1. 1 Solving the Puzzle: The Hybrid Reinsurance Pricing Method John Buchanan - Platinum Reinsurance CARe – London Casualty Pricing Approaches 16 July, 2007 CARe London-7/2007 – The Hybrid Reinsurance Pricing Method

2. 2 Agenda Typical Puzzle Improvements to Traditional Methods –Analogy to Reserving Hybrid: Experience / Exposure Method –Overriding Assumptions Testing Default Parameters US and Global Benchmarks

3. 3 Reinsurance Proposal Layer $100,000 xs $100,000 Estimated Premium: $40,000,000 GL Business –Southeast US Underwriting and Claims Info

4. 4 Traditional Methods Experience Relevant parameter defaults/overrides for: –LDFs (excess layers) –Trends (severity, frequency, exposure) –Rate changes –LOB/HzdGrp indicators Adjust for historical changes in: –Policy limits –Exposure differences o Careful “as-if” Exposure Relevant parameters defaults/overrides for: –ILFs (or ELFs, PropSOLD) –Direct loss ratios (on-level) –ALAE loads –Policy profile (LOB, HzdGrp) o Limit/subLOB allocations Adjust for expected changes in: –Rating year policy limits –Rating year exposures expected to be written

5. 5 What’s your final answer? Experience for this layer is half of the Exposure Exposure = 3.92% (1.57 mm) Experience = 1.85% (0.74 mm) Trick Question…

6. 6 Traditional Naïve Approach Naïve approach –Estimate Exposure Rate – X –Estimate Experience Rate – Y –Combine as w(X)+(1-w)Y It may be tempting to think the next step is to refine the estimate of w Not easy, but luckily, not the right next step Source: Stephen Philbrick Seminar on Ratemaking March 7-9, 2007

7. 7 Better Approach Use the Experience results of the layer, and adjacent layers to examine the Exposure rating assumptions Use the Exposure rating assumptions to help distinguish noise from signal in the Experience rating Use claim count to emphasize signal over noise – Exposure model can help provide expected frequencies Source: Stephen Philbrick Seminar on Ratemaking March 7-9, 2007

8. 8 Better Approach continued Apply forensic actuarial techniques to bring the Exposure and Experience models closer together Apply the Hybrid method to the adjusted Exposure and Experience models to arrive at the Hybrid answer Optionally, weight the answer with the Exposure indication. Ideally, the indications are now much closer, so the exact value of the weight is less important. Source: Stephen Philbrick Seminar on Ratemaking March 7-9, 2007

9. 9 Better Approach Reserving Analogy From paper submitted to CAS Variance – John Buchanan / Mike Angelina THE HYBRID REINSURANCE PRICING METHOD: A PRACTITIONER’S GUIDE

10. 10 Exposure Pricing (before investigation) Don’t look just at layer you are pricing (100 xs 100k) Look at layers below and above as well Look at Exposure burns and claim counts

11. 11 Experience Pricing (before investigation) Ditto for Experience Pricing Use same layers for easier comparison

12. 12 Exposure and Experience Comparison In this case study (CASRM 3/2007), there is an inconsistent relationship as move up the attachment points While the low layer Experience is about half of Exposure, the upper layers are about equal to Exposure Need more investigation to reconcile and help solve the puzzle

13. 13 Overall Pricing Process 1.We don't really know what exposure curve applies to a given account (e.g. we don't know that LN = 50k and CV = 400% is the true underlying distribution) 2.We have a hunch based on established curves (e.g. we postulate LN = 50k and CV = 300%) 3.We obtain some observations from a certain number of claims over a certain number of years –in the long run the results will track with the true underlying distribution in 1 but these observations will initially be compared to the hypothesis given in 2 4.If we make enough correct adjustments to the observations and underlying exposures then we will start to see a non-constant pattern in the ratios of the observed experience results to the initially selected exposure results (the Hybrid ratios). –In this example, the actual Experience will end up being heavier for the top layers 5.If credible, this lack of constant Hybrid ratios creates a pressure to fatten the tail of the exposure distribution. Making this change to the exposure curve will allow us to create a better balance (e.g. Hybrid ratios will all be closer to 100%).

14. 14 Overriding Assumptions of the Hybrid Method In theory, with perfect modeling and sufficient data the results under the Experience and Exposure methods will be identical. In practice, –if the model and parameter selections for both Experience and Exposure methods are proper and relevant, –then the results from these methods will be similar, –except for credibility and random variations. Lower layer experience helps predict higher less credible layers. Frequency is a more stable indicator than total burn estimates.

