Panjer Recursion for Aggregate Claims

The Panjer Class

Panjer recursion applies when the claim count distribution satisfies P(N=n) = (a + b/n) times P(N=n-1) for n >= 1. This (a,b,0) class includes three distributions: Poisson (a=0, b=lambda), negative binomial (a>0), and binomial (a<0). The recursion efficiently computes the aggregate loss PMF when severity is discretized. Starting from g_0 = P(S=0) = P_N(f(0)), each subsequent value g_k = P(S=k*h) is computed using previously calculated values.

Applying the Recursion

The recursion formula is g_k = (1/(1-a*f_0)) times the sum from j=1 to k of (a + b*j/k) times f_j times g_{k-j}, where f_j is the discretized severity PMF. Severity must first be discretized onto a span h, using rounding, unmodified, or matching methods. Exam STAM tests your ability to set up and execute several steps of Panjer recursion by hand, particularly for Poisson and negative binomial frequency with simple discrete or discretized severities.