We study large and moderate deviations for an insurance portfolio, with the number of claims tending to infinity, without assuming identically distributed claims. The crucial assumption is that the centered claims are bounded, and that variances are bounded below. From a general large deviations upper bound, we obtain an exponential bound for the probability of the average loss exceeding a threshold. A counterexample shows that a full large deviation principle, including also a lower bound, does not follow from our assumptions. We argue that our assumptions make sense, in particular, for life insurance portfolios and discuss how to apply our upper bound in this context. Finally, we use a moderate deviations result by Petrov (1954) to estimate the probability of exceeding a threshold that depends on portfolio size. In this asymptotic regime, the rate function that determines the asymptotic behavior is explicit and thus very easy to compute numerically, without solving an optimization problem.
(R1505) A Note on Large Deviations in Insurance Risk,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 16,
2, Article 5.
Available at: https://digitalcommons.pvamu.edu/aam/vol16/iss2/5