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We can solve Equation 5-2 for rs, again using the hat to indicate that we are dealing with an expected rate of return:10
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We can solve Equation 5-2 for rs, again using the hat to indicate that we are dealing with an expected rate of return:10
Expected rate |
Expected |
Expected growth | |
of return |
l- rate, or capital gains yield | ||
rs |
= - -Po |
g. |
(5-4) |
Thus, if you buy a stock for a price P0 = $23, and if you expect the stock to pay a dividend D1 = $1.242 one year from now and to grow at a constant rate g = 8% in the future, then your expected rate of return will be 13.4 percent:
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10The rs value in Equation 5-2 is a required rate of return, but when we transform to obtain Equation 5-4, we are finding an expected rate of return. Obviously, the transformation requires that rs = rs. This equality holds if the stock market is in equilibrium, a condition that will be discussed later in the chapter.
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In this form, we see that rs is the expected total return and that it consists of an expected dividend yield, D1/P0 = 5.4%, plus an expected growth rate or capital gains yield, g = 8%.
Suppose this analysis had been conducted on January 1, 2003, so P0 = $23 is the January 1, 2003, stock price, and D1 = $1.242 is the dividend expected at the end of 2003. What is the expected stock price at the end of 2003? We would again apply Equation 5-2, but this time we would use the year-end dividend, D2 = D1 (1 + g) = $1.242(1.08) = $1.3414:
2004
$1.3414
12/31/03
Now, note that $24.84 is 8 percent larger than P0, the $23 price on January 1, 2003:
Thus, we would expect to make a capital gain of $24.84 â€” $23.00 = $1.84 during 2003, which would provide a capital gains yield of 8 percent:
Capital gains yield2003
Capital gain _ $1.84 Beginning price $23.00
0.08
We could extend the analysis on out, and in each future year the expected capital gains yield would always equal g, the expected dividend growth rate.
Continuing, the dividend yield in 2004 could be estimated as follows:
Dividend yield2003
D2004 P12/31/03
The dividend yield for 2005 could also be calculated, and again it would be 5.4 percent. Thus, for a constant growth stock, the following conditions must hold:
1. The dividend is expected to grow forever at a constant rate, g.
2. The stock price is expected to grow at this same rate.
3. The expected dividend yield is a constant.
4. The expected capital gains yield is also a constant, and it is equal to g.
5. The expected total rate of return, rs, is equal to the expected dividend yield plus the expected growth rate: rs = dividend yield + g.
The term expected should be clarifiedâ€”it means expected in a probabilistic sense, as the "statistically expected" outcome. Thus, if we say the growth rate is expected to remain constant at 8 percent, we mean that the best prediction for the growth rate in any future year is 8 percent, not that we literally expect the growth rate to be exactly 8 percent in each future year. In this sense, the constant growth assumption is a reasonable one for many large, mature companies.
r ii j . t. What conditions must hold if a stock is to be evaluated using the constant jeu- iesi growth model?
Questions
What does the term "expected" mean when we say expected growth rate?
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