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

CAS Seminar on RatemakingCAS Seminar on Ratemaking

Introduction to Ratemaking Relativities

March 17-18, 2008

Royal Sonesta Hotel

Boston, Mass.

Presented by:

Michael J. Miller, FCAS

IntroductionIntroduction to Ratemaking to Ratemaking RelativitiesRelativities

What is the purpose of rate What is the purpose of rate relativities? relativities?

Considerations in determining Considerations in determining rating distinctionsrating distinctions

Basic methods and examplesBasic methods and examples Advanced methodsAdvanced methods

The Purpose of Rate RelativitiesThe Purpose of Rate Relativities

Example – Personal Auto:Example – Personal Auto:Overall Indicated Change for State = +10%Overall Indicated Change for State = +10%

Should everyone’s rate be increased by 10%?Should everyone’s rate be increased by 10%?

Same for youthful drivers vs. adults?Same for youthful drivers vs. adults?

Same for urban vs. rural?Same for urban vs. rural?

Same for all policy limits? Deductibles?Same for all policy limits? Deductibles?

The Purpose of Rate RelativitiesThe Purpose of Rate RelativitiesExample:Example:

Base Rate = $100 (Adult, Terr 1, No Deductible)Base Rate = $100 (Adult, Terr 1, No Deductible)

InsuredInsured AgeAge TerritoryTerritory DeductibleDeductible PremiumPremium

Male Age Male Age 4040Terr 1Terr 1$0 Ded$0 Ded

1.001.00 1.001.00 1.001.00 $100$100

Male 18Male 18Terr 2Terr 2$500 Ded$500 Ded

3.503.50 1.501.50 0.600.60 $315$315

Female 18Female 18Terr 2Terr 2$100 Ded$100 Ded

2.002.00 1.501.50 0.850.85 $255$255

Considerations in Selecting Considerations in Selecting Rate RelativitiesRate Relativities

Actuarial Actuarial (Statistical)(Statistical)

OperationalOperational SocialSocial LegalLegal

Actuarial ConsiderationsActuarial Considerations

AccuracyAccuracy Rating variable closely related to cost Rating variable closely related to cost

differencesdifferences Provides “fairest” price (fair discrimination)Provides “fairest” price (fair discrimination) Reduces Reduces adverse selectionadverse selection (discussed later) (discussed later)

HomogeneityHomogeneity Members of a class have similar expected costMembers of a class have similar expected cost Variability within class always exists – grouping Variability within class always exists – grouping

is necessary since individual lacks credibilityis necessary since individual lacks credibility

Actuarial Considerations (cont.)Actuarial Considerations (cont.)

CredibilityCredibility Class groups should be large enough to measure Class groups should be large enough to measure

costs with sufficient accuracycosts with sufficient accuracy There is a trade-off between the need to There is a trade-off between the need to

estimate costs accurately for an individual and estimate costs accurately for an individual and the need for enough data to do it.the need for enough data to do it.

ReliabilityReliability Estimated cost differences between groups Estimated cost differences between groups

should be relatively stable over timeshould be relatively stable over time This does not mean they will be the same over This does not mean they will be the same over

timetime

Adverse SelectionAdverse SelectionAdverse 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 = $100Group B expected costs = $200Your company charges $150 for bothCompetitor charges $100 for A, and $200 to B

Adverse Selection (cont.)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

Operational ConsiderationsOperational Considerations

Objective - Objective - Age & Marital Status vs “Maturity”Age & Marital Status vs “Maturity”

Administrative expense – Administrative expense – Actual mileage Actual mileage driven may be better predictor of accident driven may be better predictor of accident potential than where an insured lives, but one is potential than where an insured lives, but one is much cheaper to obtainmuch cheaper to obtain

Verifiability – Verifiability – The amount of sleep a person The amount of sleep a person has gotten in the previous 24 hours may be a has gotten in the previous 24 hours may be a significant predictor of auto accident potential. significant predictor of auto accident potential. Aside from the expense of trying to get this Aside from the expense of trying to get this information, how could it be verified?information, how could it be verified?

