BUSI 683 - MONEY AND CAPITAL
MARKETS - Stock Valuation
Question Bank - Set 3
Liberty University
Question 1
Question
A company pays an annual dividend of 3.50pershare.Iftherequiredrateofreturnis8
Solution
Step 1: Calculate the expected dividend next year using the growth rate. Step 2:
Use the Gordon Growth Model to calculate the stock price. Step 3: Substitute
the given values to find the current price of the stock.
Step 1: The expected dividend next year is given by:
D1 = D0×(1 + g)
where: - D1 is the expected dividend next year, - D0 is the current dividend,
and - gis the growth rate of dividends. Substitute the values:
D1=3.50 ×(1 + 0.04)
D1=3.50 ×1.04
D1=3.64
Step 2: The Gordon Growth Model is given by:
P0 = D1
r−g
where: - P0 is the current price of the stock, - D1 is the expected dividend next
year, - ris the required rate of return, and - gis the growth rate of dividends.
Step 3: Substitute the values into the Gordon Growth Model:
P0 = 3.64
0.08 −0.04
P0 = 3.64
0.04
P0 = 91
Therefore, the current price of the stock is 91.
Question 2
Question
A company’s stock is expected to pay a dividend of 4.00persharenextyear.Dividendsareexpectedtogrowataconstantrateof5
Solution
Let’s denote: - D0as the current dividend per share, - D1as the dividend per
share one year from now, - ras the required rate of return, and - gas the
constant growth rate of dividends.
Step 1: Calculate the current dividend per share, D0:
D1=D0×(1 + g)
4.00 = D0×(1 + 0.05)
D0=4.00
1.05
D0= 3.81
Step 2: Calculate the current value of the stock using the Dividend Discount
Model (DDM):
P0=D1
r−g
P0=4.00
0.10 −0.05
P0=4.00
0.05
P0= 80.00
Therefore, the current value of the stock is 80.00.
Question 3
Question
A company just paid a dividend of 5pershare.T hedividendsareexpectedtogrowatarateof 8
2
Solution
Step 1: Calculate the constant growth rate formula for stock valuation:
P0=D0×(1 + g)
r−g
where: - P0= current stock price - D0= most recent dividend per share - g=
growth rate of dividends - r= required rate of return
Step 2: Substitute the given values into the formula:
P0=5×(1 + 0.08)
0.12 −0.08
P0=5×1.08
0.04
P0=5.4
0.04
P0= 135
Therefore, the current stock price is 135.
Question 4
Question
A company’s stock is currently trading at $50 per share. The company is ex-
pected to pay a dividend of $2 per share next year, and dividends are expected
to grow at a constant rate of 5% per year indefinitely. If the required rate of
return is 10%, what is the intrinsic value of the stock?
Solution
Step 1: Calculate the dividend expected in the second year. Since dividends are
expected to grow at a constant rate of 5% per year, the dividend expected in
the second year will be:
$2 ×(1 + 0.05) = $2.10
Step 2: Calculate the dividend yield. The dividend yield is the expected
dividend in the next period divided by the current stock price. In this case:
Dividend Yield = $2.10
$50 = 0.042
Step 3: Calculate the constant growth rate of dividends (g). The constant
growth rate of dividends (g) is given as 5%.
3
Step 4: Use the Gordon Growth Model to calculate the intrinsic value of the
stock. The Gordon Growth Model is given by:
P0=D1
r−g
Where: - P0= Intrinsic value of the stock - D1= Expected dividend in the
next period - r= Required rate of return - g= Growth rate of dividends
Substitute the known values into the formula:
P0=$2.10
0.10 −0.05 =$2.10
0.05 = $42
Therefore, the intrinsic value of the stock is $42.
Question 5
Question
A company is expected to pay a dividend of 3persharenextyear.Dividendsareexpectedtogrowatarateof6
Solution
Step 1: Calculate the dividend in year 2 using the growth rate.
Dividend in year 2 = $3 ×(1 + 0.06) = $3.18
Step 2: Calculate the dividend discount model to find the stock value.
Stock value = Dividend in year 2
Required rate of return −Growth rate
Step 3: Substitute the values into the formula.
Stock value = $3.18
0.10 −0.06
Step 4: Simplify the expression.
Stock value = $3.18
0.04 = $79.50
Therefore, the current value of the stock is $79.50.
Question 6
Question
You are trying to determine the value of a stock using the dividend discount
model. The stock is expected to pay a dividend of
$
2 per share next year and
dividends are expected to grow at a constant rate of 5
4
Solution
Let’s denote the expected dividend next year as D1=
$
2, the growth rate of
dividends as g = 5
Stock Value = D1
r−g
Step 1: Calculate the Stock Value
Stock Value = 2
0.10 −0.05
=2
0.05
= 40
Therefore, the current value of the stock is
$
40 per share.
Question 7
Question
A company’s stock is currently trading at $80 per share. The company is ex-
pected to pay a dividend of $4 per share next year, and dividends are expected
to grow at a rate of 5
Solution
Step 1: Calculate the expected dividend for year 1. The expected dividend for
year 1 is given as D1= $4.
Step 2: Calculate the dividend growth rate. The dividend growth rate is
given as 5
Step 3: Calculate the required rate of return. The required rate of return is
given as 10
Step 4: Calculate the expected dividend in year 2 and beyond. The expected
dividend for year 2 and beyond can be calculated using the formula for the
dividend in year n, which is Dn=D1×(1 + g)n−1, where D1is the dividend in
year 1, gis the growth rate, and nis the year.
Step 5: Calculate the intrinsic value of the stock using the dividend discount
model. The intrinsic value of the stock V0can be calculated using the formula
V0=D1
r−g, where D1is the dividend in year 1, ris the required rate of return,
and gis the growth rate.
Substitute the given values into the formula: V0=$4
0.10−0.05 =$4
0.05 = $80.
Therefore, the intrinsic value of the stock is $80.
5
Question 8
Question
A company is expected to pay an annual dividend of 5pershareindefinitely.Iftherequiredrateofreturnonthestockis10
Solution
Step 1: Calculate the stock price using the Gordon Growth Model formula:
Stock Price = Dividend per share
Required rate of return
Step 2: Substitute the given values into the formula:
Stock Price = 5
0.10 = 50
Answer: The stock price is 50pershare.
Question 9
Question
A company is expected to pay a dividend of 3.50pershareoneyearf romnow.T hedividendisexpectedtogrowataconstantrateof6
Solution
Let’s denote: - D1as the dividend to be paid one year from now - ras the
required rate of return - gas the growth rate of the dividend
Step 1: Calculate the expected dividend one year from now (D1).
D1= $3.50
Step 2: Calculate the price of the stock using the Gordon Growth Model:
P0=D1
r−g
Step 3: Plug in the values into the formula:
P0=$3.50
0.12 −0.06
Step 4: Calculate the current value of the stock:
P0=$3.50
0.06 = $58.33
Therefore, the current value of the stock is
$
58.33.
6
Question 10
Question
A company pays an annual dividend of
$
3 per share, and the dividends are
expected to grow at a constant rate of 5% per year indefinitely. If the required
rate of return is 10%, what is the current value of the stock?
Solution
Step 1: Calculate the growth rate of dividends using the constant growth divi-
dend valuation formula:
g=3×0.05
3= 0.05
Step 2: Use the dividend valuation model to calculate the value of the stock:
P0=D0×(1 + g)
r−g
where: - P0is the current value of the stock, - D0is the most recent dividend per
share, - ris the required rate of return, and - gis the growth rate of dividends.
Step 3: Substitute the values into the formula:
P0=3×(1 + 0.05)
0.10 −0.05
P0=3.15
0.05
P0= 63
Therefore, the current value of the stock is
$
63 per share.
Question 11
Question
A company is expected to pay a dividend of
$
7 next year. Dividends are expected
to grow at a rate of 3
Solution
Step 1: Calculate the dividend for the next year using the given information.
Step 2: Calculate the dividend growth rate. Step 3: Use the dividend discount
model to calculate the value of the stock today.
Step 1: The dividend for the next year is
$
7.
Step 2: The dividend growth rate is 3
7
Step 3: The dividend discount model is given by:
V0=D1
r−g,
where: - V0is the value of the stock today, - D1is the dividend for the next
year, - ris the required rate of return, - gis the dividend growth rate.
Substitute the known values into the formula:
V0=7
0.10 −0.03 =7
0.07 = $100.
Therefore, the value of the stock today is
$
100.
