1. Find the tangency portfolio associated with the following securities listed in Table 1. You may assume a risk-free rate of 4% and the Table 1: Securities Securities Expected Return Beta Unsystematic Risk, JE- 1 15 1.5 500 11 1.1 625 9 0.9 800 7 0.7 600 2 3 10 1.0 600 4 5 variance of the market index is 400. Hint: To find the tangency portfolio, the first thing that should come to our mind is Equation (2.5), which asks you to solve Ew = u - ry. Hence we have to find the covariance matrix of the security returns. We can assume a market model on each security: ri = Qi+ BirM +Ei, i =1, 2, 3, 4, 5, and have Cov(ri, r; ) = Cov(a; + BirM + Ei, a; + BirM + E;) = BiBjOM + Cov(Ei, E;), where the term Cov(e;, ;) takes value 0 if i / j, and o? if i = j. Hence we can fill the covari- ance matrix E, and the result follows by solving the equation. You may also tackle this question using matrix notation. Suppose the return vector is r = (r1, r2, r3, 4, 75) , the alpha vector is a = (@1, 02, 03, 04, 05) , the beta vector is B = (B1, B2, Bs, BA, B5) , and the idiosyncratic risk vector is e = (61, (2, (3, 64, 65) , then the market model is r =atrMBte. Taking variance we have E = Var(atrMB +E) = OMBB + Var(E). Note that the matrix Var(e) is a diagonal matrix with o? on the diagonal