Venkat Anantharam, UC Berkeley, USA

Title: What Risks Lead to Ruin?

Abstract:
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. One modern scenario of particular interest that fits in this framework is the question of how to insure potential losses incurred by entities operating on the Internet. In the absence of enough data to build reasonable parametric loss models, it is 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 of probability distributions on the nonnegative integers. The insurer does not know which distribution in this set of distributions describes the marginal 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 nonparametric loss model of this type is insurable iff it contains no ``deceptive" distributions. Here the notion of “deceptive” distribution is precisely defined in information-theoretic terms. There appear to be close connections between classes of insurable probability distributions and classes of distributions studied in universal data compression.

The necessary background from information theory and risk theory will be provided during the talk. This is joint work with Narayana Santhanam, University of Hawaii, Manoa.