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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)/dy0,    (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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