Panjer Recursion and Expectation

Given a Panjer Recursion set up, with the usual properties, and supposing now that $N$ has a Poisson distribution with mean $\lambda$.

How can we derive a recursion for $E(S^k)$ where $S$ be the total amount claimed in a year.

Additional background: Consider the total amount claimed in a year on a particular risk where the number of claims $N$ has $P(N = n) = P(n)$, $n = 0, 1, 2, \dots,$ and where claims are independent and identically distributed random variables $X_1, X_2, \dots$ independent of $N$. Suppose also that the claim sizes are positive and discrete with $P(X_1 = j) = f(j)$.

• Just to be clear, is your setup $S=\sum_{i=1}^NX_i$ where $N$ is Poisson and $X_i$ are i.i.d.? And you are interested in $E(S^k)$? – Alex R. Mar 24 '12 at 16:57
• Also, if my last comment is correct, would you actually need Panjer Recursion to calculate $E(S^k)$? Afterall, $S$ has characteristic function $\exp(\lambda(\phi(t)-1))$ where $\phi(t)$ is the characteristic function of $X_1$. Differentiate this $k$ times and evaluate at $t=0$. – Alex R. Mar 24 '12 at 17:21
• Hey Sam! The question starts by asking a proof of the Panjer recursion. It then tells to formulate using Panjer Recursion, a recursion for E(S^k). And then evaluate E(s), Var(s) and Skewness. – Hassan Khan Mar 25 '12 at 2:34