Consider a portfolio optimization problem without short selling for n=3 risky securities with correlations, 12=0.10,13=0.20, and 23=0.25. Following Markowitz's critical line method, the problem can be solved as MAX:i=1xii+i=1j=1xixji 5.1.t=nnxixi=1=0 Expected returns and standard deviations are given by u1u2u3=0.100.200.12123=0.200.180.16 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 as per the lecture notes. Some partial results are as follows: Step 1: s1=0.05760+0.10000s2=a2+b2s3=0.05040+0.08000 steps. Note is being used here in place of the as per the lecture notes. Some partial results are as follows: Step 1: s1=0.05760+0.10000s2=a2+b2s3=0.05040+0.08000 Step 2: s2=0.02523+0.04862s2=0.42202+0.91743sy=a2+b3 Step 3: s1=a2+b2s2=0.32642+1.10166s2=0.406390.58677 where si is either x or 1( a slack variable) depending on whether the security is included in or excluded from the portfolio. Note that the conditions x:0,/0, and x;0 must be satisfied a) Fill in the 6 missing numbers rapresented by, a, and b; for i=1,2,3, Indicate whather each s; is a xi, or a , (6 marks). b) Identify all critical values of . Between two adjacent critical 's (including =01 indicate which of the three securities are included in the portfolios along the efficient frontier. ( 6 marks) Consider a portfolio optimization problem without short selling for n=3 risky securities with correlations, 12=0.10,13=0.20, and 23=0.25. Following Markowitz's critical line method, the problem can be solved as MAX:i=1xii+i=1j=1xixji 5.1.t=nnxixi=1=0 Expected returns and standard deviations are given by u1u2u3=0.100.200.12123=0.200.180.16 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 as per the lecture notes. Some partial results are as follows: Step 1: s1=0.05760+0.10000s2=a2+b2s3=0.05040+0.08000 steps. Note is being used here in place of the as per the lecture notes. Some partial results are as follows: Step 1: s1=0.05760+0.10000s2=a2+b2s3=0.05040+0.08000 Step 2: s2=0.02523+0.04862s2=0.42202+0.91743sy=a2+b3 Step 3: s1=a2+b2s2=0.32642+1.10166s2=0.406390.58677 where si is either x or 1( a slack variable) depending on whether the security is included in or excluded from the portfolio. Note that the conditions x:0,/0, and x;0 must be satisfied a) Fill in the 6 missing numbers rapresented by, a, and b; for i=1,2,3, Indicate whather each s; is a xi, or a , (6 marks). b) Identify all critical values of . Between two adjacent critical 's (including =01 indicate which of the three securities are included in the portfolios along the efficient frontier. ( 6 marks)
