CAS Seminar on Ratemaking

1. CAS Seminar on Ratemaking Introduction to Ratemaking Relativities (INT - 3) March 27, 2003 San Antonio Marriott Rivercenter San Antonio, Texas Presented by: Patrick B. Woods, FCAS, MAAA & Peggy Brinkmann, FCAS, MAAA

2. Introductionto Ratemaking Relativities • Why are there rate relativities? • Considerations in determining rating distinctions • Basic methods and examples • Advanced methods

3. Why are there rate relativities? • Individual Insureds differ in . . . • Risk Potential • Amount of Insurance Coverage Purchased • With Rate Relativities . . . • Each group pays its share of losses • We achieve equity among insureds (“fair discrimination”) • We avoid anti-selection

4. What is Anti-selection? • Anti-selection can result when a group can be separated into 2 or more distinct groups, but has not been. Consider a group with average cost of $150 • Subgroup A costs $100 • Subgroup B costs $200 If a competitor charges $100 to A and $200 to B, you are likely to insure B at $150. You have been selected against!

5. Considerations in setting rating distinctions • Operational • Social • Legal • Actuarial

6. Operational Considerations • Objective definition - clear who is in group • Administrative expense • Verifiability

7. Social Considerations • Privacy • Causality • Controllability • Affordability

8. Legal Considerations • Constitutional • Statutory • Regulatory

9. Actuarial Considerations • Accuracy - the variable should measure cost differences • Homogeneity - all members of class should have same expected cost • Reliability - should have stable mean value over time • Credibility - groups should be large enough to permit measuring costs

10. Basic Methods for Determining Rate Relativities • Loss ratio relativity method • Produces an indicated change in relativity • Pure premium relativity method • Produces an indicated relativity The methods produce identical results when identical data and assumptions are used.

11. Data and Data Adjustments • Policy Year or Accident Year data • Premium Adjustments • Current Rate Level • Premium Trend/Coverage Drift – generally not necessary • Loss Adjustments • Loss Development – if different by group (e.g., increased limits) • Loss Trend – if different by group • Deductible Adjustments • Catastrophe Adjustments

12. Loss Ratio Relativity Method

13. Pure Premium Relativity Method

14. Incorporating Credibility • Credibility: how much weight do you assign to a given body of data? • Credibility is usually designated by Z • Credibility weighted Loss Ratio is LR= (Z)LRclass i + (1-Z) LRstate

15. Properties of Credibility • 0 £ Z £ 1 • at Z = 1 data is fully credible (given full weight) •  Z /  E > 0 • credibility increases as experience increases •  (Z/E)/  E<0 • percentage change in credibility should decrease as volume of experience increases

16. Methods to Estimate Credibility • Judgmental • Bayesian • Z = E/(E+K) • E = exposures • K = expected variance within classes / variance between classes • Classical / Limited Fluctuation • Z = (n/k).5 • n = observed number of claims • k = full credibility standard

17. Loss Ratio Method, Continued

18. Off-Balance Adjustment Off-balance of 9.2% must be covered in base rates.

19. Expense Flattening • Rating factors are applied to a base rate which often contains a provision for fixed expenses • Example: $62 loss cost + $25 VE + $13 FE = $100 • Multiplying both means fixed expense no longer “fixed” • Example: (62+25+13) * 1.74 = $174 • Should charge: (62*1.74 + 13)/(1-.25) = $161 • “Flattening” relativities accounts for fixed expense • Flattened factor = (1-.25-.13)*1.74 + .13 = 1.61 1 - .25

20. Deductible Credits • Insurance policy pays for losses left to be paid over a fixed deductible • Deductible credit is a function of the losses remaining • Since expenses of selling policy and non claims expenses remain same, need to consider these expenses which are “fixed”

21. Deductible Credits, Continued • Deductibles relativities are based on Loss Elimination Ratios (LER’s) • The LER gives the percentage of losses removed by the deductible • Losses lower than deductible • Amount of deductible for losses over deductible • LER = (Losses <= D) + (D * # of Claims >D) Total Losses

22. Deductible Credits, Continued • F = Fixed expense ratio • V = Variable expense ratio • L = Expected loss ratio • LER = Loss Elimination Ratio • Deductible credit = L*(1-LER) + F (1 - V)

