Exercise 3. Consider N nancial assets whose return are random variables R;- with the same expectation r. The risk (stan- dard deviation} of each return is 0. Assume that you have one million dollar and you build a portfolio composed with these N asset. We denote by It; the proportion of money invested in the asset 1'. Note that necessarily 25:1 III\" 2 1. Part A. Uncorrelated returns. We assume that the returns of each asset are not correlated. 1. Let XT be the value of the portfolio at time T and X (0) the initial value of the portfolio. Prove that the - X{T)X(0} - return rate of the portfollo X{0} 18 N Z HER; 17:1 . Prove that the risk (variance) associated to this portfolio is N Vansz = 02 Zn? 1'21 1'21 . Find the optimal allocation (If?) 151'st minimizing the risk of the portfolio under the constraint [:1 1r;- : 1. Compute the value of the risk associated to the portfolio with the allocation 11* found above. What hap- pened when N goes to +00? Part B. Correlated returns. We now assume that the returns are correlated and the pairwise correlation coefcients are all equal to p E [1,1]. . . . . . 2 N 2 2 . Prove that the risk assoclated to th15 portfolio 15 a 3:1\"; + pa Ziij nix}. . Find the optimal allocation (:)15g5N minimizing the risk associated to the portfolio. . Compute the value of the risk associated to the portfolio with the allocation 1?\" found above. What hap- pened when N goes to +00? Part C. Minimizing the risk with 2 assets . Consider 2 asset with return R1 and R2 such that Var(R1) = l, Var(R2) = 2 and Cov(R1, R2) = 1. What is the portfolio that minimizes the risk