STOCHASTIC

RISK AVERSION 135
12. INCREASING AND DECREASING PROPORTIONAL RISK AVERSION
We will call a utility function [strictly] increasingly or decreasingly proportionally
risk-averse if it has a [strictly] increasing or decreasing local proportional risk
aversion function. Again the corresponding local and global properties are equivalent,
as the next theorem states.
THEOREM 6: The following conditions are equivalent.
(a") The local proportional risk aversion function r*(x) is [strictly] decreasing.
(b") The proportional risk premium n*(x,z) is a [strictly] decreasing function of x
for all z.
(c") The proportional probability premium p*(x,h) is a [strictly] decreasing function
of x for all h>0.
The same equivalences hold if "increasing" is substituted for "decreasing" throughout
and/or attention is restricted throughout to an interval, that is, if the requirement
is added that x, x + xz, x + xh, and x — xh all lie in a specified interval.
PROOF: This theorem follows upon application of Theorem 1 to ux{x)=u(x)
and u2(x) = u(kx) for arbitrary x and k.
A decreasingly risk-averse utility function may be increasingly or decreasingly
proportionally risk-averse or neither. For instance, u(x) exp[ — q~\x+by],
with 6jS0, q0 while its local
proportional risk aversion function r*(x) = x(x+b)- i [(x+b)*+l— q] is strictly
increasing if 0