Incremental Risk Vulnerability
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= Ey
Ee u '( w + y + e )
Ey+e u ' ( w + y + e )ee w y ,
where re (w + y) is as defined on page 9. Hence
^^ / ∖ ʌ / \
r( w ) — r( w )
Ey
Ee u '( w + y + e )
Ey+e u '( w + y + e )
Ey
u '( w + y ) , , .
---(—---(r-( r e( w + y )
Ey u' ( w + y )V
— r(w + y))
Condition (4) implies that the second term is positive or zero. The first term is similar
to a covariance term since the term in ( ) has zero expectation. Hence the first term is
nonnegative if the term in ( ) is single crossing downwards and re (w + y) is declining in y .
The latter is implied by condition (5). Therefore, to complete the proof we have to establish
the single crossing downward property. For notational simplicity, let Z(w + y) denote the
term in ( ),
Z(w + y)=
Ee u '( w + y + e )
a
u '( w + y )
b
with a and b being appropriately defined constants.
Differentiating with respect to y yields
Z '( w + y )
Ee u "( w + y + e )(1 + dy ) u "( w + y )
--------------------------- — -----:-----
ab
Ee u'( w + y + e ) u'( w + y ) Eeu ( w + y + e ) ∂y
---------------r e( w + y )+--;----r ( w + y )+---
ab a
For Z = 0 it follows that sgnZ`(w + y) = sgn[r(w + y) — re(w + y) + [Eeu'(w + y +
e)]-1 Eeu"(w + y + e)(∂e∕∂y)]. Hence condition (4) implies Z'(w + y) ≤ 0 at a crossing point
if e is distributed independently of y, i.e. ∂e∕∂y ≡ 0. Then only one crossing point exists,
therefore Z(w + y) is downward sloping. If the distribution of e improves with increasing y
according to second-order stochastic dominance, then Eeu"(w + y + e)(de/dy) < 0 if u111 > 0.
u`" > 0 follows from condition (5) because dre(w + y)/dy ≤ 0 holds for a small risk only if
r' < 0. Hence, at a crossing point, Z'(w + y) ≤ 0. □