Question 12
Question
A company just paid a dividend of 5pershare.T hedividendisexpectedtogrowataconstantrateof 4
Solution
Step 1: Calculate the growth rate of dividends. Step 2: Use the Gordon Growth
Model to find the current stock price.
Step 1: Calculate the growth rate of dividends. The growth rate of divi-
dends is given as 4
Step 2: Use the Gordon Growth Model to find the current stock price. The
Gordon Growth Model is given by the formula:
P=D1
r−g
where: - P is the current stock price, - D1is the expected dividend one year from
now, - r is the required rate of return, and - g is the growth rate of dividends.
Given: - D1= dividend this year ×(1 + growth rate) = 5 ×(1 + 0.04) =
$
5.20, - r = 10- g = 4
Substitute these values into the formula:
P=5.20
0.10 −0.04 =5.20
0.06 = $86.67
Therefore, the current stock price is
$
86.67.
Question 13
Question
You are considering investing in a company that pays a constant annual dividend
of 12.If therequiredrateofreturnis8%, andthedividendsareexpectedtogrowataconstantrateof5%peryear, whatisthecurrentvalueofthestock?
8
Solution
Step 1: Calculate the dividend growth rate using the formula
g=12 ×0.05
12 = 0.05
Step 2: Calculate the dividend in the next year using the formula
D1=D0×(1 + g) = 12 ×(1 + 0.05) = 12.6
Step 3: Calculate the stock price using the Gordon Growth Model formula
P0=D1
r−g
where P0is the current stock price, D1is the dividend in the next year, ris the
required rate of return, and gis the growth rate of dividends.
Step 4: Substitute the known values into the formula to find the current
stock price
P0=12.6
0.08 −0.05 =12.6
0.03 = 420
Therefore, the current value of the stock is 420.
Question 14
Question
A company’s stock is currently priced at $100 per share. The company is ex-
pected to pay a dividend of $5 per share at the end of the year. If the required
rate of return is 10%, what is the expected growth rate of the company?
Solution
Step 1: Calculate the dividend yield. The dividend yield is the dividend per
share divided by the stock price.
Dividend Yield = $5
$100 = 0.05 = 5%
Step 2: Use the Dividend Discount Model (DDM) to find the expected
growth rate. The DDM formula is:
Next year’s dividend = Current Dividend ×(1 + Growth rate)
Plugging in the values we know:
$5 = $5 ×(1 + Growth rate)
Solving for the growth rate:
1 = 1 + Growth rate
Growth rate = 0
Therefore, the expected growth rate of the company is 0%.
9
Question 15
Question
A company’s stock is currently valued at $50 per share. The company is ex-
pected to pay a dividend of $2 per share next year, with dividends expected
to grow at a constant rate of 5% per year indefinitely. If the required rate of
return on the stock is 10%, what is the intrinsic value of the stock?
Solution
Step 1: Calculate the dividend growth rate. Given that the dividend is expected
to grow at a constant rate of 5% per year, the dividend growth rate can be
calculated as follows:
g= 0.05
Step 2: Use the Gordon Growth Model to calculate the intrinsic value of the
stock. The Gordon Growth Model is given by:
P0=D1
r−g
where: - P0is the intrinsic value of the stock, - D1is the dividend expected
next year, - ris the required rate of return on the stock, and - gis the dividend
growth rate.
Substitute the given values into the formula to find the intrinsic value of the
stock:
P0=2
0.10 −0.05
P0=2
0.05
P0= 40
Therefore, the intrinsic value of the stock is $40 per share.
Question 16
Question
A company’s stock is currently trading at $50 per share. It is expected that
the company will pay a dividend of $2.50 per share next year. If dividends are
expected to grow at a constant rate of 6% per year indefinitely and the required
rate of return is 12%, what is the fair value of the stock?
10
Solution
Step 1: Calculate the expected dividend next year using the constant growth
rate formula:
D1=D0×(1 + g)
where: - D1is the dividend next year, - D0is the current dividend, and - gis
the constant growth rate.
Substitute D0= $2.50 and g= 6% = 0.06:
D1= $2.50 ×(1 + 0.06) = $2.50 ×1.06 = $2.65
Step 2: Calculate the price of the stock using the Gordon Growth Model
formula:
P0=D1
r−g
where: - P0is the fair value of the stock, - D1is the dividend next year, - ris
the required rate of return, and - gis the constant growth rate.
Substitute D1= $2.65, r= 12% = 0.12, and g= 6% = 0.06:
P0=$2.65
0.12 −0.06 =$2.65
0.06 = $44.17
Therefore, the fair value of the stock is $44.17.
Question 17
Question
ABC Corp. is expected to pay a dividend of 2.00 per share next year. Dividends
are expected to grow at a rate of 5% per year indefinitely. If the required return
on ABC Corp. stock is 12%, what is the current stock price?
Solution
Step 1: Calculate the expected dividend in the future. Given that the dividend
next year is D1= 2.00, wecancalculatethedividendinyear2as :D2=D1×(1 +
growth rate) = 2.00 ×(1 + 0.05) =2.10
Step 2: Calculate the required rate of return in decimal form. The required
rate of return is 12% which is 0.12 in decimal form.
Step 3: Calculate the price of the stock using the dividend discount model
(DDM) formula. The price of a stock can be calculated using the formula:
P0=D1
r−g
11
where: - P0= Current stock price, - D1= Dividend expected next year, - r=
Required rate of return, and - g= Growth rate of dividends.
Substitute the given values into the formula:
P0=2.00
0.12 −0.05 =2.00
0.07 =
28.57
Therefore, the current stock price of ABC Corp. is 28.57pershare.
Question 18
Question
A company is expected to pay a dividend of
$
5 per share next year. Analysts
expect the company’s dividends to grow at a rate of 3% per year indefinitely. If
the required rate of return on the stock is 10%, what is the current price of the
stock?
Solution
Step 1: Calculate the dividend expected in year 2. Given that the dividend is
expected to grow at a rate of 3% per year, the dividend expected in year 2 can
be calculated as:
D2=D1×(1 + g)
where: - D1= $5 (dividend next year) - g= 3% = 0.03 (growth rate)
D2= $5 ×(1 + 0.03) = $5.15
Step 2: Determine the price of the stock based on the growing perpetuity
formula. The price of the stock can be calculated using the growing perpetuity
formula:
P=D1
r−g
where: - D1= $5 (dividend next year) - r= 10% = 0.10 (required rate of
return) - g= 3% = 0.03 (growth rate) Substitute the values into the formula:
P=$5.15
0.10 −0.03
P=$5.15
0.07
P= $73.57
Therefore, the current price of the stock is
$
73.57.
12
Question 19
Question
A company is expected to pay a dividend of 5.00pershareoneyearfromnow.Afterthat, thedividendsareexpectedtogrowataconstantrateof8
Solution
Let’s denote: - D1= $5.00 as the dividend to be paid one year from now, -
g=8%=0.08 as the constant growth rate of dividends, - r= 12% = 0.12 as
the required rate of return, - P0as the current value of the stock.
The current value of the stock can be calculated using the Gordon Growth
Model formula:
P0=D1
r−g
Step 1: Calculate the current value of the stock using the Gordon Growth
Model formula.
P0=5.00
0.12 −0.08
P0=5.00
0.04
P0= $125.00
Therefore, the current value of the stock is $125.00.
Question 20
Question
A company’s stock is expected to pay dividends of
$
2,
$
2.50, and
$
3 for the
next three years. After that, dividends are expected to grow at a constant rate
of 5% per year indefinitely. If the required rate of return on the stock is 8%,
what is the current stock price?
Solution
Step 1: Calculate the present value of dividends for the next three years.
PV of dividends = D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
=2
(1 + 0.08)1+2.5
(1 + 0.08)2+3
(1 + 0.08)3
=2
1.08 +2.5
1.082+3
1.083
≈1.85 + 2.17 + 2.49
≈6.51
13
Step 2: Calculate the price of the stock after three years when it starts
growing at a constant rate.
Price after 3 years = D3×(1 + g)
r−g
=3×(1 + 0.05)
0.08 −0.05
=3×1.05
0.03
=3.15
0.03
= 105
Step 3: Calculate the present value of the price after three years.
PV of price after 3 years = 105
(1 + 0.08)3
=105
1.259712
≈83.48
Step 4: Calculate the current stock price by adding the present value of
dividends and the present value of the price after three years.
Current stock price = PV of dividends + PV of price after 3 years
≈6.51 + 83.48
≈89.99
Therefore, the current stock price is approximately 89.99.
