Actuarial Notes

SOA Exam ASTAM Spring 2023, Question 3

Study guide: Exam ASTAM

You are given that, under nonparametric empirical Bayes parameter estimation for the Bühlmann-Straub credibility model, X_{ij} denotes the claim frequency from risk group i in year j, and

\mathrm{Var}\left[X_{ij} \mid \theta_i\right] = \frac{v(\theta_i)}{m_{ij}}.

Part a (2 points)

  • (i) Interpret the m_{ij} term.
  • (ii) Interpret \theta_i.

Part b (3 points)

An insurer issues auto insurance policies through two different brokers, Broker A and Broker B. Claim frequency data from the last three years is given in the following table. There were no policies issued through Broker B in Year 1.

Risk Group Year 1 Year 2 Year 3 Total
Broker A Number of Claims 39 28 15 82
Number of Policies 200 150 180 530
Broker B Number of Claims n/a 16 23 39
Number of Policies n/a 50 60 110

Assume first that the insurer models the claim frequency for the two groups using Bühlmann-Straub credibility, with nonparametric empirical Bayes parameter estimation.

  • (i) Show that \hat{v} is 0.5 to the nearest 0.1. You should calculate the value to the nearest 0.001.
  • (ii) Show that \hat{a} is 0.017 to the nearest 0.001. You should calculate the value to the nearest 0.0001.

Part c (3 points)

Calculate the Bühlmann-Straub credibility estimate of the expected claim frequency for an individual policy issued through Broker B in Year 4.

Part d (3 points)

Now assume that the insurer models the claim frequency for each group using a Poisson distribution. You are given the following information:

  • The Poisson parameter for each broker is estimated by maximum likelihood, based on the data above.
  • In year 4 it is assumed that for a randomly selected policy, there is a 75% probability that the policy was issued through Broker A, and a 25% probability that the policy was issued through Broker B.
  • A randomly selected policy had two claims in Year 2 and two claims in Year 3.

(i) Show that the posterior probability that this policy was issued through Broker B is 0.86 to the nearest 0.01. You should calculate the value to the nearest 0.001.

(ii) Calculate the Bayesian estimate of the expected claim frequency for this policy in year 4.

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