01.1 01.2 (1 point) Two risky assets have return rates r, r2 that are random variables with expected values and 72, respectively. Let be the matr 02.1 022 of covariances. Because 61,2 = E[r - 71)(r2 - 12)] and 62,1 = E[(12 - 12)(r1 - )]; (1) 02,1 = 01.2 Consider the portfolio composed of these two assets with weights W1, W2. Then the variance of the portfolio is (2) Op? = E{-1 2-1 0jw;w; = w1^2*s9112+w2^2*s922^2+2*w1*22*sg12 (Use (1) to replace 02,1 with 01,2, so your answer should not involve 02.1. Type 01,1 as sg11, 01,2 as sg12, 02,2 as sg22, wi as w1, w, as w2.) The expected return rate for the portfolio is (3) Tp = w1rb1+w2 rb2 (Type 7i as rb1, 72 as rb2.) Let rf be the return rate for a risk-free asset. The portfolio F of the one fund theorem, is the combination of risky assets that maximizes (4) Ip-ry OP subject to the constraint (5) wi + w2 = 1 For any pair wi, w2 satisfying (5), it is clear that (6) r; = (wi + w2rf Subtracting (6) from (3) gives (7) 7p - r; = (-ry)w+(72 - ry)w2 Then from (7) and the square root of (2), Then from (7) and the square root of (2), rp-rf (8) = ((rb1-rf)*w1+(rb2-rf)*w2) / (w1^2*sg11^2+w2^2*sg22^2+2*w1*w2*sg12) op Maximizing (8) subject to the constraint (5) then gives the weights W7,1, W7,2 for the fund F. The method of Lagrange multipliers could be applied, but a quicker solution uses the observation that for any fixed angle 0, along the ray (w1, W2) = c(cos o, sin ) for c > 0, the value of right side of (8) depends on the angle O but is independent of c. The solution of the maximization problem may be broken up into two steps. Step 1: Find any pointi, m that maximizes (8) (ignore the constraint). This point determines the angle 0 of the maximizing ray. Step 2: Divide i, W2 by Wtot = Wi + z to find W7,1, WF,2, the point on the maximizing ray that satisfies the constraint. Results for this submission Entered Answer Preview Result correct (w1^2)*(sg11^2)+(w2^2)*(sg22^2)+2*w1*w2*sg12 wsgl + wsg22 + 2w1w2sg12 incorrect w1*rb1+w2*rb2 Wirbi + w2rb2 correct (rb1-rf)*w1+(rb2-rf)*w2 (rb rf) w1 + (rb2 rf) w2 correct 1 [(w1^2)*(sg11^2)+(w2^2)*(sg22^2)+2*w1*w2*sg12]^(1/2) (wsgl + wsg23 + 2w1w2sg12) incorrect At least one of the answers above is NOT correct