SOA Exam ASTAM Spring 2026, Question 3

Study guide: Exam ASTAM

For a medical insurance policy, the number of claims follows a Poisson distribution with $\lambda = 2$ . The amount of each claim follows an exponential distribution with $\theta = 1000$ . Let $S$ denote the aggregate claims random variable for an individual policy.

Part a (2 points)

(i) Calculate $E[S]$ .

(ii) Calculate the standard deviation of $S$ .

Part b (1 point)

During 2025, the company expects to issue 10,000 independent policies. Let $S_P$ denote the random variable representing the total claims for the portfolio.

Using the normal approximation, show that the 95th percentile of $S_P$ is 20,330,000 to the nearest 10,000. You should calculate the value to the nearest 1000.

Part c (2.5 points)

You are given the following assumptions used for calculating premiums.

The total claims for the portfolio are assumed to be equal to the 95th percentile of $S_P$ , as calculated in (b) above.
Commissions are 8% of the gross premiums.
Marginal expenses are 12% of the gross premiums.
Fixed expenses during 2025 will be 300,000 to cover the 10,000 policies.
The target profit margin, based on the above assumptions, is 5% of the gross premiums.

(i) Show that the gross premium rate per policy is 2,750 to the nearest 10. You should calculate the value to the nearest 1.

(ii) Calculate the expected profit per policy.

Part d (1.5 points)

In order to do additional analysis, the company uses a discretized severity distribution. Two methods of discretizing the severity distribution are (i) the method of rounding (mass dispersal) and (ii) local moment matching.

List one advantage and one disadvantage of each method.

Part e (3 points)

Let $Y^d$ denote the discretized severity random variable, let $h = 400$ denote the discretization step size, and let $f_k = \Pr\left[Y^d = kh\right]$ for $k = 0, 1, 2, 3 \ldots$

(i) Calculate $f_1$ to 4 decimal places using the method of rounding.

(ii) Calculate $f_1$ to 4 decimal places using the method of local moment matching, using one moment.

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