CAS Exam 8 Fall 2018, Question 13

Study guide: Exam 8

An insurance company observes the following ground-up claim experience for a book of business:

Claim counts ( $N$ ) follow a Poisson distribution with $\lambda = 5{,}000$
Claim size ( $X$ ) follows a Pareto distribution with $\alpha = 2$ and $\beta = 6{,}000$

The insurance company pays claims in excess of $10,000.

Given the following:

$\Pr(N = n) = \frac{\lambda^n e^{-\lambda}}{n!}$

$f(x) = \frac{\alpha\beta^\alpha}{(x+\beta)^{\alpha+1}} \qquad F(x) = 1 - \left(\frac{\beta}{x+\beta}\right)^\alpha$

$E[X;x] = \frac{\beta}{\alpha-1}\left[1 - \left(\frac{\beta}{x+\beta}\right)^{\alpha-1}\right]$

Part a (0.75 points)

Calculate the expected number of claims in excess of $10,000.

Assume claim size is subject to a uniform annual inflation rate of 3%.

Calculate the rate at which ground-up claim counts must change in order for there to be no change in expected annual total aggregate excess losses.