Incremental Risk Vulnerability
10
We analyse the agent’s derived risk aversion r(w) in the presence of only the y-risk and the
derived risk aversion ^(w) in the presence of the (y + e)-risk. For this purpose we define
re (w + y) as the derived risk aversion over the e-risk, given the income (w + y).
re(w + y) ≡
Ee[-u"(w + y + e)] _
Ee[u'(w + y + e)] ;
∀(w + y).
Proposition 2 provides sufficient conditions for the e-risk to raise the agent’s risk aversion.
Proposition 2 Let e be a random variable which is distributed independently of w, but per-
haps dependently on y . In case of dependence, the distribution of e improves with increasing
y according to second-order stochastic dominance.
Then
ʌ . . .
r(w) ≥ ^(w), ∀ w,
if
re(w + y) ≥ r(w + y), ∀ (w + y),
and
dre(w + y)/dy ≤ 0, ∀ (w, y).
(4)
(5)
This proposition is proved in Appendix 2. Condition (4) requires the risk aversion of an
agent with income w + y to be higher in the presence of the background risk, e. Condition
(4) rules out a subset of the second-order stochastic dominance increases in background
risk as analysed by Eeckhoudt, Gollier and Schlesinger (1996). It also rules out a simple
mean preserving spread since y2 >y1 does not imply y2 + ∆(y2) >y1 + ∆(y1). Condition
(5) requires the derived risk aversion re(w + y) to decline. For a small e-risk, condition
(5) implies declining risk aversion of u. Hence condition (5) requires this property to be
preserved under the e-risk.
Both conditions are quite natural given a utility function with declining risk aversion. The
following corollaries illustrate Proposition 2.
Corollary 6 The increase in background risk from y to (y +e) raises the derived risk aver-
sion if e is a random variable, distributed independently of y, with nonpositive expectation
and if the agent is risk vulnerable.
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