15. 15 Basic Steps of The Hybrid Method Step 1: Estimate Experience burns & counts Step 2: Estimate Exposure burns & counts Step 3: Calculate Experience/Exposure frequency ratio by attachment point Step 4: Review Hybrid frequency ratio patterns –Adjust experience or exposure models if needed and re-estimate burns (!!) Step 5: Similarly review excess severities and/or excess burns Step 6: Combine Hybrid frequency/severity results Step 7: Determine overall weight to give Hybrid

16. 16 Step 4-Review Hybrid Frequency Ratios Important Selection6.00 expos x 80.0%

17. 17 Steps 1-7: Bringing it All Together

18. 18 Example #2 (adjusting Experience for historically higher policy limits)

19. 19 Example #3 (adjusting Exposure for clash potential)

20. 20 Benefits of Hybrid Method One of main benefits is questioning Experience and Exposure Selections –To the extent credible results don’t line up, this provides pressure to the various default parameters –For example, there would be downward pressure on default exposure ILF curves or loss ratios if Exposure consistently higher than experience, and Credible experience and experience rating factors A well constructed Hybrid method can sometimes be given 100% weight if credible Can review account by account, and aggregate across accounts to evaluate pressure on industry defaults

21. 21 Test of Default Parameters Aggregate across “similar” accounts to evaluate pressure on industry defaults –May want to re-rate accounts using e.g. default rate changes, ILFs, premium allocations, LDFs, trends, etc. Each individual observation represents a cedant/attachment point exper/expos ratio Review dispersion of results and overall trend –E.g. if weighted and/or fitted exper/expos ratios are well below 100% (or e.g. 90% if give some underwriter credit) then perhaps default L/Rs overall are too high (or conversely LDFs or trends too light) –If trend is up when going from e.g. 100k to 10mm att pt, then perhaps expos curve is predicting well at lower points but is underestimating upper points

22. 22 Test of Default Parameters (cont.) Before making overall judgments, must consider –UW contract selectivity (contracts seen vs. written), –Sample size (# of cedants/years), –Impact “as-if” data (either current or historical) –Survivor bias –Systematic bias in models –“Lucky”

23. 23 Test of Default Rating Factors – Example 1 Well below 100%, pressure to reduce expos params or increase exper params…but credible??

24. 24 Test of Default Rating Factors – Example 2 Exposure curve too light with higher attachment points?

25. 25 Reinsurance Market Reinsurance business mix 1 EuropeUS / Can Property 46% 34% Motor 21% 8% Liability & WC 20% 35% Other 3% 23% Reinsurance type 2 Proportional 70% 50% Non-Proportional 30% 50% P & C Reinsurance Demand 3 $ 51 b$ 65 b Source: Tim Aman CARe-Phila: 1 Axco, 2 Estimated, 3 A M Best Co

26. 26 Exposure Benchmarks Insurance business mixEuropeUS / Can Property 24% 27% Motor 38% 41% Liability 10% 14% WC 0% 11% A&H 17% 2% Other 11% 5% Mix Source: Tim Aman CARe-Phila Axco ISO NCCI GLD (dated) Consultants Lloyds, SRe, MRe Lots of US Exposure Curves available But many sub-lines don’t have standard curves and questionable applicability to many other lines – D&O, E&O, EPLI, Umbrella, most international lines Companies need to accumulate own: difficult, credibility issues

27. 27 Global Hurricane Activity Used by permission from UK Met

28. 28 Summary Weighting of alternative methods should be viewed as the actuarial equivalent of crying “uncle”. Do not view weighting as a positive approach to coming up with an answer, but a concession that there are things going on you haven’t modeled Perfectly acceptable if the only remaining differences are noise – if not, improve the model Source: Stephen Philbrick Seminar on Ratemaking March 7-9, 2007

29. 29 Appendices More Advanced Puzzle Solving Techniques Hybrid Steps Credibility –One of the most difficult puzzle pieces

30. 30 Appendix - More advanced techniques for Solving the Puzzle Inspecting Experience/Exposure differences

31. 31 Appendix - More advanced techniques for Solving the Puzzle Pressure Indicators –years (or layers)

