24. We can use the two-stage dividend growth model for this problem, which is:
P0 = [D0(1 + g1)/(R – g1)]{1 – [(1 + g1)/(1 + R)]t}+ [(1 + g1)/(1 + R)]t[D0(1 + g2)/(R – g2)]
25. We can use the two-stage dividend growth model for this problem, which is:
P0 = [D0(1 + g1)/(R – g1)]{1 – [(1 + g1)/(1 + R)]t}+ [(1 + g1)/(1 + R)]t[D0(1 + g2)/(R – g2)]
26. a. Using the equation to calculate the price of a share of stock with the PE ratio:
P = Benchmark PE ratio × EPS
So, with a PE ratio of 21, we find:
b. First, we need to find the earnings per share next year, which will be:
EPS1 = EPS0(1 + g)
Using the equation to calculate the price of a share of stock with the PE ratio:
P1 = Benchmark PE ratio × EPS1
c. To find the implied return over the next year, we calculate the return as:
R = (P1 – P0)/P0
Notice that the return is the same as the growth rate in earnings. Assuming a stock pays no
dividends and the PE ratio is constant, this will always be true when using price ratios to
evaluate the price of a share of stock.
27. We need to find the PE ratio each year, which is:
PE1 = $49.18/$2.35 = 20.93
So, the average PE is:
First, we need to find the earnings per share next year, which will be:
EPS1 = EPS0(1 + g)
Using the equation to calculate the price of a share of stock with the PE ratio:
P1 = Benchmark PE ratio × EPS1
28. First, we need to find the earnings per share next year, which will be:
EPS1 = EPS0(1 + g)
To find the high target stock price, we need to find the average high PE ratio each year, which is:
PE1 = $27.43/$1.35 = 20.32
So, the average high PE is:
Using the equation to calculate the price of a share of stock with the PE ratio, the high target price is:
P1 = Benchmark PE ratio × EPS1
To find the low target stock price, we need to find the average low PE ratio each year, which is:
PE1 = $19.86/$1.35 = 14.71
So, the average low PE is:
Using the equation to calculate the price of a share of stock with the PE ratio, the low target price is:
P1 = Benchmark PE ratio × EPS1
29. To find the target price in five years, we first need to find the EPS in five years, which will be:
EPS5 = EPS0(1 + g)5
So, the target stock price in five years is:
P5 = Benchmark PE ratio × EPS5
30. We need to begin by finding the dividend for each year over the next five years, so:
D1 = $1.36(1 + .13) = $1.54
D2 = $1.36(1 + .13)2 = $1.74
To find the EPS in Year 5, we can use the dividends and payout ratio, which gives us:
EPS5 = D5/Payout ratio
So, the terminal stock price in Year 5 will be:
P5 = Benchmark PE ratio × EPS5
The stock price today is the present value of the dividends for the next five years, plus the present
value of the terminal stock price, discounted at the required return, or:
31. To find the target stock price, we first need to calculate the growth rate in earnings. We can use the
sustainable growth rate from a previous chapter. First, the ROE is:
ROE = Net income/Equity
We also need the retention ratio, which is one minus the payout ratio, or:
b = 1 – Dividends/Net income
So, the sustainable growth rate is:
Sustainable growth rate = (ROE × b)/(1 – ROE × b)
Now we need to find the current EPS, which is:
EPS0 = Net income/Shares outstanding
So, the EPS next year will be:
EPS1 = EPS0(1 + g)
Finally, the target share price next year is:
P1 = Benchmark PE ratio × EPS5
Challenge
32. We are asked to find the dividend yield and capital gains yield for each of the stocks. All of the
stocks have a required return of 13 percent, which is the sum of the dividend yield and the capital
W: P0 = D0(1 + g)/(R – g) = $3.75(1.10)/(.13 – .10) = $137.50
X: P0 = D0(1 + g)/(R – g) = $3.75/(.13 – 0) = $28.85
Y: P0 = D0(1 + g)/(R – g) = $3.75(1 – .05)/(.13 + .05) = $19.79
Z: P2 = D2(1 + g)/(R – g) = D0(1 + g1)2(1 + g2)/(R – g2)
In all cases, the required return is 13 percent, but this return is distributed differently between current
income and capital gains. High growth stocks have an appreciable capital gains component but a
relatively small current income yield; conversely, mature, negative-growth stocks provide a high
current income but also price depreciation over time.