Question 21
Question
A company is expected to pay a dividend of 3.00nextyear.Dividendsareexpectedtogrowatarateof5
Solution
Step 1: Calculate the dividend in one year using the dividend growth model.
Dividend in one year = D0×(1 + g) = 3.00 ×(1 + 0.05) = 3.15
Step 2: Calculate the price of the stock using the dividend discount model.
The price of the stock can be calculated as the present value of all future divi-
dends.
Price of the stock = D1
r−g
14
Where: - D1 = Dividend in one year = 3.15 −r= Required rate of return
= 12- g= Growth rate of dividends = 5
Price of the stock = 3.15
0.12 −0.05
Price of the stock = 3.15
0.07
Price of the stock = 45
Therefore, the value of the stock today is 45.
Question 22
Question
A company is expected to pay a dividend of 5 at the end of the year. Dividends
are expected to grow at a rate of 8% per year indefinitely. If the required rate
of return on the stock is 12%, what is the current value of the stock?
Solution
Step 1: Calculate the dividend at the end of Year 1. The dividend at the end
of Year 1 can be calculated as:
D1=D0×(1 + g) = 5 ×(1 + 0.08) = 5.4
Step 2: Determine the dividend yield. The dividend yield is the ratio of the
most recent dividend to the current stock price, which can be calculated using
the formula:
Dividend Yield = D1
P0
Step 3: Apply the Gordon Growth Model to find the current value of the
stock. The Gordon Growth Model is used to calculate the value of a stock based
on the present value of future dividends. The formula is:
P0=D1
r−g
where: - P0is the current value of the stock, - D1is the dividend at the end of
Year 1, - ris the required rate of return on the stock, and - gis the growth rate
of dividends.
Substitute the values we have:
P0=5.4
0.12 −0.08 =5.4
0.04 = 135
Therefore, the current value of the stock is 135.
15
Question 23
Question
A company’s stock is expected to pay a dividend of 3 dollars one year from now.
The dividend is expected to grow at a rate of 5% per year indefinitely. If the
required rate of return for the stock is 10%, what is the current value of the
stock?
Solution
Step 1: Calculate the expected dividend in year 2. Step 2: Calculate the present
value of the dividends. Step 3: Calculate the current value of the stock.
Step 1: The expected dividend in year 2 can be calculated using the growth
rate:
D2=D1×(1 + g)
D2= 3 ×(1 + 0.05)
D2= 3 ×1.05 = 3.15
Step 2: To calculate the present value of the dividends, we use the formula
for the present value of a growing perpetuity:
P V =D
r−g
where: P V = Present value of dividends, D= Dividend in year 1, r= Required
rate of return, g= Growth rate.
Plugging in the values, we get:
P V =3
0.10 −0.05
P V =3
0.05 = 60
Step 3: The current value of the stock is equal to the present value of the
dividends plus the present value of the stock one year from now:
CurrentV alue =P V +D2/(1 + r)
CurrentV alue = 60 + 3.15/(1 + 0.10)
CurrentV alue = 60 + 3.15/1.10 = 60 + 2.8636 = 62.8636
Therefore, the current value of the stock is 62.86.
Question 24
Question
A company pays an annual dividend of 3.50pershareandisexpectedtogrowataconstantrateof5
16
Solution
Step 1: Calculate the dividend growth rate. Step 2: Use the constant growth
dividend valuation model to find the current stock price.
Step 1: Calculate the dividend growth rate. Given that the dividend growth
rate is 5
Step 2: Calculate the current stock price using the constant growth dividend
valuation model. The formula for the constant growth dividend valuation model
is:
P0=D1
r−g
where: - P0is the current stock price, - D1is the dividend expected one year
from now, - ris the required rate of return, and - gis the growth rate of
dividends.
Given: - D1=D0×(1 + g)=3.50 ×(1 + 0.05) = $3.675 - r= 0.10 - g= 0.05
Substitute these values into the formula:
P0=3.675
0.10 −0.05 =3.675
0.05 = $73.50
Therefore, the current stock price is
$
73.50.
Question 25
Question
A company is expected to pay an annual dividend of 2.50forthenextfiveyears.Af terthat, dividendsareexpectedtogrowatarateof6
Solution
Let’s denote the expected dividend for year 5 as D5. Then, the dividend for
year 6 and beyond can be expressed as D6 = D5×(1+ g), where gis the growth
rate of dividends (6
Step 1: Calculate the dividend for year 5.
D5 = D4×(1 + g)
=D3×(1 + g)2
=D2×(1 + g)3
=D1×(1 + g)4
= 2.50 ×(1 + 0.06)4
≈3.5070
17
Step 2: Calculate the price of the stock at the end of year 5.
P5 = D6
r−g
=D5×(1 + g)
r−g
=3.5070 ×(1 + 0.06)
0.10 −0.06
=3.72
0.04
= 93
Step 3: Calculate the present value of all dividends.
P V =
5
X
t=1
Dt
(1 + r)t+P5
(1 + r)5
=2.50
1.10 +2.50
(1.10)2+2.50
(1.10)3+2.50
(1.10)4+93
(1.10)5
≈2.2727 + 2.0661 + 1.8783 + 1.7075 + 66.3049
≈74.2295
Therefore, the current value of the stock is approximately
$
74.23.
Question 26
Question
A company is expected to pay an annual dividend of 5.50pershareindefinitely.Iftherequiredrateofreturnis10%, whatisthevalueofthestock?
Solution
Step 1: Calculate the present value of the perpetuity using the dividend discount
model.
Stock Value = Dividend
Required Rate of Return
Step 2: Substituting the given values, we get:
Stock Value = 5.50
0.10 = 55
Step 3: Therefore, the value of the stock is 55pershare.
Question 27
Question
A company’s stock is expected to pay dividends of 2.00,2.20, and 2.40overthenextthreeyears.Afterthat, thedividendisexpectedtogrowataconstantrateof5
18
Solution
Step 1: Calculate the present value of the dividends for the first three years.
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
where: D1 = $2.00, D2 = $2.20, D3 = $2.40, r= 0.10.
P V =2.00
(1 + 0.10)1+2.20
(1 + 0.10)2+2.40
(1 + 0.10)3
P V =2.00
1.10 +2.20
1.102+2.40
1.103
P V = 1.8182 + 1.6529 + 1.5026
P V = 4.9737
Step 2: Calculate the present value of the growing perpetuity using the
Gordon Growth Model.
P V =D4
r−g
where: D4 = D3∗(1 + g) = $2.40 ∗(1 + 0.05) = $2.52, r= 0.10, g= 0.05.
P V =2.52
0.10 −0.05
P V =2.52
0.05
P V = 50.40
Step 3: Add the present values of the dividends and the growing perpetuity
to find the current value of the stock.
Current Value = P V +P V
Current Value = 4.9737 + 50.40
Current Value = 55.3737
Therefore, the current value of the stock is
$
55.37.
Question 28
Question
Company XYZ just paid a dividend of 3.50pershare.T heexpectedgrowthratef orthecompanyis5
19
Solution
Step 1: Calculate the expected dividend next year using the formula for the
dividend growth model.
D1 = D0×(1 + g)
D1=3.50 ×(1 + 0.05)
D1=3.50 ×1.05
D1 = 3.675
Step 2: Determine the price of the stock based on the dividend discount
model.
P0 = D1
r−g
P0 = 3.675
0.10 −0.05
P0 = 3.675
0.05
P0 = 73.50
Step 3: Compare the calculated price (73.50)withthecurrentpriceof thestock(70).
- Since the calculated price is higher than the current price, the stock is under-
valued based on the dividend discount model.
Question 29
Question
A company is expected to pay dividends of
$
1.50,
$
1.70, and
$
2.00 over the next
three years. After that, the dividends are expected to grow at a constant rate
of 5
Solution
Step 1: Calculate the present value of the dividends for the next three years.
Step 2: Calculate the present value of the terminal value of the stock after three
years. Step 3: Sum the present values of the dividends and terminal value to
find the current stock price.
Step 1: The present value of the three dividends can be calculated using the
formula for the present value of a series of cash flows:
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
Where: - D1= $1.50 - D2= $1.70 - D3= $2.00 - r= 0.10
Plugging in the values, we get:
20
P V =$1.50
(1 + 0.10)1+$1.70
(1 + 0.10)2+$2.00
(1 + 0.10)3
P V = $1.36 + $1.38 + $1.51
P V = $4.25
So, the present value of the dividends for the next three years is
$
4.25.
