Optimization with Inequality Constraints 1. Solve 8>< >: max x x3 3x subject to x 2 (3) and use an optimizer to verify your answer. 2. Solve 8>>>< >>>: min x1;x2;x3 (x1 2)2 + 2(x2 1)2 subject to x1; x2; x3 0 2x1 + 2x2 + 4x3 a (4) 5 Mean-Variance Optimization Consider an Investment Universe made of 3 stocks S1, S2 and S3 with the following characteristics: Covariance matrix: = 0 @ 0:010 0:002 0:001 0:002 0:011 0:003 0:001 0:003 0:020 1 A; Expected Return vector: = 0 @ 1 2 3 1 A = 0 @ 4:27% 0:15% 2:85% 1 A: 1. Find the Global Minimal Variance Portfolio. Find the Minimum Variance Portfolio (P1) with Expected Return equal to 1: 2. Find the Minimum Variance Portfolio (P2) with Expected Return equal to 2 + 3: 3. Using the Portfolios (P1) and (P2) previously found, apply the Two-fund Theorem to nd the Minimum Variance Portfolio with Expected Return equal to 1 + 2 + 3 3 : 4. Write a computer program in R, Matlab or Python to solve the 2 previous questions. Apply the Two-fund Theorem to generate and plot the Mean-Variance ecient frontier (the graph should also display the Expected Returns and Volatilities of securities S1, S2 and S3).