STOCHASTIC

the utility function u over wealth w = p'x = Y.Pi*t is concave and continuously differentiable.
If the investor is an expected utility maximizer, his decision problem is
{maxE„u(p'x)|£x, = B, xt S 0}. (1)
Suppose the return vector p has the discrete distributionpr{p = p'} — pt> 0,1 = 1,...,L < oo,
and£/>, = 1. Then (1) is
maxZO) = piu(p l 'x) + ••• + pLu(p L 'x), s.t. ^xt = B, xt S 0. (2)
(a) Show that Z is concave and continuously differentiable and
dZ(x) _ y \8u[p^x}
dx, 4* I 8 w
(b) Attempt to develop an efficient algorithm to solve (2) for large L that exploits the
fact that each of the L terms in the objective is "similar."
(c) Suppose xx is unconstrained. Show that (2) becomes
maxv(x2,...,x„) = {piu[^pl l x,+p1 1 (B-Y,x,)'] + ••• + pLu[£pi L xt + pl L {B-Y1xi)~]}
s.t. Xi £ 0, where i = 2,...,«.
(d) Show that v is concave and continuously differentiable.
(e) What is dvjdx, ? Note its similarity to the expression in (a).
(f) How does the algorithm in (b) simplify when applied to (3)?
(g) Investigate algorithmic simplifications when u is quadratic, u = log, p l > 0 for all /,
and u is exponential.
17. Under certain assumptions, such as quadratic preferences or normally distributed
random returns, the mean-variance approach is consistent with the expected utility approach.
To provide generality to the risk-return approach, it is desirable to be able to consider a
risk-return measure that is free of such restrictions on the utility function or the probability
distributions. Let 0 ( < 0) if u is strictly concave (strictly convex) and that (j> = 0 if u is
linear.
(b) The risk premium n is defined by the equation u{w— n) = Eu(w), so that tj> is related
to n via = u(w) — u(w — n). Suppose u is strictly increasing; show that ^ = 0 if and only
ifrcso.
(c) Suppose u{w) is linearly transformed into v(w) — au(w) + b. Show that 4> is transformed
to a.
(d) Let Mk, assumed finite, be the kth moment of w about w. Suppose a Taylor's expansion
of u about w exists in a neighborhood of w sufficiently large to include all w for which
there is a nonzero probability of occurrence. Show that
l
!
c=2 k { -Mk
-
where u lk, (w) is the &th derivative of u evaluated at w — w.
(e) Suppose w is normally distributed. Show that (/> is proportional to variance, i.e.,
^ = aa\ where a = - ^ ~^T n •
MIND-EXPANDING EXERCISES 351