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

so that

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,

excelent article

Thank you!

Dear Matt,

maybe you have some article or lessons in pdf about De prill’s recursion? or extension of Panjer recursion?

Thank in advance

BR

Mitja Benčina

Emil: benco.benco@gmail.com

thank you for posting this article. It really helps me

Thank you!.I appreciate the comment.

great explanation. every step is very clear