The name is absent

Incremental Risk Vulnerability

Proof: Risk vulnerability and nonpositive expectation of e imply condition (4). Since e is
independent of y and declining risk aversion is preserved under background risk, condition
(5) holds □

Next, consider the case in which the distribution of e depends on y such that the distribution
of e improves with increasing y according to a second-order stochastic dominance shift.

Corollary 7 Assume r' < 0, r'' > 0 and E(e|y) ≤ 0 ∀y. Moreover, the distribution of e
may improve with increasing y according to second-order stochastic dominance. Then the
increase in background risk replacing y by (y + e) raises the derived risk aversion.

Proof: From Gollier and Pratt(1996), r' < 0, r'' > 0 and E(e|y) ≤ 0 imply risk vulnerability
and, hence, condition (4). In Appendix 2 condition (5) is shown to hold, too □

5 Conclusion

This paper considers the effect on derived risk aversion of increases in background risk. We
first take the case of deterministic increases which are simple mean preserving spreads. We
present a necessary and sufficient condition for such an increase to raise the derived risk
aversion of an agent. Standard risk aversion and declining, convex risk aversion are shown
to be sufficient conditions.

We then analyse the effect of stochastic increases in background risk. If such an increase
is independent of the existing background risk and has a non-positive expectation, it raises
derived risk aversion if the agent is risk vulnerable. If the distribution of the increase
improves with increasing realisations of the existing background risk according to second-
order stochastic dominance and the conditional expectation of the increase is non-positive,
then the derived risk aversion of an agent with declining, convex risk aversion increases.

<<   9   10   11   12   13   14   15   >>

More intriguing information