SOA Exam ASTAM Spring 2024, Question 3

Let $S$ denote the aggregate claims for a portfolio of policies issued by ABC insurance company. You are given:

(i) The claim frequency, $N$ , belongs to the $(a,b,0)$ class with $\mathrm{E}[N] = 1.0$ and $\mathrm{Var}[N] = 0.9$

(ii) The claim severity, $X$ , also belongs to the $(a,b,0)$ class with $\mathrm{E}[X] = 10.0$ and $\mathrm{Var}[X] = 60.0$

Part a (2 points)

Calculate the mean and standard deviation of $S$ .

(i) Explain why $N$ must have a binomial distribution.

(ii) Identify the parameters of the distribution of $N$ .

(iii) Identify the distribution and parameters of $X$ .

Determine the probability generating function (pgf) of $S$ .

Using the pgf or otherwise, calculate the following probabilities:

(i) $\Pr[S = 0]$

(ii) $\Pr[S > 1]$

ABC insurance is entering a stop-loss reinsurance agreement with a deductible $d = 2$ . Calculate the net stop-loss premium.