Question 1 (Markowtiz Efficient Frontier, (15 marks)). The minimum variance portfolio problem 1 min 22 TEx s.t. et x = 1, 4+x= R where the target return must be achieved exactly, short-selling is allowed, and the covariance matrix is assumed to be positive definite. Here, e is the n-dimensional vector of all ones. Assume that there are at least two assets whose mean returns are not equal to each other. Denote three scalars A = ete-le, B = ute-le and C = ute-?. 1. Prove that AC B2 = (8 2-1(Bu Ce). (4 marks) (Eu-e)" B2 > 0. 2. Prove that AC (4 marks) Hint: You may need the fact that a positive definite matrix has an inverse which is also positive definite Let x be the minimum variance portfolio and let o = xrEx be the minimum variance. The expected value of the portfolio is u' xk, which is equal to R. (R-B/A) (C/A-B2/A2) = 1 in the (o, R)-plane. (4 marks) 3. Prove that for every R, the point (oh, R) lies on the hyperbola i 4. Explain why the efficient frontier is produced by all portfolios or having RE marks) 2,00). (3 Question 1 (Markowtiz Efficient Frontier, (15 marks)). The minimum variance portfolio problem 1 min 22 TEx s.t. et x = 1, 4+x= R where the target return must be achieved exactly, short-selling is allowed, and the covariance matrix is assumed to be positive definite. Here, e is the n-dimensional vector of all ones. Assume that there are at least two assets whose mean returns are not equal to each other. Denote three scalars A = ete-le, B = ute-le and C = ute-?. 1. Prove that AC B2 = (8 2-1(Bu Ce). (4 marks) (Eu-e)" B2 > 0. 2. Prove that AC (4 marks) Hint: You may need the fact that a positive definite matrix has an inverse which is also positive definite Let x be the minimum variance portfolio and let o = xrEx be the minimum variance. The expected value of the portfolio is u' xk, which is equal to R. (R-B/A) (C/A-B2/A2) = 1 in the (o, R)-plane. (4 marks) 3. Prove that for every R, the point (oh, R) lies on the hyperbola i 4. Explain why the efficient frontier is produced by all portfolios or having RE marks) 2,00). (3