Example 1: Answer - Southern Methodist University

[Pages:4]ECO 4368 Instructor: Saltuk Ozerturk

Common Stock Valuation and CAPM

In the last lecture, we have seen that if the expected future divi-

dends grow at a constant rate g, then the current stock price P0

is given by

P0

=

D0(1 rs

+ g) g

where D0 is the last dividend paid by the company and rs is the required rate of return on the stock. The above expression can also be written

as

P0

=

D1 rs

g

where D1 is the next expected dividend.

In the above pricing formula, the required rate of return rs comes from CAPM, i.e.,

rs = rRF + s(rM rRF )

where s is the stock's beta.

Example 1: Thames Inc.'s most recent dividend was $2:40 per share (i.e. D0 = $2:40). The dividend is expected to grow at a constant rate of 6% per year. The risk-free rate is 5% and the return on the market is 9%. If the company's beta is 1:3, what is the price of the stock today?

Answer: First, let's ...nd the required rate of return rs :

rs = rRF + s(rM rRF ) = 5% + 1:3(9% 5%) ) rs = 10:2%

Now, one can compute the current stock price as

P0

=

D0(1 + g) = 2:40(1 + 6%)

rs g

10:2% 6%

) P0 = $60:57

1

Example 2: A company's stock is currently trading for $30 a share. The stock is expected to pay a $2:40 dividend at the end of the year (i.e. D1 = $2:40) . The dividend is expected to grow at a constant rate of g = 5% a year. The risk-free rate rRF = 5% and the market risk premium (rM rRF ) is 4 %. What is the stock's beta?

Answer: First, let's ...nd the required rate of return rs :

P0

=

D1 rs g

)

rs

=

D1 P0

+

g

)

rs

=

$2:40 $30

+

5%

) rs = 13%

Now we can ...nd the stock's beta from

rs = rRF + s(rM rRF ) 13% = 5% + s(4%) ) s = 2

Example 3: XYZ corporation's stock recently paid a dividend of $4 per share (D0 = $4). The company has a constant growth rate of g = 6% and a beta equal to 1.8. The required rate of return on the market is 8% and the risk-free rate is 4%. The company is considering a change of policy, which will increase its beta coe? cient to 2. If market conditions remain unchanged, what new constant growth rate will cause the common stock price remain the same?

Answer: First, let's ...nd the current rs that corresponds to the current beta of 1.8:

rscurrent = 4% + 1:8(4%) ) rscurrent = 11:2% The current stock price is then

P0current

=

D0(1 + g) rscurrent g

=

4(1 + 11:2%

6%) 6%

) P0current = $81:54

When company beta increases to

new s

=

2;

the

new

required

rate

on

the stock becomes

rsnew

=

rRF +

new s

(rM

rRF )

) rsnew = 4% + 2(4%) = 12%

2

To keep the current stock price at P0 = $81:54; we need

P0

=

$81:54

=

D0(1 rsnew

+

g new ) gnew

)

81:54

=

4(1 + 12%

g new ) gnew

) 12% gnew = 4(1 + gnew) ) 0:12 gnew = 0:049(1 + gnew) 81:54

) 1:049gnew = 0:12 0:049 ) gnew = 6:77%

Example 4: A company paid a recent dividend of $2 per share. (D0 = $2) and it had a beta of 1.8 before a reevaluation of company's risk. Before reevaluation, company's stock price was $100. After the revaluation, the stock price dropped to $80 due to a change in its beta. The dividend growth rate g is constant and it remained the same after reevaluation.The market risk premium is 5% and the risk-free rate is 4%: What is the new beta of the company that caused the price increase?

Answer: First, let us ...nd the required return on the stock when company beta was 1.8:

rsold = 4% + 1:8(5%) = 13%

Now, one can ...nd the constant dividend growth rate g:

P0old

=

2(1 + g) $100 = 13% g

) 100(0:13 g) = 2(1 + g)

) 13 100g = 2 + 2g

) 11 = 102g ) g = 10:78%

After the increase in beta we know that the price dropped to P0new = $80; which implies that

P0new

=

2(1 + 0:1078) $80 = rsnew 0:1078

) rsnew

2(1 + 0:1078) 0:1078 =

80

) rsnew = 13:55%

3

Now ...nally, we can compute the new beta

rsnew

=

13:55% = 4% +

new s

(5%)

)

new s

=

1:91

Example 5: Suppose a company conducts a study and concludes that if they spin o? their secondary business (which is more risky), the ...rm's beta, which is currently 1.6, will decline. However, the primary business that will be kept has lower pro...t margins and this will cause the company's constant growth rate in dividends to fall from 8% to 6%. If the company is trying to maximize its current stock price, how low should the beta fall to make the spin o? the preferred option? Assume market risk premium is 4% and risk free rate is 4%.

Answer: The spin-o? is the better alternative if

P0spinof f

=

D0(1 + gspinoff ) rsspinof f gspinof f

>

D0(1 + gcurrent) rscurrent gcurrent

= P0current

which can also be written as

(1 + gspinoff ) rscurrent We know that

gcurrent > (1 + gcurrent) rsspinoff

gspinof f (1)

gspinoff = 6% and gcurrent = 8%

rsspinof f

=

4% +

spinof s

f

(4%)

rscurrent

=

4% +

current s

(4%)

=

4%

+

1:6(4%)

=

10:4%

Therefore, the condition in (1) becomes

(1 + 6%) (10:4%

8%) > (1 + 8%) 4% +

spinof s

f

(4%)

6%

OR

(1:06) (0:024) 1:08

>

0:04 +

spinof s

f

(0:04)

0:06

)

0:023 + 0:02 >

spinof s

f

(0:04)

spinof f s

<

1:09

4

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