BUSI 683 - MONEY AND CAPITAL
MARKETS - Stock Valuation
Question Bank - Set 5
Liberty University
Question 1
Question
A company’s stock is currently trading at $120 per share. The company pays
an annual dividend of $4 per share, and dividends are expected to grow at a
rate of 5% per year. If the required rate of return on the stock is 8%, what is
the stock’s intrinsic value using the Dividend Discount Model (DDM)?
Solution
Let P0be the intrinsic value of the stock, D0be the current dividend, gbe the
growth rate of dividends, and rbe the required rate of return on the stock.
Step 1: Calculate the expected dividend next year D1
D1=D0×(1 + g) = $4 ×(1 + 0.05) = $4.20
Step 2: Calculate the intrinsic value of the stock using the DDM formula:
P0=D1
r−g
P0=$4.20
0.08 −0.05
P0=$4.20
0.03
P0= $140
Therefore, the intrinsic value of the stock using the Dividend Discount Model
(DDM) is
$
140.
Question 2
Question
Suppose a company pays an annual dividend of 4.50pershareandisexpectedtogrowitsdividendsataconstantrateof 5
Solution
Let’s use the Gordon Growth Model to determine the current value of the stock:
The Gordon Growth Model formula is:
P=D0×(1 + g)
r−g
Where: - P= current price of the stock - D0= the most recent dividend per
share - r= required rate of return - g= growth rate of dividends
Step 1: Plug in the given values. Given: D0= 4.50, g= 0.05, r= 0.10
Step 2: Calculate the current value of the stock.
P=4.50 ×(1 + 0.05)
0.10 −0.05
P=4.50 ×1.05
0.05
P=4.725
0.05
P= 94.50
Therefore, the current value of the stock is 94.50pershare.
Question 3
Question
You are analyzing a company that is expected to pay a dividend of
$
2.50 per
share next year. The dividends are expected to grow at a rate of 5
Solution
Step 1: Calculate the expected dividend next year based on the growth rate.
Step 2: Use the dividend discount model to find the current value of the stock.
Step 1: Given the dividend next year (D1) =
$
2.50 and the growth rate (g)
= 5
D2 = D1×(1 + g)=2.50 ×(1 + 0.05) = 2.50 ×1.05 = $2.625
Step 2: To find the current value of the stock (P0) using the dividend dis-
count model:
P0 = D1
r−g
2
Where: - P0 = Current value of the stock - D1 = Dividend next year - r =
Required rate of return - g = Growth rate
Plugging in the values:
P0 = 2.50
0.12 −0.05 =2.50
0.07 = $35.71
Therefore, the current value of the stock is
$
35.71.
Question 4
Question
A company’s stock is expected to pay a dividend of 7.50 next year, with a
growth rate of 3%. If the required rate of return on the stock is 10%, what is
the current value of the stock?
Solution
Step 1: Calculate the dividend in two years using the growth rate.
D2=D1×(1 + g)
D2= 7.50 ×(1 + 0.03)
D2= 7.50 ×1.03 = 7.725
Step 2: Calculate the expected dividend for year 3.
D3=D2×(1 + g)
D3= 7.725 ×(1 + 0.03)
D3= 7.725 ×1.03 = 7.93875
Step 3: Use the Gordon Growth Model to find the current value of the stock.
P0=D1
r−g
P0=7.50
0.10 −0.03
P0=7.50
0.07
P0= 107.14
Therefore, the current value of the stock is 107.14.
3
Question 5
Question
Company XYZ just paid a dividend of 3pershare.T hedividendisexpectedtogrowataconstantrateof 5
Solution
Step 1: Identify the given values.
Given:
Dividend per share (D) = 3Dividendgrowthrate(g) = 5
Required rate of return (r) = 10
Step 2: Calculate the price of the stock using the Gordon Growth Model.
The Gordon Growth Model is given by:
P0=D0×(1 + g)
r−g
where:
P0= Price of the stock today
D0= Dividend per share today
Step 3: Substitute the values into the Gordon Growth Model.
Substitute D0= 3, g = 0.05, r = 0.10 into the formula:
P0=3×(1 + 0.05)
0.10 −0.05
Step 4: Calculate the price of the stock.
P0=3×1.05
0.05 =3.15
0.05 = 63
Therefore, the current price of the stock is
$
63.
Question 6
Question
Assume a company is expected to pay an annual dividend of 3.50forever, andtherequiredrateofreturnis8
4
Solution
Let D0be the current dividend, P0be the current stock price, rbe the required
rate of return, and gbe the expected growth rate of dividends. The constant
growth model formula for stock valuation is given by:
P0=D0
r−g
Step 1: Given that D0= 3.50, P0= 70, and r= 0.08, we can substitute
the values into the formula:
70 = 3.50
0.08 −g
Step 2: Solving for g, we have:
70(0.08 −g)=3.50
5.6−70g= 3.50
70g= 5.6−3.50
70g= 2.10
g=2.10
70
g= 0.03
Therefore, the expected growth rate of dividends is 3
Question 7
Question
A company’s stock is expected to pay dividends of
$
2.50,
$
3.00, and
$
3.50 over
the next three years, respectively. After that, the dividends are expected to
grow at a constant rate of 6
Solution
Step 1: Calculate the present value of the dividends for the first three years.
Step 2: Calculate the present value of the dividends after the third year. Step
3: Find the total present value of the stock.
Step 1: The present value of the dividends for the first three years can be
calculated using the formula for the present value of multiple cash flows:
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
where: D1= $2.50, D2= $3.00, D3= $3.50, r= 0.10.
5
Substitute the given values into the formula:
P V =2.50
(1 + 0.10)1+3.00
(1 + 0.10)2+3.50
(1 + 0.10)3
P V =2.50
1.10 +3.00
1.21 +3.50
1.331
P V = 2.27 + 2.48 + 2.63
P V = $7.38
Therefore, the present value of the dividends for the first three years is
$
7.38.
Step 2: The present value of the dividends after the third year can be
calculated using the Gordon Growth Model formula:
P V =Dyear 4
r−g
where: Dyear 4 =D3×(1 + g), g= 0.06.
Calculate Dyear 4:Dyear 4 = $3.50 ×(1 + 0.06), Dyear 4 = $3.50 ×1.06,
Dyear 4 = $3.71.
Substitute the values into the formula:
P V =3.71
0.10 −0.06
P V =3.71
0.04
P V = $92.75
Therefore, the present value of the dividends after the third year is
$
92.75.
Step 3: The total present value of the stock is the sum of the present value
of the dividends for the first three years and the present value of the dividends
after the third year:
T otalP V = $7.38 + $92.75
T otalP V = $100.13
Therefore, the current value of the stock is
$
100.13.
Question 8
Question
A company’s stock is expected to pay a dividend of 5persharenextyear, withdividendsexpectedtogrowatarateof6
6
Solution
Step 1: Calculate the dividend in the second year. Given that the dividends are
expected to grow at a rate of 6
D2=D1×(1 + g) = $5 ×(1 + 0.06) = $5.30
Step 2: Calculate the required rate of return (RRR). The required rate of
return (RRR) is given as 10
Step 3: Calculate the price of the stock. The price of the stock can be
calculated using the Gordon Growth Model formula:
P0=D1
RRR −g
Substitute the given values:
P0=$5.30
0.10 −0.06 =$5.30
0.04 = $132.50
Therefore, the current value of the stock is $132.50.
Question 9
Question
A company is expected to pay an annual dividend of 4.50pershareindefinitely.Iftherequiredrateof returnis10
Solution
Step 1: Calculate the value of the stock using the dividend discount model. Step
2: The formula for the dividend discount model is:
Stock Value = Dividend
Required Rate of Return
Step 3: Substitute the given values into the formula:
Stock Value = 4.50
0.10
Step 4: Solve for the stock value:
Stock Value = 45
Step 5: The current value of the stock is $45.
Question 10
Question
A company’s stock currently pays a dividend of 3.50pershare.Dividendsareexpectedtogrowatarateof 5
7
Solution
Step 1: Calculate the next dividend (D1) using the dividend growth rate for-
mula:
D1 = D0×(1 + g)
where: D0 = $3.50 (current dividend per share) g= 5% = 0.05 (dividend
growth rate)
D1 = $3.50 ×(1 + 0.05) = $3.50 ×1.05 = $3.675
Step 2: Determine the price of the stock using the Gordon Growth Model
formula:
P=D1
r−g
where: P= price of the stock r= required rate of return = 10g= dividend
growth rate = 5D1 = $3.675 (next dividend)
P=$3.675
0.10 −0.05 =$3.675
0.05 = $73.50
Therefore, the value of the stock is
$
73.50.
Question 11
Question
A company’s stock is currently trading at $120 per share. The company is
expected to pay a dividend of $6 per share next year, and dividends are expected
to grow at a constant rate of 4% per year indefinitely. If the required rate of
return on the stock is 10%, what is the intrinsic value of the stock according to
the dividend discount model?
Solution
Step 1: Calculate the expected dividend next year using the growth rate.
Expected Dividend Next Year = $6 ×(1 + 0.04) = $6.24
Step 2: Use the Gordon Growth Model to calculate the intrinsic value of the
stock.
Intrinsic Value = Expected Dividend Next Year
Required Rate of Return −Growth Rate =$6.24
0.10 −0.04 =$6.24
0.06 = $104
Therefore, the intrinsic value of the stock according to the dividend discount
model is $104 per share.
8
Question 12
Question
A company’s stock has a beta of 1.5, a dividend yield of 3
Solution
Step 1: Calculate the expected return on the market.
Market Risk Premium = Expected Market Return −Risk-Free Rate
6% = Expected Market Return −2%
Expected Market Return = 6% + 2% = 8%
Step 2: Calculate the expected return on the stock using CAPM.
Expected Stock Return = Risk-Free Rate + β(Expected Market Return −Risk-Free Rate)
Expected Stock Return = 2% + 1.5×(8% −2%)
Expected Stock Return = 2% + 1.5×6%
Expected Stock Return = 2% + 9% = 11%
Step 3: Calculate the expected dividend growth rate in dollars.