Step 2: To calculate the terminal value of the stock after three years, we can
use the Gordon Growth Model formula:
P=D4
r−g
Where: - D4is the dividend in year 4 - g= 0.05 (5
The dividend in year 4 (D4) can be calculated as:
D4=D3×(1 + g)
D4= $2.00 ×(1 + 0.05)
D4= $2.10
Plugging in the values, we get:
P=$2.10
0.10 −0.05
P=$2.10
0.05
P= $42.00
So, the terminal value of the stock after three years is
$
42.00.
Step 3: The current stock price can be found by summing the present value
of dividends for the next three years and the terminal value:
Current Stock Price = $4.25 + $42.00
Current Stock Price = $46.25
Therefore, the current stock price is
$
46.25.
Question 30
Question
Company XYZ just paid a dividend of 5pershare, andtheexpectedgrowthrateforthecompanyis8
21
Question 8
Question
A company is expected to pay an annual dividend of 5pershareindefinitely.Iftherequiredrateofreturnonthestockis10
Solution
Step 1: Calculate the stock price using the Gordon Growth Model formula:
Stock Price = Dividend per share
Required rate of return
Step 2: Substitute the given values into the formula:
Stock Price = 5
0.10 = 50
Answer: The stock price is 50pershare.
Question 9
Question
A company is expected to pay a dividend of 3.50pershareoneyearf romnow.T hedividendisexpectedtogrowataconstantrateof6
Solution
Let’s denote: - D1as the dividend to be paid one year from now - ras the
required rate of return - gas the growth rate of the dividend
Step 1: Calculate the expected dividend one year from now (D1).
D1= $3.50
Step 2: Calculate the price of the stock using the Gordon Growth Model:
P0=D1
r−g
Step 3: Plug in the values into the formula:
P0=$3.50
0.12 −0.06
Step 4: Calculate the current value of the stock:
P0=$3.50
0.06 = $58.33
Therefore, the current value of the stock is
$
58.33.
6
Question 10
Question
A company pays an annual dividend of
$
3 per share, and the dividends are
expected to grow at a constant rate of 5% per year indefinitely. If the required
rate of return is 10%, what is the current value of the stock?
Solution
Step 1: Calculate the growth rate of dividends using the constant growth divi-
dend valuation formula:
g=3×0.05
3= 0.05
Step 2: Use the dividend valuation model to calculate the value of the stock:
P0=D0×(1 + g)
r−g
where: - P0is the current value of the stock, - D0is the most recent dividend per
share, - ris the required rate of return, and - gis the growth rate of dividends.
Step 3: Substitute the values into the formula:
P0=3×(1 + 0.05)
0.10 −0.05
P0=3.15
0.05
P0= 63
Therefore, the current value of the stock is
$
63 per share.
Question 11
Question
A company is expected to pay a dividend of
$
7 next year. Dividends are expected
to grow at a rate of 3
Solution
Step 1: Calculate the dividend for the next year using the given information.
Step 2: Calculate the dividend growth rate. Step 3: Use the dividend discount
model to calculate the value of the stock today.
Step 1: The dividend for the next year is
$
7.
Step 2: The dividend growth rate is 3
7
Step 3: The dividend discount model is given by:
V0=D1
r−g,
where: - V0is the value of the stock today, - D1is the dividend for the next
year, - ris the required rate of return, - gis the dividend growth rate.
Substitute the known values into the formula:
V0=7
0.10 −0.03 =7
0.07 = $100.
Therefore, the value of the stock today is
$
100.
Question 12
Question
A company just paid a dividend of 5pershare.T hedividendisexpectedtogrowataconstantrateof 4
Solution
Step 1: Calculate the growth rate of dividends. Step 2: Use the Gordon Growth
Model to find the current stock price.
Step 1: Calculate the growth rate of dividends. The growth rate of divi-
dends is given as 4
Step 2: Use the Gordon Growth Model to find the current stock price. The
Gordon Growth Model is given by the formula:
P=D1
r−g
where: - P is the current stock price, - D1is the expected dividend one year from
now, - r is the required rate of return, and - g is the growth rate of dividends.
Given: - D1= dividend this year ×(1 + growth rate) = 5 ×(1 + 0.04) =
$
5.20, - r = 10- g = 4
Substitute these values into the formula:
P=5.20
0.10 −0.04 =5.20
0.06 = $86.67
Therefore, the current stock price is
$
86.67.
Question 13
Question
You are considering investing in a company that pays a constant annual dividend
of 12.If therequiredrateofreturnis8%, andthedividendsareexpectedtogrowataconstantrateof5%peryear, whatisthecurrentvalueofthestock?
8
Solution
Step 1: Calculate the dividend growth rate using the formula
g=12 ×0.05
12 = 0.05
Step 2: Calculate the dividend in the next year using the formula
D1=D0×(1 + g) = 12 ×(1 + 0.05) = 12.6
Step 3: Calculate the stock price using the Gordon Growth Model formula
P0=D1
r−g
where P0is the current stock price, D1is the dividend in the next year, ris the
required rate of return, and gis the growth rate of dividends.
Step 4: Substitute the known values into the formula to find the current
stock price
P0=12.6
0.08 −0.05 =12.6
0.03 = 420
Therefore, the current value of the stock is 420.
Question 14
Question
A company’s stock is currently priced at $100 per share. The company is ex-
pected to pay a dividend of $5 per share at the end of the year. If the required
rate of return is 10%, what is the expected growth rate of the company?
Solution
Step 1: Calculate the dividend yield. The dividend yield is the dividend per
share divided by the stock price.
Dividend Yield = $5
$100 = 0.05 = 5%
Step 2: Use the Dividend Discount Model (DDM) to find the expected
growth rate. The DDM formula is:
Next year’s dividend = Current Dividend ×(1 + Growth rate)
Plugging in the values we know:
$5 = $5 ×(1 + Growth rate)
Solving for the growth rate:
1 = 1 + Growth rate
Growth rate = 0
Therefore, the expected growth rate of the company is 0%.
9
Question 15
Question
A company’s stock is currently valued at $50 per share. The company is ex-
pected to pay a dividend of $2 per share next year, with dividends expected
to grow at a constant rate of 5% per year indefinitely. If the required rate of
return on the stock is 10%, what is the intrinsic value of the stock?
Solution
Step 1: Calculate the dividend growth rate. Given that the dividend is expected
to grow at a constant rate of 5% per year, the dividend growth rate can be
calculated as follows:
g= 0.05
Step 2: Use the Gordon Growth Model to calculate the intrinsic value of the
stock. The Gordon Growth Model is given by:
P0=D1
r−g
where: - P0is the intrinsic value of the stock, - D1is the dividend expected
next year, - ris the required rate of return on the stock, and - gis the dividend
growth rate.
Substitute the given values into the formula to find the intrinsic value of the
stock:
P0=2
0.10 −0.05
P0=2
0.05
P0= 40
Therefore, the intrinsic value of the stock is $40 per share.
Question 16
Question
A company’s stock is currently trading at $50 per share. It is expected that
the company will pay a dividend of $2.50 per share next year. If dividends are
expected to grow at a constant rate of 6% per year indefinitely and the required
rate of return is 12%, what is the fair value of the stock?
10
Solution
Step 1: Calculate the expected dividend next year using the constant growth
rate formula:
D1=D0×(1 + g)
where: - D1is the dividend next year, - D0is the current dividend, and - gis
the constant growth rate.
Substitute D0= $2.50 and g= 6% = 0.06:
D1= $2.50 ×(1 + 0.06) = $2.50 ×1.06 = $2.65
Step 2: Calculate the price of the stock using the Gordon Growth Model
formula:
P0=D1
r−g
where: - P0is the fair value of the stock, - D1is the dividend next year, - ris
the required rate of return, and - gis the constant growth rate.
Substitute D1= $2.65, r= 12% = 0.12, and g= 6% = 0.06:
P0=$2.65
0.12 −0.06 =$2.65
0.06 = $44.17
Therefore, the fair value of the stock is $44.17.
Question 17
Question
ABC Corp. is expected to pay a dividend of 2.00 per share next year. Dividends
are expected to grow at a rate of 5% per year indefinitely. If the required return
on ABC Corp. stock is 12%, what is the current stock price?
Solution
Step 1: Calculate the expected dividend in the future. Given that the dividend
next year is D1= 2.00, wecancalculatethedividendinyear2as :D2=D1×(1 +
growth rate) = 2.00 ×(1 + 0.05) =2.10
Step 2: Calculate the required rate of return in decimal form. The required
rate of return is 12% which is 0.12 in decimal form.
