Math for Finance Thanks.
(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 0.1 0.05 m = m2 m3 -0.05 and the covariance matrix is C13 -1 C= C11 C12 C12 C22 C13 C23 C23 C33 1 0 0 1 -1 0 0 2 [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 2Cw = lim + 12u (1) (in which u= [i]). ), combined with the two equations wimi + w2m2 + w3m3 = u and Wi+w2 + w3 = 1. (2) (a) Show that if you set 11 = 0 and drop the equation wimi + w2m2 + W3m3 = M, 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 = 2 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? (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 0.1 0.05 m = m2 m3 -0.05 and the covariance matrix is C13 -1 C= C11 C12 C12 C22 C13 C23 C23 C33 1 0 0 1 -1 0 0 2 [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 2Cw = lim + 12u (1) (in which u= [i]). ), combined with the two equations wimi + w2m2 + w3m3 = u and Wi+w2 + w3 = 1. (2) (a) Show that if you set 11 = 0 and drop the equation wimi + w2m2 + W3m3 = M, 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 = 2 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
