STOCHASTIC

COMPUTATIONAL AND REVIEW EXERCISES
1. It is of interest to determine which quadratic functions of mean and variance are
consistent with expected utility (of wealth) maximization.
(a) Show that the function E(u) = a + bR — c[_(R) 2 + aR 1 '\ is consistent, where R and
aR 2 denote mean and variance, respectively.
(b) What restrictions on the parameters a, b, and c are needed ?
(c) Show that the function £(«) = a.R—paR 2 is not consistent.
(d) Show that it is not possible to transform the function in (a) to that in (b). [Hint:
Consider the shape of the indifference curves in mean-variance space.]
2. In Eqs. (2) and (3) of Samuelson's article in Chapter 1 the use of a Taylor series approximation
is illustrated.
(a) Develop bounds on the maximum error than can occur with such an approximation
when the series is truncated at two, three, and m terms.
(b) Calculate the exact amount of this error when the series is truncated at two terms
for the case when V(W) = W-0.\W 2 + 0.0\W 3 , n = 2, and XX and X2 are uniformly
distributed over the intervals [0,1] and [—1,2], respectively.
3. Refer to Limner's paper in Part II. To determine the optimal proportions of risky assets
associated with the Tobin separation theorem, one may maximize a function of the form
f(x) = a'xKx'Hx) 112 , over some convex set K, where X is positive definite and 0 £ K.
(a) Show that / is pseudoconcave on K n {x \ a'x g 0}. [Hint: Refer to Mangasarian's
"Composition" paper in Part I.]
(b) Show that/is strictly pseudoconcave on K n {x \ a'x > 0}. [Hint: Utilize Theorem
4 in Ziemba's paper.]
(c) Show that the optimal proportions are unique if a'x > 0 and K is compact. [Hint:
Utilize Theorem 5 in Ziemba's paper.]
(d) Show that maximizing f(x) = a'xK.x'Xx) 112 is equivalent to maximizing
h(x) = log/0c) = loga'x — i logxTx.
(e) Show that/is not concave as Lintner states. [Hint: Let
H-l ~t )• a=(My -
Choose x 1 =(20,10)', x 2 = (l,l)', and X = 0.8. Then show that f{Xx l + (1-A)x 2 ) <
A/(^) + (l-A)/(^ 2 ).]
(f) Show that h is concave. [Hint: Utilize the Cauchy-Schwarz inequality, i.e.,
(w'f ) 2 ^ («'«) (v'v) for arbitrary n vectors u and v, on the Hessian matrix of ft.']
(g) When is A strictly concave?
4. The problem is to consider the effects of tax changes on portfolio selection when the
return on only one type of security is taxable, such as when one wishes to analyze the effect
of income changes between tax exempt and taxable securities. Assume the utility function
over wealth is u(w) = (1 — b) w—bw 2 , where 0 < b < 1.
(a) Show that u defines units of utility so that «(-l) = -l and w(0) = 0, and that
Eu(w) = (1— b)w— b(w 2 + o 2 ). Let the investor's initial wealth be M and suppose that
xi and x2 are the dollar amounts invested in securities 1 and 2, which provide random
returns £i and {2 per dollar invested, respectively. If t is the percentage tax rate on asset 1,
COMPUTATIONAL AND REVIEW EXERCISES 331