Step 3: Calculate the price of the stock using the dividend discount model
(DDM) formula. The price of a stock can be calculated using the formula:
P0=D1
r−g
11
where: - P0= Current stock price, - D1= Dividend expected next year, - r=
Required rate of return, and - g= Growth rate of dividends.
Substitute the given values into the formula:
P0=2.00
0.12 −0.05 =2.00
0.07 =
28.57
Therefore, the current stock price of ABC Corp. is 28.57pershare.
Question 18
Question
A company is expected to pay a dividend of
$
5 per share next year. Analysts
expect the company’s dividends to grow at a rate of 3% per year indefinitely. If
the required rate of return on the stock is 10%, what is the current price of the
stock?
Solution
Step 1: Calculate the dividend expected in year 2. Given that the dividend is
expected to grow at a rate of 3% per year, the dividend expected in year 2 can
be calculated as:
D2=D1×(1 + g)
where: - D1= $5 (dividend next year) - g= 3% = 0.03 (growth rate)
D2= $5 ×(1 + 0.03) = $5.15
Step 2: Determine the price of the stock based on the growing perpetuity
formula. The price of the stock can be calculated using the growing perpetuity
formula:
P=D1
r−g
where: - D1= $5 (dividend next year) - r= 10% = 0.10 (required rate of
return) - g= 3% = 0.03 (growth rate) Substitute the values into the formula:
P=$5.15
0.10 −0.03
P=$5.15
0.07
P= $73.57
Therefore, the current price of the stock is
$
73.57.
12
Question 19
Question
A company is expected to pay a dividend of 5.00pershareoneyearfromnow.Afterthat, thedividendsareexpectedtogrowataconstantrateof8
Solution
Let’s denote: - D1= $5.00 as the dividend to be paid one year from now, -
g=8%=0.08 as the constant growth rate of dividends, - r= 12% = 0.12 as
the required rate of return, - P0as the current value of the stock.
The current value of the stock can be calculated using the Gordon Growth
Model formula:
P0=D1
r−g
Step 1: Calculate the current value of the stock using the Gordon Growth
Model formula.
P0=5.00
0.12 −0.08
P0=5.00
0.04
P0= $125.00
Therefore, the current value of the stock is $125.00.
Question 20
Question
A company’s stock is expected to pay dividends of
$
2,
$
2.50, and
$
3 for the
next three years. After that, dividends are expected to grow at a constant rate
of 5% per year indefinitely. If the required rate of return on the stock is 8%,
what is the current stock price?
Solution
Step 1: Calculate the present value of dividends for the next three years.
PV of dividends = D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
=2
(1 + 0.08)1+2.5
(1 + 0.08)2+3
(1 + 0.08)3
=2
1.08 +2.5
1.082+3
1.083
≈1.85 + 2.17 + 2.49
≈6.51
13
Step 2: Calculate the price of the stock after three years when it starts
growing at a constant rate.
Price after 3 years = D3×(1 + g)
r−g
=3×(1 + 0.05)
0.08 −0.05
=3×1.05
0.03
=3.15
0.03
= 105
Step 3: Calculate the present value of the price after three years.
PV of price after 3 years = 105
(1 + 0.08)3
=105
1.259712
≈83.48
Step 4: Calculate the current stock price by adding the present value of
dividends and the present value of the price after three years.
Current stock price = PV of dividends + PV of price after 3 years
≈6.51 + 83.48
≈89.99
Therefore, the current stock price is approximately 89.99.
Question 21
Question
A company is expected to pay a dividend of 3.00nextyear.Dividendsareexpectedtogrowatarateof5
Solution
Step 1: Calculate the dividend in one year using the dividend growth model.
Dividend in one year = D0×(1 + g) = 3.00 ×(1 + 0.05) = 3.15
Step 2: Calculate the price of the stock using the dividend discount model.
The price of the stock can be calculated as the present value of all future divi-
dends.
Price of the stock = D1
r−g
14
Where: - D1 = Dividend in one year = 3.15 −r= Required rate of return
= 12- g= Growth rate of dividends = 5
Price of the stock = 3.15
0.12 −0.05
Price of the stock = 3.15
0.07
Price of the stock = 45
Therefore, the value of the stock today is 45.
Question 22
Question
A company is expected to pay a dividend of 5 at the end of the year. Dividends
are expected to grow at a rate of 8% per year indefinitely. If the required rate
of return on the stock is 12%, what is the current value of the stock?
Solution
Step 1: Calculate the dividend at the end of Year 1. The dividend at the end
of Year 1 can be calculated as:
D1=D0×(1 + g) = 5 ×(1 + 0.08) = 5.4
Step 2: Determine the dividend yield. The dividend yield is the ratio of the
most recent dividend to the current stock price, which can be calculated using
the formula:
Dividend Yield = D1
P0
Step 3: Apply the Gordon Growth Model to find the current value of the
stock. The Gordon Growth Model is used to calculate the value of a stock based
on the present value of future dividends. The formula is:
P0=D1
r−g
where: - P0is the current value of the stock, - D1is the dividend at the end of
Year 1, - ris the required rate of return on the stock, and - gis the growth rate
of dividends.
Substitute the values we have:
P0=5.4
0.12 −0.08 =5.4
0.04 = 135
Therefore, the current value of the stock is 135.
15
Question 23
Question
A company’s stock is expected to pay a dividend of 3 dollars one year from now.
The dividend is expected to grow at a rate of 5% per year indefinitely. If the
required rate of return for the stock is 10%, what is the current value of the
stock?
Solution
Step 1: Calculate the expected dividend in year 2. Step 2: Calculate the present
value of the dividends. Step 3: Calculate the current value of the stock.
Step 1: The expected dividend in year 2 can be calculated using the growth
rate:
D2=D1×(1 + g)
D2= 3 ×(1 + 0.05)
D2= 3 ×1.05 = 3.15
Step 2: To calculate the present value of the dividends, we use the formula
for the present value of a growing perpetuity:
P V =D
r−g
where: P V = Present value of dividends, D= Dividend in year 1, r= Required
rate of return, g= Growth rate.
Plugging in the values, we get:
P V =3
0.10 −0.05
P V =3
0.05 = 60
Step 3: The current value of the stock is equal to the present value of the
dividends plus the present value of the stock one year from now:
CurrentV alue =P V +D2/(1 + r)
CurrentV alue = 60 + 3.15/(1 + 0.10)
CurrentV alue = 60 + 3.15/1.10 = 60 + 2.8636 = 62.8636
Therefore, the current value of the stock is 62.86.
Question 24
Question
A company pays an annual dividend of 3.50pershareandisexpectedtogrowataconstantrateof5
16
Solution
Step 1: Calculate the dividend growth rate. Step 2: Use the constant growth
dividend valuation model to find the current stock price.
Step 1: Calculate the dividend growth rate. Given that the dividend growth
rate is 5
Step 2: Calculate the current stock price using the constant growth dividend
valuation model. The formula for the constant growth dividend valuation model
is:
P0=D1
r−g
where: - P0is the current stock price, - D1is the dividend expected one year
from now, - ris the required rate of return, and - gis the growth rate of
dividends.
Given: - D1=D0×(1 + g)=3.50 ×(1 + 0.05) = $3.675 - r= 0.10 - g= 0.05
Substitute these values into the formula:
P0=3.675
0.10 −0.05 =3.675
0.05 = $73.50
Therefore, the current stock price is
$
73.50.
Question 25
Question
A company is expected to pay an annual dividend of 2.50forthenextfiveyears.Af terthat, dividendsareexpectedtogrowatarateof6
Solution
Let’s denote the expected dividend for year 5 as D5. Then, the dividend for
year 6 and beyond can be expressed as D6 = D5×(1+ g), where gis the growth
rate of dividends (6
Step 1: Calculate the dividend for year 5.
D5 = D4×(1 + g)
=D3×(1 + g)2
=D2×(1 + g)3
=D1×(1 + g)4
= 2.50 ×(1 + 0.06)4
≈3.5070
17
Step 2: Calculate the price of the stock at the end of year 5.
P5 = D6
r−g
=D5×(1 + g)
r−g
=3.5070 ×(1 + 0.06)
0.10 −0.06
=3.72
0.04
= 93
Step 3: Calculate the present value of all dividends.
P V =
5
X
t=1
Dt
(1 + r)t+P5
(1 + r)5
=2.50
1.10 +2.50
(1.10)2+2.50
(1.10)3+2.50
(1.10)4+93
(1.10)5
≈2.2727 + 2.0661 + 1.8783 + 1.7075 + 66.3049
≈74.2295
Therefore, the current value of the stock is approximately
$
74.23.
