SOA Exam ASTAM Spring 2025, Question 2

Study guide: Exam ASTAM

An insurance company is analyzing a portfolio of 10 independent auto insurance policies. Drivers are classified into two categories, "Urban Drivers" and "Rural Drivers".

You are given:

(i) The annual number of claims from an "Urban Driver" follows a Poisson distribution with mean 0.5, and the annual number of claims from a "Rural Driver" follows a Poisson distribution with mean 0.3.

(ii) Four policies are issued to "Urban Drivers" and six policies are issued to "Rural Drivers".

(iii) The following table shows the distribution of claims severity, $X$ , for "Urban Drivers" and "Rural Drivers":

Amount $x$	$\Pr[X=x]$ "Urban Drivers"	$\Pr[X=x]$ "Rural Drivers"
1	0.60	0.10
2	0.30	0.40
3	0.10	0.50

You are given that the aggregate annual claims for the portfolio, $S$ , has a compound Poisson distribution with Poisson parameter $\lambda$ and associated single claim distribution $F_X$ . Let $f_X$ denote the probability function of $F_X$ .

Part a (1 point)

Assume $N_1 \sim \text{Poisson}(\lambda_1)$ and $N_2 \sim \text{Poisson}(\lambda_2)$ , and that $N_1$ and $N_2$ are independent.

Use probability generating functions to show that $N_1 + N_2 \sim \text{Poisson}(\lambda_1 + \lambda_2)$ .

Part b (2 points)

Calculate the mean and standard deviation of aggregate annual claims for an individual "Urban Driver" policy.

Part c (3 points)

(i) Calculate $\lambda$ .

(ii) Show that $f_X(1) = 0.36$ to the nearest 0.01. You should calculate the value to the nearest 0.001.

(iii) Calculate $f_X(2)$ and $f_X(3)$ .

Part d (3 points)

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

(ii) Calculate $\Pr[S = 2]$ using recursion or otherwise.

Part e (2 points)

Calculate the net stop-loss premium for the total annual claims for the portfolio with an aggregate deductible of 3.

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