Incremental Risk Vulnerability
increases in background risk, the benefit being obtaining a broader set of utility functions
that have the desired attribute.
Rothschild and Stiglitz (1970) define a mean preserving spread of an existing risk as a shift
in the probability mass from the center to the tails of the distribution. As pointed out by
Eeckhoudt, Gollier and Schlesinger (1996), this is equivalent to a second degree stochastic
dominance shift, provided the mean is fixed. To this definition we add the restriction
that the increase in background risk raises the non-tradable income in some states above
a threshold level and lowers it in some states below the threshold. We call this increase a
simple mean preserving spread.
Let y be the independent background risk with E(y) = 0, then a simple mean preserving
spread is a deterministic change in y, ∆(y), such that ∆(y) ≤ [=] [≥ ] 0 for y< [=] [> ] y0
for a given a threshold level y0, and E[∆(y)] = 0. In this case, note that the rank order of
outcomes below y0 may change, as well as the rank order of outcomes above y0 .
We introduce the concept of incremental risk vulnerability. An agent is incremental risk
vulnerable if a simple mean preserving spread in background risk makes the agent more
risk averse. In section 2 we derive a necessary and sufficient condition for incremental
risk vulnerability. It turns out that the sufficient conditions for risk vulnerability given by
Gollier and Pratt are also sufficient for incremental risk vulnerability. However, declining
risk aversion is not required. All utility functions with a negative third and a negative
fourth derivative are also incremental risk vulnerable.
In section 3, we further consider a restricted set of stochastic increases in background risk
and derive sufficient conditions for risk aversion to increase. These conditions are illustrated
by examples.
3 Characterization of Incremental Risk Vulnerability
In this section we present a necessary and sufficient condition for the utility function to
exhibit incremental risk vulnerability. The agent’s income, W , is composed of the tradable
income w and the non-tradable income y, i.e. W = w + y. The non-tradable income
represents an additive background risk. y is assumed to be distributed independently of w
and to have a zero mean. Moreover, y is assumed to be bounded from below and above,
i.e. y ∈ (y ,y). Finally, W = w + y ∈ (W, VW) is assumed. Let (Ω, F, P ) be the probability
space on which the random variables are defined.