Dividend Growth Rate (in dollars) = Dividend Yield ×Current Stock Price
Dividend Growth Rate (in dollars) = 3% ×Current Stock Price
Step 4: Calculate the price of the shares using the Dividend Discount Model
(DDM).
Expected Price = Next Dividend
Expected Return −Dividend Growth Rate
Expected Price = Current Dividend ×(1 + Dividend Growth Rate)
Expected Return −Dividend Growth Rate
Expected Price = Current Dividend ×(1 + 5%)
11% −5%
Expected Price = Current Dividend ×1.05
0.06
Thus, the expected rate of return on the stock using the Dividend Discount
Model is Current Dividend×1.05
0.06 .
Question 13
Question
A company’s stock is currently selling for Adollars per share. The company
pays an annual dividend of Ddollars per share, which is expected to grow at
a constant rate of r% per year. If the required rate of return is k%, find an
expression for the value of the stock using the dividend discount model.
9
Solution
Step 1: Calculate the dividend growth rate in decimal form. Let gbe the growth
rate in decimal form. We have g=r
100 .
Step 2: Calculate the expected dividend next year. The expected dividend
next year (D1) is D×(1 + g).
Step 3: Calculate the value of the stock. Using the dividend discount model,
the value of the stock (V0) can be calculated as:
V0=D1
k−g
Substitute the values of D1and ginto the formula:
V0=D×(1 + g)
k−g
V0=D×(1 + r
100 )
k
100 −r
100
V0=D(1 + r
100 )
k−r
100
V0=100D(1 + r
100 )
k−r
Therefore, the expression for the value of the stock using the dividend dis-
count model is 100D(1+ r
100 )
k−r.
Question 14
Question
A company’s stock is expected to pay dividends of
$
2.00,
$
2.10, and
$
2.20
over the next three years. After that, the dividends are 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 current price of the stock?
Solution
Let’s denote: - D1as the dividend in year 1 - D2as the dividend in year 2 - D3
as the dividend in year 3 - gas the annual growth rate of dividends - ras the
required rate of return on the stock
The price of the stock can be calculated using the dividend discount model
formula:
P0=D1
1 + r+D2
(1 + r)2+D3
(1 + r)3+D3·(1 + g)
(r−g)·(1 + r)3
10
Given: - D1= $2.00 - D2= $2.10 - D3= $2.20 - g= 5% = 0.05 -
r= 10% = 0.10
Step 1: Calculate the present value of dividends in the first 3 years.
P VD1=D1
1 + r=2.00
1+0.10 = $1.82
P VD2=D2
(1 + r)2=2.10
(1 + 0.10)2= $1.69
P VD3=D3
(1 + r)3=2.20
(1 + 0.10)3= $1.54
Step 2: Calculate the present value of the growing perpetuity.
P Vgrowing =D3·(1 + g)
(r−g)·(1 + r)3
=2.20 ·(1 + 0.05)
(0.10 −0.05) ·(1 + 0.10)3
=2.31
0.05 ·1.331
= $4.34
Step 3: Calculate the current price of the stock.
P0=P VD1+P VD2+P VD3+P Vgrowing
= 1.82 + 1.69 + 1.54 + 4.34
= $9.39
Therefore, the current price of the stock is
$
9.39.
Question 15
Question
A company’s stock currently pays a dividend of 3.50pershare.T hedividendisexpectedtogrowataconstantrateof 6
Solution
Step 1: Calculate the dividend growth rate g.
Dividend Growth Rate (g) = 6% = 0.06
Step 2: Use the Gordon Growth Model to calculate the current stock price.
The Gordon Growth Model formula is:
P0=D1
r−g
11
where: - P0= Current stock price - D1= Expected dividend next year - r=
Required rate of return - g= Dividend growth rate
Step 3: Calculate the expected dividend next year D1.
D1=D0×(1 + g)
D1= 3.50 ×(1 + 0.06)
D1= 3.50 ×1.06
D1= 3.71
Step 4: Plug in the values to calculate the current stock price P0.
P0=3.71
0.10 −0.06
P0=3.71
0.04
P0= 92.75
Therefore, the current value of the stock is 92.75pershare.
Question 16
Question
A company just paid a dividend of 5pershare.T hedividendisexpectedtogrowataconstantrateof 4
Solution
Step 1: Calculate the dividend growth rate. Given: Dividend = 5pershareGrowthrate =
4%
The dividend growth rate can be calculated as follows:
g=5×0.04
5= 0.04 = 4%
Step 2: Calculate the price of the stock today using the Dividend Discount
Model (DDM). The Dividend Discount Model is given by:
P0=D1
r−g
where: P0= Price of the stock today D1= Dividend expected next year r=
Required rate of return g= Growth rate
Substitute the values into the formula:
P0=5×(1 + 0.04)
0.10 −0.04
12
P0=5×1.04
0.06
P0=5.2
0.06
P0= 86.67
Thus, the current value of the stock is 86.67pershare.
Question 17
Question
A company pays an annual dividend of 5pershareandisexpectedtogrowatarateof 8
Solution
Step 1: Calculate the dividend in year 1 using the formula for the dividend in
year t:
D1=D0×(1 + growth rate) = $5 ×(1 + 0.08) = $5.40
Step 2: Calculate the price of the stock at the end of year 1 using the
dividend discount model formula:
P1=D1
r−g=$5.40
0.12 −0.08 = $135
Step 3: Calculate the current value of the stock by discounting the price at
the end of year 1 back to present value:
Current Value = P1
(1 + r)1=$135
(1 + 0.12)1=$135
1.12 ≈$120.54
Therefore, the current value of the stock is approximately
$
120.54.
Question 18
Question
A company pays an annual dividend of 3.50pershare.Iftherequiredreturnonthestockis12%andthedividendisexpectedtogrowataconstantrateof5%peryear, whatisthecurrentpriceofthestock?
Solution
Let’s denote the current price of the stock as P, the annual dividend as D, the
required return as r, and the growth rate of the dividend as g.
Step 1: Calculate the dividend growth rate (g).
g= 0.05
13
Step 2: Use the Gordon growth model to calculate the current price of the
stock. The Gordon growth model is given by:
P=D×(1 + g)
r−g
Step 3: Substitute the given values into the Gordon growth model formula.
P=3.50 ×(1 + 0.05)
0.12 −0.05
Step 4: Simplify the equation to find the current price of the stock.
P=3.50 ×1.05
0.07
P=3.675
0.07
P= 52.50
Therefore, the current price of the stock is 52.50.
Question 19
Question
A company’s stock is expected to pay a dividend of 5.00persharenextyear.Ifthedividendisexpectedtogrowatarateof 8
Solution
Step 1: Calculate the dividend per share in the second year. Given that the
dividend grows at a rate of 8
Dn=D1×(1 + g)n
where: Dn= dividend in year n,D1= dividend in the first year, g= growth
rate, n= year. Plugging in the values:
D2= 5.00 ×(1 + 0.08)2
D2= 5.00 ×(1.08)2
D2= 5.00 ×1.1664
D2= 5.832
Step 2: Calculate the expected price of the stock in the second year. The
expected price of the stock in the second year can be calculated using the for-
mula:
P1=D2
r−g
14
where: P1= price of the stock in the second year, D2= dividend in the second
year, r= required rate of return, g= growth rate. Plugging in the values:
P1=5.832
0.12 −0.08
P1=5.832
0.04
P1= 145.80
Step 3: Calculate the current value of the stock. To find the current value
of the stock, we need to find the present value of the stock in the second year
and discount it to the present value using the required rate of return.
P V =P1
(1 + r)1
Plugging in the values:
P V =145.80
(1 + 0.12)1
P V =145.80
1.12
P V = 130.54
Therefore, the current value of the stock is 130.54.
Question 20
Question
A company’s stock currently pays an annual dividend of 3.50pershare, whichisexpectedtogrowataconstantrateof5
Solution
Step 1: Calculate the dividend growth rate. Step 2: Use the Gordon Growth
Model to find the current value of the stock.
Step 1: The dividend growth rate (g) is given as 5
Step 2: Using the Gordon Growth Model:
P0=D0×(1 + g)
r−g
where: - P0= current stock price - D0= current dividend per share = 3.50−r=
requiredrateofreturn = 10−g= dividendgrowthrate = 5
Plugging in the values, we get:
P0=3.50 ×(1 + 0.05)
0.10 −0.05
15
P0=3.50 ×1.05
0.05
P0=3.675
0.05
P0= 73.50
Therefore, the current value of the stock is 73.50.
Question 21
Question
A company’s stock is expected to pay an annual dividend of 5pershareforever.If therequiredrateof returnonthestockis10
Solution
Step 1: Calculate the price of the stock using the Gordon Growth Model. Step
2: The Gordon Growth Model formula is given by:
P=D
r−g
where: P= Price of the stock, D= Annual dividend per share (given as 5), r
= Required rate of return (given as 10g= Growth rate of dividends (since the
dividend is expected to be paid forever, g= 0).
Step 3: Substituting the values into the formula:
P=5
0.10 −0
P=5
0.10
P= 50
Answer: The price of the stock is 50pershare.
Question 22
Question
A company’s stock currently pays a dividend of 5pershare.T hedividendsareexpectedtogrowataconstantrateof 7
16
Solution
Step 1: Calculate the expected dividend next year. Given: - Dividend this
year, D0=5 - Dividend growth rate, g= 7% = 0.07 - Required rate of return,
r= 12% = 0.12
The expected dividend next year, D1, can be calculated using the formula
for constant growth dividends:
D1=D0×(1 + g)
D1= 5 ×(1 + 0.07) = 5.35
Step 2: Calculate the price of the stock using the dividend discount model.
The price of the stock, P0, can be calculated using the dividend discount model:
P0=D1
r−g
Substitute the values to find the price:
P0=5.35
0.12 −0.07
P0=5.35
0.05
P0= 107
Therefore, the current price of the stock is 107pershare.
