CAS Seminar on Ratemaking

1. CAS Seminar on Ratemaking Introduction to Ratemaking Relativities March 10-11, 2005 New Orleans Marriott New Orleans, Louisiana Presented by: Brian M. Donlan, FCAS & Theresa A. Turnacioglu, FCAS

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 11.9% 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) * 1.70 = $170 • Should charge: (62*1.70 + 13)/(1-.25) = $158 • “Flattening” relativities accounts for fixed expense • Flattened factor = (1-.25-.13)*1.70 + .13 = 1.58 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 Clms>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 • Why use multivariate techniques • Minimum Bias techniques • Example • Generalized Linear Models

27. Why Use Multivariate Techniques? • One-way analyses: • Based on assumption that effects of single rating variables are independent of all other rating variables • Don’t consider the correlation or interaction between rating variables

28. Examples • Correlation: • Car value & model year • Interaction • Driving record & age • Type of construction & fire protection

29. Multivariate Techniques • Removes potential double-counting of the same underlying effects • Accounts for differing percentages of each rating variable within the other rating variables • Arrive at a set of relativities for each rating variable that best represent the experience

30. Minimum Bias Techniques • Multivariate procedure to optimize the relativities for 2 or more rating variables • Calculate relativities which are as close to the actual relativities as possible • “Close” measured by some bias function • Bias function determines a set of equations relating the observed data & rating variables • Use iterative technique to solve the equations and converge to the optimal solution

31. Minimum Bias Techniques • 2 rating variables with relativities Xi and Yj • Select initial value for each Xi • Use model to solve for each Yj • Use newly calculated Yjs to solve for each Xi • Process continues until solutions at each interval converge

32. Minimum Bias Techniques • Least Squares • Bailey’s Minimum Bias

33. Least Squares Method • Minimize weighted squared error between the indicated and the observed relativities • i.e., Min xy ∑ij wij (rij – xiyj)2 where Xi and Yj = relativities for rating variables i and j wij = weights rij = observed relativity

34. Least Squares Method Formula: Xi = ∑j wij rij Yj where Xi and Yj = relativities for rating variables i and j wij = weights rij = observed relativity

35. Bailey’s Minimum Bias • Minimize bias along the dimensions of the class system • “Balance Principle” : ∑ observed relativity = ∑ indicated relativity • i.e., ∑j wijrij = ∑j wijxiyj where Xi and Yj = relativities for rating variables i and j wij = weights rij = observed relativity

36. Bailey’s Minimum Bias Formula: Xi = ∑j wij rij where Xi and Yj = relativities for rating variables i and j wij = weights rij = observed relativity

37. Bailey’s Minimum Bias • Less sensitive to the experience of individual cells than Least Squares Method • Widely used; e.g.., ISO GL loss cost reviews

38. Type of Policy Aggregate Loss Costs at Current Level (ALCCL) Experience Ratio (ER) Class Group Class Group Light Manuf Medium Manuf HeavyManuf Light Manuf Medium Manuf HeavyManuf Mono- line 2000 250 1000 1.10 .80 .75 Multiline 4000 1500 6000 .70 1.50 2.60 A Simple Bailey’s Example- Manufacturers & Contractors

39. Bailey’s Example

40. Bailey’s Example • Start with an initial guess for relativities for one variable • e.g.., TOP: Mono = .602; Multi = 1.118 • Use TOP relativities and Baileys Minimum Bias formulas to determine the Class Group relativities

41. Bailey’s Example CGj = ∑i wij rij ∑i wij TOPi

42. Bailey’s Example • What if we continued iterating? Italic factors = newly calculated; continue until factors stop changing

43. Bailey’s Example • Apply Credibility • Balance to no overall change • Apply to current relativities to get new relativities

44. Bailey’s • Can use multiplicative or additive • All formulas shown were Multiplicative • Can be used for many dimensions • Convergence may be difficult • Easily coded in spreadsheets

45. Generalized Linear Models • Generalized Linear Models (GLM) provide a generalized framework for fitting multivariate linear models • Statistical models which start with assumptions regarding the distribution of the data • Assumptions are explicit and testable • Model provides statistical framework to allow actuary to assess results

46. Generalized Linear Models • Can be done in SAS or other statistical software packages • Can run many variables • Many Minimum bias models, are specific cases of GLM • e.g., Baileys Minimum Bias can also be derived using the Poisson distribution and maximum likelihood estimation

47. Generalized Linear Models • ISO Applications: • Businessowners, Commercial Property (Variables include Construction, Protection, Occupancy, Amount of insurance) • GL, Homeowners, Personal Auto

48. Suggested Readings • ASB Standards of Practice No. 9 and 12 • Foundations of Casualty Actuarial Science, Chapters 2 & 5 • Insurance Rates with Minimum Bias, Bailey (1963) • A Systematic Relationship Between Minimum Bias and Generalized Linear Models, Mildenhall (1999)

49. Suggested Readings • Something Old, Something New in Classification Ratemaking with a Novel Use of GLMs for Credit Insurance, Holler, et al (1999) • The Minimum Bias Procedure – A Practitioners Guide, Feldblum et al (2002) • A Practitioners Guide to Generalized Linear Models, Anderson, et al