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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Compound distributions and their applications in risk modelling

In the Panjer recursion for aggregate claims S = X1 + ... + XN, where the individual claim amounts are discrete on the positive integers (in rupee thousands), which condition on the claim number distribution is required?

The claim number distribution must belong to the (a,b,0) class, meaning p_k = (a + b/k)p_{k-1} for k ≥ 1. This covers the Poisson, binomial and negative binomial distributions, which is what makes the simple recursion for aggregate claims probabilities valid.

  1. AN belongs to the (a,b,0) class, so that p_k = (a + b/k) p_{k-1} for k ≥ 1Correct
  2. BN must be Poisson with a mean that is a whole number
  3. CN must have a finite upper bound on its support
  4. DN must be independent of the claim amounts and have the same distribution as them
  5. N must have variance exactly equal to its mean

Explanation

Panjer recursion requires the claim count to satisfy p_k = (a + b/k)p_{k-1} for k ≥ 1, which covers Poisson, binomial and negative binomial. It does not need N to be Poisson or bounded, and the claim amounts must be independent of N but need not share its distribution.

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