SOA Exam ASTAM Fall 2024, Question 5

Part a (1 point)

An insurer uses the Bühlmann Credibility Model to estimate expected claim frequency for risks in a portfolio. Let $X$ denote the number of claims from an individual risk in a single year.

You are given that $\mu(\Theta) = \mathrm{E}[X \mid \Theta]$ and that $v(\Theta) = \mathrm{Var}[X \mid \Theta]$ .

Define the Bühlmann Credibility Model parameters $\mu$ , $v$ , and $a$ in terms of the functions $\mu(\Theta)$ and $v(\Theta)$ .

Part b (1 point)

The insurer will use the semiparametric approach to estimate the Bühlmann Credibility Model parameters $\mu$ , $v$ , and $a$ .

The insurer assumes that the frequency of claims for an individual policy, conditional on an unknown parameter $\beta$ , follows a Geometric distribution with parameter $\beta$ .

(i) Show that $\mu = E[\beta]$

(ii) Show that $a = \mathrm{Var}[\beta]$

Part c (3 points)

(i) Show that $\mathrm{Var}[X] = 2\mathrm{Var}[\beta] + \mathrm{E}[\beta]\left(1 + \mathrm{E}[\beta]\right)$

(ii) Hence show that

$a = \frac{\mathrm{Var}[X] - E[X]\left(1 + E[X]\right)}{2}$

Part d (3 points)

You are given the following annual claims data for the portfolio

Number of Claims in a policy year	Number of Policies
0	2,230
1	209
2	41
3	12
4	8
Total	2,500

(i) Show that $\hat\mu = 0.14$ to the nearest 0.01. You should calculate the value to the nearest 0.0001.

(ii) Calculate the standard deviation of the claim frequency for the portfolio.

(iii) Show that $\hat a = 0.03$ to the nearest 0.01. You should calculate the value to the nearest 0.001.

(iv) Calculate $\hat v$ .

Part e (2 points)

(i) Calculate the Bühlmann Credibility estimate of the expected claim frequency for a policy which has been in force for 1 year and had 1 claim in that year.

(ii) Calculate the Bühlmann Credibility estimate of the expected claim frequency for a policy that has been in force for 5 years and had 1 claim each year in that period.

(iii) Explain why the expected claim frequency for the policy in (e)(i) is less than the expected claim frequency for the policy in (e)(ii), even though the historical claim frequencies are the same.

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