Question 23
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 rate of 5% per year indefinitely. If the required rate of return on
the stock is 10%, what is the value of the stock?
Solution
Step 1: Calculate the expected dividend in year 2. To find the dividend in year
2, we use the formula for dividend growth: D2=D1×(1 + growth rate). Given
that D1= $2 and the growth rate is 5%, we have:
D2= $2 ×(1 + 0.05) = $2.10
Step 2: Calculate the stock price. The value of a stock is equal to the present
value of all future dividends. The formula to find the present value of a growing
perpetuity is:
Stock Price = D1
r−g
17
Where: - D1= $2 (dividend in year 1) - r= 10% (required rate of return) -
g= 5% (growth rate) Plugging in the values, we get:
Stock Price = $2
0.10 −0.05 =$2
0.05 = $40
Therefore, the value of the stock is $40 per share.
Question 24
Question
A company’s stock is currently priced at $80 per share. The company pays
an annual dividend of $4 per share, which is expected to grow at a constant
rate indefinitely. The required rate of return for the stock is 10%. What is the
expected price of the stock in 5 years?
Solution
Let’s denote the expected price of the stock in 5 years as P5. We can use
the Gordon Growth Model to find the expected price. Step 1: Calculate the
dividend yield D0.
The dividend yield D0is calculated as the annual dividend divided by the
current stock price:
D0=$4
$80 = 0.05
Step 2: Calculate the growth rate of dividends g.
The growth rate of dividends gcan be calculated using the formula:
g=D0×P0
P0
= 0.05
Step 3: Use the Gordon Growth Model to find the expected price in 5 years.
The Gordon Growth Model is given by:
P5=D6
r−g
Since the dividend grows at a constant rate, the dividend in 5 years will be:
D6=D0×(1 + g)5= $4 ×(1 + 0.05)5
Now, substitute the values into the Gordon Growth Model formula:
P5=$4 ×(1 + 0.05)5
0.10 −0.05
Solving for P5gives the expected price of the stock in 5 years.
18
Question 25
Question
A company pays a yearly dividend of 2.50pershareandisexpectedtogrowataconstantrateof 5
Solution
Let’s denote the yearly dividend as D = 2.50, thegrowthrateasg = 5
P0=D1
r−g
where P0is the current stock price and D1is the dividend expected to be
received at the end of year 1.
Step 1: Find D1Since the dividend is expected to grow at a constant rate,
we can find the dividend at the end of year 1 using the formula:
D1=D×(1 + g)
D1=
2.50 ×(1 + 0.05) =2.50 ×1.05 =2.625
Step 2: Calculate the Current Stock Price Now, we can substitute
D1=2.625, r = 0.10, and g = 0.05 into the Gordon Growth Model formula to
find the current stock price:
P0=2.625
0.10 −0.05
P0=2.625
0.05 =
52.50
Therefore, the current stock price is 52.50.
Question 26
Question
A company pays a dividend of 3pershareannuallyandisexpectedtohaveagrowthrateof 5
19
Solution
Step 1: Calculate the dividend growth rate using the formula g=D1
D0
−1, where
gis the growth rate, D1is the dividend in the next year, and D0is the current
dividend.
g=3(1 + 0.05)
3−1
= 0.05
Step 2: Use the dividend discount model to find the stock price. The formula
is P=D
r−g, where Pis the stock price, Dis the dividend, ris the required rate
of return, and gis the growth rate.
P=3
0.10 −0.05
=3
0.05
= 60
Therefore, the current stock price is
$
60.
Question 27
Question
You are evaluating the stock of XYZ company, which is expected to pay a
dividend of 3.50persharenextyear.Dividendsareexpectedtogrowatarateof 7
Solution
Step 1: Calculate the dividend expected next year using the growth rate. Step
2: Calculate the price of the stock based on the dividend discount model.
Step 1: The dividend next year will be:
D1 = D0×(1 + g)
D1=3.50 ×(1 + 0.07) = 3.745
Step 2: The price of the stock can be calculated using the dividend discount
model formula:
P0 = D1
r−g
Where: P0 = Price of the stock today D1 = Dividend expected next year r=
Required rate of return g= Growth rate of dividends
Plugging in the values:
P0 = 3.745
0.10 −0.07
P0 = 3.745
0.03 = 124.83
Therefore, the value of the stock today is 124.83pershare.
20
Question 28
Question
A company’s stock is expected to pay a dividend of
$
2 per share next year.
Dividends are expected to grow at a rate of 5
Solution
Step 1: Calculate the dividend one year from now using the growth rate. Step 2:
Use the dividend one year from now and the required rate of return to calculate
the present value of the stock.
Step 1: The dividend one year from now can be calculated using the for-
mula:
D1 = D0×(1 + g)
where: - D1 = Dividend one year from now (
$
) - D0 = Dividend today (
$
2 in
this case) - g= Growth rate (5
Substitute the values:
D1 = $2 ×(1 + 0.05)
D1 = $2.10
Therefore, the dividend one year from now is
$
2.10 per share.
Step 2: The present value of the stock can be calculated using the formula
for the present value of a growing perpetuity:
P=D1
r−g
where: - P= Present value of the stock - D1 = Dividend one year from now
(
$
2.10) - r= Required rate of return (10- g= Growth rate (5
Substitute the values:
P=$2.10
0.10 −0.05
P=$2.10
0.05
P= $42
Therefore, the value of the stock today is
$
42.
Question 29
Question
A company’s stock is expected to pay a dividend of 3.50 next year. The divi-
dends are expected to grow at a rate of 6% per year indefinitely. If the required
rate of return is 12%, what is the current value of the stock?
21
Solution
Step 1: Calculate the expected dividend in year 2.
Dividend in year 2 = Dividend in year 1 ×(1 + Growth rate)
= 3.50 ×(1 + 0.06)
= 3.71
Step 2: Calculate the required rate of return as a decimal.
Required rate of return = 12% = 0.12
Step 3: Use the Gordon Growth Model to find the current value of the stock.
Current value of the stock = Dividend in year 1
Required rate of return −Growth rate
=3.50
0.12 −0.06
=3.50
0.06
= 58.33
Therefore, the current value of the stock is $58.33.
Question 30
Question
A company is expected to pay an annual dividend of 5.00pershareindefinitely.Iftherequiredrateofreturnis8
Solution
Step 1: Calculate the dividend growth rate using the formula:
g= Required rate of return −Dividend yield
Where: - The required rate of return is 8- The dividend yield is calculated
as the annual dividend divided by the current stock price. Since the current
stock price is not given, we will assume it’s Pfor now.
g= 0.08 −5
P
Step 2: Since the annual dividend is expected to be 5.00pershareindefinitely, wecanusetheGordonGrowthM odeltof indthevalueofthestock :
P=D0×(1+g)
r−g
Where: - D0is the annual dividend expected to be paid, which is 5.00 −r
is the required rate of return, which is 8- gis the growth rate we calculated in
Step 1
22
Question 2
Question
Suppose a company pays an annual dividend of 4.50pershareandisexpectedtogrowitsdividendsataconstantrateof 5
Solution
Let’s use the Gordon Growth Model to determine the current value of the stock:
The Gordon Growth Model formula is:
P=D0×(1 + g)
r−g
Where: - P= current price of the stock - D0= the most recent dividend per
share - r= required rate of return - g= growth rate of dividends
Step 1: Plug in the given values. Given: D0= 4.50, g= 0.05, r= 0.10
Step 2: Calculate the current value of the stock.
P=4.50 ×(1 + 0.05)
0.10 −0.05
P=4.50 ×1.05
0.05
P=4.725
0.05
P= 94.50
Therefore, the current value of the stock is 94.50pershare.
Question 3
Question
You are analyzing a company that is expected to pay a dividend of
$
2.50 per
share next year. The dividends are expected to grow at a rate of 5
Solution
Step 1: Calculate the expected dividend next year based on the growth rate.
Step 2: Use the dividend discount model to find the current value of the stock.
Step 1: Given the dividend next year (D1) =
$
2.50 and the growth rate (g)
= 5
D2 = D1×(1 + g)=2.50 ×(1 + 0.05) = 2.50 ×1.05 = $2.625
Step 2: To find the current value of the stock (P0) using the dividend dis-
count model:
P0 = D1
r−g
2
Where: - P0 = Current value of the stock - D1 = Dividend next year - r =
Required rate of return - g = Growth rate
Plugging in the values:
P0 = 2.50
0.12 −0.05 =2.50
0.07 = $35.71
Therefore, the current value of the stock is
$
35.71.
Question 4
Question
A company’s stock is expected to pay a dividend of 7.50 next year, with a
growth rate of 3%. If the required rate of return on the stock is 10%, what is
the current value of the stock?
Solution
Step 1: Calculate the dividend in two years using the growth rate.
D2=D1×(1 + g)
D2= 7.50 ×(1 + 0.03)
D2= 7.50 ×1.03 = 7.725
Step 2: Calculate the expected dividend for year 3.
D3=D2×(1 + g)
D3= 7.725 ×(1 + 0.03)
D3= 7.725 ×1.03 = 7.93875
Step 3: Use the Gordon Growth Model to find the current value of the stock.
P0=D1
r−g
P0=7.50
0.10 −0.03
P0=7.50
0.07
P0= 107.14
Therefore, the current value of the stock is 107.14.
3
Question 5
Question
Company XYZ just paid a dividend of 3pershare.T hedividendisexpectedtogrowataconstantrateof 5
Solution
Step 1: Identify the given values.
Given:
Dividend per share (D) = 3Dividendgrowthrate(g) = 5
Required rate of return (r) = 10
Step 2: Calculate the price of the stock using the Gordon Growth Model.
The Gordon Growth Model is given by:
P0=D0×(1 + g)
r−g
where:
P0= Price of the stock today
D0= Dividend per share today
Step 3: Substitute the values into the Gordon Growth Model.
