3.0 (20 points for this problem) (1) Consider portfolios mixing 3 risky assets K1, K2, K3. Following the book's notation, we write E[K] = m; for the mean returns and Cij = Cov(Ki, K;) for the covariance matrix. Suppose the mean returns are mi m = m2 0.1 0.05 -0.05 m3. and the covariance matrix is Cu1 C= C12 C12 C13 C22 C23 C23 C33/ = 1 0 -1 0 1 0 -1 0 2 C13 [Note: usually solving linear systems by hand is tedious. But for this problem it's (2 0 1 easy, since C-1 = 0 1 0.] Recall that mean-variance analysis characterizes the 1 0 1 minimum-variance portfolio with mean return u as the solution of 5 linear equations in 5 unknowns (W1, W2, W3, 11, 12), namely the three equations 3). 2Cw=lim + 12u (1) (in which u= [1]), combined with the two equations wimi + w2m2 + w3m3 = u and wi+w2 + W3 = 1. (2) (a) Show that if you set di = 0 and drop the equation wimi + w2m2 + w3m3 = , the resulting system of 4 equations for (W1, W2, W3, 12) describe the minimum- variance portfolio. (This requires only a sentence or two of explanation.) (b) Find the minimum variance portfolio by observing that w = c-lu, then choos- ing 12 so that wi + W2 + W3 = 1. (c) Find another solution (i.e. another portfolio with minimum variance given its mean return) by setting 12 = 0, deducing that w = *C-lm, then choosing 11 so that wi + w2 + W3 = 1. (d) We know that the efficient frontier is the upper half of a certain hyperbola. Using your answers to (b) and (c), identify the weights of the portfolios on this frontier. (Hint: use the two fund theorem.) Do any of them have w3 = 0? 3.0 (20 points for this problem) (1) Consider portfolios mixing 3 risky assets K1, K2, K3. Following the book's notation, we write E[K] = m; for the mean returns and Cij = Cov(Ki, K;) for the covariance matrix. Suppose the mean returns are mi m = m2 0.1 0.05 -0.05 m3. and the covariance matrix is Cu1 C= C12 C12 C13 C22 C23 C23 C33/ = 1 0 -1 0 1 0 -1 0 2 C13 [Note: usually solving linear systems by hand is tedious. But for this problem it's (2 0 1 easy, since C-1 = 0 1 0.] Recall that mean-variance analysis characterizes the 1 0 1 minimum-variance portfolio with mean return u as the solution of 5 linear equations in 5 unknowns (W1, W2, W3, 11, 12), namely the three equations 3). 2Cw=lim + 12u (1) (in which u= [1]), combined with the two equations wimi + w2m2 + w3m3 = u and wi+w2 + W3 = 1. (2) (a) Show that if you set di = 0 and drop the equation wimi + w2m2 + w3m3 = , the resulting system of 4 equations for (W1, W2, W3, 12) describe the minimum- variance portfolio. (This requires only a sentence or two of explanation.) (b) Find the minimum variance portfolio by observing that w = c-lu, then choos- ing 12 so that wi + W2 + W3 = 1. (c) Find another solution (i.e. another portfolio with minimum variance given its mean return) by setting 12 = 0, deducing that w = *C-lm, then choosing 11 so that wi + w2 + W3 = 1. (d) We know that the efficient frontier is the upper half of a certain hyperbola. Using your answers to (b) and (c), identify the weights of the portfolios on this frontier. (Hint: use the two fund theorem.) Do any of them have w3 = 0