2. CAS Seminar on Ratemaking Introduction to Ratemaking Relativities March 17-18, 2008 Royal Sonesta Hotel Boston, Mass. Presented by: Michael J. Miller, FCAS

3. Introduction to Ratemaking Relativities What is the purpose of rate relativities? What is the purpose of rate relativities? Considerations in determining rating distinctions Considerations in determining rating distinctions Basic methods and examples Basic methods and examples Advanced methods Advanced methods

4. The Purpose of Rate Relativities Example – Personal Auto: Overall Indicated Change for State = +10% Should everyone’s rate be increased by 10%? Same for youthful drivers vs. adults? Same for urban vs. rural? Same for all policy limits? Deductibles?

5. The Purpose of Rate Relativities Example: Base Rate = $100 (Adult, Terr 1, No Deductible) InsuredAgeTerritoryDeductiblePremium Male Age 40 Terr 1 $0 Ded 1.001.001.00$100 Male 18 Terr 2 $500 Ded 3.501.500.60$315 Female 18 Terr 2 $100 Ded 2.001.500.85$255

6. Considerations in Selecting Rate Relativities Actuarial (Statistical) Actuarial (Statistical) Operational Operational Social Social Legal Legal

7. Actuarial Considerations Accuracy Accuracy Rating variable closely related to cost differences Rating variable closely related to cost differences Provides “fairest” price (fair discrimination) Provides “fairest” price (fair discrimination) Reduces adverse selection (discussed later) Reduces adverse selection (discussed later) Homogeneity Homogeneity Members of a class have similar expected cost Members of a class have similar expected cost Variability within class always exists – grouping is necessary since individual lacks credibility Variability within class always exists – grouping is necessary since individual lacks credibility

8. Actuarial Considerations (cont.) Credibility Credibility Class groups should be large enough to measure costs with sufficient accuracy Class groups should be large enough to measure costs with sufficient accuracy There is a trade-off between the need to estimate costs accurately for an individual and the need for enough data to do it. There is a trade-off between the need to estimate costs accurately for an individual and the need for enough data to do it. Reliability Reliability Estimated cost differences between groups should be relatively stable over time Estimated cost differences between groups should be relatively stable over time This does not mean they will be the same over time This does not mean they will be the same over time

9. Adverse Selection Adverse selection can result when a group can be accurately separated into 2 or more distinct groups, but has not been. Consider the following scenario: Group A expected costs = $100 Group B expected costs = $200 Your company charges $150 for both Competitor charges $100 for A, and $200 to B

10. Adverse Selection (cont.) At the outset, your company is collecting enough to cover expected costs for both groups. Life is good. All of your insureds in Group A learn about your competitor’s lower rate and switch. Your company is left with all of Group B at a $150 rate. You have been selected against! Typically this process happens gradually

11. Operational Considerations Objective - Age & Marital Status vs “Maturity” Objective - Age & Marital Status vs “Maturity” Administrative expense – Actual mileage driven may be better predictor of accident potential than where an insured lives, but one is much cheaper to obtain Administrative expense – Actual mileage driven may be better predictor of accident potential than where an insured lives, but one is much cheaper to obtain Verifiability – The amount of sleep a person has gotten in the previous 24 hours may be a significant predictor of auto accident potential. Aside from the expense of trying to get this information, how could it be verified? Verifiability – The amount of sleep a person has gotten in the previous 24 hours may be a significant predictor of auto accident potential. Aside from the expense of trying to get this information, how could it be verified?

12. Social Considerations Privacy – certain personal info is off limits Privacy – certain personal info is off limits Causality – as opposed to correlation Causality – as opposed to correlation Controllability – something the insured can impact (e.g. install sprinklers in commercial property, non-smoker in health insurance) Controllability – something the insured can impact (e.g. install sprinklers in commercial property, non-smoker in health insurance) Affordability – balance with availability (e.g. hospitals closing ER and OB due to high cost of med mal insurance for these classes) Affordability – balance with availability (e.g. hospitals closing ER and OB due to high cost of med mal insurance for these classes)

13. Legal Considerations Choice of rating variable may be prohibited by law at many levels (e.g. Federal, State). Some examples: Race Race Gender (always in Health ins, sometimes in other lines – even auto) Gender (always in Health ins, sometimes in other lines – even auto) Income Income

14. Basic Methods for Determining Rate Relativities Loss ratio relativity method Loss ratio relativity method Compare “actual” LR to expected LR to produce an indicated change in relativity Pure premium relativity method Develop expected cost per unit of exposure to produce indicated relativity The methods produce identical results when identical data and assumptions are used.

