Incremental Risk Vulnerability
a non-random negative y which then necessitates declining risk aversion. Since we only
consider fair background risks, declining risk aversion is not implied by incremental risk
vulnerability.
The main result of this paper is the following proposition which presents a necessary and
sufficient condition for a marginal simple mean preserving spread in background risk to
raise derived risk aversion, i.e. ∂r(w)/∂s > 0.
Proposition 1 (Derived Risk Aversion and Simple Mean Preserving Spreads in Back-
ground Risk)
If u'(W) > 0 and u"(W) < 0, then for any simple mean preserving spread in background
risk,
∂r(w)/∂s > [=][<] 0, ∀(w,y,s) ^⇒
u'"(W2) - u'"(W1) < [=][>] - r(W)[u"(W2) - u"(W1)],
∀ ( W, W1 ,W2), W <W1 ≤ W ≤ W2 < TW ,W2 - W1 <y - y .
Proof: See Appendix 1.
Proposition 1 allows us to analyze the effect of any simple mean preserving spread in an
independent background risk. Since a finite increase in background risk is the sum of
marginal increases, the sufficiency condition in Proposition 1 also holds for finite increases
in background risk.
In order to interpret the necessary and sufficient condition under which a simple mean
preserving spread in a background risk will raise the risk aversion of the derived utility
function, first consider the special case in which background risk changes from zero to a
small positive level. This is the case analyzed previously by Gollier and Pratt (1996) and
by Keenan and Snow (2003). In this case, we have
Corollary 1 Starting with no background risk, for any marginal increase in background
risk,
^(w) > [=][ <] r (w) if and only if -W < [=][>]θ, ∀ W
where θ(W) ≡ u",(W)/u'(W).