STOCHASTIC

MIND-EXPANDING EXERCISES
1. Suppose u is the exponential utility function «(tv) = 1— e~ ow , a> 0, and that wealth
w = T."=iiiXi = i'x, where (t is the return per dollar invested in /' and xt is the level of
investment /. Suppose that the random vector £ is joint-normally distributed with finite
mean | and variance-covariance matrix X (all of whose elements are assumed finite). Show
that the problem of maximizing the expected utility is equivalent to maximizing the concave
quadratic function !'*—(aj2)x"Lx.
2. Consider an investor who wishes to allocate his resources B among investments
i = l,...,/i, which have returns per dollar invested of (d,...,i„) = £, where the £, have the
joint-normal distribution FOJi,...,£„). Let the return i? = L"= i £1 **• For any given
x = (JCI, ...,*„)> R has a normal distribution fx(R) which has mean 1,'x and variance x"Lx,
where £ and £ are the mean vector and variance-covariance matrix of 0.
(d) What does the result of (c) mean ?
(e) Show that there is no utility function u(R) that is consistent in an expected utility
sense to the decision rule obtained from (b), if a ^ 0.
(f) Show that the preference ordering can be represented by the utility function
«(«)
(g) What does a = 0 mean ?
— 00 if R < A and a = 0
R otherwise.
3. The Tobin separation theorem as described in Lintner's paper in Part II shows that
the efficient surface is a ray if there exists a risk-free asset. It is of interest to know if the
efficient surface converges to that ray if there is one asset that becomes less and less risky.
Suppose there are n risky assets that have mean returns |, and positive-definite variancecovariance
matrix E, and a risky asset /' = 0 that is independent of 1 = 1,..., n having mean {0
and variance /?. One may calculate the mean-variance efficient surface by computing
Kcfi = <
It 0
0 E
0
e'x = 1, i'x a a, x S 0 (i)
for all a, where x = (x0,x1,...,x„)' are the investment allocations in i = 0,...,«,
e = (1,1,...,1), and ! = (&,!i,...,£,)•
(a) Show that /(a, /?) is strictly concave in a for all fi > 0, assuming that x*(a) is one-to-one.
[Hint: Begin by showing that the matrix in the objective of (i) is positive definite.]
(b) Prove that limA_0 /(