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
More intriguing information
1. The name is absent2. The name is absent
3. Estimating the Impact of Medication on Diabetics' Diet and Lifestyle Choices
4. The name is absent
5. The name is absent
6. The name is absent
7. Eigentumsrechtliche Dezentralisierung und institutioneller Wettbewerb
8. Cancer-related electronic support groups as navigation-aids: Overcoming geographic barriers
9. The resources and strategies that 10-11 year old boys use to construct masculinities in the school setting
10. A Rare Case Of Fallopian Tube Cancer