1. Commercial Property Size of Loss Distributions Glenn Meyers Insurance Services Office, Inc. Casualty Actuaries in Reinsurance June 15, 2000 Boston, Massachusetts

2. Outline Data Classification Strategy –Amount of Insurance –Occupancy Class Mixed Exponential Model –“Credibility” Considerations Limited Classification Information Program Demonstration Goodness of Fit Tests Comparison with Ludwig Tables

3. Separate Tables For Commercial Property (AY 1991-95) Sublines – BG1 (Fire and Lightning) – BG2 (Wind and Hail) – SCL (Special Causes of Loss) Coverages –Building –Contents –Building + Contents –Building + Contents + Time Element

4. Exposures Reported separately for building and contents losses Model is based on combined building and contents exposure –Even if time element losses are covered

5. Classification Strategy Amount of Insurance –Big buildings have larger losses –How much larger? Occupancy Class Group –Determined by data availability Not used –Construction Class –Protection Class

6. Potential Credibility Problems Over 600,000 Occurrences 59 AOI Groupings 21 Occupancy Groups The groups could be “grouped” but: –Boundary discontinuities –We have another approach

7. The Mixed Exponential Size of Loss Distribution i ’s vary by subline and coverage w i ’s vary by AOI and occupancy group in addition to subline and coverage

8. The Mixed Exponential Size of Loss Distribution i = mean of the ith exponential distribution For higher i ’s, a higher severity class will tend to have higher w i ’s.

9. The Fitting Strategy for each Subline/Coverage Fit a single mixed exponential model to all occurrences Choose the w i ’s and i ’s that maximize the likelihood of the model. Toss out the w i ’s but keep the i ’s The w i ’s will be determined by the AOI and the occupancy group.

10. Back to the Credibility Problem

12. Varying W i ’s by AOI Prior expectations Larger AOIs will tend to have higher losses In mixed exponential terminology, the AOI’s will tend to have higher w i ’s for the higher i ’s. How do we make this happen?

13. Solution Let W 1i ’s be the weights for a given AOI. Let W 2i ’s be the weights for a given higher AOI. Given the W 1i ’s, determine the W 2i ’s as follows.

14. Step 1 Choose 0  d 11  1 Shifting the weight from 1st exponential to the 2nd exponential increases the expected claim cost.

15. Step 2 Choose 0  d 12  1 Shifting the weight from 2nd exponential to the 3rd exponential increases the expected claim cost.

16. Step 3 and 4 Similar Step 5 — Choose 0  d 15  1 Shifting the weight from 5th exponential to the last exponential increases the expected claim cost.

17. Several AOI Groups Choose W’s for lowest AOI Group

18. Then choose d’s to Construct W’s for the 2nd AOI Group

19. Then choose d’s to Construct W’s for the 3rd AOI Group

20. Then choose d’s to Construct W’s for the 4th AOI Group

21. Continue choosing d’s and constructing W’s until the end.

22. Estimating W’s (for the 1st AOI Group) and d’s (for the rest) Let: F k (x) = CDF for kth AOI Group (x h+1, x h ) be the hth size of loss group n hk = number of occurrences for h and k Then the log-likelihood of data is given by:

23. Estimating W’s (for the 1st AOI Group) and d’s (for the rest) Choose W’s and d’s to maximize log- likelihood 59 AOI Groups 5 parameters per AOI Group 295 parameters! Too many!

24. Parameter Reduction Fit W’s for AOI=1, and d’s for AOI=10, 100, 1,000, 10,000, 100,000 and 1,000,000. Note AOI coded in 1,000’s The W’s are obtained by linear interpolation on log(AOI)’s The interpolated W’s go into the log- likelihood function. 35 parameters -- per occupancy group

25. On to Occupancy Groups Let W be a set of W’s that is used for all AOI amounts for an occupancy group. Let X be the occurrence size data for all AOI amounts for an occupancy group. Let L[ X|W ] be the likelihood of X given W i.e. the probability of X given W

26. There’s No Theorem Like Bayes’ Theorem Let be n parameter sets. Then, by Bayes’ Theorem:

27. Bayesian Results Applied to an AOI and Occupancy Group Let be the ith weight that W k assigns to the AOI/Occupancy Group. Then the w i ‘s for the AOI/Occupancy Group is:

28. What Does Bayes’ Theorem Give Us? Before –A time consuming search for parameters –Credibility problems If we can get suitable W k ’s we can reduce our search to n W ’s. If we can assign prior Pr{ W k }’s we can solve the credibility problem.