23. Example: Loss Elimination Ratio

24. Example: Expenses Use same expense allocation as overall indications.

25. Example: Deductible Credit

26. Advanced Techniques • Multivariate techniques • Bailey’s Minimum Bias • Generalized Linear Models • Curve fitting

27. Why Use Multivariate Techniques? • Many rating variables are correlated • Different variables, when viewed one at a time, may be “double counting” the same underlying effect • Using a multivariate approach removes potential double-counting and can account for interaction effects

28. Age Group Exposures Pure Premium Car Size Car Size Large Medium Small Large Medium Small 1 100 1200 500 100 310 840 2 300 500 400 470 1460 2530 A Simple Example

29. Class Exposures Pure Premium Relativity Large car 400 380 1.00 Medium car 1700 650 1.70 Small car 900 1590 4.20 Age Group 1 1800 450 1.00 Age Group 2 1200 1570 3.50 One-Way Relativities

30. Age Group Multi-Way Relativities One-way Relativities Car Size Car Size Large Medium Small Large Medium Small 1 1.00 3.10 8.40 1.00 1.70 4.20 2 4.70 14.60 25.30 3.50 6.00 14.60 Multi-way vs. One-way

31. When to use Multivariate? • Can use Multivariate techniques for entire rating plan, or for particular variables that are correlated or have interaction effects • Example of correlation • Value of car and Model Year • Examples of interaction effects • Driving record and Age • Type of construction and Fire protection

32. Bailey’s Minimum Bias • To get toward multivariate but still have simple method to calculate premiums • Can have credibility issues with many cells • Can use either Loss Ratio or Pure Premium methods • Can assume multiplicative and/or additive relationships of rating variables and dependent variable

33. Bailey’s Example • Start with initial guess at factors for one variable

34. Age Group Exposures Theoretical Premium Car Size Car Size Large Medium Small Large Medium Small 1 100 1200 500 10000 120000 50000 2 300 500 400 105000 175000 140000 Bailey’s Example: Step 1A • What would the premiums be, assuming base rate = $100 and this rating plan?

35. Bailey’s Example: Step 1B • What should the factors for car size be, given the rating factors for age group?

36. Age Group Exposures Theoretical Premium Car Size Car Size Large Medium Small Large Medium Small 1 100 1200 500 10000 336000 285000 2 300 500 400 30000 140000 228000 Bailey’s Example: Step 2A • What would the premiums be, assuming base rate = $100 and this rating plan?

37. Bailey’s Example: Step 2B • What should the factors for age group be, given the rating factors for car size?

38. Bailey’s Example: Steps 3-6 • What if we continued iterating this way? Italic factors = newly calculated; continue until factors stop changing

39. Age Group Multi-Way Relativities Bailey Relativities Car Size Car Size Large Medium Small Large Medium Small 1 1.00 3.10 8.40 1.00 2.90 5.80 2 4.70 14.60 25.30 3.60 10.40 20.10 Bailey’s Example: Results

40. Bailey’s Minimum Bias • Bailey Relativities get much closer to multi-way relativities than univariate approach • Premium calculation by multiplying factors vs. table lookup for multi-way • This example assumed two multiplicative factors, but approach can be modified for more variables and/or additive rating plans

41. Generalized Linear Models • Generalized Linear Models (GLM) is a generalized framework for fitting multivariate linear models • Bailey’s method is a specific case of GLM • Factors can be estimated with SAS or other statistical software packages

42. Curve Fitting • Can calculate certain type of relativities using smooth curves • Fit exposure data to a curve • Determine a functional relationship of loss data and exposure data • Taking derivative of this function and relating the value at any given point to a base point produces relativity

43. Curve Fitting • HO Policy Size Relativities • Assume the distribution of exposures by amount of insurance is log normal • Assume the cumulative loss distribution has a functional relationship to the cumulative exposure distribution

44. Curve Fitting • Let r = amount of insurance • f (r) is density of exposures at r • = exposures at r / total exposures • g (r) is density of losses at r • = losses at r / total losses • F(A) and G (A) are the cumulative functions of f and g

45. Curve Fitting • F (A) and G (A) are cumulative functions of f and g • G (A) = H[ F (A)] • Then dG (A)/dF (A) = g(a)/f(a) = (losses at A / total losses) (exposures at A / total exposures) • = pure premium at A/ total pure premium

46. Suggested Readings • ASB Standard of Practice No. 9 • ASB Standard of Practice No. 12 • Foundations of Casualty Actuarial Science, Chapters 2 and 5 • Insurance Rates with Minimum Bias, Bailey (1963) • Something Old, Something New in Classification Ratemaking with a Novel Use of GLMs for Credit Insurance, Holler, Sommer, and Trahair (1999)