The insurance of lower probability events

We consider a model in which the agent faces two independent risks of loss with different probabilities of occurrence and (possibly) different levels of potential loss. We show that it is optimal to select a deductible for th e low probability event that is not larger than the optimal deductible for the other risk. This result holds for any preference functional that satisf ies the second-order stochastic dominance property. It tends to support the view that insurance is the most appropriate risk management tool for low-f requency risks. When the expected loss is the same for the two risks, i.e. when the low probability event is also "catastrophic," it is never optimal not to insure the catastrophic risk when some insurance is purchased for th e other risk. We also obtain some additional properties of the optimal insu rance strategy in the case of expected utility, or in the case of Yaari' a dual theory.