29. Finding Suitable W k ’s Select three Occupancy Class Group “Groups” For each “Group” –Fit W’s varying by AOI –Find W’s corresponding to scale change Scale factors from 0.500 to 2.000 by 0.025 183 W k ’s for each Subline/Coverage

30. Graph of Log-Likelihoods

31. Prior Probabilities Set: Final formula becomes: Can base update prior on Pr{ W k | X }.

32. The Classification Data Availability Problem Focus on Reinsurance Treaties –Primary insurers report data in bulk to reinsurers –Property values in building size ranges –Some classification, state and deductible information Reinsurers can use ISO demographic information to estimate effect of unreported data.

33. Database Behind PSOLD 30,000+ records (for each coverage/line combination) containing: Severity model parameters Amount of insurance group –59 AOI groups Occupancy class group State Number of claims applicable to the record

34. Constructing a Size of Loss Distribution Consistent with Available Data Using ISO Demographic Data Select relevant data Selection criteria can include: –Occupancy Class Group(s) –Amount of Insurance Range(s) –State(s) Supply premium for each selection Each state has different occupancy/class demographics

35. Constructing a Size of Loss Distribution for a “Selection” Record output - Layer Average Severity Combine all records in selection: LAS Selection = Wt Average(LAS Records ) Use the record’s claim count as weights

36. Constructing a Size of Loss Distribution for a “Selection” Where:  i = ith overall weight parameter w ij = ith weight parameter for the jth record C j = Claim weight for the jth record

37. The Combined Size of Loss Distribution for Several “Selections” Claim Weights for a “selection” are proportional to Premium  Claim Severity LAS Combined = Wt Average(LAS Selection ) Using the “selection” total claim weights The definition of a “selection” is flexible

38. The Combined Size of Loss Distribution for Several “Selections” Calculate  i ’s for groups for which you have pure premium information. Calculate the average severity for jth group

39. The Combined Size of Loss Distribution for Several “Selections” Calculate the group claim weights Calculate the weights for the treaty size of loss distribution

40. The Deductible Problem The above discussion dealt with ground up coverage. Most property insurance is sold with a deductible –A lot of different deductibles We need a size of loss distribution net of deductibles

41. Size of Loss Distributions Net of Deductibles Remove losses below deductible Subtract deductible from loss amount Relative Frequency

42. Size of Loss Distributions Net of Deductibles Combine over all deductibles LAS Combined Post Deductible Equals Wt Average(LAS Specific Deductible ) Weights are the number of claims over each deductible.

43. Size of Loss Distributions Net of Deductibles For an exponential distribution: Net severity Need only adjust frequency -- i.e. w i ’s

44. Adjusting the w i ’s D j  jth deductible amount  ij W i

45. Goodness of Fit - Summary 16 Tables Fits ranged from good to very good Model LAS was not consistently over or under the empirical LAS for any table Model unlimited average severity –Over empirical 8 times –Under empirical 8 times

46. A Major Departure from Traditional Property Size of Loss Tabulations Tabulate by dollars of insured value Traditionally, property size of loss distributions have been tabulated by % of insured value.

47. Fitted $ Average Severity against Insured Value

48. Fitted Average Severity as % of Insured Value Blow up this area

49. Fitted Average Severity as % of Insured Value Eventually, assuming that loss distributions based on a percentage of AOI will produce layer costs that are too high.

50. PSOLD Demonstration No Information Size of Building Information Size + Class Information Size + Class + Location Information

51. Comparison with Ludwig Tables Tabulated by % of amount of insurance Organized by occupancy class and amount of insurance –Broader AOI classes –Broader occupancy classes Fewer occurrances No model A very good paper

52. Comparison with Ludwig Tables Ludwig — Exhibit 15 (all classes) Matched insured value ranges Obtained % of insured value distributions from PSOLD –assuming low end of range –assuming high end of range Results on Spreadsheet

53. What’s new for the next review? Include data through 1998 Fewer exclusions of loss information –Recall that we excluded claims if exposure and class information were missing. –Include claims if we trust the losses and use Bayesian techniques to spread losses to possible class and exposure groups. Include HPR classes