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A B C D E F G H I J 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 A B C D E F G H I J 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 A B C D E F G H I J 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 A B C D E F G H I J 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 A B C D E F G H I J 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.