The name is absent



Incremental Risk Vulnerability


16


= 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.



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