SOA Exam ASTAM Sample Question 23

Study guide: Exam ASTAM

In a Bühlmann credibility model, you are given that for a portfolio of short-term insurance contracts, the number of claims $X_1, \ldots, X_T$ , conditional on $\Theta$ , are independent and identically distributed with

hypothetical mean: $\mu(\Theta) = E\left[X_t \mid \Theta\right]$

and

process variance: $\nu(\Theta) = Var\left[X_t \mid \Theta\right]$

Denote the structural parameters by $\mu$ = pure premium, $v$ = expected process variance, and $a$ = variance of the hypothetical mean.

Part a

Prove that the unconditional variance of $X_t$ is the sum of the expected process variance and the variance of the hypothetical mean, that is,

 $Var\left[X_t\right] = \nu + a.$

Part b

In the context above, give an expression for the Bühlmann credibility factor $Z$ for the number of claims for a policy observed for $T$ years.

Part c

An insurance company analyzes its portfolio of insurance policies using the Bühlmann credibility model to estimate claim frequency. It assumes that claim frequency for a particular policy, conditional on the mean parameter $\lambda$ , is Poisson.

Show that the following equations are true:

(i) $\mu = E[\lambda]$

(ii) $Var\left[X_t\right] = E[\lambda] + Var[\lambda]$

(iii) $a = Var\left[X_t\right] - E\left[X_t\right]$

Part d

You are given the following claim frequencies for the insurance company's portfolio with one year experience for each policyholder:

Number of claims	Number of policyholders
0	8,416
1	1,427
2	153
3	3
4	1
Total	10,000

Calculate estimates for the structural parameters $\mu$ , $\nu$ , and $a$ .

Part e

Calculate the estimated Bühlmann credibility factor $Z$ . Use it to calculate the estimated credibility premium for the number of claims for each policyholder.

Part f

Explain why only one year's experience per policyholder is needed to estimate Bühlmann credibility factors.

Answer this question and see the worked solution