Part A: 1. Consider a portfolio optimization problem without short selling for n = 3 risky securities with correlations, P12 = 0.10, P13 = 0.20, and P23 = 0.25. Following Markowitz's critical line method, the problem can be solved as MAX: -1 2-1 Xilli + $1=1h=1 Xixj Wij S.T. 21=1 Xi = 1 Xi 20 Expected returns and standard deviations are given by ui [0.10] 0.20 u2 01 [0.20 02] = 0.18 L0.16 luz] 10.12] For this portfolio problem, the efficient frontier can be constructed in a small number of steps. Note 1 is being used here in place of the n as per the lecture notes. Some partial results are as follows: Step 1: S1 = -0.05760+ 0.10000 S2 = A2 + b22 S3 = -0.05040 + 0.08000 Step 2: S1 = -0.02523 + 0.048622 S2 = 0.42202 + 0.91743 S3 = a3 + b31 Step 3: S1 = a1 + bid S2 = 0.32642 + 1.10166 S3 = 0.40639-0.586771 where s; is either xior 8; (a slack variable) depending on whether the security is included in or excluded from the portfolio. Note that the conditions xi 2 0,8;20, and x; 8;20 must be satisfied a) Fill in the 6 missing numbers represented by, aj, and b; for i = 1,2,3. Indicate whether each s; is a xi or a di. (6 marks). b) Identify all critical values of 1. Between two adjacent critical ls (including 1 = EO), indicate which of the three securities are included in the portfolios along the efficient frontier. (6 marks) Part A: 1. Consider a portfolio optimization problem without short selling for n = 3 risky securities with correlations, P12 = 0.10, P13 = 0.20, and P23 = 0.25. Following Markowitz's critical line method, the problem can be solved as MAX: -1 2-1 Xilli + $1=1h=1 Xixj Wij S.T. 21=1 Xi = 1 Xi 20 Expected returns and standard deviations are given by ui [0.10] 0.20 u2 01 [0.20 02] = 0.18 L0.16 luz] 10.12] For this portfolio problem, the efficient frontier can be constructed in a small number of steps. Note 1 is being used here in place of the n as per the lecture notes. Some partial results are as follows: Step 1: S1 = -0.05760+ 0.10000 S2 = A2 + b22 S3 = -0.05040 + 0.08000 Step 2: S1 = -0.02523 + 0.048622 S2 = 0.42202 + 0.91743 S3 = a3 + b31 Step 3: S1 = a1 + bid S2 = 0.32642 + 1.10166 S3 = 0.40639-0.586771 where s; is either xior 8; (a slack variable) depending on whether the security is included in or excluded from the portfolio. Note that the conditions xi 2 0,8;20, and x; 8;20 must be satisfied a) Fill in the 6 missing numbers represented by, aj, and b; for i = 1,2,3. Indicate whether each s; is a xi or a di. (6 marks). b) Identify all critical values of 1. Between two adjacent critical ls (including 1 = EO), indicate which of the three securities are included in the portfolios along the efficient frontier. (6 marks)