Substitute D0= 3, g = 0.05, r = 0.10 into the formula:
P0=3×(1 + 0.05)
0.10 −0.05
Step 4: Calculate the price of the stock.
P0=3×1.05
0.05 =3.15
0.05 = 63
Therefore, the current price of the stock is
$
63.
Question 6
Question
Assume a company is expected to pay an annual dividend of 3.50forever, andtherequiredrateof returnis8
4
Solution
Let D0be the current dividend, P0be the current stock price, rbe the required
rate of return, and gbe the expected growth rate of dividends. The constant
growth model formula for stock valuation is given by:
P0=D0
r−g
Step 1: Given that D0= 3.50, P0= 70, and r= 0.08, we can substitute
the values into the formula:
70 = 3.50
0.08 −g
Step 2: Solving for g, we have:
70(0.08 −g)=3.50
5.6−70g= 3.50
70g= 5.6−3.50
70g= 2.10
g=2.10
70
g= 0.03
Therefore, the expected growth rate of dividends is 3
Question 7
Question
A company’s stock is expected to pay dividends of
$
2.50,
$
3.00, and
$
3.50 over
the next three years, respectively. After that, the dividends are expected to
grow at a constant rate of 6
Solution
Step 1: Calculate the present value of the dividends for the first three years.
Step 2: Calculate the present value of the dividends after the third year. Step
3: Find the total present value of the stock.
Step 1: The present value of the dividends for the first three years can be
calculated using the formula for the present value of multiple cash flows:
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
where: D1= $2.50, D2= $3.00, D3= $3.50, r= 0.10.
5
Substitute the given values into the formula:
P V =2.50
(1 + 0.10)1+3.00
(1 + 0.10)2+3.50
(1 + 0.10)3
P V =2.50
1.10 +3.00
1.21 +3.50
1.331
P V = 2.27 + 2.48 + 2.63
P V = $7.38
Therefore, the present value of the dividends for the first three years is
$
7.38.
Step 2: The present value of the dividends after the third year can be
calculated using the Gordon Growth Model formula:
P V =Dyear 4
r−g
where: Dyear 4 =D3×(1 + g), g= 0.06.
Calculate Dyear 4:Dyear 4 = $3.50 ×(1 + 0.06), Dyear 4 = $3.50 ×1.06,
Dyear 4 = $3.71.
Substitute the values into the formula:
P V =3.71
0.10 −0.06
P V =3.71
0.04
P V = $92.75
Therefore, the present value of the dividends after the third year is
$
92.75.
Step 3: The total present value of the stock is the sum of the present value
of the dividends for the first three years and the present value of the dividends
after the third year:
T otalP V = $7.38 + $92.75
T otalP V = $100.13
Therefore, the current value of the stock is
$
100.13.
Question 8
Question
A company’s stock is expected to pay a dividend of 5persharenextyear, withdividendsexpectedtogrowatarateof6
6
Solution
Step 1: Calculate the dividend in the second year. Given that the dividends are
expected to grow at a rate of 6
D2=D1×(1 + g) = $5 ×(1 + 0.06) = $5.30
Step 2: Calculate the required rate of return (RRR). The required rate of
return (RRR) is given as 10
Step 3: Calculate the price of the stock. The price of the stock can be
calculated using the Gordon Growth Model formula:
P0=D1
RRR −g
Substitute the given values:
P0=$5.30
0.10 −0.06 =$5.30
0.04 = $132.50
Therefore, the current value of the stock is $132.50.
Question 9
Question
A company is expected to pay an annual dividend of 4.50pershareindefinitely.Iftherequiredrateofreturnis10
Solution
Step 1: Calculate the value of the stock using the dividend discount model. Step
2: The formula for the dividend discount model is:
Stock Value = Dividend
Required Rate of Return
Step 3: Substitute the given values into the formula:
Stock Value = 4.50
0.10
Step 4: Solve for the stock value:
Stock Value = 45
Step 5: The current value of the stock is $45.
Question 10
Question
A company’s stock currently pays a dividend of 3.50pershare.Dividendsareexpectedtogrowatarateof 5
7
Solution
Step 1: Calculate the next dividend (D1) using the dividend growth rate for-
mula:
D1 = D0×(1 + g)
where: D0 = $3.50 (current dividend per share) g= 5% = 0.05 (dividend
growth rate)
D1 = $3.50 ×(1 + 0.05) = $3.50 ×1.05 = $3.675
Step 2: Determine the price of the stock using the Gordon Growth Model
formula:
P=D1
r−g
where: P= price of the stock r= required rate of return = 10g= dividend
growth rate = 5D1 = $3.675 (next dividend)
P=$3.675
0.10 −0.05 =$3.675
0.05 = $73.50
Therefore, the value of the stock is
$
73.50.
Question 11
Question
A company’s stock is currently trading at $120 per share. The company is
expected to pay a dividend of $6 per share next year, and dividends are expected
to grow at a constant rate of 4% per year indefinitely. If the required rate of
return on the stock is 10%, what is the intrinsic value of the stock according to
the dividend discount model?
Solution
Step 1: Calculate the expected dividend next year using the growth rate.
Expected Dividend Next Year = $6 ×(1 + 0.04) = $6.24
Step 2: Use the Gordon Growth Model to calculate the intrinsic value of the
stock.
Intrinsic Value = Expected Dividend Next Year
Required Rate of Return −Growth Rate =$6.24
0.10 −0.04 =$6.24
0.06 = $104
Therefore, the intrinsic value of the stock according to the dividend discount
model is $104 per share.
8
Question 12
Question
A company’s stock has a beta of 1.5, a dividend yield of 3
Solution
Step 1: Calculate the expected return on the market.
Market Risk Premium = Expected Market Return −Risk-Free Rate
6% = Expected Market Return −2%
Expected Market Return = 6% + 2% = 8%
Step 2: Calculate the expected return on the stock using CAPM.
Expected Stock Return = Risk-Free Rate + β(Expected Market Return −Risk-Free Rate)
Expected Stock Return = 2% + 1.5×(8% −2%)
Expected Stock Return = 2% + 1.5×6%
Expected Stock Return = 2% + 9% = 11%
Step 3: Calculate the expected dividend growth rate in dollars.
Dividend Growth Rate (in dollars) = Dividend Yield ×Current Stock Price
Dividend Growth Rate (in dollars) = 3% ×Current Stock Price
Step 4: Calculate the price of the shares using the Dividend Discount Model
(DDM).
Expected Price = Next Dividend
Expected Return −Dividend Growth Rate
Expected Price = Current Dividend ×(1 + Dividend Growth Rate)
Expected Return −Dividend Growth Rate
Expected Price = Current Dividend ×(1 + 5%)
11% −5%
Expected Price = Current Dividend ×1.05
0.06
Thus, the expected rate of return on the stock using the Dividend Discount
Model is Current Dividend×1.05
0.06 .
Question 13
Question
A company’s stock is currently selling for Adollars per share. The company
pays an annual dividend of Ddollars per share, which is expected to grow at
a constant rate of r% per year. If the required rate of return is k%, find an
expression for the value of the stock using the dividend discount model.
9
Solution
Step 1: Calculate the dividend growth rate in decimal form. Let gbe the growth
rate in decimal form. We have g=r
100 .
Step 2: Calculate the expected dividend next year. The expected dividend
next year (D1) is D×(1 + g).
Step 3: Calculate the value of the stock. Using the dividend discount model,
the value of the stock (V0) can be calculated as:
V0=D1
k−g
Substitute the values of D1and ginto the formula:
V0=D×(1 + g)
k−g
V0=D×(1 + r
100 )
k
100 −r
100
V0=D(1 + r
100 )
k−r
100
V0=100D(1 + r
100 )
k−r
Therefore, the expression for the value of the stock using the dividend dis-
count model is 100D(1+ r
100 )
k−r.
Question 14
Question
A company’s stock is expected to pay dividends of
$
2.00,
$
2.10, and
$
2.20
over the next three years. After that, the dividends are 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 current price of the stock?
Solution
Let’s denote: - D1as the dividend in year 1 - D2as the dividend in year 2 - D3
as the dividend in year 3 - gas the annual growth rate of dividends - ras the
required rate of return on the stock
The price of the stock can be calculated using the dividend discount model
formula:
P0=D1
1 + r+D2
(1 + r)2+D3
(1 + r)3+D3·(1 + g)
(r−g)·(1 + r)3
10
Given: - D1= $2.00 - D2= $2.10 - D3= $2.20 - g= 5% = 0.05 -
r= 10% = 0.10
Step 1: Calculate the present value of dividends in the first 3 years.
P VD1=D1
1 + r=2.00
1+0.10 = $1.82
P VD2=D2
(1 + r)2=2.10
(1 + 0.10)2= $1.69
P VD3=D3
(1 + r)3=2.20
(1 + 0.10)3= $1.54
Step 2: Calculate the present value of the growing perpetuity.
P Vgrowing =D3·(1 + g)
(r−g)·(1 + r)3
=2.20 ·(1 + 0.05)
(0.10 −0.05) ·(1 + 0.10)3
=2.31
0.05 ·1.331
= $4.34
Step 3: Calculate the current price of the stock.
P0=P VD1+P VD2+P VD3+P Vgrowing
= 1.82 + 1.69 + 1.54 + 4.34
= $9.39
Therefore, the current price of the stock is
$
9.39.
Question 15
Question
A company’s stock currently pays a dividend of 3.50pershare.T hedividendisexpectedtogrowataconstantrateof 6
Solution
Step 1: Calculate the dividend growth rate g.
Dividend Growth Rate (g) = 6% = 0.06
Step 2: Use the Gordon Growth Model to calculate the current stock price.
The Gordon Growth Model formula is:
P0=D1
r−g
11
where: - P0= Current stock price - D1= Expected dividend next year - r=
Required rate of return - g= Dividend growth rate
Step 3: Calculate the expected dividend next year D1.