Question 26
Question
A company is expected to pay an annual dividend of 5.50pershareindefinitely.Iftherequiredrateofreturnis10%, whatisthevalueofthestock?
Solution
Step 1: Calculate the present value of the perpetuity using the dividend discount
model.
Stock Value = Dividend
Required Rate of Return
Step 2: Substituting the given values, we get:
Stock Value = 5.50
0.10 = 55
Step 3: Therefore, the value of the stock is 55pershare.
Question 27
Question
A company’s stock is expected to pay dividends of 2.00,2.20, and 2.40overthenextthreeyears.Afterthat, thedividendisexpectedtogrowataconstantrateof5
18
Solution
Step 1: Calculate the present value of the dividends for the first three years.
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
where: D1 = $2.00, D2 = $2.20, D3 = $2.40, r= 0.10.
P V =2.00
(1 + 0.10)1+2.20
(1 + 0.10)2+2.40
(1 + 0.10)3
P V =2.00
1.10 +2.20
1.102+2.40
1.103
P V = 1.8182 + 1.6529 + 1.5026
P V = 4.9737
Step 2: Calculate the present value of the growing perpetuity using the
Gordon Growth Model.
P V =D4
r−g
where: D4 = D3∗(1 + g) = $2.40 ∗(1 + 0.05) = $2.52, r= 0.10, g= 0.05.
P V =2.52
0.10 −0.05
P V =2.52
0.05
P V = 50.40
Step 3: Add the present values of the dividends and the growing perpetuity
to find the current value of the stock.
Current Value = P V +P V
Current Value = 4.9737 + 50.40
Current Value = 55.3737
Therefore, the current value of the stock is
$
55.37.
Question 28
Question
Company XYZ just paid a dividend of 3.50pershare.T heexpectedgrowthratef orthecompanyis5
19
Solution
Step 1: Calculate the expected dividend next year using the formula for the
dividend growth model.
D1 = D0×(1 + g)
D1=3.50 ×(1 + 0.05)
D1=3.50 ×1.05
D1 = 3.675
Step 2: Determine the price of the stock based on the dividend discount
model.
P0 = D1
r−g
P0 = 3.675
0.10 −0.05
P0 = 3.675
0.05
P0 = 73.50
Step 3: Compare the calculated price (73.50)withthecurrentpriceof thestock(70).
- Since the calculated price is higher than the current price, the stock is under-
valued based on the dividend discount model.
Question 29
Question
A company is expected to pay dividends of
$
1.50,
$
1.70, and
$
2.00 over the next
three years. After that, the dividends are expected to grow at a constant rate
of 5
Solution
Step 1: Calculate the present value of the dividends for the next three years.
Step 2: Calculate the present value of the terminal value of the stock after three
years. Step 3: Sum the present values of the dividends and terminal value to
find the current stock price.
Step 1: The present value of the three dividends can be calculated using the
formula for the present value of a series of cash flows:
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
Where: - D1= $1.50 - D2= $1.70 - D3= $2.00 - r= 0.10
Plugging in the values, we get:
20
P V =$1.50
(1 + 0.10)1+$1.70
(1 + 0.10)2+$2.00
(1 + 0.10)3
P V = $1.36 + $1.38 + $1.51
P V = $4.25
So, the present value of the dividends for the next three years is
$
4.25.
Step 2: To calculate the terminal value of the stock after three years, we can
use the Gordon Growth Model formula:
P=D4
r−g
Where: - D4is the dividend in year 4 - g= 0.05 (5
The dividend in year 4 (D4) can be calculated as:
D4=D3×(1 + g)
D4= $2.00 ×(1 + 0.05)
D4= $2.10
Plugging in the values, we get:
P=$2.10
0.10 −0.05
P=$2.10
0.05
P= $42.00
So, the terminal value of the stock after three years is
$
42.00.
Step 3: The current stock price can be found by summing the present value
of dividends for the next three years and the terminal value:
Current Stock Price = $4.25 + $42.00
Current Stock Price = $46.25
Therefore, the current stock price is
$
46.25.
Question 30
Question
Company XYZ just paid a dividend of 5pershare, andtheexpectedgrowthrateforthecompanyis8
21
Question 8
Question
A company is expected to pay an annual dividend of 5pershareindefinitely.Iftherequiredrateofreturnonthestockis10
Solution
Step 1: Calculate the stock price using the Gordon Growth Model formula:
Stock Price = Dividend per share
Required rate of return
Step 2: Substitute the given values into the formula:
Stock Price = 5
0.10 = 50
Answer: The stock price is 50pershare.
Question 9
Question
A company is expected to pay a dividend of 3.50pershareoneyearf romnow.T hedividendisexpectedtogrowataconstantrateof6
Solution
Let’s denote: - D1as the dividend to be paid one year from now - ras the
required rate of return - gas the growth rate of the dividend
Step 1: Calculate the expected dividend one year from now (D1).
D1= $3.50
Step 2: Calculate the price of the stock using the Gordon Growth Model:
P0=D1
r−g
Step 3: Plug in the values into the formula:
P0=$3.50
0.12 −0.06
Step 4: Calculate the current value of the stock:
P0=$3.50
0.06 = $58.33
Therefore, the current value of the stock is
$
58.33.
6
Question 10
Question
A company pays an annual dividend of
$
3 per share, and the dividends are
expected to grow at a constant rate of 5% per year indefinitely. If the required
rate of return is 10%, what is the current value of the stock?
Solution
Step 1: Calculate the growth rate of dividends using the constant growth divi-
dend valuation formula:
g=3×0.05
3= 0.05
Step 2: Use the dividend valuation model to calculate the value of the stock:
P0=D0×(1 + g)
r−g
where: - P0is the current value of the stock, - D0is the most recent dividend per
share, - ris the required rate of return, and - gis the growth rate of dividends.
Step 3: Substitute the values into the formula:
P0=3×(1 + 0.05)
0.10 −0.05
P0=3.15
0.05
P0= 63
Therefore, the current value of the stock is
$
63 per share.
Question 11
Question
A company is expected to pay a dividend of
$
7 next year. Dividends are expected
to grow at a rate of 3
Solution
Step 1: Calculate the dividend for the next year using the given information.
Step 2: Calculate the dividend growth rate. Step 3: Use the dividend discount
model to calculate the value of the stock today.
Step 1: The dividend for the next year is
$
7.
Step 2: The dividend growth rate is 3
7
Step 3: The dividend discount model is given by:
V0=D1
r−g,
where: - V0is the value of the stock today, - D1is the dividend for the next
year, - ris the required rate of return, - gis the dividend growth rate.
Substitute the known values into the formula:
V0=7
0.10 −0.03 =7
0.07 = $100.
Therefore, the value of the stock today is
$
100.
Question 12
Question
A company just paid a dividend of 5pershare.T hedividendisexpectedtogrowataconstantrateof 4
Solution
Step 1: Calculate the growth rate of dividends. Step 2: Use the Gordon Growth
Model to find the current stock price.
Step 1: Calculate the growth rate of dividends. The growth rate of divi-
dends is given as 4
Step 2: Use the Gordon Growth Model to find the current stock price. The
Gordon Growth Model is given by the formula:
P=D1
r−g
where: - P is the current stock price, - D1is the expected dividend one year from
now, - r is the required rate of return, and - g is the growth rate of dividends.
Given: - D1= dividend this year ×(1 + growth rate) = 5 ×(1 + 0.04) =
$
5.20, - r = 10- g = 4
Substitute these values into the formula:
P=5.20
0.10 −0.04 =5.20
0.06 = $86.67
Therefore, the current stock price is
$
86.67.
Question 13
Question
You are considering investing in a company that pays a constant annual dividend
of 12.If therequiredrateofreturnis8%, andthedividendsareexpectedtogrowataconstantrateof5%peryear, whatisthecurrentvalueofthestock?
8
Solution
Step 1: Calculate the dividend growth rate using the formula
g=12 ×0.05
12 = 0.05
Step 2: Calculate the dividend in the next year using the formula
D1=D0×(1 + g) = 12 ×(1 + 0.05) = 12.6
Step 3: Calculate the stock price using the Gordon Growth Model formula
P0=D1
r−g
where P0is the current stock price, D1is the dividend in the next year, ris the
required rate of return, and gis the growth rate of dividends.
Step 4: Substitute the known values into the formula to find the current
stock price
P0=12.6
0.08 −0.05 =12.6
0.03 = 420
Therefore, the current value of the stock is 420.
