The name is absent



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 s0,

∂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. How does an infant acquire the ability of joint attention?: A Constructive Approach
2. The name is absent
3. The name is absent
4. The name is absent
5. On the Relation between Robust and Bayesian Decision Making
6. The name is absent
7. The name is absent
8. EXECUTIVE SUMMARIES
9. The name is absent
10. Improving Business Cycle Forecasts’ Accuracy - What Can We Learn from Past Errors?