Incremental Risk Vulnerability
declining, it follows that the first term in (3) is positive. Now consider the second term:
∂[u'(W)/Ey [u`(W)]]/∂y ∆(y) is positive for y 1 and negative for y2 and has zero expectation.
Therefore a declining r implies that the second term is positive. Hence a sufficient condition
for ∂^(w)/∂s > 0 is a declining and convex r □
Although corollaries 3 and 4 use the property of declining risk aversion, this property is
clearly not required for incremental risk vulnerability, as already noted by Gollier and Pratt.
Corollary 5 : For every utility function with u"'(W) < 0 and u""(W) ≤ 0 a simple mean
preserving spread in background risk raises derived risk aversion.
Proof: u""(W) ≤ 0 implies that the left hand side of the condition in Proposition 1 is non-
positive. u"'(W) < 0 implies that the right hand side is positive □
A utility function with u111 (W) < 0 exhibits negative prudence and increasing risk aversion.
Yet this utility function has the property of incremental risk vulnerability if the fourth
derivative is also negative. In terms of equation (3), the second term is now negative, but
it is overcompensated by a strongly positive first term due to strong convexity of r.
An example of a utility function with the properties stated in corollary 5 is the HARA-
function
u(W) =
A+
Wγ
1-γ
, where γ ∈ (1, 2),W < A(γ - 1)
4 Stochastic Increases in Background Risk and Risk Aver-
sion
A simple mean preserving spread in background risk is a deterministic change relating ∆(y)
to y. A natural generalization is to consider a stochastic change e such that y is replaced
by (y + e) with e being distributed independently of w, but perhaps dependently on y.
In the case of dependence, the distribution of e is assumed to improve with increasing y
according to second-order stochastic dominance, i.e. the distribution of e conditional on y
second-order stochastically dominates the distribution conditional on a smaller y. It will be
assumed throughout that this improvement can be captured by the differential ∂e∕∂y. This
differential is zero in the case of independence. We, first, derive sufficient conditions on e
and on absolute risk aversion to ensure an increase in derived risk aversion and, second,
illustrate these conditions.