Question 14
Question
A company’s stock is currently priced at $100 per share. The company is ex-
pected to pay a dividend of $5 per share at the end of the year. If the required
rate of return is 10%, what is the expected growth rate of the company?
Solution
Step 1: Calculate the dividend yield. The dividend yield is the dividend per
share divided by the stock price.
Dividend Yield = $5
$100 = 0.05 = 5%
Step 2: Use the Dividend Discount Model (DDM) to find the expected
growth rate. The DDM formula is:
Next year’s dividend = Current Dividend ×(1 + Growth rate)
Plugging in the values we know:
$5 = $5 ×(1 + Growth rate)
Solving for the growth rate:
1 = 1 + Growth rate
Growth rate = 0
Therefore, the expected growth rate of the company is 0%.
9
Question 15
Question
A company’s stock is currently valued at $50 per share. The company is ex-
pected to pay a dividend of $2 per share next year, with dividends expected
to grow at a constant rate of 5% per year indefinitely. If the required rate of
return on the stock is 10%, what is the intrinsic value of the stock?
Solution
Step 1: Calculate the dividend growth rate. Given that the dividend is expected
to grow at a constant rate of 5% per year, the dividend growth rate can be
calculated as follows:
g= 0.05
Step 2: Use the Gordon Growth Model to calculate the intrinsic value of the
stock. The Gordon Growth Model is given by:
P0=D1
r−g
where: - P0is the intrinsic value of the stock, - D1is the dividend expected
next year, - ris the required rate of return on the stock, and - gis the dividend
growth rate.
Substitute the given values into the formula to find the intrinsic value of the
stock:
P0=2
0.10 −0.05
P0=2
0.05
P0= 40
Therefore, the intrinsic value of the stock is $40 per share.
Question 16
Question
A company’s stock is currently trading at $50 per share. It is expected that
the company will pay a dividend of $2.50 per share next year. If dividends are
expected to grow at a constant rate of 6% per year indefinitely and the required
rate of return is 12%, what is the fair value of the stock?
10
Solution
Step 1: Calculate the expected dividend next year using the constant growth
rate formula:
D1=D0×(1 + g)
where: - D1is the dividend next year, - D0is the current dividend, and - gis
the constant growth rate.
Substitute D0= $2.50 and g= 6% = 0.06:
D1= $2.50 ×(1 + 0.06) = $2.50 ×1.06 = $2.65
Step 2: Calculate the price of the stock using the Gordon Growth Model
formula:
P0=D1
r−g
where: - P0is the fair value of the stock, - D1is the dividend next year, - ris
the required rate of return, and - gis the constant growth rate.
Substitute D1= $2.65, r= 12% = 0.12, and g= 6% = 0.06:
P0=$2.65
0.12 −0.06 =$2.65
0.06 = $44.17
Therefore, the fair value of the stock is $44.17.
Question 17
Question
ABC Corp. is expected to pay a dividend of 2.00 per share next year. Dividends
are expected to grow at a rate of 5% per year indefinitely. If the required return
on ABC Corp. stock is 12%, what is the current stock price?
Solution
Step 1: Calculate the expected dividend in the future. Given that the dividend
next year is D1= 2.00, wecancalculatethedividendinyear2as :D2=D1×(1 +
growth rate) = 2.00 ×(1 + 0.05) =2.10
Step 2: Calculate the required rate of return in decimal form. The required
rate of return is 12% which is 0.12 in decimal form.
Step 3: Calculate the price of the stock using the dividend discount model
(DDM) formula. The price of a stock can be calculated using the formula:
P0=D1
r−g
11
where: - P0= Current stock price, - D1= Dividend expected next year, - r=
Required rate of return, and - g= Growth rate of dividends.
Substitute the given values into the formula:
P0=2.00
0.12 −0.05 =2.00
0.07 =
28.57
Therefore, the current stock price of ABC Corp. is 28.57pershare.
Question 18
Question
A company is expected to pay a dividend of
$
5 per share next year. Analysts
expect the company’s dividends to grow at a rate of 3% per year indefinitely. If
the required rate of return on the stock is 10%, what is the current price of the
stock?
Solution
Step 1: Calculate the dividend expected in year 2. Given that the dividend is
expected to grow at a rate of 3% per year, the dividend expected in year 2 can
be calculated as:
D2=D1×(1 + g)
where: - D1= $5 (dividend next year) - g= 3% = 0.03 (growth rate)
D2= $5 ×(1 + 0.03) = $5.15
Step 2: Determine the price of the stock based on the growing perpetuity
formula. The price of the stock can be calculated using the growing perpetuity
formula:
P=D1
r−g
where: - D1= $5 (dividend next year) - r= 10% = 0.10 (required rate of
return) - g= 3% = 0.03 (growth rate) Substitute the values into the formula:
P=$5.15
0.10 −0.03
P=$5.15
0.07
P= $73.57
Therefore, the current price of the stock is
$
73.57.
12
Question 19
Question
A company is expected to pay a dividend of 5.00pershareoneyearfromnow.Afterthat, thedividendsareexpectedtogrowataconstantrateof8
Solution
Let’s denote: - D1= $5.00 as the dividend to be paid one year from now, -
g=8%=0.08 as the constant growth rate of dividends, - r= 12% = 0.12 as
the required rate of return, - P0as the current value of the stock.
The current value of the stock can be calculated using the Gordon Growth
Model formula:
P0=D1
r−g
Step 1: Calculate the current value of the stock using the Gordon Growth
Model formula.
P0=5.00
0.12 −0.08
P0=5.00
0.04
P0= $125.00
Therefore, the current value of the stock is $125.00.
Question 20
Question
A company’s stock is expected to pay dividends of
$
2,
$
2.50, and
$
3 for the
next three years. After that, dividends are expected to grow at a constant rate
of 5% per year indefinitely. If the required rate of return on the stock is 8%,
what is the current stock price?
Solution
Step 1: Calculate the present value of dividends for the next three years.
PV of dividends = D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
=2
(1 + 0.08)1+2.5
(1 + 0.08)2+3
(1 + 0.08)3
=2
1.08 +2.5
1.082+3
1.083
≈1.85 + 2.17 + 2.49
≈6.51
13
Step 2: Calculate the price of the stock after three years when it starts
growing at a constant rate.
Price after 3 years = D3×(1 + g)
r−g
=3×(1 + 0.05)
0.08 −0.05
=3×1.05
0.03
=3.15
0.03
= 105
Step 3: Calculate the present value of the price after three years.
PV of price after 3 years = 105
(1 + 0.08)3
=105
1.259712
≈83.48
Step 4: Calculate the current stock price by adding the present value of
dividends and the present value of the price after three years.
Current stock price = PV of dividends + PV of price after 3 years
≈6.51 + 83.48
≈89.99
Therefore, the current stock price is approximately 89.99.
Question 21
Question
A company is expected to pay a dividend of 3.00nextyear.Dividendsareexpectedtogrowatarateof5
Solution
Step 1: Calculate the dividend in one year using the dividend growth model.
Dividend in one year = D0×(1 + g) = 3.00 ×(1 + 0.05) = 3.15
Step 2: Calculate the price of the stock using the dividend discount model.
The price of the stock can be calculated as the present value of all future divi-
dends.
Price of the stock = D1
r−g
14
Where: - D1 = Dividend in one year = 3.15 −r= Required rate of return
= 12- g= Growth rate of dividends = 5
Price of the stock = 3.15
0.12 −0.05
Price of the stock = 3.15
0.07
Price of the stock = 45
Therefore, the value of the stock today is 45.
Question 22
Question
A company is expected to pay a dividend of 5 at the end of the year. Dividends
are expected to grow at a rate of 8% per year indefinitely. If the required rate
of return on the stock is 12%, what is the current value of the stock?
Solution
Step 1: Calculate the dividend at the end of Year 1. The dividend at the end
of Year 1 can be calculated as:
D1=D0×(1 + g) = 5 ×(1 + 0.08) = 5.4
Step 2: Determine the dividend yield. The dividend yield is the ratio of the
most recent dividend to the current stock price, which can be calculated using
the formula:
Dividend Yield = D1
P0
Step 3: Apply the Gordon Growth Model to find the current value of the
stock. The Gordon Growth Model is used to calculate the value of a stock based
on the present value of future dividends. The formula is:
P0=D1
r−g
where: - P0is the current value of the stock, - D1is the dividend at the end of
Year 1, - ris the required rate of return on the stock, and - gis the growth rate
of dividends.