Social ConsiderationsSocial Considerations

Privacy – Privacy – certain personal info is off limitscertain personal info is off limits Causality – Causality – as opposed to correlationas opposed to correlation Controllability – Controllability – something the insured something the insured

can impact (e.g. install sprinklers in can impact (e.g. install sprinklers in commercial property, non-smoker in health commercial property, non-smoker in health insurance)insurance)

Affordability – Affordability – balance with availability balance with availability (e.g. hospitals closing ER and OB due to high (e.g. hospitals closing ER and OB due to high cost of med mal insurance for these classes)cost of med mal insurance for these classes)

Legal ConsiderationsLegal Considerations

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

Basic Methods for Determining Basic Methods for Determining Rate RelativitiesRate Relativities

Loss ratio relativity methodLoss 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.

Data and Data AdjustmentsData and Data Adjustments

Policy Year or Accident Year dataPolicy Year or Accident Year data Premium Adjustments Premium Adjustments (LR method)(LR method)

Current Rate LevelCurrent Rate Level Premium Trend/Coverage Drift (not typical)Premium Trend/Coverage Drift (not typical)

Loss AdjustmentsLoss 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”)

Loss Ratio Relativity MethodLoss Ratio Relativity Method

ClassClass Premium Premium @CRL@CRL

Trended & Trended & Developed Developed

LossesLosses

Loss Loss RatioRatio

Loss Loss Ratio Ratio

AdjustmAdjustmtt

Current Current RelativitRelativit

yy

New New RelativitRelativit

yy

11$1,168,12$1,168,12

55$759,281$759,281 0.60.6

551.001.00 1.001.00 1.001.00

22$2,831,50$2,831,50

00$1,472,71$1,472,71

990.50.522

0.800.80 2.002.00 1.601.60

Pure Premium Relativity MethodPure Premium Relativity Method

ClassClass ExposuresExposures Trended & Trended & Developed Developed

LossesLosses

Pure Pure PremiumPremium

Pure Pure Premium Premium RelativitRelativit

yy

11 6,1956,195 $759,281$759,281 $123$123 1.001.00

22 7,7707,770 $1,472,719$1,472,719 $190$190 1.551.55

Incorporating CredibilityIncorporating Credibility

Credibility: how much predictive Credibility: how much predictive weight do you assign to a given body weight do you assign to a given body of data?of data?

Credibility is usually designated by Z Credibility is usually designated by Z Credibility Weighted Loss Ratio: Credibility Weighted Loss Ratio:

LR= (Z) * LRLR= (Z) * LRclassclass + (1-Z) * LR + (1-Z) * LRcomplement complement

Methods to Estimate Methods to Estimate CredibilityCredibility

JudgmentalJudgmental BayesianBayesian

Z = E/(E+K)Z = E/(E+K) E = exposuresE = exposures K = expected variance within classes / K = expected variance within classes /

variance between classesvariance between classes Classical / Limited FluctuationClassical / Limited Fluctuation

Z = (n/Z = (n/kk)).5 .5

n = observed number of claimsn = observed number of claims kk = full credibility standard = full credibility standard

Loss Ratio Method, ContinuedLoss Ratio Method, Continued

ClassClass Loss Loss RatioRatio

CredibilitCredibilityy

CredibilitCredibility y

Weighted Weighted Loss Loss RatioRatio

Loss Loss Ratio Ratio

AdjustmAdjustmtt

Current Current RelativityRelativity

New New RelativitRelativit

yy

11 0.650.65 0.500.50 0.610.61 1.001.00 1.001.00 1.001.00

22 0.520.52 0.900.90 0.520.52 0.850.85 2.002.00 1.701.70

TotalTotal 0.560.56

Off-Balance AdjustmentOff-Balance AdjustmentClassClass Premium Premium

@CRL@CRLCurrent Current

RelativityRelativityPremium @ Premium @ Base Class Base Class

RatesRates

Proposed Proposed RelativityRelativity

Proposed Proposed PremiumPremium

11 $1,168,125$1,168,125 1.001.00 $1,168,12$1,168,1255

1.001.00 $1,168,12$1,168,1255

22 $2,831,500$2,831,500 2.002.00 $1,415,75$1,415,7500

1.701.70 $2,406,77$2,406,7755

TotaTotall

$3,999,625$3,999,625 $3,574,90$3,574,9000

Impact on Current Premium (“Off-Balance”)Impact on Current Premium (“Off-Balance”) -11.9%-11.9%

Off-balance of 11.9% must be covered in base rates. (How?)

