Incremental Risk Vulnerability
17
Proof of Corollary 7
We need to show that condition (5) holds if the distribution of e improves with increasing
y according to second-order stochastic dominance. Since
u u w u u( w + y + e ) , ,
r e( W + y ) = Ee ----•---— r ( W + y + e ) ,
EEe u ' ( w + y + e ) J
|
dr e( w + y ) |
= Ee |
u'(w + y + e) dr(w + y + e)’ |
|
dy |
Ee u! ( w + y + e ) dy |
+ er∖ dw + y + e) \r ( w + y + e )
LdyVEe u'( w + y + e .
The first term is a ”risk-adjusted” expectation ofdr(W+y+e)/dy.Ife were distributed inde-
pendently of y, then r' < 0 would imply a negative expectation. This is reinforced for r' < 0
and r'' > 0 if the distribution of e improves according to second-order stochastic dominance.
Now consider the second term. Using the proof technique of Gollier and Pratt (1996,
p. 1122) it follows that this term is negative if it is for every binomial distribution of e.
Suppose that e is distributed independently of y. Then u'' < 0 and u"' > 0 imply that
uz(w + y + e)/Eeu'(w + y + e) declines [increases] in y for the lower [higher] realization of e.
Hence r' < 0 implies that the second term is negative. This is reinforced if the distribution of
e improves according to second-order stochastic dominance. Hence dre(w + y)/d(w +y) ≤ 0
□