1. Simulating Aggregate Loss (AL) distributions A national bank collects data on petty thefts by its employees/customers on a daily basis from all regions and aggregates them at the corporate level. After some preliminary analysis, the bank finds that the number of losses is independent of the dollar loss per event. The bank decides to adopt frequency-severity analysis to model this aspect of operational risk, and accordingly set aside reserves (on a daily basis) for petty thefts. Suppose the arrival of number of theft events is known to be a Poisson distribution with Mean 3. This means the frequency of events (N) follows the following distribution: Pr(X = n) = 12" n! a> 0;n= 0,1,2,... Simultaneously, the loss severity per incident (X) follows the distribution given below: Severity Amount Probability 500 50% 1000 20% 1500 12% 2000 10% 2500 7% 3000 1% 1 a. Compute the analytical values of the aggregate loss statistics (mean, variance, skewness). b. Simulate the aggregate loss distribution 20,000 times and compute the mean, variance, and skewness statistics from the simulated distribution. Do they match the analytical values? You may use the SKEW function in EXCEL instead of computing the third moment for this exercise. C. Report the maximum aggregate loss you observe among these 20,000 simulations d. Compute the following: i. Probability (AL) = 0; ii. Probability (AL) = 9,000; iii. Probability (AL) = 15, 000; If operational risk reserve is going to be set at 95% level of AL distribution, what should it be? At 97.5%? At 99%? Which of these three levels is/are more stable estimate/s (to check stability, press the recalculation key (SHIFT+F9, or F9), and the estimates should not bounce around much!)? f. Repeat part 'e' above with 5000 simulations. Which of the three levels (99%, 97.5%, 95%) is/are more stable? Comment on what you learn from this comparison (Note the comment part is important, if the group does not comment it stands to lose points). e. 1. Simulating Aggregate Loss (AL) distributions A national bank collects data on petty thefts by its employees/customers on a daily basis from all regions and aggregates them at the corporate level. After some preliminary analysis, the bank finds that the number of losses is independent of the dollar loss per event. The bank decides to adopt frequency-severity analysis to model this aspect of operational risk, and accordingly set aside reserves (on a daily basis) for petty thefts. Suppose the arrival of number of theft events is known to be a Poisson distribution with Mean 3. This means the frequency of events (N) follows the following distribution: Pr(X = n) = 12" n! a> 0;n= 0,1,2,... Simultaneously, the loss severity per incident (X) follows the distribution given below: Severity Amount Probability 500 50% 1000 20% 1500 12% 2000 10% 2500 7% 3000 1% 1 a. Compute the analytical values of the aggregate loss statistics (mean, variance, skewness). b. Simulate the aggregate loss distribution 20,000 times and compute the mean, variance, and skewness statistics from the simulated distribution. Do they match the analytical values? You may use the SKEW function in EXCEL instead of computing the third moment for this exercise. C. Report the maximum aggregate loss you observe among these 20,000 simulations d. Compute the following: i. Probability (AL) = 0; ii. Probability (AL) = 9,000; iii. Probability (AL) = 15, 000; If operational risk reserve is going to be set at 95% level of AL distribution, what should it be? At 97.5%? At 99%? Which of these three levels is/are more stable estimate/s (to check stability, press the recalculation key (SHIFT+F9, or F9), and the estimates should not bounce around much!)? f. Repeat part 'e' above with 5000 simulations. Which of the three levels (99%, 97.5%, 95%) is/are more stable? Comment on what you learn from this comparison (Note the comment part is important, if the group does not comment it stands to lose points). e