Assume that there are two risky assets and one risk-free asset in the investment universe. Denote by u1 and 61 (/2 and 02) the mean return and the standard deviation of returns of the first (second) risky asset. The covariance between the returns of the two risky assets is denoted by 012, and r denotes the risk-free rate of return. We consider an investor that constructs a portfolio of all available assets. Recall that the mean return and the variance of returns of the investor's portfolio are given by Hp = W1( M1 - r) + w2( 12 - r) + r, op - wiof + 2w1w2012 + w202, where w1 and w2 are the weights of the first and the second risky asset, respectively, in the portfolio. The investor's goal is to select the optimal risky portfolio by maximizing the mean-variance utility function max U - Hp 21 , W2 where A is the investor's risk aversion coefficient. a) Without using linear algebra, derive the analytical solutions for the weights of the optimal risky portfolio and show that the weights are given by W1 1 (12 - r)01 - (M1 - 1)012 A 0102 - 012 W2 A 0102 - 012 b) Determine the weight of the risk-free asset in the investor's overall portfolio. c) Show that the weights of the risky assets in the tangency portfolio are given by WI (#1 - 7)(05- 012) + (12 -7) (09 - 012) W2 - (412 -1)01 - (41 - T)012 (#1 - 7)(03 - 012) + (12 - 7) (07 -012) d) Assume now that the returns on the two risky assets are independent. Demon- strate that in this case, the weights of the risky assets in the tangency portfolio are given by 201 1 142 2 Under which conditions are both weights positive? 2 W R 5 .C