Stocks can be evaluated in two ways: (1) find the stock price directly by calculating the present value of the expected future dividends, or (2) find the stock price indirectly by first calculating the value of the entire corporation, which is the the present value of the firm's expected future free cash flows, and then subtracting the value of the debt and preferred stock to find the total value of the common equity. Only the first approach (the dividend model) is discussed in this chapter. The second approach (the corporate valuation model) is described in Chapter 13.
The value of any financial asset is the present value of the future cash flows provided by the asset. When an 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 dependent upon the dividends the next investor expects to earn, and so on for different generations of investors. Thus, the stock's value ultimately depends on the cash dividends the company is expected to provide and the discount rate used to find the present value of those dividends.
P0 =D1 D2 DN
( 1 + rs ) ( 1 + rs ) 2 ( 1 + rs ) N
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 that can be used to find the PV of the dividend stream, provided it is growing at a constant rate.
VALUING A CONSTANT GROWTH STOCK (Section 7.6)
In the constant growth model, we assume that the dividend and stock will grow forever at a constant growth rate. Naturally, assuming a constant growth rate for the rest of eternity is a rather bold assumption. However, considering the implications of imperfect information, information asymmetry, and general uncertainty, the assumption of constant growth is often reasonable. It is reasonable to guess that a given stock will experience ups and downs throughout its life. By assuming constant growth, we are trying to find the average of the good times and the bad times, and we assume that we will see both scenarios over the firm's life. In addition to a constant growth rate, we also need the estimated long-term required return for the stock, and it too must be constant. If these variables are constant, our price equation for common stock simplifies to the following expression:
In this equation, the long-run growth rate (g) can be approximated by multiplying the firm's return on assets by the retention ratio. Generally speaking, the long-run growth rate of a firm is likely to fall between 5% and 8% a year.
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, assuming it has a required return of 13.4%?
D0 =
rs =
P0 =D1 D0 (1 + g)
( rs – g ) ( rs – g )
P0 =
How sensitive is the stock's price to changes in the dividend, the growth rate, and rs? We can construct a series of data tables and a graph to examine this question.
in D0 Dividend, D0
-30% -20% -10% 0% 10% 20% 30%
$0
$2
$4
$6
$8
$10
$12
Changes in Dividends, Req's Return, and Growth: Effect on Stock Price
PV of dividends in Years 1 through 5 = $4.98 Current stock price = $23.00
21.6%
78.4%
For most stock, the percentage of the current price that is due to long-term cash flows is over 80%.
This chart shows that the stock price has a positive relationship with the dividend and the growth rate, and a negative relationship with the required return. Furthermore, we see that the dividend has a linear relationship with price, while the growth rate seems to have a quadratic relationship. The relationship between required 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 other words, required return is the value driver in this valuation technique. However, the final effects also depend on the amount of change in each of the three variables. If the required return and dividend are expected to be stable, but the dividend growth rate is expected to change significantly, then the growth rate will be the primary determinant of the stock price.
Managers often claim that stock prices are "short-term" in nature in the sense that they reflect what is happening in the near-term and ignore the long-term. We can use the results for the constant growth model to shed light on this claim.
The first step is to forecast the dividends for the next 5 years. Then we find the present value of these dividends and compare that PV with the current stock price, which reflects the PV of all future dividends.
P0 =
D0 =
rs =
Percent of current stock price due to dividends in Years 1 through 5 =
Percent of current stock price due to dividends beyond Year 5 =
A B C D E F G H93949596979899100101102103104105106107108109110111112113114115
EXAMPLE: EXPECTED RATE OF RETURN ON A CONSTANT GROWTH STOCK
$23.00
$1.242 g 8%
13.40%Dividend Yield + Capital Gains Yield = Expected Rate of Return
Dividend yield 5.40%
EXAMPLE: EXPECTED PRICE IN THE FUTUREWhat is the expected price of this stock in five years?
N = 5Using the growth rate we find that:
$33.79 =B152*(1+B154)^B163
EXPECTED RATE OF RETURN ON A CONSTANT GROWTH STOCK (Section 7.7)
Using the constant growth equation introduced earlier, we can re-work the equation to solve for rs. In doing so, we are now solving for an expected return. The expression we are left is:
rs =D1
P0
This expression tells us that the expected return on a stock comprises two components. First, it consists of the expected dividend yield, which is simply the next expected dividend divided by the current price. The second component of the expected return is the expected capital gains yield. The expected capital gains yield is the expected annual price appreciation of the stock, and is given by g. This shows us the dual role of g in the constant growth rate model. Not only does g indicate expected dividend growth, but it is also the expected stock price growth rate.
