1. 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:S.T.i=1nxii+i=1nj=1nxixjiji=1nxi=1xi0 Expected returns and standard deviations are given by 123=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 Step 2: s1=0.02523+0.04862s2=0.42202+0.91743s3=a3+b3 Step 3: s1=a1+b1s2=0.32642+1.10166s3=0.406390.58677 where si is either xi or i (a slack variable) depending on whether the security is included in or excluded from the portfolio. Note that the conditions xi0,i0, and xii0 must be satisfied a) Fill in the 6 missing numbers represented by, ai, and bi for i=1,2,3. Indicate whether each si is a xi or a i. (6 marks). b) Identify all critical values of . Between two adjacent critical 's (including =0 ), indicate which of the three securities are included in the portfolios along the efficient frontier. (6 marks) 1. Consider an eight security universe with a constant correlation across all eight securities of =0.50. The return of the risk free security is rf=2%. The returns and standard deviations for the eight securities are as follows: 12345678=0.160.140.140.110.110.100.100.0612345678=0.100.100.080.100.050.080.050.08 a) If short selling is allowed, determine the optimal allocation across each security in the construction of the risky portfolio ( 6 marks). b) If short selling is not allowed determine the optimal allocation across each security in the construction of the risky portfolio. (6 marks). Consider a portfolio selection problem without short sales based on four securities in two industrial sectors (or groups) A and B. Securities 1 and 2 belong to sector A while securities 3 and 4 belong to sector B. The correlation amongst security returns within sector A is constant at A=0.60 while the correlation amongst security returns within sector B is constant at B=0.40. The returns of securities between sectors A and B are uncorrelated. The risk free return is rf=3% while the returns and standard deviations of the four securities are as follows: 1234=.05.09.12081234=.05101205 a) Determine the relevant "cutoff rate" applicable to each security (6 marks). b) Determine the optimal allocation of investment funds for each security. (6 marks)