Question

Stock A has an expected return of 5% and standard deviation of 10%. Stock B has an expected return of 10% and standard deviation of 15%. The correlation between the two stocks’ returns is 0.70. If you wanted to form a portfolio of these two stocks and wanted that portfolio to have an expected return of 8%, what weights would you put on each stock? Show your work (“algebra”). What would be the standard deviation of this portfolio?

Answer 1

Assume that % invested in stock A is X . Thus % invested in stock B = (1- X%)

Expected retrun = (% invested in stock A x Return of stock A) + (% invested in stock B x Return of stock B)

8% = X(5%) + (1-X)(10%)

0.08 = 0.05X + 0.1 - 0.1X

0.08 = 0.1 -0.05X

-0.02 = -0.05X

X = 40%

Thus % of amount invested in stock A = 40%

and % of amount invested in stock B = 60%

standard deviation of portfolio = [(W₁² x Standard deviation of Stock A²) +( W₂²x Standard deviation of Stock B²) + W₁x W₂ x Standard deviation of Stock A x  Standard deviation of Stock B x r]^0.5

W_{1 =} % of amount invested in stock A = 40%

W_{2 =} % of amount invested in stock B = 60%

Standard deviation of Stock A = 10%

Standard deviation of Stock B = 15%

r = correlation between the two stocks’ returns = 0.70

Thus standard deviation of portfolio = [(0.4² x 10²) + (0.6² x 15²) + (0.4 x 0.6 x 10 x 15 x 0.7)]^0.5

=[(0.16 x 100) + (0.36 x 225) + (25.20)]^0.5

=[16 + 81 + 25.20]^0.5

=122^0.5

=11.05

Thus standard deviation of portfolio = 11.05%