Practical Applications of Credibility Theory

1. Practical Applications of Credibility Theory Tom Rhodes, FSA, MAAA, FCA AVP & Actuarial Director MIB Solutions February 16, 2007

2. Practical Applications of Credibility Theory • Importance to Practicing Actuaries • Credibility Methods • Formula and Variance • Real Life Mortality Example • Selection of Method • Elements of Credibility Theory • Prior A/E Ratios • Company A/E Ratios • Estimate A/E Ratios • Lapse Application • Mortality Applications (VM-20, pricing, risk mgmt)

3. Importance of Credibility Theory • Company results vary substantially from the intercompany averages • Credibility Theory provides a statistically sound method of measuring variability of company results from intercompany results. • Credibility Theory provides company specific estimates of mortality and lapse based on company results, intercompany results and the measured variability.

4. Importance of Credibility Theory Prospect of Principles-Based Reserving raises its importance among life and annuity actuaries • NAIC’s Life and Health Actuarial Task Force • New Valuation Law and Valuation Manual (VM) • VM-20 includes credibility practices in valuation of life insurance products • Companies set reserves based on own experience • Academy’s Credibility Practice Note highlights • Determine the level of reliance that can be placed on company experience • Determine assumptions for modeling company cash flows

5. Credibility Theory Practices Report • The SOA’s Committee on Life Insurance Research, Financial Reporting and Product Development Sections sponsored a report on application and adoption of credibility theory for life insurance and annuities. • MIB Solutions team of Stuart Klugman, Tom Rhodes, Marianne Purushotham and Stacy Gill produced Credibility Theory Practices report: http://www.soa.org/research/life/research-credibility-theory-pract.aspx

6. Credibility Theory Practices ReportTheory, Practical Methodology and More! Theory: Formulas developed for two established statistical credibility methods, Limited Fluctuation and Bühlmann Empirical Bayesian Practical Methodology: For Limited Fluctuation and Bühlmann Empirical Bayesian, formulas for ‘Z’ applied to sample of small, medium and large companies: • Mortality and lapse results • Excel files implementing formulas included • Appendices document this Excel file implementation • Companies can perform Limited Fluctuation method in-house Survey and Bibliography Survey Bibliography

7. Credibility MethodsFormula • Credibility formulas use a credibility factor ‘Z’ varies from 0 to 1. • The standard form of credibility formula: Estimate A/E Ratio = Z × (Company A/E Ratio) + (1-Z) x (Prior A/E Ratio) Addresses Academy Practice Note Highlights • Determine the level of reliance that can be placed on company experience (Z) • Determine modeling assumptions for company cash flows (Estimate A/E Ratio)

8. Credibility Methods Variance • Total Variance = Process Variance + Variance of Hypothetical Means • Limited Fluctuation uses Process Variance • Bühlmann Empirical Bayesian uses both Process Variance + Variance of Hypothetical Means

11. Real Life Mortality Example • Experience is from ten companies that participated in the SOA 2004-05 experience study • Companies were selected in order to give a mixture of large, medium and small size companies. • To maintain strict confidentiality of individual company information, only a portion of each company’s mortality and lapse data was used.

12. Real Life Mortality Example • The variability of the Company A/E Ratios around the Prior A/E Ratio of the inter-company study is evident • In general, Credibility Factor ‘Z’ increases with increasing number of deaths. • Difference between estimate A/E ratios and company A/E ratios varies substantially among different companies

13. Real Life Mortality ExampleNS Preferred Class Structure of 2 (NS PCS 2)Limited Fluctuation Method

14. Real Life Mortality ExampleNS Preferred Class Structure of 2 (NS PCS 2) Bühlmann Empirical Bayesian Method

15. Elements of Credibility Theory • Prior A/E Ratios • Company A/E Ratios • Estimate A/E Ratios

16. Prior A/E Ratio in Credibility Theory Methods Credibility method for PCS 2 NS • Sample company experience was compared to expected basis 100% of 2001 VBT • Admittedly NOT a preferred class table • Intercompany study adjusts • From 100% of 2001 VBT expected basis goes down to 65.6% of 2001 VBT • 65.6% of 2001 VBT is the ‘Prior A/E Ratio’ used in credibility formula

17. Prior A/E Ratio in Credibility Theory Methods • Fixed Table as Prior A/E Ratio • Limited Fluctuation Method • Could use fixed table as Prior A/E Ratio • For example 100% 2001 VBT • Intercompany experience does not modify fixed table • Choice of fixed table can drive results, actuarial judgment is necessary • For example 65% of 2001 VBT for preferred

18. Prior A/E RatioActuarial Judgment Needed The selection of the Prior A/E Ratio affects estimate resulting from credibility method Prior 65.6% , Company 112.4% , Estimate 70.4% If one varies the Prior A/E assumption, one varies the Estimate A/E The selection of the Prior A/E Ratio assumption requires actuarial judgment

19. Practical Sources of Prior A/E Ratios • Fixed table (RR table, pricing assumption as % of standard table) • Pivot tables from SOA Experience Reports • Important to select appropriate mortality segment to correspond with assumption • Subsets of industry studies for product/underwriting class/peer group

20. Company A/E Ratios • In individual company experience study • Select mortality segment consistent with mortality segment of Prior A/E Ratio • Use policy by policy experience results • Mandatory company studies under both NAIC Model Regulation 815 and proposed Valuation Law/Valuation Manual • Statistical agent for Life needs to be named, comparable to statistical agent for P&C

