SOA Exam ASTAM Sample Question 9

Study guide: Exam ASTAM

An insurer manages a portfolio of $r$ commercial insurance policies.

Let $X_{ij}$ denote the aggregate claims from insurance policy $i$ in year $j$ , for $j = 1, 2, \ldots, n$ .

You are given the following information.

(i) Each $X_{ij}$ is the average loss from $m_{i,j}$ different risks, that is $X_{ij} = \dfrac{1}{m_{i,j}} \displaystyle\sum_{l=1}^{m_{ij}} Y_{ijl}$
(ii) The individual risks, $Y_{i,j,l}$ , are dependent on an unknown parameter, $\theta_i$ .
(iii) $Y_{i,j,l} \mid \theta_i$ are i.i.d for $l = 1, 2, \ldots, m_{i,j}$ , and for $j = 1, 2, \ldots, n$ .
(iv) $E\left[Y_{i,j,l} \mid \theta_i\right] = \mu(\theta_i)$ and $\operatorname{Var}\left[Y_{i,j,l} \mid \theta_i\right] = v(\theta_i)$
(v) $\theta_i$ are i.i.d, with $\mu = E\left[\mu(\theta_i)\right]$ , $v = E\left[v(\theta_i)\right]$ , for $i = 1, 2, \cdots, r$ .

Part a

(i) Write down the assumptions of the Bühlmann-Straub model, in terms of $\mu(\theta_i)$ and $v(\theta_i)$ .
(ii) Show that the assumptions for the Bühlmann-Straub model are valid in this case.

(i) Write down the formula for $Z_i$ , the credibility factor for insurance policy $i$ , defining all the terms.
(ii) Explain why it is reasonable that a larger value of $\hat{a}$ generates a larger value for $Z_i$ .
(iii) Explain why it is reasonable that a larger value of $\hat{v}$ generates a smaller value for $Z_i$ .
(iv) Explain why it is reasonable that a larger value of $m_i$ generates a larger value for $Z_i$ .

The estimator for $\mu$ is $\hat{\mu} = \dfrac{\sum_{i=1}^{r} Z_i \bar{X}_i}{\sum_{i=1}^{r} Z_i}$

(i) Show that that the estimator for $\mu$ is unbiased.
(ii) The weights used in the estimator for $\mu$ are proportional to $\operatorname{Var}[\bar{X}_i]$ (as printed). Explain why this weighting is preferred to other possible weightings.