1. Consider a portfolio optimization problem without short selling for n-3 risky securities with correlations, 12- 0.10, ,-0.20, and Pa-0.25. Following Markowitz's critical line method, the problem can be solved as MAX: xi 0 01] [0.20 92-10, 18 3 0.16 111] [0.10 "21-10.20 #3] 10.12 For this portfolio problem, the efficient frontier can be constructed in a small number of steps. Note 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 S3 :0.05040 + 0.08000 Step 2 s:-0.02523 + 0.04862 s2= 0.42202 + 0.91743 Step 3: S1a1 bil s2= 0.32642 + 1.10166 S3-0.40639-0.58677 where si is either xi or (a slack variable) depending on whether the security is included in or excluded from the portfolio. Note that the conditions xi20, 20, and Xi 20 must be satisfied a) Fill in the 6 missing numbers represented by, ai and bi for i 1,2,3. Indicate whether each s is a xi or a 6L (6 marks) Identify all critical values of . Between two adjacent critical A's (including -0), indicate which of the three securities are included in the portfolios along the efficient frontier. (6 marks) b) 1. Consider a portfolio optimization problem without short selling for n-3 risky securities with correlations, 12- 0.10, ,-0.20, and Pa-0.25. Following Markowitz's critical line method, the problem can be solved as MAX: xi 0 01] [0.20 92-10, 18 3 0.16 111] [0.10 "21-10.20 #3] 10.12 For this portfolio problem, the efficient frontier can be constructed in a small number of steps. Note 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 S3 :0.05040 + 0.08000 Step 2 s:-0.02523 + 0.04862 s2= 0.42202 + 0.91743 Step 3: S1a1 bil s2= 0.32642 + 1.10166 S3-0.40639-0.58677 where si is either xi or (a slack variable) depending on whether the security is included in or excluded from the portfolio. Note that the conditions xi20, 20, and Xi 20 must be satisfied a) Fill in the 6 missing numbers represented by, ai and bi for i 1,2,3. Indicate whether each s is a xi or a 6L (6 marks) Identify all critical values of . Between two adjacent critical A's (including -0), indicate which of the three securities are included in the portfolios along the efficient frontier. (6 marks) b)