Multivariate TechniquesMultivariate Techniques

Univariate (One-Way) Analyses:Univariate (One-Way) Analyses: Based on assumption that effects of Based on assumption that effects of

single rating variables are independent single rating variables are independent of all other rating variablesof all other rating variables

Multivariate Analyses:Multivariate Analyses: Give consideration to the correlation or Give consideration to the correlation or

interaction between rating variablesinteraction between rating variables

Bailey’s MethodBailey’s Method Least Squares MethodLeast Squares Method Generalized Linear Model (GLM) Generalized Linear Model (GLM)

MethodMethod

Multivariate Techniques (cont.)Multivariate Techniques (cont.)

Example: Bailey’s MethodExample: Bailey’s Method

2 rating variables with relativities X2 rating variables with relativities X ii and Yand Yjj

Select initial value for each XSelect initial value for each X ii

Use model to solve for each YUse model to solve for each Yjj

Use new YUse new Yjjs to solve for each Xs to solve for each Xii

Process continues until solutions at Process continues until solutions at each interval converge each interval converge

Bailey’s Minimum BiasBailey’s Minimum Bias

““Balance Principle” :Balance Principle” :∑ ∑ observed relativity = ∑ indicated relativity observed relativity = ∑ indicated relativity

i.e., ∑i.e., ∑jj w wijijrrijij = ∑ = ∑j j wwijijxxiiyyjj

where where XXii and Y and Yjj = relativities for rating variables i and j = relativities for rating variables i and j

wwijij = weights = weights

rrijij = observed relativity = observed relativity


Formula:Formula:

XXii = = ∑∑jj w wij ij rrijij

wherewhere XXii and Y and Yj j = relativities for rating = relativities for rating

variables i and jvariables i and j



∑ ∑jj w wij ij YYjj


Less sensitive to the experience of Less sensitive to the experience of individual cells than Least Squares individual cells than Least Squares MethodMethod

Widely used; e.g.., ISO GL loss cost Widely used; e.g.., ISO GL loss cost reviewsreviews

A Simple Bailey’s Example- A Simple Bailey’s Example- Manufacturers & ContractorsManufacturers & Contractors

Type of

Policy

Aggregate Loss Costs at Current

Level (wwijij)

Experience Ratio

Class Group Class Group (CGj)

Light Manu

f

Medium

Manuf

Heavy

Manuf

Light Manu

f

Medium

Manuf

Heavy

Manuf

Mono-line

2000 250 1000 1.10 .80 .75

Multiline

4000 1500 6000 .70 1.50 2.60

SW = 1.61SW = 1.61

Bailey’s ExampleBailey’s Example

Experience Ratio RelativitiesExperience Ratio Relativities

Class Group (CGClass Group (CGjj)) StatewideStatewide

Type of Type of Policy (TOPPolicy (TOPii))

Light Light ManufManuf

Medium Medium ManufManuf

Heavy Heavy ManufManuf

MonolineMonoline .683.683 .497.497 .466.466 .602.602

MultilineMultiline .435.435 .932.932 1.6151.615 1.1181.118


Initial guess for relativities of one Initial guess for relativities of one variable variable (e.g., TOP: Mono = .602; Multi = (e.g., TOP: Mono = .602; Multi = 1.118)1.118)

Use TOP relativities and Bailey’s Use TOP relativities and Bailey’s Minimum Bias formulas to determine Minimum Bias formulas to determine the Class Group (CG) relativitiesthe Class Group (CG) relativities


CGCGjj = = ∑∑ii w wij ij rrijij

∑ ∑ii w wij ij TOPTOPii

Class GroupClass Group Bailey’s OutputBailey’s Output

Light ManufLight Manuf .547.547

Medium ManufMedium Manuf .833.833

Heavy ManufHeavy Manuf 1.3891.389


What if we continued iterating?What if we continued iterating?