33. a. Using the constant growth model, the price of the stock paying annual dividends will be:
P0 = D0(1 + g)/(R – g)
b. If the company pays quarterly dividends instead of annual dividends, the quarterly dividend
will be one-fourth of the annual dividend, or:
To find the equivalent annual dividend, we must assume that the quarterly dividends are
The effective annual dividend will be the FVA of the quarterly dividend payments at the
effective quarterly required return. In this case, the effective annual dividend will be:
Now, we can use the constant growth model to find the current stock price as:
Note that we cannot find the quarterly effective required return and growth rate to find the
value of the stock. This would assume the dividends increased each quarter, not each year.
34. Here we have a stock with supernormal growth, but the dividend growth changes every year for the
first four years. We can find the price of the stock in Year 3 since the dividend growth rate is constant
after the third dividend. The price of the stock in Year 3 will be the dividend in Year 4, divided by the
required return minus the constant dividend growth rate. So, the price in Year 3 will be:
The price of the stock today will be the PV of the first three dividends, plus the PV of the stock price
in Year 3, so:
35. Here we want to find the required return that makes the PV of the dividends equal to the current
stock price. The equation for the stock price is:
We need to find the roots of this equation. Using spreadsheet, trial and error, or a calculator with a
root solving function, we find that:
36. Even though the question concerns a stock with a constant growth rate, we need to begin with the
equation for two-stage growth given in the chapter, which is:
P0 =
D0(1+ g1)
R – g1
[
1 –
(
1+ g1
1+ R
)
t
]
+
Pt
(1+ R)t
We can expand the equation (see Problem 37 for more detail) to the following:
P0 =
D0(1+ g1)
R – g1
[
1 –
(
1+ g1
1+ R
)
t
]
+
(
1+ g1
1+ R
)
t
D0(1+ g2)
R – g2
Since the growth rate is constant, g1 = g2 , so:
P0 =
D0(1+ g )
R – g
[
1 –
(
1+ g
1+ R
)
t
]
+
(
1+ g
1+ R
)
t
Since we want the first t dividends to constitute one-half of the stock price, we can set the two terms
on the right hand side of the equation equal to each other, which gives us:
D0(1 +g)
R – g
[
1 –
(
1+ g
1+ R
)
t
]
=
(
1+ g
1+ R
)
t
D0(1+ g )
R – g
Since
D0(1 +g)
R – g
appears on both sides of the equation, we can eliminate this, which leaves:
1 –
(
1 +g
1 +R
)
t
=
(
1+ g
1+ R
)
t
Solving this equation, we get:
1 =
(
1 +g
1 +R
)
t
+
(
1+ g
1+ R
)
t
1 = 2
(
1 +g
1 +R
)
t
1/2 =
(
1+ g
1+ R
)
t
t ln
(
1 +g
1 +R
)
= ln(.5)
t =
ln(. 5)
ln
(
1+ g
1+ R
)
This expression will tell you the number of dividends that constitute one-half of the current stock
price.
37. To find the value of the stock with two-stage dividend growth, consider that the present value of the
P0 = PV of t dividends + PV(Pt)
Using g1 to represent the first growth rate and substituting the equation for the present value of a
growing annuity, we get:
P0 = D1
[
1 –
(
1+ g1
1+ R
)
t
R – g1
]
+ PV(Pt)
Since the dividend in one year will increase at g1, we can rewrite the expression as:
P0 = D0(1 + g1)
[
1 –
(
1+ g1
1+ R
)
t
R – g1
]
+ PV(Pt)
Now we can rewrite the equation again as:
P0 =
D0(1+ g1)
R – g1
[
1 –
(
1 + g1
1 + R
)
t
]
+ PV(Pt)
To find the price of the stock at Time t, we can use the constant dividend growth model, or:
Pt =
Dt +1
R – g2
The dividend at t + 1 will have grown at g1 for t periods, and at g2 for one period, so:
So, we can rewrite the equation as:
Pt =
D(1+ g1)t(1+ g2)
R – g2
Next, we can find value today of the future stock price as:
PV(Pt) =
D(1+ g1)t(1+ g2)
R – g2
×
1
(1+ R)t
which can be written as:
PV(Pt) =
(
1+ g1
1+ R
)
t
×
D(1+ g2)
R – g2
Substituting this into the stock price equation, we get:
P0 =
D0(1+ g1)
R – g1
[
1 –
(
1+ g1
1+ R
)
t
]
+
(
1+ g1
1+ R
)
t
×
D(1+ g2)
R – g2
In this equation, the first term on the right-hand side is the present value of the first t dividends, and
the second term is the present value of the stock price when constant dividend growth forever
begins.