STOCHASTIC

(d) Develop a similar result to (c) when the risk-free asset has return (1 +r), r > 0, for
each dollar invested.
19. In Lintner's paper it is shown how to compute the efficient frontier by solving a
linear complementary problem of the form w = Xz—a, z'w = 0, z & 0, w 2: 0, where X is the
positive-definite variance-covariance matrix of returns and a is the mean return vector net
of the risk-free rate of return.
(a) Show that solving the linear complementary problem is equivalent to solving the
strictly concave quadratic program
{maxa'z-\z"Lz\ z g 0}. (1)
[Hint: Examine the Kuhn-Tucker conditions of (1).]
(b) Develop a modification of the simplex method to solve this quadratic program.
20. In Lintner's article it is shown that under the normality and existence of riskless asset
assumptions, the investor's decision problem may be decomposed into a two-stage process.
In stage 1, a choice is made of the optimal proportions of the risky assets, and in stage 2
optimal proportions of the risky asset and the riskless asset are chosen.
(a) Show that the proportions of the risky assets that are optimal are unique if the
variance-covariance matrix X of the risky assets is positive definite. [Hint: Begin by
showing that the efficient surface is strictly concave.]
(b) Illustrate graphically the various situations that may occur if X is not positivedefinite.
(c) How does one calculate these "different" ratios when X is positive semidefinite ?
(d) Is there a circumstance in which it would be useful to know more than one of these
ratios ?
21. Referring to the Vickson article, prove
(a) If/(A) = aX' + bX" with a, b, a, and jS constant (a # /?), and if
CI(