Question

a. Fill in the missing values in the table. b. Is the stock of Firm A correctly priced according to the capital-asset-pricing model (CAPM)? What about the stock ofFirm B? Firm C? If these securities are not correctly priced, what is your investment recommendation for someone with a well-diversified portfolio?
You have been provided the following data on the securities of three firms, the market portfolio, and the risk-free asset:

Security Expected Return Standard Deviation Correlation Beta
Firm A 0.13 0.12 ? 0.9
Firm B 0.16 ? 0.4 1.1
Firm C 0.25 0.24 0.75 ?
The market portfolio (S&P500) 0.15 0.1 ? ?
The risk-free asset (U.S. T-Bill) 0.05 ? ? ?

a. Fill in the missing values in the table.
b. Is the stock of Firm A correctly priced according to the capital-asset-pricing model (CAPM)? What about the stock of Firm B? Firm C? If these securities are notcorrectly priced, what is your investment recommendation for someone with a well-diversified portfolio?

Answer 1

a. Let b_i = the betaof Security i

s_i = the standard deviation of Security i

s_m = the standard deviation of the market

r_i,m = the correlation between returns on Security i and the market

(i) b_i= (r_i,_m)(s_i) / s_m

0.9 = (r_i,_m)(0.12) / 0.10

r_i,_m = 0.75

(ii) b_i= (r_i,_m)(s_i) / s_m

1.1 = (0.4)(s_i) / 0.10

s_i = 0.275

(iii) b_i= (r_i,_m)(s_i) /s_m

= (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) = r_f + b[E(r_m) – r_f]

whereE(r) = the expectedreturn on the stock

r_f = the risk-free rate

b = the stock’s beta

E(r_m) = the expected return on the market portfolio

Firm A

r_f = 0.05

b = 0.9

E(r_m) = 0.15

E(r) = r_f + b[E(r_m) – r_f]

= 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 isonly13%. Therefore, Firm A’s stock is overpriced, and you should sell it.

Firm B

r_f = 0.05

b = 1.1

E(r_m) = 0.15

E(r) = r_f + b[E(r_m) – r_f]

= 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 also16%. Therefore, Firm A’s stock is correctly priced.

Firm C

r_f = 0.05

b = 1.8

E(r_m) = 0.15

E(r) = r_f + b[E(r_m) – r_f]

= 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 is25%. Therefore, Firm A’s stock is underpriced, and you should buy it.