Reinsurance: Suppose we have a portfolio of policies where

I need help from someone who might know a little about actuarial studies and reinsurance. The question I am having problems with is:

Suppose that we have a portfolio of insurance policies where claims amouts have a distribution with CDF:

FX(x)=1-(100/(100+x))^a , x>0,

for a>0 ie. Pareto distributed with parameters a and 100.

Now suppose that the number of claims in one year, N, is Poisson distributed with mean lambda. We are considering two forms of reinsurance for this portfolio, the first a typical excess-of-loss reinsurance scheme with retention level M and the second a so-called "largest claim" reinsurance scheme, whereby the reinsurer pays for the entirety of L=max{X1,...,XN}, the largest claim during the year.

Find the CDF of L. Also, show that E(L) exists if and only if a>1.

Is this from Mahler's Study Notes?

Who is Mahler?

I'll take that as a "no". Just checking. I wouldn't want to be supplying solutions for a current exam.

Someone may know off the top of their head. I'll have to look it up. Is there a deadline?

Really, though, you have shown no personal effort, other than to type in the problem. What else have you?

I am just a uni student from Australia. I am not doing any exams (our syllabus might be different from that of the US), just having trouble doing one of the extra revision questions.

Here are my thoughts:

If the underlying claim amounts, X, have a distribution with pdf fX(x; θ) and CDF FX(x; θ), then the CDF of the distribution of what the reinsurer pays i think can be found using the relationship:

Prθ(Z ≤ z|Z > 0) = Prθ(X ≤ z +M|X >M)
= Prθ(M <X ≤ z +M)/Prθ(X >M)
= [FX(z +M; θ) − FX(M; θ)]/[1 − FX(M; θ)]

where M is the retention level and Z is the amt which must be paid by the reinsurer.

Where I am having trouble is understand what FX(z +M; θ) is in the formula and how I am suppose to use this formula with L=......(given in the question) to find the CDF.

However given that the underlying claim amounts are Pareto distributed, doesn't the density of the claim amounts for which the reinsurer is liable changes to some other function.

I am probably way of the mark here. Any help would be much appreciated.

Sorry I haven't gotten back to this.

So far, I am only coming up with the arduous process of quantizing the Pareto Distribution and calculating an explicit convolustion with the Poisson. This seems entirely possible but far, far too tedious and time-consuming for an exam question (possibly even if there were no other qeustions on the exam). I'm certain there is a clear and concise solution, it just hasn't struck me, yet. I do have a tendency to do things the hardest way that can be imagined.