D1=D0×(1 + g)
D1= 3.50 ×(1 + 0.06)
D1= 3.50 ×1.06
D1= 3.71
Step 4: Plug in the values to calculate the current stock price P0.
P0=3.71
0.10 −0.06
P0=3.71
0.04
P0= 92.75
Therefore, the current value of the stock is 92.75pershare.
Question 16
Question
A company just paid a dividend of 5pershare.T hedividendisexpectedtogrowataconstantrateof 4
Solution
Step 1: Calculate the dividend growth rate. Given: Dividend = 5pershareGrowthrate =
4%
The dividend growth rate can be calculated as follows:
g=5×0.04
5= 0.04 = 4%
Step 2: Calculate the price of the stock today using the Dividend Discount
Model (DDM). The Dividend Discount Model is given by:
P0=D1
r−g
where: P0= Price of the stock today D1= Dividend expected next year r=
Required rate of return g= Growth rate
Substitute the values into the formula:
P0=5×(1 + 0.04)
0.10 −0.04
12
P0=5×1.04
0.06
P0=5.2
0.06
P0= 86.67
Thus, the current value of the stock is 86.67pershare.
Question 17
Question
A company pays an annual dividend of 5pershareandisexpectedtogrowatarateof 8
Solution
Step 1: Calculate the dividend in year 1 using the formula for the dividend in
year t:
D1=D0×(1 + growth rate) = $5 ×(1 + 0.08) = $5.40
Step 2: Calculate the price of the stock at the end of year 1 using the
dividend discount model formula:
P1=D1
r−g=$5.40
0.12 −0.08 = $135
Step 3: Calculate the current value of the stock by discounting the price at
the end of year 1 back to present value:
Current Value = P1
(1 + r)1=$135
(1 + 0.12)1=$135
1.12 ≈$120.54
Therefore, the current value of the stock is approximately
$
120.54.
Question 18
Question
A company pays an annual dividend of 3.50pershare.Iftherequiredreturnonthestockis12%andthedividendisexpectedtogrowataconstantrateof5%peryear, whatisthecurrentpriceofthestock?
Solution
Let’s denote the current price of the stock as P, the annual dividend as D, the
required return as r, and the growth rate of the dividend as g.
Step 1: Calculate the dividend growth rate (g).
g= 0.05
13
Step 2: Use the Gordon growth model to calculate the current price of the
stock. The Gordon growth model is given by:
P=D×(1 + g)
r−g
Step 3: Substitute the given values into the Gordon growth model formula.
P=3.50 ×(1 + 0.05)
0.12 −0.05
Step 4: Simplify the equation to find the current price of the stock.
P=3.50 ×1.05
0.07
P=3.675
0.07
P= 52.50
Therefore, the current price of the stock is 52.50.
Question 19
Question
A company’s stock is expected to pay a dividend of 5.00persharenextyear.Ifthedividendisexpectedtogrowatarateof 8
Solution
Step 1: Calculate the dividend per share in the second year. Given that the
dividend grows at a rate of 8
Dn=D1×(1 + g)n
where: Dn= dividend in year n,D1= dividend in the first year, g= growth
rate, n= year. Plugging in the values:
D2= 5.00 ×(1 + 0.08)2
D2= 5.00 ×(1.08)2
D2= 5.00 ×1.1664
D2= 5.832
Step 2: Calculate the expected price of the stock in the second year. The
expected price of the stock in the second year can be calculated using the for-
mula:
P1=D2
r−g
14
where: P1= price of the stock in the second year, D2= dividend in the second
year, r= required rate of return, g= growth rate. Plugging in the values:
P1=5.832
0.12 −0.08
P1=5.832
0.04
P1= 145.80
Step 3: Calculate the current value of the stock. To find the current value
of the stock, we need to find the present value of the stock in the second year
and discount it to the present value using the required rate of return.
P V =P1
(1 + r)1
Plugging in the values:
P V =145.80
(1 + 0.12)1
P V =145.80
1.12
P V = 130.54
Therefore, the current value of the stock is 130.54.
Question 20
Question
A company’s stock currently pays an annual dividend of 3.50pershare, whichisexpectedtogrowataconstantrateof5
Solution
Step 1: Calculate the dividend growth rate. Step 2: Use the Gordon Growth
Model to find the current value of the stock.
Step 1: The dividend growth rate (g) is given as 5
Step 2: Using the Gordon Growth Model:
P0=D0×(1 + g)
r−g
where: - P0= current stock price - D0= current dividend per share = 3.50−r=
requiredrateofreturn = 10−g= dividendgrowthrate = 5
Plugging in the values, we get:
P0=3.50 ×(1 + 0.05)
0.10 −0.05
15
P0=3.50 ×1.05
0.05
P0=3.675
0.05
P0= 73.50
Therefore, the current value of the stock is 73.50.
Question 21
Question
A company’s stock is expected to pay an annual dividend of 5pershareforever.If therequiredrateof returnonthestockis10
Solution
Step 1: Calculate the price of the stock using the Gordon Growth Model. Step
2: The Gordon Growth Model formula is given by:
P=D
r−g
where: P= Price of the stock, D= Annual dividend per share (given as 5), r
= Required rate of return (given as 10g= Growth rate of dividends (since the
dividend is expected to be paid forever, g= 0).
Step 3: Substituting the values into the formula:
P=5
0.10 −0
P=5
0.10
P= 50
Answer: The price of the stock is 50pershare.
Question 22
Question
A company’s stock currently pays a dividend of 5pershare.T hedividendsareexpectedtogrowataconstantrateof 7
16
Solution
Step 1: Calculate the expected dividend next year. Given: - Dividend this
year, D0=5 - Dividend growth rate, g= 7% = 0.07 - Required rate of return,
r= 12% = 0.12
The expected dividend next year, D1, can be calculated using the formula
for constant growth dividends:
D1=D0×(1 + g)
D1= 5 ×(1 + 0.07) = 5.35
Step 2: Calculate the price of the stock using the dividend discount model.
The price of the stock, P0, can be calculated using the dividend discount model:
P0=D1
r−g
Substitute the values to find the price:
P0=5.35
0.12 −0.07
P0=5.35
0.05
P0= 107
Therefore, the current price of the stock is 107pershare.
Question 23
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 rate of 5% per year indefinitely. If the required rate of return on
the stock is 10%, what is the value of the stock?
Solution
Step 1: Calculate the expected dividend in year 2. To find the dividend in year
2, we use the formula for dividend growth: D2=D1×(1 + growth rate). Given
that D1= $2 and the growth rate is 5%, we have:
D2= $2 ×(1 + 0.05) = $2.10
Step 2: Calculate the stock price. The value of a stock is equal to the present
value of all future dividends. The formula to find the present value of a growing
perpetuity is:
Stock Price = D1
r−g
17
Where: - D1= $2 (dividend in year 1) - r= 10% (required rate of return) -
g= 5% (growth rate) Plugging in the values, we get:
Stock Price = $2
0.10 −0.05 =$2
0.05 = $40
Therefore, the value of the stock is $40 per share.
Question 24
Question
A company’s stock is currently priced at $80 per share. The company pays
an annual dividend of $4 per share, which is expected to grow at a constant
rate indefinitely. The required rate of return for the stock is 10%. What is the
expected price of the stock in 5 years?
Solution
Let’s denote the expected price of the stock in 5 years as P5. We can use
the Gordon Growth Model to find the expected price. Step 1: Calculate the
dividend yield D0.
The dividend yield D0is calculated as the annual dividend divided by the
current stock price:
D0=$4
$80 = 0.05
Step 2: Calculate the growth rate of dividends g.
The growth rate of dividends gcan be calculated using the formula:
g=D0×P0
P0
= 0.05
Step 3: Use the Gordon Growth Model to find the expected price in 5 years.
The Gordon Growth Model is given by:
P5=D6
r−g
Since the dividend grows at a constant rate, the dividend in 5 years will be:
D6=D0×(1 + g)5= $4 ×(1 + 0.05)5
Now, substitute the values into the Gordon Growth Model formula:
P5=$4 ×(1 + 0.05)5
0.10 −0.05
Solving for P5gives the expected price of the stock in 5 years.
18
Question 25
Question
A company pays a yearly dividend of 2.50pershareandisexpectedtogrowataconstantrateof 5
Solution
Let’s denote the yearly dividend as D = 2.50, thegrowthrateasg = 5
P0=D1
r−g
where P0is the current stock price and D1is the dividend expected to be
received at the end of year 1.
Step 1: Find D1Since the dividend is expected to grow at a constant rate,
we can find the dividend at the end of year 1 using the formula:
D1=D×(1 + g)
D1=
2.50 ×(1 + 0.05) =2.50 ×1.05 =2.625
Step 2: Calculate the Current Stock Price Now, we can substitute
D1=2.625, r = 0.10, and g = 0.05 into the Gordon Growth Model formula to
find the current stock price:
P0=2.625
0.10 −0.05
P0=2.625
0.05 =
52.50
Therefore, the current stock price is 52.50.
Question 26
Question
A company pays a dividend of 3pershareannuallyandisexpectedtohaveagrowthrateof 5
19
Solution
Step 1: Calculate the dividend growth rate using the formula g=D1
D0
−1, where
gis the growth rate, D1is the dividend in the next year, and D0is the current
dividend.
g=3(1 + 0.05)
3−1
= 0.05
Step 2: Use the dividend discount model to find the stock price. The formula
is P=D
r−g, where Pis the stock price, Dis the dividend, ris the required rate
of return, and gis the growth rate.
P=3
0.10 −0.05
=3
0.05
= 60
Therefore, the current stock price is
$
60.
Question 27
Question
You are evaluating the stock of XYZ company, which is expected to pay a
dividend of 3.50persharenextyear.Dividendsareexpectedtogrowatarateof 7
Solution
Step 1: Calculate the dividend expected next year using the growth rate. Step
2: Calculate the price of the stock based on the dividend discount model.
