You have been provided the following data on the securities of three firms and t
ID: 2737769 • Letter: Y
Question
You have been provided the following data on the securities of three firms and the market:
Security
E[Rj]
sj
rjM
bj
Firm A
0.13
.12
?
.90
Firm B
0.16
?
0.40
1.10
Firm C
0.25
0.24
0.75
?
Market
0.15
0.10
1
1
Risk-free
0.05
0
0
0
Assume the CAPM holds true.
A.) Fill in the missing values in the table.
B.) What is your investment recommendation on each asset? Buy or sell?
C.) Suppose that you are currently holding a portfolio consisting of Firm B only. If you increase your portfolio weight on Firm B by 0.2 (or 20%) and borrow the needed money at the risk-free rate, what will be the new standard deviation of your portfolio?
PLEASE SHOW FORMULAS
Security
E[Rj]
sj
rjM
bj
Firm A
0.13
.12
?
.90
Firm B
0.16
?
0.40
1.10
Firm C
0.25
0.24
0.75
?
Market
0.15
0.10
1
1
Risk-free
0.05
0
0
0
Explanation / Answer
a. Let bi = the beta of Security i
si = the standard deviation of Security i
sm = the standard deviation of the market
ri,m = the correlation between returns on Security i and the market
(i) bi = (ri,m)(si) / sm
0.9 = (ri,m)(0.12) / 0.10
ri,m = 0.75
(ii) bi = (ri,m)(si) / sm
1.1 = (0.4)(si) / 0.10
si = 0.275
(iii) bi = (ri,m)(si) / sm
= (0.75)(0.24) / 0.10
= 1.8
(iv) The market has a correlation of 1 with itself.
(v) The beta of the market is 1.
(vi) The risk-free asset has 0 standard deviation.
(vii) The risk-free asset has 0 correlation with the market portfolio.
(viii) The beta of the risk-free asset is 0.
b. According to the Capital Asset Pricing Model:
E(r) = rf + b[E(rm) – rf]
where E(r) = the expected return on the stock
rf = the risk-free rate
b = the stock’s beta
E(rm) = the expected return on the market portfolio
Firm A
rf = 0.05
b = 0.9
E(rm) = 0.15
E(r) = rf + b[E(rm) – rf]
= 0.05 + 0.9(0.15 – 0.05)
= 0.14
According to the CAPM, the expected return on Firm A’s stock should be 14%. However, the expected return on Firm A’s stock given in the table is only13%. Therefore, Firm A’s stock is overpriced, and you should sell it.
Firm B
rf = 0.05
b = 1.1
E(rm) = 0.15
E(r) = rf + b[E(rm) – rf]
= 0.05 + 1.1(0.15 – 0.05)
= 0.16
According to the CAPM, the expected return on Firm B’s stock should be 16%. The expected return on Firm B’s stock given in the table is also 16%. Therefore, Firm A’s stock is correctly priced.
Firm C
rf = 0.05
b = 1.8
E(rm) = 0.15
E(r) = rf + b[E(rm) – rf]
= 0.05 + 1.8(0.15 – 0.05)
= 0.23
According to the CAPM, the expected return on Firm C’s stock should be 23%. However, the expected return on Firm C’s stock given in the table is 25%. Therefore, Firm A’s stock is underpriced, and you should buy it.
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