STOCHASTIC

8. Consider an investor having a utility function for wealth u(w) which is strictly increasing
in w. Recall that the risk premium n(w,z) of a random return z given initial wealth w is
defined by the equation
Let
(a) Show that if h is any constant,
u[w+Ez — n(w,z)~\ = Eu(w + z).
n(w+h,z)
P(t; w)
and /?(?)
= n(w,z+h).
u"{t)
be the absolute, partial relative, and relative measures of risk aversion, respectively.
(b) Show that > P/1(r) = K'[K _1 (0] is concave, linear, or convex according as A(t) is
nondecreasing, constant, or nonincreasing.
(c) Show that Y,(/) = «~'(0«'[«"' (f)] —»«'[«"'(')] is concave, linear, or convex
according as P(t; w) is nondecreasing, constant, or nonincreasing in /.
(d) Show that 4'B(/) = « -1 (0«'[«"'(')] is concave, linear, or convex according as
R(t) is nondecreasing, constant, or nonincreasing.
(e) Show that statements (bHd) may be altered to strictly concave, linear, strictly convex
as ... is strictly increasing, constant, or strictly decreasing.
Suppose that z is a random variable with distribution function F defined on an interval
[a, b]. We are interested in the behavior of the risk premium n(w,z) as z and w are subject
to scale changes. If z is replaced by Xz and X — constant, we restrict a and X in such a way that
P[w+Xz < 0] = 0, so that the individual can lose at most his total initial wealth.
(f) Show that (8l8X)[n(w,Xz)IX] > 0, =0, or