Step 1: The dividend next year will be:
D1 = D0×(1 + g)
D1=3.50 ×(1 + 0.07) = 3.745
Step 2: The price of the stock can be calculated using the dividend discount
model formula:
P0 = D1
r−g
Where: P0 = Price of the stock today D1 = Dividend expected next year r=
Required rate of return g= Growth rate of dividends
Plugging in the values:
P0 = 3.745
0.10 −0.07
P0 = 3.745
0.03 = 124.83
Therefore, the value of the stock today is 124.83pershare.
20
Question 28
Question
A company’s stock is expected to pay a dividend of
$
2 per share next year.
Dividends are expected to grow at a rate of 5
Solution
Step 1: Calculate the dividend one year from now using the growth rate. Step 2:
Use the dividend one year from now and the required rate of return to calculate
the present value of the stock.
Step 1: The dividend one year from now can be calculated using the for-
mula:
D1 = D0×(1 + g)
where: - D1 = Dividend one year from now (
$
) - D0 = Dividend today (
$
2 in
this case) - g= Growth rate (5
Substitute the values:
D1 = $2 ×(1 + 0.05)
D1 = $2.10
Therefore, the dividend one year from now is
$
2.10 per share.
Step 2: The present value of the stock can be calculated using the formula
for the present value of a growing perpetuity:
P=D1
r−g
where: - P= Present value of the stock - D1 = Dividend one year from now
(
$
2.10) - r= Required rate of return (10- g= Growth rate (5
Substitute the values:
P=$2.10
0.10 −0.05
P=$2.10
0.05
P= $42
Therefore, the value of the stock today is
$
42.
Question 29
Question
A company’s stock is expected to pay a dividend of 3.50 next year. The divi-
dends are expected to grow at a rate of 6% per year indefinitely. If the required
rate of return is 12%, what is the current value of the stock?
21
Solution
Step 1: Calculate the expected dividend in year 2.
Dividend in year 2 = Dividend in year 1 ×(1 + Growth rate)
= 3.50 ×(1 + 0.06)
= 3.71
Step 2: Calculate the required rate of return as a decimal.
Required rate of return = 12% = 0.12
Step 3: Use the Gordon Growth Model to find the current value of the stock.
Current value of the stock = Dividend in year 1
Required rate of return −Growth rate
=3.50
0.12 −0.06
=3.50
0.06
= 58.33
Therefore, the current value of the stock is $58.33.
Question 30
Question
A company is expected to pay an annual dividend of 5.00pershareindefinitely.Iftherequiredrateofreturnis8
Solution
Step 1: Calculate the dividend growth rate using the formula:
g= Required rate of return −Dividend yield
Where: - The required rate of return is 8- The dividend yield is calculated
as the annual dividend divided by the current stock price. Since the current
stock price is not given, we will assume it’s Pfor now.
g= 0.08 −5
P
Step 2: Since the annual dividend is expected to be 5.00pershareindefinitely, wecanusetheGordonGrowthM odeltof indthevalueofthestock :
P=D0×(1+g)
r−g
Where: - D0is the annual dividend expected to be paid, which is 5.00 −r
is the required rate of return, which is 8- gis the growth rate we calculated in
Step 1
22
Question 2
Question
Suppose a company pays an annual dividend of 4.50pershareandisexpectedtogrowitsdividendsataconstantrateof 5
Solution
Let’s use the Gordon Growth Model to determine the current value of the stock:
The Gordon Growth Model formula is:
P=D0×(1 + g)
r−g
Where: - P= current price of the stock - D0= the most recent dividend per
share - r= required rate of return - g= growth rate of dividends
Step 1: Plug in the given values. Given: D0= 4.50, g= 0.05, r= 0.10
Step 2: Calculate the current value of the stock.
P=4.50 ×(1 + 0.05)
0.10 −0.05
P=4.50 ×1.05
0.05
P=4.725
0.05
P= 94.50
Therefore, the current value of the stock is 94.50pershare.
Question 3
Question
You are analyzing a company that is expected to pay a dividend of
$
2.50 per
share next year. The dividends are expected to grow at a rate of 5
Solution
Step 1: Calculate the expected dividend next year based on the growth rate.
Step 2: Use the dividend discount model to find the current value of the stock.
Step 1: Given the dividend next year (D1) =
$
2.50 and the growth rate (g)
= 5
D2 = D1×(1 + g)=2.50 ×(1 + 0.05) = 2.50 ×1.05 = $2.625
Step 2: To find the current value of the stock (P0) using the dividend dis-
count model:
P0 = D1
r−g
2
Where: - P0 = Current value of the stock - D1 = Dividend next year - r =
Required rate of return - g = Growth rate
Plugging in the values:
P0 = 2.50
0.12 −0.05 =2.50
0.07 = $35.71
Therefore, the current value of the stock is
$
35.71.
Question 4
Question
A company’s stock is expected to pay a dividend of 7.50 next year, with a
growth rate of 3%. If the required rate of return on the stock is 10%, what is
the current value of the stock?
Solution
Step 1: Calculate the dividend in two years using the growth rate.
D2=D1×(1 + g)
D2= 7.50 ×(1 + 0.03)
D2= 7.50 ×1.03 = 7.725
Step 2: Calculate the expected dividend for year 3.
D3=D2×(1 + g)
D3= 7.725 ×(1 + 0.03)
D3= 7.725 ×1.03 = 7.93875
Step 3: Use the Gordon Growth Model to find the current value of the stock.
P0=D1
r−g
P0=7.50
0.10 −0.03
P0=7.50
0.07
P0= 107.14
Therefore, the current value of the stock is 107.14.
3
Question 5
Question
Company XYZ just paid a dividend of 3pershare.T hedividendisexpectedtogrowataconstantrateof 5
Solution
Step 1: Identify the given values.
Given:
Dividend per share (D) = 3Dividendgrowthrate(g) = 5
Required rate of return (r) = 10
Step 2: Calculate the price of the stock using the Gordon Growth Model.
The Gordon Growth Model is given by:
P0=D0×(1 + g)
r−g
where:
P0= Price of the stock today
D0= Dividend per share today
Step 3: Substitute the values into the Gordon Growth Model.
Substitute D0= 3, g = 0.05, r = 0.10 into the formula:
P0=3×(1 + 0.05)
0.10 −0.05
Step 4: Calculate the price of the stock.
P0=3×1.05
0.05 =3.15
0.05 = 63
Therefore, the current price of the stock is
$
63.
Question 6
Question
Assume a company is expected to pay an annual dividend of 3.50forever, andtherequiredrateof returnis8
4
Solution
Let D0be the current dividend, P0be the current stock price, rbe the required
rate of return, and gbe the expected growth rate of dividends. The constant
growth model formula for stock valuation is given by:
P0=D0
r−g
Step 1: Given that D0= 3.50, P0= 70, and r= 0.08, we can substitute
the values into the formula:
70 = 3.50
0.08 −g
Step 2: Solving for g, we have:
70(0.08 −g)=3.50
5.6−70g= 3.50
70g= 5.6−3.50
70g= 2.10
g=2.10
70
g= 0.03
Therefore, the expected growth rate of dividends is 3
Question 7
Question
A company’s stock is expected to pay dividends of
$
2.50,
$
3.00, and
$
3.50 over
the next three years, respectively. After that, the dividends are expected to
grow at a constant rate of 6
Solution
Step 1: Calculate the present value of the dividends for the first three years.
Step 2: Calculate the present value of the dividends after the third year. Step
3: Find the total present value of the stock.
Step 1: The present value of the dividends for the first three years can be
calculated using the formula for the present value of multiple cash flows:
P V =D1
(1 + r)1+D2
(1 + r)2+D3
(1 + r)3
where: D1= $2.50, D2= $3.00, D3= $3.50, r= 0.10.
5
Substitute the given values into the formula:
P V =2.50
(1 + 0.10)1+3.00
(1 + 0.10)2+3.50
(1 + 0.10)3
P V =2.50
1.10 +3.00
1.21 +3.50
1.331
P V = 2.27 + 2.48 + 2.63
P V = $7.38
Therefore, the present value of the dividends for the first three years is
$
7.38.
Step 2: The present value of the dividends after the third year can be
calculated using the Gordon Growth Model formula:
P V =Dyear 4
r−g
where: Dyear 4 =D3×(1 + g), g= 0.06.
Calculate Dyear 4:Dyear 4 = $3.50 ×(1 + 0.06), Dyear 4 = $3.50 ×1.06,
Dyear 4 = $3.71.
Substitute the values into the formula:
P V =3.71
0.10 −0.06
P V =3.71
0.04
P V = $92.75
Therefore, the present value of the dividends after the third year is
$
92.75.
Step 3: The total present value of the stock is the sum of the present value
of the dividends for the first three years and the present value of the dividends
after the third year:
T otalP V = $7.38 + $92.75
T otalP V = $100.13
Therefore, the current value of the stock is
$
100.13.
Question 8
Question
A company’s stock is expected to pay a dividend of 5persharenextyear, withdividendsexpectedtogrowatarateof6
6
Solution
Step 1: Calculate the dividend in the second year. Given that the dividends are
expected to grow at a rate of 6
D2=D1×(1 + g) = $5 ×(1 + 0.06) = $5.30
Step 2: Calculate the required rate of return (RRR). The required rate of
return (RRR) is given as 10
Step 3: Calculate the price of the stock. The price of the stock can be
calculated using the Gordon Growth Model formula:
P0=D1
RRR −g
Substitute the given values:
P0=$5.30
0.10 −0.06 =$5.30
0.04 = $132.50
Therefore, the current value of the stock is $132.50.
Question 9
Question
A company is expected to pay an annual dividend of 4.50pershareindefinitely.Iftherequiredrateofreturnis10
Solution
Step 1: Calculate the value of the stock using the dividend discount model. Step
2: The formula for the dividend discount model is:
Stock Value = Dividend
Required Rate of Return
Step 3: Substitute the given values into the formula:
Stock Value = 4.50
0.10
Step 4: Solve for the stock value:
Stock Value = 45
Step 5: The current value of the stock is $45.
