and predict future dividends. Then, we must estimate a constant long-term growth rate that
162   the firm is expected to grow at. Generally, we assume that after a certain point of time, all firms begin to grow at a
163   rather constant rate. Of course, the difficulty in this framework is estimating the short-term growth rate, how long the
164   short-term growth will hold, and the long-term growth rate.
165
166   Specifically, we will predict as many future dividends as we can and discount them back to the present. Then we will
167   treat all dividends to be received after the convention of constant growth rate with the Gordon constant growth model
168   described above. The point in time when the dividend begins to grow constantly is called the horizon date. When we
169   calculate the constant growth dividends, we solve for a terminal value (or a continuing value) as of the horizon date. The
170   terminal value can be summarized as:
171
172                                             D N+1
TV N =               PN =
173                                            (ks-g)
174
175   This condition holds true, where N is the terminal date. The terminal value can be described as the expected value of
176   the firm in the time period corresponding to the horizon date.
177
178   PROBLEM
179   A company's stock just paid a \$1.15 dividend, which is expected to grow at 30% for the next three years. After three
180   years the dividend is expected to grow constantly at 8% forever. The stock's required return is 13.4%, what is the
181   price of the stock today?
182
183          D0                \$1.15
184          ks                13.4%
185          gs                 30%      Short-run g; for Years 1-3 only.
186          gL                 8%       Long-run g; for Year 4 and all following years.
187                                  30%                                       8%
188         Year                 0            1              2              3            4
189       Dividend             \$1.15        1.495         1.9435        2.52655      2.7287
190
191   PV of dividends
192            \$1.3183
193             1.5113
194             1.7326                                                                                           2.7287
195            \$4.5622                                                     50.5310    = Terminal value =
196          \$34.6512                                                                                             5.4%        = k - gL
197          \$39.2135 = P0
198
199
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200   USING FREE CASH FLOWS TO VALUE A STOCK
201
202   Recall from Chapters 2 and 4, free cash flow represents the amount of cash generated in a given year minus the amount
203   of cash needed to finance the additional capital expenditures and working capital needed to support future growth.
204   Specifically, we said that free cash flow was equal to net operating profit after taxes minus net investment in operating
205   capital. Since these are funds available to both stockholders and bondholders, they should be discounted at the WACC.
206   We find that using free cash flows to value stock is quite like using dividends. We will attempt to predict free cash
207   flows as far as we can, and then assume a rate of constant growth. We can use Gordon's constant growth model at this
208   point. So far, this method is nearly identical to the non-constant growth method used above. Upon using the Gordon
209   model and discounting the predicted free cash flows, we find that we have solved for the market value of the firm's
210   assets. To find the market value of equity, we must subtract out the market value of debt which is generally assumed to
211   be just the book value of debt. Once we have the market value of equity, we need only to divide by the number of shares
212   outstanding to find the stock price.
213
214   PROBLEM
215   Use the FCF method to value the stock of a firm that has a WACC of 10%, a long-run growth rate of 7%, and the
216   forecasted free cash flow (FCF) as shown on the following time line. The market value of the debt is \$904 million, and
217   50 million shares of common stock are outstanding.
218
219   WACC                     10%
220   Long-run g                7%
221   MV of debt               \$904
222   No. of shares             50
223
224         Year
225         2000               2001            2002          2003           2004         2005        2006         2007          2008
226           0                  1               2             3              4            5           6            7             8
227   FREE CF's                 \$8.4            \$8.9         \$9.8           \$34.6       \$63.7       \$96.9        \$103.7        \$111.0
228   PV of FCF's              \$7.64           \$7.36         \$7.36         \$23.63       \$39.55      \$54.70       \$53.21
229
230   PV of FCF1-7 =                                                       \$193.45                                                        \$111.0
231   TV at Year 7 of FCF after Year 7 = FCF8/(WACC - g):                                                        \$3,698.63              0.10 - 0.07
232   PV at of TV at Year 0 = TV/(1+WACC)^7:                              \$1,897.98
233
234   Sum = Value of the Total Corporation                                  \$2,091.44
235   Less: MV of Debt and Preferred                                           904.00
236   Value of Common Equity                                                 1,187.44
237   Divide by No. of Shares                                                      50
238   Value per Share = Value of Common Equity/No. Shares:                     \$23.75
239
240
241
A                  B               C              D              E             F          G           H       I   J
242   Now, we are going to assume in that beginning in the fourth year, the free cash flows are to grow by 10% less than
243   previously predicted. Our reconstructed free cash flow schedule becomes:
244                                                              -10%
245                           Year         Old FCF      New FCF
246           1               2001                 \$8.4           \$8.4
247           2               2002                 \$8.9           \$8.9
248           3               2003                 \$9.8           \$9.8
249           4               2004                \$34.6          \$31.1
250           5               2005                \$63.7          \$57.3
251           6               2006                \$96.9          \$87.2
252           7               2007               \$103.7          \$93.3
253           8               2008               \$111.0          \$99.9
254
255   We will assume that the long-term growth rate and WACC will be the same as previously assumed. From this
256   information, we can do the following calculations.
257
258   Total PV of New FCF's, Years 1-7                        \$176.34
259   FCF 8                                                     \$99.86                  \$99.86
260   TV at Year 7                                           \$3,328.77        =
261   PV of TV                                               \$1,708.19                    3%       = WACC- gL
262
263   Market Value of Total Company                         \$1,884.53
264   Less: MV of Debt                                          \$904
265   Market Value of Equity                                 \$980.53
266
267   No. of Shares                                                50
268
269   Value Per Share                                          \$19.61     versus \$23.75 under original assumptions.
270
271   Therefore, a 10% reduction in some of the cash flows leads to a 17.4% deline in the value per share.

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