Incremental Risk Vulnerability
1 new bit for introduction
In this paper, we consider a particular set of increases in an independent background risk.
In reality all investors face some level of background risk, so a relevant question is how they
would react to an increase in such a risk. We look for conditions on utility functions such
that an agent becomes more averse to a market risk given an increase in background risk.
This question has been considered previously by Kimball (1993) and by Eeckhoudt, Gollier
and Schlesinger (1996). Kimball considers the set of ’patent increases’ in background risk1.
He shows that standard risk aversion is a sufficient condition for ’incremental risk vulner-
ability’ for this set of increases. Eeckhoudt, Gollier and Schlesinger (1996), on the other
hand, consider the larger sets of first order and second order stochastic dominance increases
in background risk. They find that the conditions on utility functions are quite restrictive
and exclude many commonly assumed utility functions.
In this paper, we follow Kimball’s general approach by looking at the effect of particular
sets of increases in background risk. We begin with a set of non-stochastic mean-preserving
spread increases that we term ‘simple increases’. For this set, we derive a necessary and
sufficient condition for incremental risk vulnerability. We find in this case, a condition on
utility which is weaker than standard risk aversion.
An interesting subset of our simple increases is the set of monotonic increases in background
risk. This set is interesting, first, because it reflects the property that background risks in-
crease in scale. Second, this case generalizes naturally to the set of stochastic increases that
‘improve’ according to nth order stochastic dominance. Kimball (1993) has shown that, in
the case of stochastic increases that improve according to third-order stochastic dominance,
standard risk aversion is a sufficient condition for incremental risk vulnerability. Consider-
ing the case of n-th order improvements, we find a sufficient condition for incremental risk
vulnerability, which is less restrictive than standard risk aversion.
2 Introduction
Many economic decisions are made in a context where some of the risks are tradable, while
others are not. These non-tradable or background risks are not controllable by the decision-
maker and yet influence the agent’s risk-taking behavior with respect to the tradable claims.
1 Patent increases are those such that an agent who is more risk averse than another, always requires a
larger risk premium to bear the increased risk than the other. See Kimball (1993), p 603.