Question 10
Question
A company’s stock currently pays a dividend of 3.50pershare.Dividendsareexpectedtogrowatarateof 5
7
Solution
Step 1: Calculate the next dividend (D1) using the dividend growth rate for-
mula:
D1 = D0×(1 + g)
where: D0 = $3.50 (current dividend per share) g= 5% = 0.05 (dividend
growth rate)
D1 = $3.50 ×(1 + 0.05) = $3.50 ×1.05 = $3.675
Step 2: Determine the price of the stock using the Gordon Growth Model
formula:
P=D1
r−g
where: P= price of the stock r= required rate of return = 10g= dividend
growth rate = 5D1 = $3.675 (next dividend)
P=$3.675
0.10 −0.05 =$3.675
0.05 = $73.50
Therefore, the value of the stock is
$
73.50.
Question 11
Question
A company’s stock is currently trading at $120 per share. The company is
expected to pay a dividend of $6 per share next year, and dividends are expected
to grow at a constant rate of 4% per year indefinitely. If the required rate of
return on the stock is 10%, what is the intrinsic value of the stock according to
the dividend discount model?
Solution
Step 1: Calculate the expected dividend next year using the growth rate.
Expected Dividend Next Year = $6 ×(1 + 0.04) = $6.24
Step 2: Use the Gordon Growth Model to calculate the intrinsic value of the
stock.
Intrinsic Value = Expected Dividend Next Year
Required Rate of Return −Growth Rate =$6.24
0.10 −0.04 =$6.24
0.06 = $104
Therefore, the intrinsic value of the stock according to the dividend discount
model is $104 per share.
8
Question 12
Question
A company’s stock has a beta of 1.5, a dividend yield of 3
Solution
Step 1: Calculate the expected return on the market.
Market Risk Premium = Expected Market Return −Risk-Free Rate
6% = Expected Market Return −2%
Expected Market Return = 6% + 2% = 8%
Step 2: Calculate the expected return on the stock using CAPM.
Expected Stock Return = Risk-Free Rate + β(Expected Market Return −Risk-Free Rate)
Expected Stock Return = 2% + 1.5×(8% −2%)
Expected Stock Return = 2% + 1.5×6%
Expected Stock Return = 2% + 9% = 11%
Step 3: Calculate the expected dividend growth rate in dollars.
Dividend Growth Rate (in dollars) = Dividend Yield ×Current Stock Price
Dividend Growth Rate (in dollars) = 3% ×Current Stock Price
Step 4: Calculate the price of the shares using the Dividend Discount Model
(DDM).
Expected Price = Next Dividend
Expected Return −Dividend Growth Rate
Expected Price = Current Dividend ×(1 + Dividend Growth Rate)
Expected Return −Dividend Growth Rate
Expected Price = Current Dividend ×(1 + 5%)
11% −5%
Expected Price = Current Dividend ×1.05
0.06
Thus, the expected rate of return on the stock using the Dividend Discount
Model is Current Dividend×1.05
0.06 .
Question 13
Question
A company’s stock is currently selling for Adollars per share. The company
pays an annual dividend of Ddollars per share, which is expected to grow at
a constant rate of r% per year. If the required rate of return is k%, find an
expression for the value of the stock using the dividend discount model.
9
Solution
Step 1: Calculate the dividend growth rate in decimal form. Let gbe the growth
rate in decimal form. We have g=r
100 .
Step 2: Calculate the expected dividend next year. The expected dividend
next year (D1) is D×(1 + g).
Step 3: Calculate the value of the stock. Using the dividend discount model,
the value of the stock (V0) can be calculated as:
V0=D1
k−g
Substitute the values of D1and ginto the formula:
V0=D×(1 + g)
k−g
V0=D×(1 + r
100 )
k
100 −r
100
V0=D(1 + r
100 )
k−r
100
V0=100D(1 + r
100 )
k−r
Therefore, the expression for the value of the stock using the dividend dis-
count model is 100D(1+ r
100 )
k−r.
Question 14
Question
A company’s stock is expected to pay dividends of
$
2.00,
$
2.10, and
$
2.20
over the next three years. After that, the dividends are 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 current price of the stock?
Solution
Let’s denote: - D1as the dividend in year 1 - D2as the dividend in year 2 - D3
as the dividend in year 3 - gas the annual growth rate of dividends - ras the
required rate of return on the stock
The price of the stock can be calculated using the dividend discount model
formula:
P0=D1
1 + r+D2
(1 + r)2+D3
(1 + r)3+D3·(1 + g)
(r−g)·(1 + r)3
10
Given: - D1= $2.00 - D2= $2.10 - D3= $2.20 - g= 5% = 0.05 -
r= 10% = 0.10
Step 1: Calculate the present value of dividends in the first 3 years.
P VD1=D1
1 + r=2.00
1+0.10 = $1.82
P VD2=D2
(1 + r)2=2.10
(1 + 0.10)2= $1.69
P VD3=D3
(1 + r)3=2.20
(1 + 0.10)3= $1.54
Step 2: Calculate the present value of the growing perpetuity.
P Vgrowing =D3·(1 + g)
(r−g)·(1 + r)3
=2.20 ·(1 + 0.05)
(0.10 −0.05) ·(1 + 0.10)3
=2.31
0.05 ·1.331
= $4.34
Step 3: Calculate the current price of the stock.
P0=P VD1+P VD2+P VD3+P Vgrowing
= 1.82 + 1.69 + 1.54 + 4.34
= $9.39
Therefore, the current price of the stock is
$
9.39.
Question 15
Question
A company’s stock currently pays a dividend of 3.50pershare.T hedividendisexpectedtogrowataconstantrateof 6
Solution
Step 1: Calculate the dividend growth rate g.
Dividend Growth Rate (g) = 6% = 0.06
Step 2: Use the Gordon Growth Model to calculate the current stock price.
The Gordon Growth Model formula is:
P0=D1
r−g
11
where: - P0= Current stock price - D1= Expected dividend next year - r=
Required rate of return - g= Dividend growth rate
Step 3: Calculate the expected dividend next year D1.
D1=D0×(1 + g)
D1= 3.50 ×(1 + 0.06)
D1= 3.50 ×1.06
D1= 3.71
Step 4: Plug in the values to calculate the current stock price P0.
P0=3.71
0.10 −0.06
P0=3.71
0.04
P0= 92.75
Therefore, the current value of the stock is 92.75pershare.
Question 16
Question
A company just paid a dividend of 5pershare.T hedividendisexpectedtogrowataconstantrateof 4
Solution
Step 1: Calculate the dividend growth rate. Given: Dividend = 5pershareGrowthrate =
4%
The dividend growth rate can be calculated as follows:
g=5×0.04
5= 0.04 = 4%
Step 2: Calculate the price of the stock today using the Dividend Discount
Model (DDM). The Dividend Discount Model is given by:
P0=D1
r−g
where: P0= Price of the stock today D1= Dividend expected next year r=
Required rate of return g= Growth rate
Substitute the values into the formula:
P0=5×(1 + 0.04)
0.10 −0.04
12
P0=5×1.04
0.06
P0=5.2
0.06
P0= 86.67
Thus, the current value of the stock is 86.67pershare.
Question 17
Question
A company pays an annual dividend of 5pershareandisexpectedtogrowatarateof 8
Solution
Step 1: Calculate the dividend in year 1 using the formula for the dividend in
year t:
D1=D0×(1 + growth rate) = $5 ×(1 + 0.08) = $5.40
Step 2: Calculate the price of the stock at the end of year 1 using the
dividend discount model formula:
P1=D1
r−g=$5.40
0.12 −0.08 = $135
Step 3: Calculate the current value of the stock by discounting the price at
the end of year 1 back to present value:
Current Value = P1
(1 + r)1=$135
(1 + 0.12)1=$135
1.12 ≈$120.54
Therefore, the current value of the stock is approximately
$
120.54.
Question 18
Question
A company pays an annual dividend of 3.50pershare.Iftherequiredreturnonthestockis12%andthedividendisexpectedtogrowataconstantrateof5%peryear, whatisthecurrentpriceofthestock?
Solution
Let’s denote the current price of the stock as P, the annual dividend as D, the
required return as r, and the growth rate of the dividend as g.
Step 1: Calculate the dividend growth rate (g).
g= 0.05
13
Step 2: Use the Gordon growth model to calculate the current price of the
stock. The Gordon growth model is given by:
P=D×(1 + g)
r−g
Step 3: Substitute the given values into the Gordon growth model formula.
P=3.50 ×(1 + 0.05)
0.12 −0.05
Step 4: Simplify the equation to find the current price of the stock.
P=3.50 ×1.05
0.07
P=3.675
0.07
P= 52.50
Therefore, the current price of the stock is 52.50.
Question 19
Question
A company’s stock is expected to pay a dividend of 5.00persharenextyear.Ifthedividendisexpectedtogrowatarateof 8
Solution
Step 1: Calculate the dividend per share in the second year. Given that the
dividend grows at a rate of 8
Dn=D1×(1 + g)n
where: Dn= dividend in year n,D1= dividend in the first year, g= growth
rate, n= year. Plugging in the values:
D2= 5.00 ×(1 + 0.08)2
D2= 5.00 ×(1.08)2
D2= 5.00 ×1.1664
D2= 5.832
Step 2: Calculate the expected price of the stock in the second year. The
expected price of the stock in the second year can be calculated using the for-
mula:
P1=D2
r−g
14
where: P1= price of the stock in the second year, D2= dividend in the second
year, r= required rate of return, g= growth rate. Plugging in the values:
P1=5.832
0.12 −0.08
P1=5.832
0.04
P1= 145.80
Step 3: Calculate the current value of the stock. To find the current value
of the stock, we need to find the present value of the stock in the second year
and discount it to the present value using the required rate of return.
P V =P1
(1 + r)1
Plugging in the values:
P V =145.80
(1 + 0.12)1
P V =145.80
1.12
P V = 130.54
Therefore, the current value of the stock is 130.54.
