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(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
(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(K, K;) for the covariance matrix. Suppose the mean returns are m1 = 0.1 m = m2 = 0.05 m3 -0.05 and the covariance matrix is C11 C12 C13 1 - C = C12 C22 C23 0 1 0 C13 C23 C33, -1 0 2 [Note: usually solving linear systems by hand is tedious. But for this problem it's (201 easy, since C-1010] Recall that mean-variance analysis characterizes the 1 0 1 minimum-variance portfolio with mean return as the solution of 5 linear equations in 5 unknowns (w1, W2, W3, A1, A2), namely the three equations 2Cw1m+2u (in which u = [1]), combined with the two equations (1) w1m1+w2m2 + w3m3 = and w + w2+ w3 = 1. (2) (a) Show that if you set = 0 and drop the equation wm +w2m2 + w3m3 = , the resulting system of 4 equations for (w1, W2, W3, A2) describe the minimum- variance portfolio. (This requires only a sentence or two of explanation.) (b) Find the minimum variance portfolio by observing that w = 2C-u, then choos- ing 2 so that w + w2 + w3 = 1. (c) Find another solution (i.e. another portfolio with minimum variance given its mean return) by setting 2 = 0, deducing that w = 11C-1m, then choosing A so that ww+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?
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