Incremental Risk Vulnerability
12
Appendix 1
Proof of Proposition 1
From the definition of r(w),
r( w ) =
Ey [-u "( W )]
Ey [ u, ( W )]
(6)
we have the following condition. For any distribution of y and for any s ≥ 0,
∂r(w)/∂s > [=][<] 0 <⇒ f (w,y,s) > [=][<] 0, (7)
where f (w,y,s) is defined as
f (w,y,s) ≡ Ey [∆(y) {-u"'(W) - u"(W)^(w)}] . (8)
Necessity
We now show that
f (w,y,s) > [=][<] 0=⇒
u'"(W2) - u"'(W1 ) < [=][>] -r(W) [u"(W2) - u"(W1)] ,∀ W1 ≤ W ≤ W2
Consider a background risk with three possible outcomes, y0, y 1, and y2, such that
y 1 < y0 < y2 and ∆(y 1) < ∆(y0) = 0 < ∆(y2). Define
Wi = w + yi + s∆(yi), i = 0, 1, 2,
and let qi denote the probability of the outcome yi. For the special case of such a risk,
equation (8) can be written as
f (w,y,s) = q 1∣∆(y 1)| {-u'"(W2) + u111 (W1 ) - [u"(W2) - u"(W1)]^(w)} (9)
since
2
E [∆( y )]= £q i ∆( y i) = 0
i=0