SOA Exam ASTAM Fall 2025, Question 6
Study guide: Exam ASTAM
Claims from an insurance portfolio have a compound distribution. The primary distribution is negative binomial, with parameters r = 5 and \beta = 3. The secondary distribution is gamma with parameters \alpha = 0.035 and \theta = 36{,}000.
Part a (2 points)
(i) Show that the mean of the aggregate distribution is 19,000 to the nearest 1,000. You should calculate the value to the nearest 10.
(ii) Show that the standard deviation of the aggregate distribution is 28,000 to the nearest 1,000. You should calculate the value to the nearest 10.
Part b (2 points)
The insurer first estimates the aggregate distribution using a Normal approximation.
(i) Calculate the 95% Value at Risk (VaR) of the aggregate claims using the Normal approximation.
(ii) Calculate the 95% Expected Shortfall of the aggregate claims using the Normal approximation.
Part c (3 points)
The insurer then estimates the aggregate distribution using a lognormal approximation, with parameters \mu^* and \sigma^* fitted by matching moments.
(i) Show that \sigma^* = 1.1 to the nearest 0.1. You should calculate the value to the nearest 0.001.
(ii) Calculate the 95% Value at Risk (VaR) of the aggregate claims using the lognormal approximation.
(iii) Calculate the 95% Expected Shortfall of the aggregate claims using the lognormal approximation.
Part d (2 points)
(i) State with reasons which of these two approximations is expected to give a more accurate approximation of the 95% Expected Shortfall in this case.
(ii) Explain briefly how a more accurate estimate of the Expected Shortfall could be determined based on a discretized severity (secondary) distribution.
Answer this question and see the worked solution
Concepts tested
- Aggregate Loss Model
- Normal Approximation
- Lognormal Distribution
- Risk Measure
- Severity Discretization
- Negative Binomial Distribution
- Moment