Question 20
Question
A company’s stock currently pays an annual dividend of 3.50pershare, whichisexpectedtogrowataconstantrateof5
Solution
Step 1: Calculate the dividend growth rate. Step 2: Use the Gordon Growth
Model to find the current value of the stock.
Step 1: The dividend growth rate (g) is given as 5
Step 2: Using the Gordon Growth Model:
P0=D0×(1 + g)
r−g
where: - P0= current stock price - D0= current dividend per share = 3.50−r=
requiredrateofreturn = 10−g= dividendgrowthrate = 5
Plugging in the values, we get:
P0=3.50 ×(1 + 0.05)
0.10 −0.05
15
P0=3.50 ×1.05
0.05
P0=3.675
0.05
P0= 73.50
Therefore, the current value of the stock is 73.50.
Question 21
Question
A company’s stock is expected to pay an annual dividend of 5pershareforever.If therequiredrateof returnonthestockis10
Solution
Step 1: Calculate the price of the stock using the Gordon Growth Model. Step
2: The Gordon Growth Model formula is given by:
P=D
r−g
where: P= Price of the stock, D= Annual dividend per share (given as 5), r
= Required rate of return (given as 10g= Growth rate of dividends (since the
dividend is expected to be paid forever, g= 0).
Step 3: Substituting the values into the formula:
P=5
0.10 −0
P=5
0.10
P= 50
Answer: The price of the stock is 50pershare.
Question 22
Question
A company’s stock currently pays a dividend of 5pershare.T hedividendsareexpectedtogrowataconstantrateof 7
16
Solution
Step 1: Calculate the expected dividend next year. Given: - Dividend this
year, D0=5 - Dividend growth rate, g= 7% = 0.07 - Required rate of return,
r= 12% = 0.12
The expected dividend next year, D1, can be calculated using the formula
for constant growth dividends:
D1=D0×(1 + g)
D1= 5 ×(1 + 0.07) = 5.35
Step 2: Calculate the price of the stock using the dividend discount model.
The price of the stock, P0, can be calculated using the dividend discount model:
P0=D1
r−g
Substitute the values to find the price:
P0=5.35
0.12 −0.07
P0=5.35
0.05
P0= 107
Therefore, the current price of the stock is 107pershare.
Question 23
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 rate of 5% per year indefinitely. If the required rate of return on
the stock is 10%, what is the value of the stock?
Solution
Step 1: Calculate the expected dividend in year 2. To find the dividend in year
2, we use the formula for dividend growth: D2=D1×(1 + growth rate). Given
that D1= $2 and the growth rate is 5%, we have:
D2= $2 ×(1 + 0.05) = $2.10
Step 2: Calculate the stock price. The value of a stock is equal to the present
value of all future dividends. The formula to find the present value of a growing
perpetuity is:
Stock Price = D1
r−g
17
Where: - D1= $2 (dividend in year 1) - r= 10% (required rate of return) -
g= 5% (growth rate) Plugging in the values, we get:
Stock Price = $2
0.10 −0.05 =$2
0.05 = $40
Therefore, the value of the stock is $40 per share.
Question 24
Question
A company’s stock is currently priced at $80 per share. The company pays
an annual dividend of $4 per share, which is expected to grow at a constant
rate indefinitely. The required rate of return for the stock is 10%. What is the
expected price of the stock in 5 years?
Solution
Let’s denote the expected price of the stock in 5 years as P5. We can use
the Gordon Growth Model to find the expected price. Step 1: Calculate the
dividend yield D0.
The dividend yield D0is calculated as the annual dividend divided by the
current stock price:
D0=$4
$80 = 0.05
Step 2: Calculate the growth rate of dividends g.
The growth rate of dividends gcan be calculated using the formula:
g=D0×P0
P0
= 0.05
Step 3: Use the Gordon Growth Model to find the expected price in 5 years.
The Gordon Growth Model is given by:
P5=D6
r−g
Since the dividend grows at a constant rate, the dividend in 5 years will be:
D6=D0×(1 + g)5= $4 ×(1 + 0.05)5
Now, substitute the values into the Gordon Growth Model formula:
P5=$4 ×(1 + 0.05)5
0.10 −0.05
Solving for P5gives the expected price of the stock in 5 years.
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Question 25
Question
A company pays a yearly dividend of 2.50pershareandisexpectedtogrowataconstantrateof 5
Solution
Let’s denote the yearly dividend as D = 2.50, thegrowthrateasg = 5
P0=D1
r−g
where P0is the current stock price and D1is the dividend expected to be
received at the end of year 1.
Step 1: Find D1Since the dividend is expected to grow at a constant rate,
we can find the dividend at the end of year 1 using the formula:
D1=D×(1 + g)
D1=
2.50 ×(1 + 0.05) =2.50 ×1.05 =2.625
Step 2: Calculate the Current Stock Price Now, we can substitute
D1=2.625, r = 0.10, and g = 0.05 into the Gordon Growth Model formula to
find the current stock price:
P0=2.625
0.10 −0.05
P0=2.625
0.05 =
52.50
Therefore, the current stock price is 52.50.
Question 26
Question
A company pays a dividend of 3pershareannuallyandisexpectedtohaveagrowthrateof 5
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Solution
Step 1: Calculate the dividend growth rate using the formula g=D1
D0
−1, where
gis the growth rate, D1is the dividend in the next year, and D0is the current
dividend.
g=3(1 + 0.05)
3−1
= 0.05
Step 2: Use the dividend discount model to find the stock price. The formula
is P=D
r−g, where Pis the stock price, Dis the dividend, ris the required rate
of return, and gis the growth rate.
P=3
0.10 −0.05
=3
0.05
= 60
Therefore, the current stock price is
$
60.
Question 27
Question
You are evaluating the stock of XYZ company, which is expected to pay a
dividend of 3.50persharenextyear.Dividendsareexpectedtogrowatarateof 7
Solution
Step 1: Calculate the dividend expected next year using the growth rate. Step
2: Calculate the price of the stock based on the dividend discount model.
Step 1: The dividend next year will be:
D1 = D0×(1 + g)
D1=3.50 ×(1 + 0.07) = 3.745
Step 2: The price of the stock can be calculated using the dividend discount
model formula:
P0 = D1
r−g
Where: P0 = Price of the stock today D1 = Dividend expected next year r=
Required rate of return g= Growth rate of dividends
Plugging in the values:
P0 = 3.745
0.10 −0.07
P0 = 3.745
0.03 = 124.83
Therefore, the value of the stock today is 124.83pershare.
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Question 28
Question
A company’s stock is expected to pay a dividend of
$
2 per share next year.
Dividends are expected to grow at a rate of 5
Solution
Step 1: Calculate the dividend one year from now using the growth rate. Step 2:
Use the dividend one year from now and the required rate of return to calculate
the present value of the stock.
Step 1: The dividend one year from now can be calculated using the for-
mula:
D1 = D0×(1 + g)
where: - D1 = Dividend one year from now (
$
) - D0 = Dividend today (
$
2 in
this case) - g= Growth rate (5
Substitute the values:
D1 = $2 ×(1 + 0.05)
D1 = $2.10
Therefore, the dividend one year from now is
$
2.10 per share.
Step 2: The present value of the stock can be calculated using the formula
for the present value of a growing perpetuity:
P=D1
r−g
where: - P= Present value of the stock - D1 = Dividend one year from now
(
$
2.10) - r= Required rate of return (10- g= Growth rate (5
Substitute the values:
P=$2.10
0.10 −0.05
P=$2.10
0.05
P= $42
Therefore, the value of the stock today is
$
42.
Question 29
Question
A company’s stock is expected to pay a dividend of 3.50 next year. The divi-
dends are expected to grow at a rate of 6% per year indefinitely. If the required
rate of return is 12%, what is the current value of the stock?
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Solution
Step 1: Calculate the expected dividend in year 2.
Dividend in year 2 = Dividend in year 1 ×(1 + Growth rate)
= 3.50 ×(1 + 0.06)
= 3.71
Step 2: Calculate the required rate of return as a decimal.
Required rate of return = 12% = 0.12
Step 3: Use the Gordon Growth Model to find the current value of the stock.
Current value of the stock = Dividend in year 1
Required rate of return −Growth rate
=3.50
0.12 −0.06
=3.50
0.06
= 58.33
Therefore, the current value of the stock is $58.33.
Question 30
Question
A company is expected to pay an annual dividend of 5.00pershareindefinitely.Iftherequiredrateofreturnis8
Solution
Step 1: Calculate the dividend growth rate using the formula:
g= Required rate of return −Dividend yield
Where: - The required rate of return is 8- The dividend yield is calculated
as the annual dividend divided by the current stock price. Since the current
stock price is not given, we will assume it’s Pfor now.
g= 0.08 −5
P
Step 2: Since the annual dividend is expected to be 5.00pershareindefinitely, wecanusetheGordonGrowthM odeltof indthevalueofthestock :
P=D0×(1+g)
r−g
Where: - D0is the annual dividend expected to be paid, which is 5.00 −r
is the required rate of return, which is 8- gis the growth rate we calculated in
Step 1
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Step 3: Substitute the values into the formula:
P=
5×(1 + 0.08−5
P
1+ 5
P
)
0.08 −5
P
Step 4: Solve for P by cross multiplying:
P(0.08 −5
P) = 5(1 + 0.08 −5
P
1 + 5
P
)
0.08P−5 = 5(1 + 0.08P−5
P+ 5 )
0.08P−5 = 5 + 5(0.08P−5)
P+ 5
5(0.08P−5) = P(5 + 0.08P−5)
0.4P−25 = P2+ 0.08P2−5P
Step 5: Simplify and rearrange the equation into a quadratic equation:
0.08P2−5P−25 = 0
Step 6: Solve the quadratic equation using the quadratic formula:
P=−(−5) ±p(−5)2−4∗0.08 ∗(−25)
2∗0.08
Solving for P will give two possible stock prices, and we choose the positive
value since price cannot be negative.
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