SOA Exam ASTAM Sample Question 21

Study guide: Exam ASTAM

For all policies in an auto insurance portfolio, the number of accidents in a year follows a Poisson distribution with unknown parameter $\theta$ . The accident rate depends on the driving quality of the policyholder as described in the table below. The driving quality is not known in advance to the insurer.

Quality	Number of accidents per year	Proportion of population
Good	Poisson(1)	0.6
Fair	Poisson(2)	0.3
Poor	Poisson(5)	0.1

Part a

For a random sample of 20 drivers from the population, calculate the expected number of accidents.

Part b

Calculate the standard deviation of the aggregate claims cost per year for a policyholder selected at random, assuming that for all claims the severity distribution has mean 1000 and standard deviation 2000.

Part c

A policyholder has a claims experience of 3 claims (in total) in the past 3 years. Determine the posterior distribution of $\theta$ for this policyholder.

Part d

Calculate the Bayesian estimate of annual claim frequency for this policyholder.

Part e

Identify the following functions/parameters used to calculate the Bühlmann credibility estimate of claim frequency, in the context of the parameters and distributions given above.

(i) $\mu(\theta)$ ; $v(\theta)$

(ii) $\mu$ ; $v$ ; $a$

Part f

Calculate the Bühlmann Credibility estimate of claim frequency for the policyholder in part (c).

Part g

Your boss says, "It looks like this policyholder is a low-risk driver, so the correct risk premium should assume that they fall in the 'good driver' category." Comment on the appropriateness of this suggestion.

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