ECO 4368
Instructor: Saltuk Ozerturk
Common Stock Valuation and CAPM
• In the last lecture, we have seen that if the expected future dividends grow at a constant rate g, then the current stock price P0
is given by
D0 (1 + g)
P0 =
rs − 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
D1
P0 =
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 + β s (rM − rRF )
where β 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 find the required rate of return rs :
rs = rRF + β s (rM − rRF ) = 5% + 1.3(9% − 5%)
⇒ rs = 10.2%
Now, one can compute the current stock price as
2.40(1 + 6%)
D0 (1 + g)
=
rs − g
10.2% − 6%
⇒ P0 = $60.57
P0 =
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 find the required rate of return rs :
D1
D1
$2.40
+ g ⇒ rs =
⇒ rs =
+ 5%
rs − g
P0
$30
⇒ rs = 13%
P0 =
Now we can find the stock’s beta from
rs = rRF + β s (rM − rRF )
13% = 5% + β s (4%) ⇒ β 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 coefficient 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 find 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
D0 (1 + g)
4(1 + 6%)
=
rscurrent − g
11.2% − 6%
current
⇒ P0
= $81.54
P0current =
When company beta increases to β new
= 2, the new required rate on
s
the stock becomes
rsnew = rRF + β new
s (rM − rRF )
new
⇒ rs = 4% + 2(4%) = 12%
2
To keep the current stock price at P0 = $81.54, we need
D0 (1 + gnew )
4(1 + g new )
⇒
81.54
=
rsnew − g new
12% − g new
4(1 + gnew )
⇒ 0.12 − gnew = 0.049(1 + g new )
⇒ 12% − g new =
81.54
⇒ 1.049g new = 0.12 − 0.049 ⇒ g new = 6.77%
P0 = $81.54 =
• 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 find the required return on the stock when company beta was 1.8:
rsold = 4% + 1.8(5%) = 13%
Now, one can find the constant dividend growth rate g:
2(1 + g)
13% − g
⇒ 100(0.13 − g) = 2(1 + g)
⇒ 13 − 100g = 2 + 2g
⇒ 11 = 102g ⇒ g = 10.78%
P0old = $100 =
After the increase in beta we know that the price dropped to P0new =
$80, which implies that
2(1 + 0.1078)
rsnew − 0.1078
2(1 + 0.1078)
⇒ rsnew − 0.1078 =
80
⇒ rsnew = 13.55%
P0new = $80 =
3
Now finally, we can compute the new beta
rsnew = 13.55% = 4% + β new
s (5%)
new
⇒ β s = 1.91
• Example 5: Suppose a company conducts a study and concludes that
if they spin off their secondary business (which is more risky), the
firm’s beta, which is currently 1.6, will decline. However, the primary
business that will be kept has lower profit 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 off the preferred option? Assume
market risk premium is 4% and risk free rate is 4%.
Answer: The spin-off is the better alternative if
P0spinof f =
D0 (1 + gspinof f )
D0 (1 + gcurrent )
>
= P0current
spinof f
current − g current
spinof
f
r
rs
−g
s
which can also be written as
¡
¡
¢
¢
(1 + gspinof f ) rscurrent − g current > (1 + gcurrent ) rsspinof f − g spinof f
(1)
We know that
g spinof f = 6% and gcurrent = 8%
f
(4%)
rsspinof f = 4% + β spinof
s
current
(4%) = 4% + 1.6(4%) = 10.4%
rscurrent = 4% + β s
Therefore, the condition in (1) becomes
¢
¡
f
(4%) − 6%
(1 + 6%) (10.4% − 8%) > (1 + 8%) 4% + β spinof
s
OR
(1.06) (0.024)
1.08
f
β spinof
s
f
> 0.04 + β spinof
(0.04) − 0.06
s
f
⇒ 0.023 + 0.02 > β spinof
(0.04)
s
< 1.09
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