21. Credibility Theory Lapse Application Many lapses, ‘Z’ is high Company Experience Rules

22. Application of Credibility Methods to Lapse • Used term policies in the NS PCS 2 for the 7 companies studied for mortality sample • Recent LIMRA study is basis for expected lapse • Actual experience of 7 companies produces mean used for Prior A/E Ratio

23. Term Lapse for NS PCS 2Limited Fluctuation Method

24. Term Lapse for NS PCS 2 Bühlmann Empirical Bayesian Method

25. Credibility Theory and VM-20 • NAIC’s Life and Health Actuarial Task Force • New Valuation Law and Valuation Manual (VM) • VM-20 covers valuation of life insurance products • Credibility theory not used for less than 30 deaths • Credibility theory used for 30 or greater deaths in credibility set of mortality segments

26. VM-20: For 30 or greater deaths in credibility set of mortality segments Determine prudent estimate mortality by: • Selecting credibility method • Selecting industry basic table (UCS method) • Determine mortality from experience studies and credibility method • Determine margin • Use credibility mortality plus margin to select industry basic table with higher mortality

27. Selecting credibility method • Either the Limited Fluctuation method or the Bühlmann Empirical Bayesian method are options. • Limited Fluctuation method • Requires only your company’s data • Can use Excel tables from MIB Solutions report • Can be done in-house • Bühlmann Empirical Bayesian method • May give higher ‘Z’ than Limited Fluctuation method • Requires data from multiple companies • Can only be done by statistical agent • For this example, use Limited Fluctuation Method

28. VM-20: For 30 or greater deaths in credibility set of mortality segments Determine prudent estimate mortality by: • Selecting credibility method • Selecting industry basic table (UCS method) • Determine mortality from experience studies and credibility method • Determine margin • Use credibility mortality plus margin to select industry basic table with higher mortality

29. Selecting industry basic table (UCS method) • For each mortality segment • Apply the UCS method for mortality segment • Select the appropriate RR table

30. VM-20: For 30 or greater deaths in credibility set of mortality segments Determine prudent estimate mortality by: • Selecting credibility method • Selecting industry basic table (UCS method) • Determine mortality from experience studies and credibility method • Determine margin • Use credibility mortality plus margin to select industry basic table with higher mortality

31. Determine Mortality from Experience Studies and Credibility Method • Determine mortality segments • RR table for Prior A/E Ratio • Company A/E Ratio calculated using policy by policy results from experience study • Use MIB Solutions report • Needed formulas, Appendices serve as guide • Excel files provide template to do calculations • Apply actuarial judgment

32. Determine Mortality from Experience Studies and Credibility Method Apply Actuarial Judgment • Start with credibility model’s Estimate A/E Ratio • Credibility model produces an estimate of flat percentage of base mortality table • Flat percentages may be modified for higher older age Qx’s • Mortality curve varied by product characteristics ,e.g., term mortality after level period • Other actuarial judgment changes possible

33. VM-20: For 30 or greater deaths in credibility set of mortality segments Determine prudent estimate mortality by: • Selecting credibility method • Selecting industry basic table (UCS method) • Determine mortality from experience studies and credibility method • Determine margin • Use credibility mortality plus margin to select industry basic table with higher mortality

34. Margins For credibility theory methods, margin considerations: • Reliability of Company Experience Studies • Reliability of Prior A/E Ratio (If a fixed table and not adjusted by intercompany study, future mortality improvement eventually makes it out of date) • Difference of Company and Estimate A/E Ratios: • Company E has 46.2% Company and 63.6% Estimate • Company F has 84.9% Company and 68.2% Estimate • Future mortality trends • Refer to SOA ‘s ‘Analysis of Methods for Determining Margins for Uncertainty under a Principles-Based Framework for Life Insurance and Annuity Products’

35. VM-20: For 30 or greater deaths in credibility set of mortality segments Determine prudent estimate mortality by: • Selecting credibility method • Selecting industry basic table (UCS method) • Determine mortality from experience studies and credibility method • Determine margin • Use credibility mortality plus margin to select industry basic table with higher mortality

36. Credibility TheoryOther Mortality Applications • For evaluating pricing assumptions • Use your company’s pricing assumptions as percentage of expected table as Prior A/E Ratio • Use company experience study for Co A/E Ratio • Limited Fluctuation method and Excel files from MIB Solution’s report to produce Estimate A/E Ratio • Modify results using actuarial judgment

37. Credibility TheoryOther Mortality Applications • For risk management application • Use your best guess as Prior A/E Ratio • Use company experience study for Co A/E Ratio • Limited Fluctuation method and Excel files from MIB Solution’s report to produce Estimate A/E Ratio • Modify results using actuarial judgment

38. Practical Applications of Credibility Theory • Important – Co A/E varies from Intercompany A/E • Credibility Methods applied with actuarial judgment • Prior A/E Ratios • Company A/E Ratios • Estimate A/E Ratios • Selection of Credibility Method • Limited Fluctuation can be done in-house by company using MIB Solutions’ Report • Lapse Application – Company results rule, insist on high quality company lapse experience studies • Mortality Application - VM-20, evaluating pricing assumptions, risk management studies

39. Invitations I.MIB Solutions is seeking current SOA contributor to ILEC study to develop practical implementation techniques for Credibility Theory: • Use that company’s already submitted data • Co-author practical credibility techniques paper • Involvement will • Improve knowledge of credibility techniques • Provide roadmap for applying in your company II. If you have questions on implementing credibility theory, call me for free advice!

40. Questions? trhodes@mib.com 917-400-0213