What Risks Lead to Ruin

Insurance transfers losses associated with risks to the insurer for a price, the premium. We adopt the collective risk approach. Namely, we abstract the problem to include just two agents: the insured and the insurer. We are interested in scenarios where the underlying model for the loss distribution is not very well known, and the potential losses can also be quite high, which is of potential interest, for instance, in insuring against loss-of-use risk for services on offer over the Internet, where models for the statistics of the loss are not well established. It is then natural to adopt a nonparametric formulation. Considering a natural probabilistic framework for the insurance problem, assuming independent and identically distributed (i.i.d.) losses, we derive a necessary and sufficient condition on nonparametric loss models such that the insurer remains solvent despite the losses taken on.

In more detail, we model the loss at each time by a nonnegative integer. An insurer’s scheme is defined by the premium demanded by the insurer from the insured at each time as a function of the loss sequence observed up to that time. The insurer is allowed to wait for some period before beginning to insure the process, but once insurance commences, the insurer is committed to continue insuring the process. All that the insurer knows is that the loss sequence is a realization from some i.i.d. process with marginal law in some set $P$ of probability distributions on the nonnegative integers. The insurer does not know which $p \in{\cal P}$ describes the distribution of the loss sequence. The insurer goes bankrupt when the loss incurred exceeds the built up buffer of reserves from premiums charged so far.

We show that a finite support nonparametric loss model of this type is insurable if it contains no “deceptive” distributions. Here the notion of ``deceptive” distribution is precisely defined in information-theoretic terms. Note that, even though we assume a finite support for each $p \in{\cal P}$, there is no absolute bound assumed on the possible loss at any time. The necessary background from information theory and risk theory as well as some motivation for the problem formulation will be provided during the talk (joint work with Narayana Prasad Santhanam (University of Hawaii, Manoa)).