You buy a stock for $23, and you expect the next annual dividend to be $1.242. Furthermore, you expect the dividend to 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 stock?
P0
D1
rs
5.40% + 8% = 13.40%
P5 =
VALUING NONCONSTANT GROWTH STOCKS (Section 7.8)
For many companies, it is unreasonable to assume that they grow at a constant growth rate. Hence, valuation 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 constant long-term growth rate at which the firm is expected to grow. Generally, we assume that after a certain point of time, all firms begin to grow at a rather constant rate. Of course, the difficulty in this framework is estimating the short-term growth rate, how long the short-term growth will hold, and the long-term growth rate.
$4.5622 = PV of nonconstant dividends $50.5310 = Horizon value = ──────
$34.6512 = PV of horizon value 5.4%
$39.2135
÷ = $10.00 ÷ 10.00%
= $100.00
Specifically, we will predict as many future dividends as we can and discount them back to the present. Then we will treat all dividends to be received after the convention of constant growth rate with the Gordon constant
growth model described above. The point in time when the dividend begins to grow constantly is called the horizon date. When we calculate the constant growth dividends, we solve for a terminal value (or a continuing
value) as of the horizon date. The terminal value can be summarized as:
TVN = PN =DN+1 DN (1 + g)
( rs – g ) ( rs – g )
This condition holds true, where N is the terminal date. The terminal value can be described as the expected value of the firm in the time period corresponding to the horizon date.
A company's stock just paid a $1.15 dividend, which is expected to grow at 30% for the next three years. After three years the dividend is expected to grow constantly at 8% forever. The stock's required return is 13.4%, what is the price of the stock today?
D0 =
rs =
gs =
gL = Long-run gL; for all years after Year 3.
PV of dividends discounted at rs
= P0 rs – gL
PREFERRED STOCK (Section 7.11)
Consider an issue of preferred stock that pays a $10 dividend and has a required return of 10%. What is the price of this preferred stock?
Years to Maturity (N): 50Annual Dividend (PMT): $10Par value (FV): $100
8%
$124.47
Changes in Equilibrium Stock PricesSmall changes in the market's expectations can cause large changes in stock price!
Original New8% 7%
4% 3%
2 1
16.00% 10.00%
5% 6%
$2.8571 $2.8571
Price of Stock i $27.27 $75.71
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 preferred stock?
Stocks can be evaluated in two ways: (1) find the stock price directly by calculating the present value of the expected future dividends, or (2) find the stock price indirectly by first calculating the value of the entire corporation, which is the the present value of the firm's expected future free cash flows, and then subtracting the value of the debt and preferred stock to find the total value of the common equity. Only the first approach (the dividend model) is discussed in this chapter. The second approach (the corporate valuation model) is
The value of any financial asset is the present value of the future cash flows provided by the asset. When an 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 dependent upon the dividends the next investor expects to earn, and so on for different generations of investors. Thus, the stock's value ultimately depends on the cash dividends the company is expected to
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 that
In the constant growth model, we assume that the dividend and stock will grow forever at a constant growth rate. Naturally, assuming a constant growth rate for the rest of eternity is a rather bold assumption. However, considering the implications of imperfect information, information asymmetry, and general uncertainty, the assumption of constant growth is often reasonable. It is reasonable to guess that a given stock will experience ups and downs throughout its life. By assuming constant growth, we are trying to find the average of the good times and the bad times, and we assume that we will see both scenarios over the firm's life. In addition to a constant growth rate, we also need the estimated long-term required return for the stock, and it too must be constant. If these variables are constant, our price equation for common stock simplifies to the following
In this equation, the long-run growth rate (g) can be approximated by multiplying the firm's return on assets by the retention ratio. Generally speaking, the long-run growth rate of a firm is likely to fall between 5% and 8% a
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
? We can construct a
-30% -20% -10% 0% 10% 20% 30%
$0
$2
$4
$6
$8
$10
$12
Changes in Dividends, Req's Return, and Growth: Effect on Stock Price
For most stock, the percentage of the current price that is due to long-term cash flows is over 80%.
This chart shows that the stock price has a positive relationship with the dividend and the growth rate, and a negative relationship with the required return. Furthermore, we see that the dividend has a linear relationship with price, while the growth rate seems to have a quadratic relationship. The relationship between required 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 other words, required return is the value driver in this valuation technique. However, the final effects also depend on the amount of change in each of the three variables. If the required return and dividend are expected to be stable, but the dividend growth rate is expected to change significantly, then the growth rate will be the
Managers often claim that stock prices are "short-term" in nature in the sense that they reflect what is happening in the near-term and ignore the long-term. We can use the results for the constant growth model to
The first step is to forecast the dividends for the next 5 years. Then we find the present value of these dividends and compare that PV with the current stock price, which reflects the PV of all future dividends.
