1. Approximation of Aggregate Losses Dmitry Papush Commercial Risk Reinsurance Company CAS Seminar on Reinsurance June 7, 1999 Baltimore, MD

2. Approximation of an Aggregate Loss Distribution Usual Frequency - Severity Approach: Analyze Number of Claims Distribution and Claim Size Distribution separately, then convolute.

3. The Problem How to approximate an Aggregate Loss Distribution if there is no individual claim data available? - What type of distribution to use?

4. Method Used ÀChoose severity and frequency distributions ÁSimulate number of claims and individual claims amounts and the corresponding aggregate loss ÂRepeat many times (5,000) to obtain a sample of Aggregate Losses ÃFor different Distributions calculate Method of Moments parameter estimators using the simulated sample of Aggregate Losses ÄTest the Goodness of fit

5. Assumptions for Frequency and Severity Used Frequency: Negative Binomial Severity: Five parameter Pareto, Lognormal Layers: 0 - $250K(Low Retention) 0 - $1000K (High Retention) $750K xs $250K (Working Excess) $4M xs $1M (High Excess)

6. Distributions Used to Approximate Aggregate Losses À Lognormal Á Normal  Gamma

7. Gamma Distribution  -  *x  -1 *exp(-x/  ) f(x) =  (  ) Mean =  *  Variance =  *  2

8. Goodness of Fit Tests ÀPercentile Matching ÁLimited Expected Loss Costs

9. Example 1. Small Book of Business, Low Retention: Expected Number of Claims = 50, Layer: 0 - $250K, Severity - 5 Parameter Pareto

10. Example 1.

11. Example 2. Large Book of Business, Low Retention: Expected Number of Claims = 500, Layer: 0 - $250K, Severity - 5 Parameter Pareto

12. Example 2.

13. Example 3. Small Book of Business, High Retention: Expected Number of Claims = 50, Layer: 0 - $1,000K, Severity - 5 Parameter Pareto

14. Example 3.

15. Example 4. Large Book of Business, High Retention: Expected Number of Claims = 500, Layer: 0 - $1,000K, Severity - 5 Parameter Pareto

16. Example 4.

17. Example 5. Working Excess Layer: Layer: $750K xs $250K, Expected Number of Claims = 20, Severity - 5 Parameter Pareto

18. Example 5.

19. Example 6. High Excess Layer: Layer: $4M xs $1M, Expected Number of Claims = 10, Severity - 5 Parameter Pareto

20. Example 6.

21. Example 7. High Excess Layer: Layer: $4M xs $1M, Expected Number of Claims = 10, Severity - Lognormal

22. Example 7.

23. Results ÀNormal works only for very large number of claims ÁLognormal is too severe on the tail ÂGamma is the best approximation out of the three