Please answer all parts (a)-(d).

Formula 3.5:

(4 pts) Hyperbola - The canonical form of a shifted hyperbola in the (0,4) plane is 02/cz - ( Mo)2/cm = 1. (a) Derive the expression for the 2 asymptotes of above shifted hyperbola (u intercept, slopes). (b) Formula 3.5 shows that the weights for any portfolio on the MVF are a linear function of p, w* (u) = api + b. Using this, show that the variance can be expressed as o(u) = Au? + Bu + C for all points on the MVF, and express the parameters A, B, C in terms of a, b, C. Hint: For any u, the MVF's variance is o(q) = (aji + b)^C(au + b) Au? + BH +C (c) Complete the square above to show that MVF can be written as 02 /cz ( Mob/c = 1 and hence is a hyperbola, and express Co, Cramo in terms of A, B, C. (d) For the 2 risky asset case, Ri ~ (41,01), R2 (141,01), R2 ~ (112, 02) with corre- lation pa u = Whi + (1 - w)M2, and hence w is a linear function of + 1 12 112 Mi-12 and the variance can be expressed as o2 = Au? + Bu + C. Express A, B, C and the asymptotes in terms of H1,H2,01,02, p. w* (u) = au + b (4 pts) Hyperbola - The canonical form of a shifted hyperbola in the (0,4) plane is 02/cz - ( Mo)2/cm = 1. (a) Derive the expression for the 2 asymptotes of above shifted hyperbola (u intercept, slopes). (b) Formula 3.5 shows that the weights for any portfolio on the MVF are a linear function of p, w* (u) = api + b. Using this, show that the variance can be expressed as o(u) = Au? + Bu + C for all points on the MVF, and express the parameters A, B, C in terms of a, b, C. Hint: For any u, the MVF's variance is o(q) = (aji + b)^C(au + b) Au? + BH +C (c) Complete the square above to show that MVF can be written as 02 /cz ( Mob/c = 1 and hence is a hyperbola, and express Co, Cramo in terms of A, B, C. (d) For the 2 risky asset case, Ri ~ (41,01), R2 (141,01), R2 ~ (112, 02) with corre- lation pa u = Whi + (1 - w)M2, and hence w is a linear function of + 1 12 112 Mi-12 and the variance can be expressed as o2 = Au? + Bu + C. Express A, B, C and the asymptotes in terms of H1,H2,01,02, p. w* (u) = au + b