Substitute the values we have:
P0=5.4
0.12 −0.08 =5.4
0.04 = 135
Therefore, the current value of the stock is 135.
15
Question 23
Question
A company’s stock is expected to pay a dividend of 3 dollars one year from now.
The dividend is expected to grow at a rate of 5% per year indefinitely. If the
required rate of return for the stock is 10%, what is the current value of the
stock?
Solution
Step 1: Calculate the expected dividend in year 2. Step 2: Calculate the present
value of the dividends. Step 3: Calculate the current value of the stock.
Step 1: The expected dividend in year 2 can be calculated using the growth
rate:
D2=D1×(1 + g)
D2= 3 ×(1 + 0.05)
D2= 3 ×1.05 = 3.15
Step 2: To calculate the present value of the dividends, we use the formula
for the present value of a growing perpetuity:
P V =D
r−g
where: P V = Present value of dividends, D= Dividend in year 1, r= Required
rate of return, g= Growth rate.
Plugging in the values, we get:
P V =3
0.10 −0.05
P V =3
0.05 = 60
Step 3: The current value of the stock is equal to the present value of the
dividends plus the present value of the stock one year from now:
CurrentV alue =P V +D2/(1 + r)
CurrentV alue = 60 + 3.15/(1 + 0.10)
CurrentV alue = 60 + 3.15/1.10 = 60 + 2.8636 = 62.8636
Therefore, the current value of the stock is 62.86.
Question 24
Question
A company pays an annual dividend of 3.50pershareandisexpectedtogrowataconstantrateof5
16
Solution
Step 1: Calculate the dividend growth rate. Step 2: Use the constant growth
dividend valuation model to find the current stock price.
Step 1: Calculate the dividend growth rate. Given that the dividend growth
rate is 5
Step 2: Calculate the current stock price using the constant growth dividend
valuation model. The formula for the constant growth dividend valuation model
is:
P0=D1
r−g
where: - P0is the current stock price, - D1is the dividend expected one year
from now, - ris the required rate of return, and - gis the growth rate of
dividends.
Given: - D1=D0×(1 + g)=3.50 ×(1 + 0.05) = $3.675 - r= 0.10 - g= 0.05
Substitute these values into the formula:
P0=3.675
0.10 −0.05 =3.675
0.05 = $73.50
Therefore, the current stock price is
$
73.50.
Question 25
Question
A company is expected to pay an annual dividend of 2.50forthenextfiveyears.Af terthat, dividendsareexpectedtogrowatarateof6
Solution
Let’s denote the expected dividend for year 5 as D5. Then, the dividend for
year 6 and beyond can be expressed as D6 = D5×(1+ g), where gis the growth
rate of dividends (6
Step 1: Calculate the dividend for year 5.
D5 = D4×(1 + g)
=D3×(1 + g)2
=D2×(1 + g)3
=D1×(1 + g)4
= 2.50 ×(1 + 0.06)4
≈3.5070
17
Step 2: Calculate the price of the stock at the end of year 5.
P5 = D6
r−g
=D5×(1 + g)
r−g
=3.5070 ×(1 + 0.06)
0.10 −0.06
=3.72
0.04
= 93
Step 3: Calculate the present value of all dividends.
P V =
5
X
t=1
Dt
(1 + r)t+P5
(1 + r)5
=2.50
1.10 +2.50
(1.10)2+2.50
(1.10)3+2.50
(1.10)4+93
(1.10)5
≈2.2727 + 2.0661 + 1.8783 + 1.7075 + 66.3049
≈74.2295
Therefore, the current value of the stock is approximately
$
74.23.
Question 26
Question
A company is expected to pay an annual dividend of 5.50pershareindefinitely.Iftherequiredrateofreturnis10%, whatisthevalueofthestock?
Solution
Step 1: Calculate the present value of the perpetuity using the dividend discount
model.
Stock Value = Dividend
Required Rate of Return
Step 2: Substituting the given values, we get:
Stock Value = 5.50
0.10 = 55
Step 3: Therefore, the value of the stock is 55pershare.
Question 27
Question
A company’s stock is expected to pay dividends of 2.00,2.20, and 2.40overthenextthreeyears.Afterthat, thedividendisexpectedtogrowataconstantrateof5
18
Solution
Step 1: Calculate the present value of the dividends for the first three years.
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
where: D1 = $2.00, D2 = $2.20, D3 = $2.40, r= 0.10.
P V =2.00
(1 + 0.10)1+2.20
(1 + 0.10)2+2.40
(1 + 0.10)3
P V =2.00
1.10 +2.20
1.102+2.40
1.103
P V = 1.8182 + 1.6529 + 1.5026
P V = 4.9737
Step 2: Calculate the present value of the growing perpetuity using the
Gordon Growth Model.
P V =D4
r−g
where: D4 = D3∗(1 + g) = $2.40 ∗(1 + 0.05) = $2.52, r= 0.10, g= 0.05.
P V =2.52
0.10 −0.05
P V =2.52
0.05
P V = 50.40
Step 3: Add the present values of the dividends and the growing perpetuity
to find the current value of the stock.
Current Value = P V +P V
Current Value = 4.9737 + 50.40
Current Value = 55.3737
Therefore, the current value of the stock is
$
55.37.
Question 28
Question
Company XYZ just paid a dividend of 3.50pershare.T heexpectedgrowthratef orthecompanyis5
19
Solution
Step 1: Calculate the expected dividend next year using the formula for the
dividend growth model.
D1 = D0×(1 + g)
D1=3.50 ×(1 + 0.05)
D1=3.50 ×1.05
D1 = 3.675
Step 2: Determine the price of the stock based on the dividend discount
model.
P0 = D1
r−g
P0 = 3.675
0.10 −0.05
P0 = 3.675
0.05
P0 = 73.50
Step 3: Compare the calculated price (73.50)withthecurrentpriceof thestock(70).
- Since the calculated price is higher than the current price, the stock is under-
valued based on the dividend discount model.
Question 29
Question
A company is expected to pay dividends of
$
1.50,
$
1.70, and
$
2.00 over the next
three years. After that, the dividends are expected to grow at a constant rate
of 5
Solution
Step 1: Calculate the present value of the dividends for the next three years.
Step 2: Calculate the present value of the terminal value of the stock after three
years. Step 3: Sum the present values of the dividends and terminal value to
find the current stock price.
Step 1: The present value of the three dividends can be calculated using the
formula for the present value of a series of cash flows:
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
Where: - D1= $1.50 - D2= $1.70 - D3= $2.00 - r= 0.10
Plugging in the values, we get:
20
P V =$1.50
(1 + 0.10)1+$1.70
(1 + 0.10)2+$2.00
(1 + 0.10)3
P V = $1.36 + $1.38 + $1.51
P V = $4.25
So, the present value of the dividends for the next three years is
$
4.25.
Step 2: To calculate the terminal value of the stock after three years, we can
use the Gordon Growth Model formula:
P=D4
r−g
Where: - D4is the dividend in year 4 - g= 0.05 (5
The dividend in year 4 (D4) can be calculated as:
D4=D3×(1 + g)
D4= $2.00 ×(1 + 0.05)
D4= $2.10
Plugging in the values, we get:
P=$2.10
0.10 −0.05
P=$2.10
0.05
P= $42.00
So, the terminal value of the stock after three years is
$
42.00.
Step 3: The current stock price can be found by summing the present value
of dividends for the next three years and the terminal value:
Current Stock Price = $4.25 + $42.00
Current Stock Price = $46.25
Therefore, the current stock price is
$
46.25.
Question 30
Question
Company XYZ just paid a dividend of 5pershare, andtheexpectedgrowthrateforthecompanyis8
21
Solution
Step 1: Calculate the dividend growth rate. Step 2: Use the dividend discount
model to find the current stock price.
Step 1: To calculate the dividend growth rate, we use the formula:
Dividend Growth Rate = Dividend per Share at Year 1 −Dividend per Share at Year 0
Dividend per Share at Year 0
Given that the dividend per share at Year 1 is 5 ×(1 + 0.08) =5.40, we can
now calculate the growth rate:
Dividend Growth Rate = 5.40 −5
5=0.40
5= 0.08 = 8%
Step 2: Using the dividend discount model formula:
Stock Price = Dividend per Share at Year 1
Required Rate of Return - Dividend Growth Rate
Substitute the values into the formula:
Stock Price = 5.40
0.12 −0.08 =5.40
0.04 = 135
Therefore, the current stock price for Company XYZ is 135pershare.
22
