3. (40 points) Consider the optimization problem max Rp(wi,w2) = piwi + p2w2 subject to (3) Vow +02w + 2P120102W1W2 = OT Wi+w2 = 1 where we are given two securities with Expected Returns pi and P2, volatilities 01 and 02 and correlation P12. In equation (3), of is positive number called the target risk. 2 (a) Solve the problem (3) using a Lagrangian approach. You will denote the solution the optimal solution by w* (or) and the optimal value of the problem by Rp(wi(or), wh(t)) by Rp(or). (b) Assume that pi 5%, p2 = 10%, 01 = 10%,02 = 20%, p12 = -0.5, and given a grid of ot in the range (2%,30%] by step of 0.5%, plot the efficient frontier, namely the graph of the mapping of Rplor). That graphs maps of from the x-axis into Rp(ot) on the y-axis. (c) Given an arbitrary PT, solve min owi + 02w2 + 2p12 subject to (4) Piwi + P2w2 = PT += 1 1 1 = + (d) Plot the Efficient Frontier for problem (4) and compare to the result of question 3(b)