Plz solve question 6 of return s. Hnd an expression tor n t Assume there is a ri

Question

Plz solve question 6

of return s. Hnd an expression tor n t Assume there is a risk-free asset with rate of the Xi's and 's. 6. (Simpleland) In Simpleland there are only two risky stocks, A and B, whose detai listed in Table 7.4. TABLE 7.4 DETAILS OF STOCKS A AND B Expected Standard deviation Price per sharerate of return Number of shares outstanding 100 150 of return 15% 9% Stock A Stock B $1.50 $2.00 15% 12% Furthermore, the correlation coefficient between the returns of stocks A and B is PAB = 3. There is also a risk-free asset, and Simpleland satisfies the CAPM exactly. (a) What is the expected rate of return of the market portfolio? (b) What is the standard deviation of the market portfolio? (c) What is the beta of stock A? (d) What is the risk-free rate in Simpleland?

Explanation / Answer

Capital of A = 100*1.5 = $150

Capital of B = 150*2 = $300

Total capital = $450

wA = 150/450 = 1/3, wB = 2/3

Cov(rA, rB) = 0.15*0.09*(1/3)

Return on portfolio = (1/3)*15 + (2/3)*12 = 13%

Variance of portfolio = (1/3)^2*0.15^2 + (2/3)^2*0.09^2 + 2*(1/3)*(2/3)*0.15*0.09*(1/3)

Variance of portfolio = 0.0081 = (0.0900)^2

Standard deviation of portfolio = 0.0900 = 9%

This portfolio is in fact market portfolio consisting only stock A and B.

A = Cov(rA, rM) / Var(rM)

Cov(rA, rM) = Cov(rA, wA*rA + wB*rB) = wA*Cov(rA, rA) + wB*Cov(rA, rB)

Cov(rA, rM) = wA*Var(rA) + wB*Cov(rA, rB) = (1/3)*0.15^2 + (2/3)*0.15*0.09*(1/3) = 0.0105

A = 0.0105/0.0081 = 1.296296 = 1.30

Using CAPM for stock A, rA = rF + A(rM - rF) = rF(1 - A) + A*rM

rF = (rA - A*rM)/( 1 - A)

rF = (15 - 1.30*13)/(1 - 1.30) = 6.33%

Risk-free rate = 6.33%

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