1. C-2: Loss Simulation C-2: Loss Simulation

2. Statistical Analysis in Risk Management – Two main approaches: – Maximum probable loss (or MPY) if $5 million is the maximum probable loss at the _______percent level, then the firm’s losses will be less than $_____million with probability 0.95. Same concept as “Value at risk”

3. When to Use the Normal Distribution – Most loss distributions are not normal – From the __________ theorem, using the normal distribution will nevertheless be appropriate when – Example where it might be appropriate:

4. Using the Normal Distribution Important property – If Losses are normally distributed with – Then Probability (Loss < ) = 0.95 Probability (Loss < ) = 0.99

5. Using the Normal Distribution - An Example – Worker compensation losses for Stallone Steel sample mean loss per worker = $_____ sample standard deviation per worker = $20,000 number of workers = ________ – Assume total losses are normally distributed with mean = $3 million standard deviation = – Then maximum probable loss at the 95 percent level is $3 million + = $6.3 million

6. A Limitation of the Normal Distribution Applies only to aggregate losses, not _______losses Thus, it cannot be used to analyze decisions about per occurrence deductibles and limits

7. Monte Carlo Simulation – Overcomes some of the shortcomings of the normal distribution approach – Overview: Make assumptions about distributions for ________ and _______ of individual losses Randomly draw from each distribution and calculate the firm’s total losses under alternative risk management strategies Redo step two many times to obtain a distribution for total losses

8. A. Total Loss Profile A. Total Loss Profile 1. E(L) forecast a. single best estimate ………. b. variations from this number will occur, therefore … 2. Example for a large company.(next slide) mode, median expected = $ Pr(L) > $11,500,000 = Pr(L) > $14,000,000 =

10. 3. Uses of Total Loss Profile a. Evaluate and loss limits b. c. d. MPL (MPY)

11. B. Monte Carlo Steps B. Monte Carlo Steps 1. Select frequency distribution 2. Select severity distribution 3. Draw from ________ distribution => N 1 losses 4. Draw N 1 severity values from severity distribution 5. Repeat steps____and ____ for 1000 or more iterations

12. Iteration Number 1 2 1,000 N i 70 23 … 43 S 1 $ 600 $ 94,000 $ _____ S 2 $ 18,400 $ 150 $ 970 … S 10 $ _____ $ 2,600 $ 500 … S 23 $ 19,500 $ 1,350 $ 32,150 … S 43 $ 3,750 NA $182,000 … S 70 $ 54,000 NA NA Total $ $ $

13. Rank Order the Total Losses IterationPercentileTotal Losses 1 0.1$ 143,000. 100 10 1,790,000. 500 50 2,280,000. 700 70 ________. 900 90 3,130,000. 950 95 ________. 1,000 100 3,970,000

15. D. Interpretation of Results D. Interpretation of Results 1. Look at summary statistics: mean, sigma, percentiles 2. 3.

16. Within LimitsAt Limits,000 X BARSigmaX BARSigma 1 - 10 $ $ $ $ 10 25$ 612 $ 88 $ 2,655 $ 176 25 - 50 $ 326 $ 92 $ 2,981 $ 239 50 - 75 $ 128 $ 55 $ 3,109 $ 275 75 - 100 $ 65 $ 41 $ 3,174 $ 298 100 - 150 $ 60 $ 53 $ 3,234 $ 325 150 - 200 $ 26 $ 32 $ 3,260 $ 340 200 - 250 $ 15 $ 23 $ 3,275 $ 350 250 - 500 $ 23 $ 60 $ 3,298 $ 370 500 - 1,000 $ 9 $ 62 $ 3,307 $ 400 > 1,000 $ 1 $ 8 $ 3,307 $ 404 $

17. E. Aggregates – Recap using text E. Aggregates – Recap using text

18. Simulation Example - Assumptions – Claim frequency follows a Poisson distribution Important property: Poisson distribution gives the probability of 0 claims, 1 claim, 2 claims, etc.

19. Simulation Example - Assumptions – Claim severity follows a expected value = standard deviation = note skewness

20. Simulation Example - Assumptions

21. Simulation Example - Alternative Strategies Policy 1 2 3 Per Occurrence Deductible$500,000$1,000,000none Per Occurrence Policy Limit$5,000,000$5,000,000none Aggregate Deductiblenonenone $6,000,000 Aggregate Policy Limitnonenone $10,000,000 Premium$780,000$415,000$165,000

22. Simulation Example - Results

23. StatisticPolicy 1: Policy 2:Policy 3:No insurance Mean value of retained losses$______ $2,716 $2,925 $3,042 Standard deviation of retained losses 1,065 1,293 1,494 1,839 Maximum probable retained loss at 95% level4,2545,003 ______ 6,462 Maximum value of retained losses 11,325 12,125 7,899 18,898 Probability that losses exceed policy limits 1.1% 0.7% 0.1% n.a. Probability that retained losses  $6 million 99.7% ____% 99.9% 92.7% Premium $780 $415 $165 $0 Mean total cost 3,194 3,131 3,090 3,042 Maximum probable total cost at 95% level 5,034 5,418 6,165 6,462