1. PEBELS: Policy Exposure Based Excess Loss Smoothing Marquis J. Moehring Liberty Mutual Group

2. Liberty Mutual Insurance Outline 1.Background 2.Goal 3.PEBELS Defined 4.PEBELS Derived (PPR Generalized) 5.Applications 6.Summary

3. Liberty Mutual Insurance My Challenge Strong Regional Focus State/Program Large Loss Provisions Low Credibility High Heterogeneity

4. Liberty Mutual Insurance This Should be Easier No applicable method in literature ILFs for Liability ELFs for Workers Compensation Nothing for Commercial Property or Homewners!

5. Liberty Mutual Insurance Goal of PEBELS PEBELS = Property Large Loss Exposure Segmentation Meet my challenge New applications! Deceptively difficult 1)No clear limit 2)Multiple non-linearities 3)Additional nuances 4)Practical considerations

6. Liberty Mutual Insurance PEBELS Defined

7. Liberty Mutual Insurance Exposure Curve

8. Liberty Mutual Insurance PEBELS Derived

9. Liberty Mutual Insurance PEBELS Derived

10. Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value Retention + Limit as a % of Insured value Exposure Factor Subject Premium Expected Loss Ratio Expected Primary Losses Expected Reinsurer Losses 20-10060167%833%0%682,00065%443,3000 100-25017557%286%26%161,00065%104,65027,209 250-1,00062516%80%41%285,00065%185,25075,953 1,000-2,0001,5007%33% 1,156,00065%751,400247,962 Grand Total 2,284,00065%1,484,600351,124

11. Liberty Mutual Insurance PEBELS Derived

12. Liberty Mutual Insurance Per Policy Generalization Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value Retention + Limit as a % of Insured value Exposure Factor Subject Premium Expected Loss Ratio Expected Primary Losses Expected Reinsurer Losses 20-100 60167%833%0%682,00065%443,3000 100-250 17557%286%26%161,00065%104,65027,209 250- 1,000 62516%80%41%285,00065%185,25075,953 1,000- 2,000 1,5007%33% 1,156,00065%751,400247,962 Grand Total 2,284,00065%1,484,600351,124

13. Liberty Mutual Insurance PEBELS Derived

14. Liberty Mutual Insurance Heterogeneity Generalization Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value Retention + Limit as a % of Insured value Exposure Factor Subject Premium Expected Loss Ratio Expected Primary Losses Expected Reinsurer Losses 20-10060167%833%0%682,000 65% 443,3000 100-25017557%286%26%161,000 65% 104,65027,209 250-1,00062516%80%41%285,000 65% 185,25075,953 1,000-2,0001,5007%33% 1,156,000 65% 751,400247,962 Grand Total 2,284,00065%1,484,600351,124

15. Liberty Mutual Insurance Heterogeneity Generalization State:House X65.0% Y Z40.0%

16. Liberty Mutual Insurance PEBELS Derived

17. Liberty Mutual Insurance Heterogeneity Generalization Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value Retention + Limit as a % of Insured value Exposure Factor Subject Premium Expected Loss Ratio Expected Primary Losses Expected Reinsurer Losses 20-10060167%833% 0% 682,00065%443,3000 100-25017557%286% 26% 161,00065%104,65027,209 250-1,00062516%80% 41% 285,00065%185,25075,953 1,000-2,0001,5007%33% 1,156,00065%751,400247,962 Grand Total 2,284,00065%1,484,600351,124

18. Liberty Mutual Insurance Exposure Curve

19. Liberty Mutual Insurance Exposure Curve Dispersed: e.g. Estate / University Campus

20. Liberty Mutual Insurance Exposure Curve Dispersed: e.g. Estate / University Campus Concentrated: e.g. Barn / Minimart

21. Liberty Mutual Insurance PEBELS Derived

22. Liberty Mutual Insurance PEBELS Derived

23. Liberty Mutual Insurance Applications Indications Motivated PEBELS Allocate large losses to state and program –Low credibility –High heterogeneity in underlying exposures

24. Liberty Mutual Insurance Applications Adjusted Modeled Catastrophe AALs Traditionally assume AAL linear with IV This contradicts –Theory presented –Ludwig’s study of Hurricane Hugo Implies bias between Personal & Commercial Can adjust AALs with PEBELS

25. Liberty Mutual Insurance Applications Predictive Models Hypothesize that PEBELS More predictive of large loss than IV Most predictive for highly skewed perils Most predictive in severity/excess models

26. Liberty Mutual Insurance Applications

27. Liberty Mutual Insurance Applications

28. Liberty Mutual Insurance Summary PEBELS = Property Large Loss Exposure Segmentation Only game in town Quantifies messy non-linearities Multiple applications –Indications –Catastrophe Modeling –Risk Segmentation

30. Liberty Mutual Insurance Historical vs. Prospective Selecting exposure profile for the application? Prospective (current inforce) –Catastrophe modeling –Reinsurance quotes Historical (“earned” over experience period) –Loss ratio ratemaking –Revised per risk reinsurance exposure rating

31. Liberty Mutual Insurance Historical vs. Prospective Loss ratio ratemaking examples 1)State in run-off scenario 2)State newly entered scenario Both scenarios lead to skewed state indications Even small shifts will distort indications

32. Liberty Mutual Insurance Credibility

33. Liberty Mutual Insurance Appendix Misc. Topics Exposure curve considerations Data limitations and NLE Methods in common usage