Incremental Risk Vulnerability
Eeckhoudt and Kimball (1992) and Meyer and Meyer (1998) demonstrate this for the de-
mand for insurance, Franke, Stapleton and Subrahmanyam (1998) for portfolio choice. A
central question, in this context, is whether an additive background risk makes the agent
more risk averse.
Gollier and Pratt (1996) answer this question by considering an agent who starts without
background risk and then faces an independent background risk. They introduce the concept
of risk vulnerability and show that risk vulnerability is equivalent to the notion that an
undesirable risk can never be made desirable by the presence of an independent, unfair
risk. Furthermore, the background risk makes the agent more risk averse. Hence, such a
background risk reduces the agent’s demand for a risky asset, given a choice between a risky
and a risk-free asset. Gollier and Pratt derive a necessary and sufficient condition for risk
vulnerability. They show that a sufficient condition for risk vulnerability is either that the
absolute risk aversion of the agent is declining and convex or that the agent is standard risk
averse in the sense of Kimball (1993). In a recent paper Keenan and Snow (2003) relate
Gollier and Pratt’s condition of local risk vulnerability to compensated increases in risk,
introduced by Diamond and Stiglitz (1974). They show that the introduction of a small
fair background risk increases risk aversion of agents more, the higher is their index of local
risk vulnerability.
Usually, agents have to bear some background risk, but the level of this risk may change.
Therefore the relevant question is not so much whether the presence of background risk
makes the agent more risk averse, but whether an increase in this background risk makes
the agent more risk averse. Kimball (1993) analyzes patent increases in background risk.
He shows that such an increase raises the risk aversion of an agent if it raises the expected
marginal utility conditional on his tradable income and if the agent is standard risk averse.
Kimball argues that the background risk X is patently more risky than the background risk
x if X can be obtained from x by adding a random variable v such that the distribution of
v conditional on x improves for increasing x according to third-order stochastic dominance.
Eeckhoudt, Gollier and Schlesinger (1996) consider this issue in the context of increases in
an independent background risk that exhibit second order stochastic dominance. Given this
broad set of increases in background risk they derive necessary and sufficient conditions,
which leave room only for a small set of utility functions. Finally, Eichner and Wagener
(2003) discuss the conditions on two-parameter, mean-variance preferences such that the
agent is variance vulnerable, i.e. an increase in the variance of an independent background
risk induces the agent to take less tradable risk.
Intuitively, there must be an inverse relation between the set of admissible increases in
background risk considered and the set of utility functions that exhibit the characteristic of
increased risk aversion. Therefore, in this article we consider a smaller, but plausible set of