STOCHASTIC

Consider problem (1) and suppose that Z is positive definite and /?, p, and n are such that
p'x £ P is always binding.
(f) Show that x* = aj + a^and/GS) = b1 + b2P+b20 2 , where
ax = T.- l {dp-ec)j(c(adc-b)), a2 = T.- 1 (ac 2 e-bp)l(c(adc-bj),
bi = ai"Lau b2 = 2ai'a2, b3 = a2'a2,
a = e'£ -1 /?, b = e"L~ l e, c=/5'E~V> and d = p''L' 1 e.
[Hint: Develop the Kuhn-Tucker conditions for (1), then utilize the constraints to eliminate
the Lagrange multipliers.]
(g) Consider the parametric problem Z(t) = {min^Cx) \Ax = t,Bx = b], where g is a
convex function, A and B are constant matrices, b is a constant vector, and t is a vector
parameter. Show that Z is a convex function. [Hint: Choose t 1 and ? 2 from
{t\3x: Ax— t, Bx = b) and let x 1 and x 2 be optimal values when t = t l and t = t 2 ,
respectively, and so on.]
(h) When is Z strictly convex ? [Hint: Refer to Exercise I-ME-18.]
(i) Referring to (f), utilize (h) to show that/is strictly convex.
(j) Interpret the results in (f). What does it mean to have minimum variance equal to a
quadratic function of mean return?
15. (A diversification experiment) Consider the model in Exercise 14. Estimate the mean
and variance-covariance matrix for a group of m securities. Devise a preferential ordering
scheme so that securities with higher numbers are less preferred, such as the scheme i = 1
for maxi fijcr, 2 , etc.
(a) Calculate x„* and/(/?)„ for selected /? values, where ji(lgn£ m) refers to the optimal
portfolio when the n most preferred securities are considered to be the portfolio universe.
(b) Let
, _ /(A.-/(/?)».
" /(/?).-/(«,/
Show that (Ai = 1,