As I mentioned in my post on the Cramer-Lundberg model, I’ve been trying to teach myself some insurance maths. One of the books I’ve been consulting is David Dickson’s Insurance Risk and Ruin. One of the sections was on computationally useful formulae for distribution of the aggregate claims process
where are the i.i.d. individual claims. The formula I want to share with you today is the Panjer recursion formula due to Harry Panjer. Given a “suitable” claims number process and nonnegative-integer-distributed claims , the probability mass function of the compound process can be computed recursively.
We begin by specifying what “suitable” above mathematically means. A discrete random variable is said to belong to belong to the class if its probability mass function can be calculated recursively by
where are fixed constants. We will show that there are precisely three distributions in the class: the Poisson, the binomial, and the negative binomial (where we allow noninteger parameters) distributions.
We assume that , which is indicated by the in . Observe that , so it is necessary that . We proceed by examining the cases for . If , then and hence for all and has a degenerate distribution at . If , then , so that
Since , we obtain that
We conclude that, if , then is a Poisson distribution with parameter .
Lastly, consider the case and . Repeatedly applying the recursion formula, we obtain the identity
Set , so that
I claim that . By the ratio test, it suffices to show .
which is a contradiction since the harmonic series diverges. Recall that a negative binomial distribution satisfies
where . Since , for all , we see that , so that .
Lastly, consider the case where and . I claim that there exists a positive integer such that , so that for . Indeed, since and , for all sufficiently large . If such a does not exist, then choosing to be minimal such that , we obtain , which is a contradiction. We can write
Set . Since and for , we see that
By the binomial formula, , so that . Since any positive number can be written as , where , we have that
which shows that .
If has distribution of class, then there is a closed-form expression for its generating function . Denote the generating function of by . Since generating functions convergence uniformly for , we can differentiate term-by-term to obtain
Rearranging, we can write . For sufficiently small , we can divide both sides by and solve the differential equation to obtain
Since , we conclude that .
Panjer recursion gives us a formula for computing the probability distribution function of the aggregate claims when the counting random variable has distribution of class and the individual claim has discrete distribution with probability function . We restrict our attention to the case the individual claims are distributed on the nonnegative integers, so that is also distributed on the nonnegative integers. Let denote the probability distribution function of . Since is independent of the ,
We know that the probability generating function of is given by
Differentiating with respect to , we obtain by the chain rule that
Substituting in the identity with , we obtain
Substituting in the series expressions for the generating functions, we obtain
Multiplying both sides by , we have
where we use the fact that the product of convergent power series is the Cauchy product. Since a power series is uniquely determined by its coefficients, we can compare coefficients on both sides to obtain the formula
Rearranging, we obtain
for all . Note that the RHS of the second identity only contains terms with , hence defines a recursive formula for . The preceding formula is called the Panjer recursion formula first introduced by Harry Panjer in 1981. One might ask how we a priori know that , but this follows from our examination of class distributions above and our hypothesis that belongs to the class.
We have established a recursive formula for the probability mass function. Does an analogous formula exist for the cumulative distribution function (cdf)? According to one of the books I’m consulting, a recursion formula for the cdf does not exist, in general. However, an exception to this statement is when has a geometric distribution . As the geometric distribution is just a distribution, we have that , , and for . Hence,