SOA Exam ASTAM Fall 2023, Question 1

Study guide: Exam ASTAM

An insurance company sells a 1-year insurance policy covering dental costs. In 2023, the policy has no deductible and no policy limit.

The number of dental visits is assumed to follow a Poisson distribution with an expected value of 1.9. The cost of each dental visit is assumed to follow a Pareto distribution with parameters of $\alpha = 4$ and $\theta = 900$ . The number of dental visits and cost of each dental visit are assumed to be mutually independent.

Part a (2 points)

Let $S$ represent the aggregate payments in 2023 under this policy.

(i) Show that $E[S] = 570$ .
(ii) Calculate the standard deviation of $S$ .

Part b (1 point)

In 2024, the cost of each dental visit is expected to increase uniformly by 20% due to inflation. The number of dental visits is assumed to have the same distribution in 2024 as in 2023.

Explain why the distribution of the cost of each dental visit in 2024 is Pareto, with parameters $\alpha = 4$ and $\theta = 1080$ .

Part c (4 points)

The insurer is considering imposing an ordinary deductible per dental visit, $d$ , that would result in the insurer's expected aggregate payments being the same in 2024 as in 2023.

(i) Show that $d = 70$ to the nearest 10. You should calculate the value to the nearest 0.1.
(ii) Calculate the expected number of claim payments if the deductible in (c)(i) is applied, assuming that the number of dental visits is not affected by the deductible.
(iii) Briefly describe 2 ways in which the introduction of a deductible might impact policyholder behavior.

Part d (2 points)

The insurer is instead considering imposing an ordinary deductible of 50 along with a maximum covered loss of $L$ on each claim payment in 2024. Calculate the value of $L$ that would result in the insurer's expected aggregate payments to the insurer (as printed) being the same in 2024 as in 2023.

Part e (1 point)

The insurer wants to estimate the aggregate loss distribution by discretizing the 2023 severity distribution, using the Method of Rounding with a span of 100.

Determine the probability that is assigned to a claim severity of 0.

Part f (1 point)

Calculate $f_S(0)$ .

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