1) Consider portfolios mixing 3 risky assets K1,K2,K3. Following the book's notation, we write E[Ki]=mi for the mean returns and cij=Cov(Ki,Kj) for the covariance matrix. Suppose the mean returns are m=m1m2m3=0.10.050.05 and the covariance matrix is C=c11c12c13c12c22c23c13c23c33=101010102. [Note: usually solving linear systems by hand is tedious. But for this problem it's easy, since C1=201010101.] Recall that mean-variance analysis characterizes the minimum-variance portfolio with mean return as the solution of 5 linear equations in 5 unknowns (w1,w2,w3,1,2), namely the three equations 2Cw=1m+2u (in which u=111 ), combined with the two equations w1m1+w2m2+w3m3=andw1+w2+w3=1. (a) Show that if you set 1=0 and drop the equation w1m1+w2m2+w3m3=, the resulting system of 4 equations for (w1,w2,w3,2) describe the minimumvariance portfolio. (This requires only a sentence or two of explanation.) (b) Find the minimum variance portfolio by observing that w=22C1u, then choosing 2 so that w1+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=21C1m, then choosing 1 so that w1+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[Ki]=mi for the mean returns and cij=Cov(Ki,Kj) for the covariance matrix. Suppose the mean returns are m=m1m2m3=0.10.050.05 and the covariance matrix is C=c11c12c13c12c22c23c13c23c33=101010102. [Note: usually solving linear systems by hand is tedious. But for this problem it's easy, since C1=201010101.] Recall that mean-variance analysis characterizes the minimum-variance portfolio with mean return as the solution of 5 linear equations in 5 unknowns (w1,w2,w3,1,2), namely the three equations 2Cw=1m+2u (in which u=111 ), combined with the two equations w1m1+w2m2+w3m3=andw1+w2+w3=1. (a) Show that if you set 1=0 and drop the equation w1m1+w2m2+w3m3=, the resulting system of 4 equations for (w1,w2,w3,2) describe the minimumvariance portfolio. (This requires only a sentence or two of explanation.) (b) Find the minimum variance portfolio by observing that w=22C1u, then choosing 2 so that w1+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=21C1m, then choosing 1 so that w1+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
