The name is absent

Incremental Risk Vulnerability

Corollary 3 Standard risk aversion is a sufficient condition for derived risk aversion to
increase with a simple mean preserving spread in background risk.

Proof: Standard risk aversion requires both positive, decreasing absolute risk aversion and
positive, decreasing absolute prudence. Further, r'(W) < 0 ⇒ p(W) > r(W) and hence
u"'(W) > 0. It follows that the condition in the Proposition for an increase in the derived
risk aversion can be written as ²

u^m(W2) - u"'(W1 )
u"(W2) - u"(W1)

Since p(W1) > r(W1), a sufficient condition is that the ratio of the square brackets exceeds
1. This, in turn, follows from decreasing absolute prudence, p'(W) < 0. Hence, standard
risk aversion is a sufficient condition □

Gollier and Pratt (1996) showed not only that standard risk aversion is sufficient for risk
vulnerability, but so also is declining and convex absolute risk aversion r(w). The next
corollary shows that the latter condition is also sufficient for incremental risk vulnerability.

Corollary 4 Declining and convex absolute risk aversion is a sufficient condition for de-
rived risk aversion to increase with a simple mean preserving spread in background risk.

Proof: From

As shown in the appendix, it suffices to consider a three-point distribution of background
risk (y1, y0, y2) with y1 < 0,y2 > 0,y1 <y0 <y2 and ∆(y0)=0,∆(y1) < 0, ∆(y2) > 0.
The first term in equation (3) is positive whenever r is declining and convex. This follows
since E(∆(y)) = 0 and ∆(y₂) > ∆(y ₁) implies that E[r'(W)∆(y)] ≥ 0. Since u'(W) is

²Note that whenever r'(W) has the same sign for all W, the three-state condition in the Proposition (i.e.
the condition on W, W1 , and W2) can be replaced by a two-state condition (a condition on W1 and W2).

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