(a) Portfolio with an expected return of 8%
Let wA and wB represent the portfolio weights of stocks A and B.
Expected portfolio return:
E(Rp) = wA(6%) + wB(10%)
Since wA + wB = 1:
8% = (1 - wB)(6%) + wB(10%)
8 = 6 + 4wB
wB = 0.50
wA = 0.50
Therefore, invest 50% in stock A and 50% in stock B.
Calculate the portfolio standard deviation:
σp = √[wA²σA² + wB²σB² + 2(wA)(wB)(σA)(σB)(ρAB)]
σp = √[(0.5)²(0.12)² + (0.5)²(0.20)² + 2(0.5)(0.5)(0.12)(0.20)(0.4)]
σp = √0.0184
σp = 0.13565
Answer (a): Invest 50% in each stock. Expected return = 8%; standard deviation = 13.56%.
(b) Portfolio with a standard deviation of 15%
Using the same portfolio standard deviation formula:
0.15² = wA²(0.12)² + wB²(0.20)² + 2(wA)(wB)(0.12)(0.20)(0.4)
Substitute wA = 1 - wB:
0.0225 = (1 - wB)²(0.0144) + wB²(0.04) + 0.0192wB(1 - wB)
Expand and simplify:
0.0352wB² - 0.0096wB - 0.0081 = 0
Using the quadratic formula:
wB = [0.0096 ± √(0.0096² + 4(0.0352)(0.0081))] / 0.0704
The two solutions are approximately:
wB = 0.6384 or wB = -0.3657
Assuming no short selling, use the positive solution:
wB = 0.6384
wA = 1 - 0.6384 = 0.3616
Calculate the expected return:
E(Rp) = 0.3616(6%) + 0.6384(10%)
E(Rp) ≈ 8.55%
Answer (b): Invest approximately 36.16% in stock A and 63.84% in stock B. The portfolio's expected return is approximately 8.55%, with a standard deviation of 15%.
Note: If short selling is permitted, the negative solution provides another possible portfolio with the same standard deviation.
