SOA Exam ASTAM Spring 2026, Question 4

Study guide: Exam ASTAM

Let $N$ denote the claim frequency random variable for an automobile insurance policy. The distribution of $N$ is assumed to depend on the driving habits of the policyholder. To model this, the insurer assumes that $N$ depends on an unknown risk parameter $\Lambda$ , where $\Lambda$ varies by policyholder.

You are given that

$N \mid \Lambda \sim \text{Poisson}(\Lambda)$
The prior distribution of $\Lambda$ is a Gamma distribution, with parameters $\alpha = 2$ and $\theta = 0.2$ .

Part a (2 points)

(i) Calculate $E[N]$

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

Part b (2 points)

You are given the following claims history for an individual policyholder.

Calendar Year	Policy Year	Number of Accidents
2023	1	0
2024	2	2
2025	3	0

Show that the posterior distribution of $\Lambda$ is a Gamma distribution with parameters $\alpha^* = 4$ and $\theta^* = \frac{1}{8} = 0.125$ .

Part c (3 points)

(i) Show that the predictive distribution of claim frequency for this policyholder is Negative Binomial.

(ii) Calculate E[N] under the predictive distribution.

Part d (2 points)

Now suppose the claim frequency is estimated using Buhlmann credibility.

(i) Write down the values for the Buhlmann parameters, $v$ , $a$ , and $\mu$ .

(ii) Calculate the Buhlmann estimate of the claim frequency.

Part e (1 point)

Your colleague suggests that the best estimate of the annual claim frequency for this policyholder is the estimate from the data, $\bar{X} = \frac{2}{3}$ . Critique this suggestion.

Answer this question and see the worked solution