Incremental Risk Vulnerability
Proof: Let W2 — W1 → dW. In this case, u"'(W2) — u"'(W1) → u""(W)dW. Similarly
u''(W2) — u"(W1) → u"'(W)dW.
Hence, the condition in the Proposition yields, in this case, u""(W) < [=][>] — r(W)u'"(W).
This is equivalent to ∂θ∕∂W < [=][>] 0, ∀ W □
In Corollary 1, θ(W) = u"'(W)/u'(W) is a combined prudence/risk aversion measure. This
measure is defined by the product of the coefficient of absolute prudence and the coefficient
of absolute risk aversion. The corollary says that for a small background risk derived risk
aversion exceeds [is equal to] [is smaller than] risk aversion if and only if θ(W) decreases
[stays constant] [increases] with W. Hence, it is significant that neither decreasing prudence
nor decreasing absolute risk aversion is necessary for derived risk aversion to exceed risk
aversion. However, the combination of these conditions is sufficient for the result to hold,
since the requirement is that the product of the two must be decreasing. The condition
in corollary 1 is thus weaker than standard risk aversion, which is characterized by both
absolute risk aversion and absolute prudence being positive and decreasing. Note that
the condition in this case is the same as the ’local risk vulnerability’ condition derived by
Gollier and Pratt (1996). Local risk vulnerability is r" > 2rr', which is equivalent to θ' < 0.
Keenan and Snow (2003) define — θ' as the local risk vulnerability index. They show for a
small background risk that the difference between derived risk aversion and risk aversion
increases in this index.
Since an interior maximum of r(w) implies r'(w) = 0 and r"(w) < 0, it rules out local risk
vulnerability. Therefore, we have
Corollary 2 Risk vulnerability and incremental risk vulnerability rule out al l utility func-
tions with an interior maximum of absolute risk aversion.
An alternative way to interpret Corollary 1 and Proposition 1 is to assume u"' > 0. In
this case, Corollary 1 states that a marginal increase in background risk, starting with
no background risk, makes the agent more risk averse if and only if temperance t(W) =
—u""(W)/u"'(W) exceeds risk aversion r(W) everywhere. Proposition 1 states that a simple
mean preserving spread in background risk makes an agent more risk averse if and only if
— [u"'(W2) — u'"(Wι)]/[u''(W2) — u''(W1)] > r(W), for W1 ≤ W ≤ W2. The left hand side
of this inequality can be interpreted as an average temperance over the range [W1, W2]. In
their analysis of second order stochastic dominance shifts in background risk, Eeckoudt,
Gollier and Schlesinger (1996) find the much stronger condition t(W) ≥ r(W'),∀(W,W').
We now apply Proposition 1 to show that standard risk aversion is sufficient for incremental
risk vulnerability.