Question

1. Your client wants to construct a portfolio from two stocks (Stocks Y and Z) and has asked you to determine the portfolio weights for each stock that will generate the portfolio with the smallest standard deviation. Use the relevant information in the table below to calculate the standard deviation for each portfolio weighting scenario and indicate the scenario that you should recommend to your client.

	Stock Y	Stock Z
Portfolio Weighting Scenario 1	60%	40%
Portfolio Weighting Scenario 2	70%	30%
Portfolio Weighting Scenario 3	80%	20%
Standard Deviation	9%	6%
Covariance Between Stocks Y and Z	-0.003

Answer 1

Portfolio Weighting Scenario 1

Portfolio Variance = w1212 + w2222 + 2w1w2Cov(Y,Z)

w1 = weight of stock Y; w2 = weight of stock Z; 1 = standard deviation of stock Y; 2 = standard deviation of stock Z; Cov(Y,Z) = covariance between stock Y and Z

Portfolio Variance = 0.62*0.092 + 0.42*0.062 + 2*0.6*0.4*-0.003 = 0.36*0.0081‬ + 0.16*0.0036 - 0.00144 = 0.002916‬ + 0.000576‬ - 0.00144 = 0.002052‬

Portfolio Standard deviation = (Portfolio Variance)1/2 = (0.002052‬)1/2 = 0.002052‬0.5 = 0.0453 or 4.53%

Portfolio Weighting Scenario 2

Portfolio Variance = 0.72*0.092 + 0.32*0.062 + 2*0.7*0.3*-0.003 = 0.49*0.0081‬ + 0.09*0.0036 - 0.00126 = 0.003969‬ + 0.000324‬‬ - 0.00126 = 0.003033

Portfolio standard deviation = (0.003033‬)1/2 = 0.003033​​​​​​​​​​​​​​0.5 = 0.0551 or 5.51%

Portfolio Weighting Scenario 3

Portfolio Variance = 0.82*0.092 + 0.22*0.062 + 2*0.8*0.2*-0.003 = 0.64*0.0081‬ + 0.04*0.0036 - 0.00096‬ = 0.005184‬‬ + 0.000144‬‬ - 0.00096‬ = 0.004368‬

Portfolio standard deviation = (0.004368‬‬)1/2 = 0.004368‬​​​​​​​​​​​​​​0.5 = 0.0661 or 6.61%

Portfolio Weighting Scenario 1 is recommended because portfolio weight of stock Y's 60% and stock Z's 40% will generate the portfolio with the smallest standard deviation of 4.53%.