SOA Exam ASTAM Spring 2023, Question 4
Study guide: Exam ASTAM
A company is modeling the severity of flood losses. The following table shows the largest 20 values from a sample of 750 observations in decreasing order, with their ranks.
| Rank | 750 | 749 | 748 | 747 | 746 | 745 | 744 | 743 | 742 | 741 |
|---|---|---|---|---|---|---|---|---|---|---|
| Loss | 28,250 | 14,782 | 12,427 | 9,676 | 7,005 | 4,322 | 4,057 | 3,773 | 3,504 | 3,123 |
| Rank | 740 | 739 | 738 | 737 | 736 | 735 | 734 | 733 | 732 | 731 |
|---|---|---|---|---|---|---|---|---|---|---|
| Loss | 2,924 | 2,780 | 2,656 | 2,483 | 2,441 | 2,357 | 2,067 | 2,045 | 2,008 | 1,977 |
For convenience, you are also given that the total of the largest 10 losses is 90,919.
Part a (3 points)
- (i) Estimate the 98.4% Value at Risk (VaR) of the losses, using the smoothed empirical method.
- (ii) Estimate the 98.4% Expected Shortfall of the losses.
- (iii) Calculate the empirical Mean Excess Loss,
\hat{e}(x), atx = 3{,}000.
Part b (1 point)
The actuary decides to estimate the risk measures using the Generalized Pareto Distribution (GPD) for the excess loss distribution. She has created the following plot of the empirical mean excess loss function. (The last five points are excluded).
The actuary proposes a threshold of d = 2{,}100 for fitting the GPD. Her colleague points out that there is a gradient change at 1,500. He states that this would be a better threshold because it provides more data to fit the GPD.
Critique the colleague's suggestion.
Part c (3 points)
The MLE estimators with d = 2{,}100 are \hat{\xi} = 0.68 and \hat{\beta} = 2{,}000.
- (i) Show that the 98.4% VaR of the loss using the fitted GPD is 2,700 to the nearest 100. You should calculate the value to the nearest 1.
- (ii) Calculate the 98.4% Expected Shortfall (ES) using the fitted GPD.
Part d (2 points)
- (i) Explain why the VaR estimates in part (a) and in part (c) are similar, while the Expected Shortfall estimates are quite different.
- (ii) Recommend which Expected Shortfall estimate the company should use for assessing its potential risk. Justify your answer.
Answer this question and see the worked solution