SOA Exam ASTAM Spring 2025, Question 6
Study guide: Exam ASTAM
For a portfolio of 100 independent policies, the annual claim amount on a policy, conditional on the risk parameter \Theta = \theta, has an exponential distribution with mean \theta.
An actuary assumes that the prior distribution of \Theta is an Inverse Gamma distribution, denoted IG(\alpha, \beta), with density function:
\pi(\theta) = \frac{\beta^{\alpha} e^{-\left(\frac{\beta}{\theta}\right)}}{\theta^{\alpha+1}\Gamma(\alpha)}, \quad \theta, \alpha, \beta > 0
You are given that E\left[\Theta^k\right] = \dfrac{\beta^k\, \Gamma(\alpha - k)}{\Gamma(\alpha)},\ k < \alpha.
Let x_i denote the observed claim amount from the i^{\text{th}} policy, i = 1, \ldots, 100, in Year 1. The actuary first uses the Bayesian approach.
Part a (2 points)
Show that the posterior distribution of \Theta is IG\left(\alpha + 100, \beta + \sum_{i=1}^{100} x_i\right).
Part b (1 point)
You are given that \alpha = 30, \beta = 10{,}000, and \sum_{i=1}^{100} x_i = 50{,}000.
Calculate the Bayesian estimate of the annual claim amount of a randomly selected policy in Year 2.
Part c (3 points)
The actuary decides to approximate the Bayesian premium using the Bühlmann credibility model.
(i) Show that the expected value of the hypothetical means, \mu, is 350 to the nearest 50. You should calculate the value to the nearest 1.
(ii) Show that the expected value of the process variance, v, is 123,150 to the nearest 50. You should calculate the value to the nearest 1.
(iii) Show that the variance of the hypothetical means, a, is 4,250 to the nearest 50. You should calculate the value to the nearest 1.
Part d (2 points)
For estimating the annual claim amount of a randomly selected policy from the portfolio in Year 2:
(i) Calculate the Bühlmann credibility factor.
(ii) Determine the Bühlmann credibility estimate.
Part e (1 point)
Explain why the Bayesian and Bühlmann estimates are the same in this case.
Answer this question and see the worked solution