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CAS Seminar on Ratemaking Introduction to Ratemaking Relativities March 13-14, 2006 Salt Lake City Marriott Salt Lake City, Utah Presented by: Brian M. Donlan, FCAS & Theresa A. Turnacioglu, FCAS

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

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

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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!

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Considerations in setting rating distinctions Operational Operational Social Social Legal Legal Actuarial Actuarial

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

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Social Considerations Privacy Privacy Causality Causality Controllability Controllability Affordability Affordability

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Legal Considerations Constitutional Constitutional Statutory Statutory Regulatory Regulatory

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

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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.

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

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Loss Ratio Relativity Method Class Premium @CRL Losses Loss Ratio Loss Ratio Relativity 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

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

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Incorporating Credibility Credibility: how much weight do you assign to a given body of data? Credibility: how much 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 is LR= (Z)LR class i + (1-Z) LR state Credibility weighted Loss Ratio is LR= (Z)LR class i + (1-Z) LR state

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

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

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

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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 Off-balance of 11.9% must be covered in base rates.

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Expense Flattening Rating factors are applied to a base rate which often contains a provision for fixed expenses 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” 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 “Flattening” relativities accounts for fixed expense –Flattened factor = (1-.25-.13)*1.70 +.13 = 1.58 1 -.25

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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”

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Deductible Credits, Continued 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 –Amount of deductible for losses over deductible LER = ( Losses D) LER = ( Losses D) Total Losses Total Losses

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Deductible Credits, Continued 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)

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Example: Loss Elimination Ratio Loss Size # of Claims Total Losses Average Loss Losses Net of Deductible $100$200$500 0 to 100 50030,00060000 101 to 200 35054,25015519,25000 201 to 500 550182,625332127,62572,6250 501 + 335375,1251120341,625308,125207,625 Total1,735642,000370488,500380,750207,625 Loss Eliminated 153,500261,250434,375 L.E.R.0.2390.407.677

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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.

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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

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Advanced Techniques Multivariate techniques Multivariate techniques –Why use multivariate techniques –Minimum Bias techniques –Example Generalized Linear Models Generalized Linear Models

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Why Use Multivariate Techniques? One-way analyses: 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

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

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

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

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Minimum Bias Techniques 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 newly calculated Y j s to solve for each X i Use newly calculated 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

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Minimum Bias Techniques Least Squares Least Squares Bailey’s Minimum Bias Bailey’s Minimum Bias

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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

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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

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Bailey’s Minimum Bias Minimize bias along the dimensions of the class system Minimize bias along the dimensions of the class system “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

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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

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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

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A Simple Bailey’s Example- Manufacturers & Contractors Type of Policy Aggregate Loss Costs at Current Level (ALCCL) Experience Ratio (ER) Class Group 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

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Bailey’s Example Experience Ratio Relativities Class Group Statewide Type of Policy Light Manuf Light Manuf Medium Manuf Heavy Manuf Monoline.683.497.466.602 Multiline.435.9321.6151.118

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

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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

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

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

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

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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 Statistical models which start with assumptions regarding the distribution of the data 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

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Generalized Linear Models Can be done in SAS or other statistical software packages Can be done in SAS or other statistical software packages Can run many variables Can run many variables Many Minimum bias models, are specific cases of GLM 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

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

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

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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) The Minimum Bias Procedure – A Practitioners Guide, Feldblum et al (2002) The Minimum Bias Procedure – A Practitioners Guide, Feldblum et al (2002) A Practitioners Guide to Generalized Linear Models, Anderson, et al A Practitioners Guide to Generalized Linear Models, Anderson, et al

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