15. Data and Data Adjustments Policy Year or Accident Year data Policy Year or Accident Year data Premium Adjustments (LR method) Premium Adjustments (LR method) Current Rate Level Current Rate Level Premium Trend/Coverage Drift (not typical) Premium Trend/Coverage Drift (not typical) Loss Adjustments Loss Adjustments Loss Development (project to ultimate) Loss Development (project to ultimate) Loss Trend (project to same time period) Loss Trend (project to same time period) Coverage Adjustments (diff Ded’s, Limits?) Coverage Adjustments (diff Ded’s, Limits?) Catastrophe Adjustments (“Shock Losses”) Catastrophe Adjustments (“Shock Losses”)

16. Loss Ratio Relativity Method Class Premium @CRL Trended & Developed Losses Loss Ratio Loss Ratio Adjustmt Current Relativity New Relativity 1$1,168,125$759,2810.651.001.001.00 2$2,831,500$1,472,7190.520.802.001.60

17. Pure Premium Relativity Method ClassExposures Trended & Developed Losses Pure Premium Pure Premium Relativity 16,195$759,281$1231.00 27,770$1,472,719$1901.55

18. Incorporating Credibility Credibility: how much predictive weight do you assign to a given body of data? Credibility: how much predictive weight do you assign to a given body of data? Credibility is usually designated by Z Credibility is usually designated by Z Credibility Weighted Loss Ratio: LR= (Z) * LR class + (1-Z) * LR complement Credibility Weighted Loss Ratio: LR= (Z) * LR class + (1-Z) * LR complement

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

20. Loss Ratio Method, Continued Class Loss Ratio Credibility Credibility Weighted Loss Ratio Loss Ratio Adjustmt Current Relativity New Relativity 10.650.500.611.001.001.00 20.520.900.520.852.001.70 Total0.56

21. Off-Balance Adjustment Class Premium @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$1,415,7501.70$2,406,775 Total$3,999,625$3,574,900 Impact on Current Premium (“Off-Balance”) -11.9% Off-balance of 11.9% must be covered in base rates. (How?)

22. Multivariate Techniques Univariate (One-Way) Analyses: Univariate (One-Way) Analyses: Based on assumption that effects of single rating variables are independent of all other rating variables Based on assumption that effects of single rating variables are independent of all other rating variables Multivariate Analyses: Multivariate Analyses: Give consideration to the correlation or interaction between rating variables Give consideration to the correlation or interaction between rating variables

23. Bailey’s Method Bailey’s Method Least Squares Method Least Squares Method Generalized Linear Model (GLM) Method Generalized Linear Model (GLM) Method Multivariate Techniques (cont.) Multivariate Techniques (cont.)

24. Example: Bailey’s Method 2 rating variables with relativities X i and Y j 2 rating variables with relativities X i and Y j Select initial value for each X i Select initial value for each X i Use model to solve for each Y j Use model to solve for each Y j Use new Y j s to solve for each X i Use new Y j s to solve for each X i Process continues until solutions at each interval converge Process continues until solutions at each interval converge

25. Bailey’s Minimum Bias “Balance Principle” : “Balance Principle” : ∑ observed relativity = ∑ indicated relativity i.e., ∑ j w ij r ij = ∑ j w ij x i y j i.e., ∑ j w ij r ij = ∑ j w ij x i y jwhere X i and Y j = relativities for rating variables i and j X i and Y j = relativities for rating variables i and j w ij = weights w ij = weights r ij = observed relativity r ij = observed relativity

26. Bailey’s Minimum Bias Formula: X i = ∑ j w ij r ij X i = ∑ j w ij r ij where X i and Y j = relativities for rating variables i and j X i and Y j = relativities for rating variables i and j w ij = weights w ij = weights r ij = observed relativity r ij = observed relativity ∑ j w ij Y j ∑ j w ij Y j

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

28. A Simple Bailey’s Example- Manufacturers & Contractors Type of Policy w ij Aggregate Loss Costs at Current Level (w ij ) Experience Ratio Class Group Class Group (CG j ) Light Manuf Medium Manuf Heavy Manuf Light Manuf Medium Manuf Heavy Manuf Mono- line 200025010001.10.80.75 Multiline400015006000.701.502.60 SW = 1.61

29. Bailey’s Example Experience Ratio Relativities Class Group (CG j ) Statewide Type of Policy (TOP i ) Light Manuf Light Manuf Medium Manuf Heavy Manuf Monoline.683.497.466.602 Multiline.435.9321.6151.118

30. Bailey’s Example Initial guess for relativities of one variable (e.g., TOP: Mono =.602; Multi = 1.118) Initial guess for relativities of one variable (e.g., TOP: Mono =.602; Multi = 1.118) Use TOP relativities and Bailey’s Minimum Bias formulas to determine the Class Group (CG) relativities Use TOP relativities and Bailey’s Minimum Bias formulas to determine the Class Group (CG) relativities

31. Bailey’s Example CG j = ∑ i w ij r ij ∑ i w ij TOP i ∑ i w ij TOP i Class Group Bailey’s Output Light Manuf.547 Medium Manuf.833 Heavy Manuf 1.389

