Incremental Risk Vulnerability
15
f *(w,y,s) = qι∆(yι) [-u'"(Wι) - u"(Wɪ)r] + q2∆(y2) [-u'"(W2) - u"(W2)r] (16)
Since q 1∆(yɪ) + q2∆(y2) = 0, it follows that (14) can be rewritten as
f*(w,y,s) = qι∆(yɪ) [(-u’"(Wɪ) - u"(Wɪ)f) - (-u’"(W2) - u"(W2)f)] (17)
Hence
u'"(W2) - u'"(Wɪ) < -f [u"(W2) - u"(Wɪ)] (18)
is a sufficient condition for f* > 0, given f.
As shown in equation (10), Γ is a convex combination of r(Wα), r(Wɪ) and r(W2) with
Wɪ < Wα < W2, hence f ∈ {r(W)|W ∈ [Wɪ,W2]}. Hence, a sufficient condition for (18) is
that
u'"(W2) - u'"(Wɪ) < -r(W) [u"(W2) - u"(Wɪ)] (19)
for all {Wɪ ≤ W ≤ W2} as given by the condition of Proposition 1.
Alternatively, suppose that q0 replaces either qɪ or q2 in the optimal basis. In this case the
above argument remains the same with qα instead of either qɪ or q2, in equation (16).
b) By an analogous argument, it can be shown that ∂r(w)/∂s < [=] 0 is equivalent to
u'"(W2) - u"'(Wɪ) > [=] - r(W)[u''(W2) - u''(Wɪ)] ∀ {Wɪ ≤ W ≤ W2} □
Appendix 2
Proof of Proposition 2
We need to show that conditions (4) and (5) are sufficient for r(w) - ^(w) ≥ 0. Ev [∙] denotes
expectations over v. From the definition of the twice derived risk aversion, ^,
ʌ . __
r(^ w ) = Ey+e
u '( w + y + e ) , .
---v ,z r---_ ( w + y + e )
Ey+e u ' ( w + y + e ) v
ee u /( w + y + e ) E Г u /( w + y + e ) r ( w + y + e A
Ey+e u ' ( w + y + e ) e t Ee u '( w + y + e ) ʃ