CAS Seminar on Ratemaking Introduction to Ratemaking Relativities March 17-18, 2008 Royal Sonesta...
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Transcript of CAS Seminar on Ratemaking Introduction to Ratemaking Relativities March 17-18, 2008 Royal Sonesta...
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
Bailey’s Minimum BiasBailey’s Minimum Bias
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
wwijij = weights = weights
rrijij = observed relativity = observed relativity
∑ ∑jj w wij ij YYjj
Bailey’s Minimum BiasBailey’s Minimum Bias
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
Bailey’s ExampleBailey’s Example
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
Bailey’s ExampleBailey’s Example
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
Bailey’s ExampleBailey’s Example
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
wwijij = weights = weights
rrijij = observed relativity = observed relativity
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
wwijij = weights = weights
rrijij = observed relativity = observed relativity
∑∑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
Deductible Credits (cont.)Deductible Credits (cont.)
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
Deductible Credits (cont.)Deductible Credits (cont.)
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