2. CAS Seminar on Ratemaking Introduction to Ratemaking Relativities (INT - 3) March 11, 2004 Wyndham Franklin Plaza Hotel Philadelphia, Pennsylvania Presented by: Francis X. Gribbon, FCAS & Julie A. Jordan, FCAS

3. Introduction to Ratemaking Relativities Why are there rate relativities? Considerations in determining rating distinctions Basic methods and examples Advanced methods

4. 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

5. 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!

6. Considerations in setting rating distinctions Operational Social Legal Actuarial

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

8. Social Considerations Privacy Causality Controllability Affordability

9. Legal Considerations Constitutional Statutory Regulatory

10. 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

11. 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.

12. 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

13. Loss Ratio Relativity Method ClassPremium @CRL LossesLoss Ratio Loss Ratio Relativity Current Relativity New Relativity 1$1,168,125$759,2810.651.00 2$2,831,500$1,472,7190.520.802.001.60

14. Pure Premium Relativity Method ClassExposuresLossesPure Premium Pure Premium Relativity 16,195$759,281$1231.00 27,770$1,472,719$1901.55

15. 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)LR class i + (1-Z) LR state

16. Properties of Credibility 0   – 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

17. 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

18. Loss Ratio Method, Continued ClassLoss Ratio CredibilityCredibility Weighted Loss Ratio Loss Ratio Relativity Current Relativity New Relativity 10.650.500.601.00 20.520.900.520.872.001.74 Total0.56

19. Off-Balance Adjustment ClassPremium @CRL Current Relativity Premium @ Base Class Rates Proposed Relativity Proposed Premium 1$1,168,1251.00$1,168,1251.00$1,168,125 2$2,831,5002.00$1,415,7501.74$2,463,405 Total$3,999,625$3,631,530 Off-balance of 9.2% must be covered in base rates.

20. 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

21. 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”

22. 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) Total Losses

23. 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)

24. Example: Loss Elimination Ratio Loss Size# of Claims Total Losses Average Loss Losses Net of Deductible $100$200$500 0 to 10050030,00060000 101 to 20035054,25015519,25000 201 to 500550182,625332127,62572,6250 501 +335375,1251120341,625308,125207,625 Total1,735642,000370488,500380,750207,625 Loss Eliminated153,500261,250434,375 L.E.R.0.2390.407.677

25. Example: Expenses TotalVariableFixed Commissions 15.5% 0.0% Other Acquisition 3.8%1.9% Administrative 5.4%0.0%5.4% Unallocated Loss Expenses 6.0%0.0%6.0% Taxes, Licenses & Fees 3.4% 0.0% Profit & Contingency 4.0% 0.0% Other Costs 0.5% 0.0% Total 38.6%25.3%13.3% Use same expense allocation as overall indications.

26. Example: Deductible Credit DeductibleCalculationFactor $100 (.614)*(1-.239) +.133 (1-.253) 0.804 $200 (.614)*(1-.407) +.133 (1-.253) 0.665 $500 (.614)*(1-.677) +.133 (1-.253) 0.444

27. Advanced Techniques Multivariate techniques – Bailey’s Minimum Bias – Generalized Linear Models Curve fitting

28. 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

29. A Simple Example Age Group ExposuresPure Premium Car Size LargeMediumSmallLargeMediumSmall 11001200500100310840 230050040047014602530

30. One-Way Relativities ClassExposuresPure Premium Relativity Large car4003801.00 Medium car17006501.70 Small car90015904.20 Age Group 118004501.00 Age Group 2120015703.50

31. Multi-way vs. One-way Age Group Multi-Way RelativitiesOne-way Relativities Car Size LargeMediumSmallLargeMediumSmall 11.003.108.401.001.704.20 24.7014.6025.303.506.0014.60

32. 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

33. 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

34. Bailey’s Example Start with initial guess at factors for one variable ClassPure PremiumRelativity Age group 1$4501.00 Age group 2$15703.50

35. Bailey’s Example: Step 1A What would the premiums be, assuming base rate = $100 and this rating plan? Age Group ExposuresTheoretical Premium Car Size LargeMediumSmallLargeMediumSmall 110012005001000012000050000 2300500400105000175000140000

36. Bailey’s Example: Step 1B What should the factors for car size be, given the rating factors for age group? Car sizeTheoretical Premium Theoretical Loss Ratio Loss Ratio Relativity Large1150001.301.00 Medium2950003.702.80 Small1900007.505.70

37. Bailey’s Example: Step 2A What would the premiums be, assuming base rate = $100 and this rating plan? Age Group ExposuresTheoretical Premium Car Size LargeMediumSmallLargeMediumSmall 1100120050010000336000285000 230050040030000140000228000

38. Bailey’s Example: Step 2B What should the factors for age group be, given the rating factors for car size? Age groupTheoretical Premium Theoretical Loss Ratio Loss Ratio Relativity Age group 16310001.301.00 Age group 23980004.703.70

39. Bailey’s Example: Steps 3-6 What if we continued iterating this way? ClassStep 1Step 2Step 3Step 4Step 5Step 6 Large Car1.00 Medium Car2.80 2.90 Small Car5.70 5.80 Age Group 11.00 Age Group 23.503.70 3.60 Italic factors = newly calculated; continue until factors stop changing

40. Bailey’s Example: Results Age Group Multi-Way RelativitiesBailey Relativities Car Size LargeMediumSmallLargeMediumSmall 11.003.108.401.002.905.80 24.7014.6025.303.6010.4020.10

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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)