Incremental Risk Vulnerability
Definition 1 (A Simple Mean Preserving Spread in Background Risk)
Let y be a background risk with E(y) = 0. Then a simple mean preserving spread in the
background risk changes y to y + s∆(y), with E(∆(y)) = 0, where ∆(y) ≤ [=] [≥ ] 0 for
y< [=] [> ] y0 ,and s ≥ 0 denotes the scale of the increase.
The agent’s utility function is u(W). We assume that the utility function is state-independent,
strictly increasing, strictly concave, and four times differentiable on W ∈ (W, W). We as-
sume that there exist integrable functions on ω ∈ Ω, u0 and u 1 such that
u0(ω) ≤ u(W) ≤ u1(ω)
We also assume that similar conditions hold for the derivatives u'(W),u''(W) and u'"(W).
The agent’s expected utility, conditional on w, is given by the derived utility function, as
defined by Kihlstrom et al. (1981) and Nachman (1982):
ν(w)=Ey[u(W)] ≡ E[u(w + y)|w]
(1)
where Ey indicates an expectation taken over different outcomes of y. Thus, the agent with
background risk and a von Neumann-Morgenstern concave utility function u(W) acts like an
individual without background risk and a concave utility function ν(w). The coefficient of
absolute risk aversion is defined as r(W) = — u"(W)/u1 (W) and the coefficient of absolute
prudence as p(W) = — u111 (W)/u"(W). The absolute risk aversion of the agents derived
utility function is defined as the negative of the ratio of the second derivative to the first
derivative of the derived utility function with respect to w, i.e.,
^( w ) = —
ν"(w) _ Ey [u"(W)]
ν '( w ) Ey [ u ' ( W )]
(2)
It is worth noting that, in the absence of background risk, ^(w) is equal to r(w), the
coefficient of absolute risk aversion of the original utility function.
We are now in a position to define incremental risk vulnerability.
Definition 2 (Incremental Risk Vulnerability)
An agent is incremental risk vulnerable if a simple mean preserving spread in background
risk increases the agent’s derived risk aversion for all w.
This definition also includes the case in which the agent initially has no background risk.
This case is analyzed by Gollier and Pratt (1996). Hence incremental risk vulnerability
implies risk vulnerability subject to E[∆(y)] = E[y] = 0. Gollier and Pratt allow also for