32. Bailey’s Example What if we continued iterating? Step 1 Step 2 Step 3 Step 4 Step 5 Light Manuf.547.547.534.534.533 Medium Manuf.833.833.837.837.837 Heavy Manuf 1.3891.3891.3971.3971.397 Monoline.602.727.727.731.731 Multiline1.1181.0901.0901.0901.090 circled factors = newly calculated; continue until factors stop changing

33. Bailey’s Can be used for multiplicative or additive rating relativities Can be used for multiplicative or additive rating relativities Can be used for many dimensions (convergence may be difficult) Can be used for many dimensions (convergence may be difficult) Easily coded in spreadsheets Easily coded in spreadsheets

34. Least Squares Method Minimize weighted squared error between the indicated and the observed relativities Minimize weighted squared error between the indicated and the observed relativities i.e., Min xy ∑ ij w ij (r ij – x i y j ) 2 i.e., Min xy ∑ ij w ij (r ij – x i y j ) 2where X i and Y j = relativities for rating variables i and j X i and Y j = relativities for rating variables i and j w ij = weights w ij = weights r ij = observed relativity r ij = observed relativity

35. Least Squares Method Formula: X i = ∑ j w ij r ij Y j X i = ∑ j w ij r ij Y j where X i and Y j = relativities for rating variables i and j X i and Y j = relativities for rating variables i and j w ij = weights w ij = weights r ij = observed relativity r ij = observed relativity ∑ j w ij ( Y j ) 2

36. Generalized Linear Models Generalized Linear Models (GLM) provide a generalized framework for fitting multivariate linear models Generalized Linear Models (GLM) provide a generalized framework for fitting multivariate linear models User-friendly software allows for ease of changing assumptions, adding variables, testing results, visual depiction of actual and expected results User-friendly software allows for ease of changing assumptions, adding variables, testing results, visual depiction of actual and expected results Methodology based on Maximum Likelihood Methodology based on Maximum Likelihood

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

38. Suggested Readings ASB Standards of Practice No. 9 and 12 ASB Standards of Practice No. 9 and 12 Foundations of Casualty Actuarial Science, Chapters 3 (Ratemaking) & 6 (Risk Classification) Foundations of Casualty Actuarial Science, Chapters 3 (Ratemaking) & 6 (Risk Classification) Insurance Rates with Minimum Bias, Bailey (1963) Insurance Rates with Minimum Bias, Bailey (1963) The Minimum Bias Procedure – A Practitioners Guide, Feldblum et al (2002) The Minimum Bias Procedure – A Practitioners Guide, Feldblum et al (2002)

39. Suggested Readings Something Old, Something New in Classification Ratemaking with a Novel Use of GLMs for Credit Insurance, Holler, et al (1999) Something Old, Something New in Classification Ratemaking with a Novel Use of GLMs for Credit Insurance, Holler, et al (1999) A Practitioners Guide to Generalized Linear Models, Anderson, et al A Practitioners Guide to Generalized Linear Models, Anderson, et al A Systematic Relationship Between Minimum Bias and Generalized Linear Models, Mildenhall (1999) A Systematic Relationship Between Minimum Bias and Generalized Linear Models, Mildenhall (1999)

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

42. Deductible Credits (cont.) Deductibles relativities are based on Loss Elimination Ratios (LER’s) Deductibles relativities are based on Loss Elimination Ratios (LER’s) The LER gives the percentage of losses removed by the deductible The LER gives the percentage of losses removed by the deductible Losses lower than deductible Losses lower than deductible Amount of deductible for losses over deductible Amount of deductible for losses over deductible LER = ( Losses D) LER = ( Losses D) Total Losses Total Losses

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

44. Deductible Credits (cont.) Example: Loss Elimination Ratio Loss Size # of Claims Total Losses Average Loss Loss Eliminated by Deductible $100$200$500 0 to 100 50030,0006030,00030,00030,000 101 to 200 35054,25015535,00054,25054,250 201 to 500 550182,62533255,000110,000182,625 501 + 335375,125112033,50067,000167,500 Total1,735642,000370153,500261,250434,375 L.E.R.0.2390.407.677

45. Deductible Credits (cont.) Example: Expenses TotalVariableFixed Commissions15.5%15.5%0.0% Other Acquisition 3.8%1.9%1.9% Administrative5.4%0.0%5.4% Unallocated Loss Expenses 6.0%0.0%6.0% Taxes, Licenses & Fees 3.4%3.4%0.0% Profit & Contingency 4.0%4.0%0.0% Other Costs 0.5%0.5%0.0% Total38.6%25.3%13.3% Use same expense allocation as overall indications.

46. Deductible Credits (cont.) Calculation of Deductible Factor 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

47. Deductible Credits (cont.) Calculations above are at one point in time Calculations above are at one point in time Need to perform calculation for deductible credits with each periodic review Need to perform calculation for deductible credits with each periodic review Compare results of most recent review to factors in place to decide if deductible factors need to change Compare results of most recent review to factors in place to decide if deductible factors need to change