2 Problem 2: Markowitz Mean-Variance with Risky Securities (30 points) Consider an Investment Universe made of 3 securities $1, $2 and S3 with the following characteristics: 10% 2% 1% P1 4.5% Covariance matrix: > = 2% 11% 3% ; Expected Return vector: p = P2 0.30% 1% 3% 20% 2.85% 1. (5 points) Using the method of Lagrange multipliers, find (P.), the Global Minimal Variance Portfolio, which is the solution of min w Ew WER3 (3) s.t. eTw = 1. where e = (1, 1, 1)T 2. (5 points) Using the method of Lagrange multipliers, find (P1), the Markowitz Mean-Variance Portfolio with Expected Return equal to 2. Find also (P2), the Markowitz Mean-Variance Portfolio with Expected Return equal to p2 + p3. 3. (5 points) Using the Portfolios (Pi) and (P2) previously found, apply the Two-fund Theorem to find (P3), the Markowitz Mean-Variance Portfolio with Expected Return equal to Pitztes . 3