SOA Exam ASTAM Sample Question 6
Study guide: Exam ASTAM
Individual losses from a risk follow a Pareto distribution with parameters \alpha and \lambda. Insurance covers these losses up to a policy limit of M. There is no deductible.
Part a
Show that the expected claim severity is
\frac{\lambda}{\alpha - 1}\left(1 - \left(\frac{\lambda}{\lambda + M}\right)^{\alpha - 1}\right)
Part b
A sample of 100 claims is available. Of these, three were above the policy limit of 20,000. The remaining 97 were below the policy limit and the precise values, x_1, x_2, \ldots, x_{97}, are known.
Derive and simplify as far as possible, expressions for the maximum likelihood estimators \hat{\alpha} and \hat{\lambda}.
Part c
You are given that \hat{\lambda} = 10{,}683, and that
\sum_{i=1}^{97}\log\left(\hat{\lambda} + x_i\right) = 925.314; \quad \sum_{i=1}^{97}\frac{1}{\hat{\lambda} + x_i} = 0.007150; \quad \sum_{i=1}^{97}\frac{1}{\left(\hat{\lambda} + x_i\right)^2} = 5.488\times 10^{-7}
Show that \hat{\alpha} = 3.4 to the nearest 0.1. You should calculate the value to the nearest 0.01.
Part d
Calculate the approximate asymptotic variance-covariance matrix for \hat{\alpha} and \hat{\lambda}.
Answer this question and see the worked solution
Concepts tested
- Maximum Likelihood Estimation
- Pareto Distribution
- Policy Limit
- Limited Expected Value
- Censoring
- Fisher Information