The name is absent

Incremental Risk Vulnerability

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 q_i denote the probability of the outcome y_i. 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

E [∆( y )]= £q i ∆( y i) = 0

i=0