32. 32 Basic Steps of The Hybrid Method Step 1: Estimate Experience burns & counts –Select base attachment points/layers above the reporting data threshold –Estimate total excess burns using projection factors –Estimate excess counts using frequency trends, claim count LDFs –Calculate implied severities Step 2: Estimate Exposure burns & counts –Use same attachment points/layers as Experience –Estimate total burns and bifurcate between counts, average severities Step 3: Calculate Experience/Exposure frequency ratio by attachment point –Estimate overall averages using number of claims/variability Step 4: Review frequency ratio patterns –Adjust experience or exposure models if needed and re-estimate burns (!!) –Select indicated experience/exposure frequency ratio(s) Step 5: Similarly review excess severities and/or excess burns Step 6: Combine Hybrid frequency/severity results –Using experience adjusted exposure frequencies and severities Step 7: Determine overall weight to give Hybrid

33. 33 Estimation of Hybrid Counts Preview Steps 1 to 4 A: Select base attachment points above data threshold –Example: threshold=150k; reins layers=500x500k, 1x1mm –Select 200k, 250k, 350k, 500k, 750k, 1mm attachment points B: Calculate experience counts –At lower attachment points, year by year patterns should be variable about some mean –For example, if upward trend, then perhaps: Overdeveloping or trending later years C: Calculate exposure counts for comparison D: Review experience/exposure frequency patterns –Should be relatively stable until credibility runs out –Double back to methods if not –Select frequency ratios to estimate Hybrid counts

34. 34 Step 1a: Experience Counts and Burns Sublayer $150,000 xs 350,000

35. 35 Step 1b: Review Experience Counts Year Variability: >350,000 Attachment Apparently random pattern around selection of #=12.05 Note: Claim counts are on-leveled

36. 36 Step 1c: Review Experience Counts Year Variability: > 1,000,000 Attachment Credibility runs out; indication is #=.36

37. 37 Step 1-Recap: Estimation of Experience Burns, Counts and Implied Severities To be compared to exposure counts

38. 38 Step 2: Estimation of Exposure Burns Bifurcated Between Counts and Severities 12.05 exper / 15.34 expos = 78.6%

39. 39 Step 3: Calculate Experience/Exposure Frequency Ratios and Base Layer Weights 12.05 exper / 15.34 expos = 78.6%

40. 40 Step 4a: Review Exper/Expos Frequencies Attachment Point Pattern: 200k…1mm Expos and Exper count ratios relatively consistent through 350k- IF experience very credible, then perhaps pressure to reduce exposure L/R; check out spikes

41. 41 Step 4-Recap: Select Exper/Expos Frequency Ratio For Hybrid Claim Count Estimate Important Selection6.00 expos x 80.0%

42. 42 Step 5: Selected Severity Unrealistic experience severity

43. 43 Step 6: Selected Overall Hybrid Burn Hybrid: Experience adjusted Exposure count & severity… 100% credibility to burn??

44. 44 Classical Credibility Weighting Estimate separate Experience and Exposure burns Select credibility weights using combination of: –Formulaic Approach Expected # of Claims / Variability Exposure ROL (or burn on line) –Questionnaire Approach Apriori Neutral vs. Experience vs. Exposure Patrik/Mashitz paper –Judgment Need to check that burn patterns make sense –i.e. higher layer ROL < lower ROL –similar to Miccolis ILF consistency test

45. 45 Classical Credibility Weighting Credibility weights judgmentally selected

46. 46 Assessing Credibility of Exposure Method Assess confidence in: –Exposure curve selected –Exposure profile –Source of hazard or sub-line information –Prediction of next years primary loss ratio –Percentage of non-modeled exposure, clash, etc. –Company strategy and ability to realize strategy Possibly take questionnaire / scoring approach to mechanize (Patrik/Mashitz)

47. 47 Assessing Credibility of Experience Method Assess confidence due to: –Overall volume of claims –Volume of claims within layer (lucky or unlucky?) –Stability of year by year experience results – “ layer to layer experience results –Source of loss development, trend factors, historical rate changes and deviations –Changes in historical profile limits –Appropriateness of any claims or divisions that may have been removed (or “as-if’d”) Experience score compared to exposure score to determine credibility weight

48. 48 Increase Credibility by Reducing Variability Above figure from iconic Philbrick CAS paper In this case, A represents Experience rating average (with indicated process noise), while B represents Exposure Goal will be to bring A and B closer together thereby reducing parameter variance, with any remaining difference being process noise