Step 1Step 1 Step 2Step 2 Step 3Step 3 Step 4Step 4 Step 5Step 5

Light ManufLight Manuf .547.547 .547.547 .534.534 .534.534 .533.533

Medium Manuf Medium Manuf .833.833 .833.833 .837.837 .837.837 .837.837

Heavy ManufHeavy Manuf 1.3891.389 1.3891.389 1.3971.397 1.3971.397 1.3971.397

MonolineMonoline .602.602 .727.727 .727.727 .731.731 .731.731

MultilineMultiline 1.1181.118 1.0901.090 1.0901.090 1.0901.090 1.0901.090

circled factors = newly calculated; continue until factors stop changing

Bailey’sBailey’s

Can be used for multiplicative or Can be used for multiplicative or additive rating relativitiesadditive rating relativities

Can be used for many dimensions Can be used for many dimensions (convergence may be difficult)(convergence may be difficult)

Easily coded in spreadsheetsEasily coded in spreadsheets

Least Squares MethodLeast Squares Method

Minimize weighted squared error between Minimize weighted squared error between the indicated and the observed relativitiesthe indicated and the observed relativities

i.e., Min i.e., Min xy xy ∑ ∑ijij w wijij (r (rijij – x – xiiyyjj))22

wherewhere

XXii and Y and Yjj = relativities for rating variables i and j = relativities for rating variables i and j



Least Squares MethodLeast Squares Method

Formula:Formula:

XXii = = ∑∑jj w wij ij rrij ij YYjj

wherewhere XXii and Y and Yjj = relativities for rating variables i = relativities for rating variables i

and jand j



∑∑j j w wij ij ( ( YYjj))22

Generalized Linear ModelsGeneralized Linear Models

Generalized Linear Models (GLM) provide a Generalized Linear Models (GLM) provide a generalized framework for fitting generalized framework for fitting multivariate linear modelsmultivariate linear models

User-friendly software allows for ease of User-friendly software allows for ease of changing assumptions, adding variables, changing assumptions, adding variables, testing results, visual depiction of actual testing results, visual depiction of actual and expected resultsand expected results

Methodology based on Maximum Methodology based on Maximum LikelihoodLikelihood

Generalized Linear ModelsGeneralized Linear Models

ISO Applications:ISO Applications: Businessowners, Commercial Property Businessowners, Commercial Property

(Variables include Construction, (Variables include Construction, Protection, Occupancy, Amount of Protection, Occupancy, Amount of insurance) insurance)

GL, Homeowners, Personal AutoGL, Homeowners, Personal Auto

Suggested ReadingsSuggested Readings

ASB Standards of Practice No. 9 and 12ASB Standards of Practice No. 9 and 12 Foundations of Casualty Actuarial Foundations of Casualty Actuarial

Science, Science, Chapters 3 (Ratemaking) & 6 Chapters 3 (Ratemaking) & 6 (Risk Classification)(Risk Classification)

Insurance Rates with Minimum Bias, Insurance Rates with Minimum Bias, Bailey (1963)Bailey (1963)

The Minimum Bias Procedure – A The Minimum Bias Procedure – A Practitioners Guide, Feldblum et al Practitioners Guide, Feldblum et al (2002)(2002)

Suggested ReadingsSuggested Readings

Something Old, Something New in Something Old, Something New in Classification Ratemaking with a Classification Ratemaking with a Novel Use of GLMs for Credit Novel Use of GLMs for Credit Insurance, Holler, et al (1999)Insurance, Holler, et al (1999)

A Practitioners Guide to Generalized A Practitioners Guide to Generalized Linear Models, Anderson, et alLinear Models, Anderson, et al

A Systematic Relationship Between A Systematic Relationship Between Minimum Bias and Generalized Minimum Bias and Generalized Linear Models, Mildenhall (1999)Linear Models, Mildenhall (1999)

Deductible CreditsDeductible Credits

Insurance policy pays for losses left Insurance policy pays for losses left to be paid over a fixed deductibleto be paid over a fixed deductible

Deductible credit is a function of Deductible credit is a function of the losses remainingthe losses remaining

Since expenses of selling policy and Since expenses of selling policy and non claims expenses remain same, non claims expenses remain same, need to consider these expenses need to consider these expenses which are “fixed”which are “fixed”

Deductible Credits (cont.)Deductible Credits (cont.)

