SOA Exam ASTAM Spring 2026, Question 6
Study guide: Exam ASTAM
A reinsurance company is analyzing the tail risk of cyber losses. The following table provides the largest 20 values from a sample of 500 cyber losses, in decreasing order.
| Rank | 500 | 499 | 498 | 497 | 496 | 495 | 494 | 493 | 492 | 491 |
|---|---|---|---|---|---|---|---|---|---|---|
Loss X |
21,355 | 12,495 | 10,751 | 8,450 | 7,548 | 7,530 | 6,354 | 5,668 | 5,617 | 5,498 |
ln(X) |
9.969 | 9.433 | 9.283 | 9.042 | 8.929 | 8.927 | 8.757 | 8.643 | 8.634 | 8.612 |
| Rank | 490 | 489 | 488 | 487 | 486 | 485 | 484 | 483 | 482 | 481 |
Loss X |
5,388 | 5,326 | 5,247 | 5,184 | 5,092 | 5,088 | 4,953 | 4,668 | 4,635 | 4,469 |
ln(X) |
8.592 | 8.580 | 8.565 | 8.553 | 8.535 | 8.535 | 8.508 | 8.448 | 8.441 | 8.405 |
You are given that the sum of the natural log values of the largest 20 losses is 175.391.
Part a (2 points)
(i) Calculate the empirical 99% VaR.
(ii) Calculate the empirical 99% Expected Shortfall (ES).
Part b (3.5 points)
The data is fitted to the Generalized Pareto Distribution (GPD) with a threshold of d = 4{,}953. At this threshold, the maximum likelihood estimates (MLE) of the GPD parameters are \hat{\xi} = 0.74 and \hat{\beta} = 1{,}100.
(i) Show that the 99% VaR of the cyber losses using the fitted GPD is 7,000 to the nearest 100. You should calculate the value to the nearest 1.
(ii) Calculate the 99% ES of the cyber losses using the fitted GPD.
Part c (1 point)
Show that the Hill estimator of \alpha = \xi^{-1}, denoted \hat{\alpha}^H, is 3.1 to the nearest 0.1. You should calculate the value to the nearest 0.01.
Part d (3.5 points)
The Hill estimator for the tail survival function has the form \hat{S}^H(x) = 0.032\left(\dfrac{x}{d}\right)^{-\hat{\alpha}^H}, where the threshold is d = 4{,}953.
(i) Show that the 99% VaR of the cyber losses using the Hill estimator of the survival function is 7,200 to the nearest 100. You should calculate the value to the nearest 1.
(ii) Calculate the 99% ES of the cyber losses, using the Hill estimator of the survival function.
Part e (1 point)
State with reasons whether you would recommend using the empirical estimate from part (a)(ii), the MLE GPD estimate from part (b)(ii), or the Hill estimator of the survival function estimate from part (d)(ii) for evaluating the 99% Expected Shortfall.
Answer this question and see the worked solution