# Compound Poisson approximation

The compound Poisson approximation assumes that the probability of payout for an individual policy is fairly small. This is usually true, but has the advantage over the de Pril method in allowing that the payout distribution is a random variable rather than a fixed amount.

Let nj be the number of policies with probability of claim pj. The number of payouts in this stratum is therefore Binomial(nj,pj). If nj is large and pj is small, the Binomial is well approximated by a Poisson(nj*pj) = Poisson(li) distribution. The additive property of the Poisson distribution tells us that the frequency distribution for payouts over all groups of lines of insurance is given by:

And the total number of claims = Poisson(lall).

The probability that one of these claims, randomly selected, comes from stratum j is given by:

Let Fj(x) be the cumulative distribution function for the claim size of stratum j. The probability that a random claim is less than or equal to some value x is therefore:

Thus we can consider the aggregate distribution for the total claims to have a frequency distribution equal to Poisson(lall) and a severity distribution given by F(x).