Deductibles relativities are based Deductibles relativities are based on Loss Elimination Ratios (LER’s)on Loss Elimination Ratios (LER’s)

The LER gives the percentage of The LER gives the percentage of losses removed by the deductiblelosses removed by the deductible Losses lower than deductibleLosses lower than deductible Amount of deductible for losses over deductibleAmount of deductible for losses over deductible

LER = (LER = (Losses<= D)+(D * # of Clms>D)Losses<= D)+(D * # of Clms>D) Total LossesTotal Losses


F = Fixed expense ratio F = Fixed expense ratio V = Variable expense ratioV = Variable expense ratio L = Expected loss ratioL = Expected loss ratio LER = Loss Elimination RatioLER = Loss Elimination Ratio

Deductible credit = Deductible credit = L*(1-LER) + FL*(1-LER) + F (1 - V) (1 - V)

Deductible Credits (cont.)Deductible Credits (cont.)Example: Loss Elimination RatioExample: Loss Elimination Ratio

Loss SizeLoss Size # of # of ClaimsClaims

Total Total LossesLosses

Average Average LossLoss

Loss Eliminated by Loss Eliminated by DeductibleDeductible

$100$100 $200$200 $500$500

0 to 1000 to 100 500500 30,00030,000 6060 30,00030,000 30,00030,000 30,00030,000

101 to 200101 to 200 350350 54,25054,250 155155 35,00035,000 54,25054,250 54,25054,250

201 to 500201 to 500 550550 182,625182,625 332332 55,00055,000 110,00110,0000

182,62182,6255

501 +501 + 335335 375,125375,125 11201120 33,50033,500 67,00067,000 167,50167,5000

TotalTotal 1,7351,735 642,000642,000 370370 153,50153,5000

261,25261,2500

434,37434,3755

L.E.R.L.E.R. 0.2390.239 0.4070.407 .677.677

Deductible Credits (cont.)Deductible Credits (cont.)Example: ExpensesExample: Expenses

TotalTotal VariableVariable FixedFixed

CommissionsCommissions 15.5%15.5% 15.5%15.5% 0.0%0.0%

Other AcquisitionOther Acquisition 3.8%3.8% 1.9%1.9% 1.9%1.9%

AdministrativeAdministrative 5.4%5.4% 0.0%0.0% 5.4%5.4%

Unallocated Loss Unallocated Loss ExpensesExpenses 6.0%6.0% 0.0%0.0% 6.0%6.0%

Taxes, Licenses & Taxes, Licenses & FeesFees 3.4%3.4% 3.4%3.4% 0.0%0.0%

Profit & ContingencyProfit & Contingency 4.0%4.0% 4.0%4.0% 0.0%0.0%

Other CostsOther Costs 0.5%0.5% 0.5%0.5% 0.0%0.0%

TotalTotal 38.6%38.6% 25.3%25.3% 13.3%13.3%

Use same expense allocation as overall indications.

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

DeductibleDeductible CalculationCalculation FactorFactor

$100$100 (.614)*(1-.239) + .133(.614)*(1-.239) + .133 (1-.253) (1-.253) 0.8040.804

$200$200 (.614)*(1-.407) + .133(.614)*(1-.407) + .133 (1-.253) (1-.253) 0.6650.665

$500$500 (.614)*(1-.677) + .133(.614)*(1-.677) + .133 (1-.253) (1-.253) 0.4440.444


Calculations above are at one point in Calculations above are at one point in timetime

Need to perform calculation for Need to perform calculation for deductible credits with each periodic deductible credits with each periodic reviewreview

Compare results of most recent Compare results of most recent review to factors in place to decide if review to factors in place to decide if deductible factors need to changedeductible factors need to change

Documents

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