SOA Exam ASTAM Fall 2023, Question 4
Study guide: Exam ASTAM
For a portfolio of m independent policies, the number of claims in a year for the i^{th} policy, denoted I_i, is a Bernoulli random variable with parameter q.
Let n denote the total observed number of claims from the portfolio in one year so that n = \sum_{i=1}^{m} I_i.
Part a (3 points)
- (i) Write down the likelihood function in terms of
q,n, andm. - (ii) Show that the maximum likelihood estimate of
qis the sample mean,\hat{q} = \dfrac{n}{m}.
Part b (2 points)
An actuary decides to use a Bayesian approach to model the heterogeneity of the policies. They decide to use the Beta(2, \beta) distribution, with the following density function, as the prior distribution for q:
\pi(q) = \beta(\beta + 1)\,q\,(1 - q)^{\beta - 1} \quad \text{where} \quad 0 \le q \le 1;\ \beta > 0Show that the posterior distribution of q is Beta(n + 2,\ m + \beta - n).
Part c (2 points)
The Bayes estimate of q under the squared error loss function can be expressed in credibility form as Z \times \text{sample mean} + (1 - Z) \times \text{prior mean}.
Show that Z = \dfrac{m}{m + \beta + 2}.
Part d (3 points)
The actuary uses the above Bayesian approach and distributions to analyze two portfolios, A and B, described in the following table:
| Portfolio | Number of policies | Number of claims | \beta |
|---|---|---|---|
| A | 100 | 20 | 20 |
| B | 100 | 10 | 10 |
- (i) For each portfolio, calculate the credibility factor
Z. - (ii) For each portfolio, calculate the Bayes estimate of
q. - (iii) Interpret the impact of the value of
\betaon the credibility factor.
Answer this question and see the worked solution