Using the constant growth equation introduced earlier, we can re-work the equation to solve for rs. In doing so,
This expression tells us that the expected return on a stock comprises two components. First, it consists of the expected dividend yield, which is simply the next expected dividend divided by the current price. The second component of the expected return is the expected capital gains yield. The expected capital gains yield is the expected annual price appreciation of the stock, and is given by g. This shows us the dual role of g in the constant growth rate model. Not only does g indicate expected dividend growth, but it is also the expected
You buy a stock for $23, and you expect the next annual dividend to be $1.242. Furthermore, you expect the dividend to grow at a constant rate of 8%. What is the expected rate of return on the stock, and what is the
For many companies, it is unreasonable to assume that they grow at a constant growth rate. Hence, valuation 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 constant long-term growth rate at which the firm is expected to grow. Generally, we assume that after a certain point of time, all firms begin to grow at a rather constant rate. Of course, the difficulty in this framework is estimating the short-term growth rate, how long the short-term growth will hold, and the long-term growth rate.
Specifically, we will predict as many future dividends as we can and discount them back to the present. Then we will treat all dividends to be received after the convention of constant growth rate with the Gordon constant
growth model described above. The point in time when the dividend begins to grow constantly is called the horizon date. When we calculate the constant growth dividends, we solve for a terminal value (or a continuing
value) as of the horizon date. The terminal value can be summarized as:
This condition holds true, where N is the terminal date. The terminal value can be described as the expected
A company's stock just paid a $1.15 dividend, which is expected to grow at 30% for the next three years. After three years the dividend is expected to grow constantly at 8% forever. The stock's required return is 13.4%,
Consider an issue of preferred stock that pays a $10 dividend and has a required return of 10%. What is the
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
I225
226
227228229230231232
233234235236237238239240241242
243
244
245
246
247
248249
SECTION 7.5SOLUTIONS TO SELF-TEST
$3.00
$50.00
$52.00
Exp. dividend yield 6.0% =B6/B7Exp. capital gains yield 4.0% =(B8-B7)/B7Exp. total return 10.0% =C10+C11
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
P0
Expected P1
0.06$0.04 $0.10
= $50, and the expected P at t=1 is equal to $52, what are the stock’s expected dividend yield,
SECTION 7.6SOLUTIONS TO SELF-TEST
$2.00 g 4%
12%
Stock price $25.00 25
$2.00 g 0%
12%
Stock price $16.67 16.6666666666667
A stock is expected to pay a dividend of $2 at the end of the year. The required rate of return is rs = 12%. What would the stock’s price be if the growth rate were 4%?
D1
rs
A stock is expected to pay a dividend of $2 at the end of the year. The required rate of return is rs = 12%. What would the stock’s price be if the growth rate were 0%?
D1
rs
A stock is expected to pay a dividend of $2 at the end of the year. The required rate of return is rs = 12%. What would the
A stock is expected to pay a dividend of $2 at the end of the year. The required rate of return is rs = 12%. What would the
SECTION 7.7SOLUTIONS TO SELF-TEST
$4.00 g 5%
9%
$4.20 4.2
Stock price $105.00 105
Expected dividend yield 4.00% 0.04
Expected capital gains yield 5.00% 0.05
Alternatively, you know that the capital gains yield is equal to the growth rate.
Expected capital gains yield = growth rate = 5.00%
Expected dividend yield 4.00%
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
rs
Expected D1
Because the total return is rs, the dividend yield is rs minus the capital gains yield:
SECTION 7.8SOLUTIONS TO SELF-TEST
$5.00
20%
10%
5%
10%Year
1 2
D1 D2Expected dividends $6.00 $6.60 6 $6.60
$138.60 138.6
PV of expected dividends $10.91 $10.91
$114.55 $114.55
$125.45 $125.45
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
g0 to 1
g1 to 2
gn
rs
D1 D2
Expected P2
PV of expected P2
Expected P0
D36.93
) = 10%, and the constant rate = 5%. What are the expected dividends for Year 1 and Year 2? What is the
SECTION 7.11SOLUTIONS TO SELF-TEST
$5.00
8%
$62.50 $62.50
A preferred stock has an annual dividend of $5. The required return is 8%. What is the Vps?
Dps
rps
Vps
A preferred stock has an annual dividend of $5. The required return is 8%. What is the Vps?
