CAS Seminar on CAS Seminar on RatemakingRatemaking
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
IntroductionIntroduction to to Ratemaking Ratemaking RelativitiesRelativities
Why are there rate relativities? Why are there rate relativities? Considerations in determining Considerations in determining
rating distinctionsrating distinctions Basic methods and examplesBasic methods and examples Advanced methodsAdvanced methods
Why are there rate Why are there rate relativities?relativities?
Individual Insureds differ in . . .Individual Insureds differ in . . .– Risk PotentialRisk Potential– Amount of Insurance Coverage Amount of Insurance Coverage
PurchasedPurchased
With Rate Relativities . . . With Rate Relativities . . . – Each group pays its share of losses Each group pays its share of losses – We achieve equity among insureds We achieve equity among insureds
(“fair discrimination”)(“fair discrimination”)– We avoid anti-selectionWe avoid anti-selection
What is Anti-selection?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 $150Subgroup A costs $100Subgroup 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!
Considerations in Considerations in setting rating setting rating distinctionsdistinctions
OperationalOperational SocialSocial LegalLegal ActuarialActuarial
Operational Operational ConsiderationsConsiderations
Objective definition - clear who is Objective definition - clear who is in groupin group
Administrative expenseAdministrative expense VerifiabilityVerifiability
Social ConsiderationsSocial Considerations
PrivacyPrivacy CausalityCausality ControllabilityControllability AffordabilityAffordability
Legal ConsiderationsLegal Considerations
ConstitutionalConstitutional StatutoryStatutory RegulatoryRegulatory
Actuarial Actuarial ConsiderationsConsiderations Accuracy - the variable should Accuracy - the variable should
measure cost differencesmeasure cost differences Homogeneity - all members of class Homogeneity - all members of class
should have same expected costshould have same expected cost Reliability - should have stable mean Reliability - should have stable mean
value over timevalue over time Credibility - groups should be large Credibility - groups should be large
enough to permit measuring costsenough to permit measuring costs
Basic Methods for Basic Methods for Determining Rate Determining Rate RelativitiesRelativities
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.
Data and Data Data and Data AdjustmentsAdjustments Policy Year or Accident Year dataPolicy Year or Accident Year data Premium AdjustmentsPremium Adjustments
– Current Rate LevelCurrent Rate Level– Premium Trend/Coverage Drift – generally not Premium Trend/Coverage Drift – generally not
necessarynecessary
Loss AdjustmentsLoss Adjustments– Loss Development – if different by group (e.g., Loss Development – if different by group (e.g.,
increased limits)increased limits)– Loss Trend – if different by groupLoss Trend – if different by group– Deductible AdjustmentsDeductible Adjustments– Catastrophe AdjustmentsCatastrophe Adjustments
Loss Ratio Relativity Loss Ratio Relativity MethodMethod
ClasClasss
Premium Premium @CRL@CRL
LossesLosses Loss Loss RatiRati
oo
Loss Loss Ratio Ratio
RelativitRelativityy
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$1,472,7$1,472,7
19190.50.522
0.800.80 2.002.00 1.601.60
Pure Premium Pure Premium Relativity MethodRelativity Method
ClasClasss
ExposuresExposures LossesLosses Pure Pure PremiuPremiu
mm
Pure Pure PremiuPremiu
m m RelativitRelativit
yy
11 6,1956,195 $759,281$759,281 $123$123 1.001.00
22 7,7707,770 $1,472,7$1,472,71919 $190$190 1.551.55
Incorporating Incorporating CredibilityCredibility Credibility: how much weight do Credibility: how much weight do
you assign to a given body of you assign to a given body of data?data?
Credibility is usually designated Credibility is usually designated by Z by Z
Credibility weighted Loss Ratio is Credibility weighted Loss Ratio is LR= (Z)LRLR= (Z)LRclass iclass i + (1-Z) LR + (1-Z) LRstate state
Properties of CredibilityProperties of Credibility
0 0 – at Z = 1 data is fully credible (given full at Z = 1 data is fully credible (given full
weight)weight) Z / Z / E > 0 E > 0
– credibility increases as experience increasescredibility increases as experience increases (Z/E)/ (Z/E)/ E<0 E<0
– percentage change in credibility should percentage change in credibility should decrease as volume of experience increasesdecrease as volume of experience increases
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 = (Z = (nn//kk)).5 .5
– n = observed number of claimsn = observed number of claims– kk = full credibility standard = full credibility standard
Loss Ratio Method, Loss Ratio Method, ContinuedContinued
ClassClass Loss Loss RatioRatio
CredibilitCredibilityy
CredibilitCredibility y
WeighteWeighted Loss d Loss RatioRatio
Loss Loss Ratio Ratio
RelativitRelativityy
Current Current RelativitRelativit
yy
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
TotaTotall
0.560.56
Off-Balance Off-Balance AdjustmentAdjustment
ClassClass Premium Premium @CRL@CRL
Current Current RelativityRelativity
Premium @ Premium @ Base Class Base Class
RatesRates
Proposed Proposed RelativityRelativity
Proposed Proposed PremiumPremium
11$1,168,12$1,168,12
551.001.00 $1,168,1$1,168,1
2525 1.001.00 $1,168,12$1,168,1255
22$2,831,50$2,831,50
002.002.00 $1,415,7$1,415,7
5050 1.701.70 $2,406,77$2,406,7755
TotaTotall
$3,999,62$3,999,6255
$3,574,90$3,574,9000
Off-balance of 11.9% must be covered in base rates.
Expense FlatteningExpense Flattening
Rating factors are applied to a base rate Rating factors are applied to a base rate which often contains a provision for fixed which often contains a provision for fixed expensesexpenses– Example: $62 loss cost + $25 VE + $13 FE = $100Example: $62 loss cost + $25 VE + $13 FE = $100
Multiplying both means fixed expense no Multiplying both means fixed expense no longer “fixed”longer “fixed”– Example: (62+25+13) * 1.70 = $170Example: (62+25+13) * 1.70 = $170– Should charge: (62*1.70 + 13)/(1-.25) = $158Should charge: (62*1.70 + 13)/(1-.25) = $158
““Flattening” relativities accounts for fixed Flattening” relativities accounts for fixed expense expense – Flattened factor = Flattened factor = (1-.25-.13)*1.70 + .13 (1-.25-.13)*1.70 + .13 = 1.58= 1.58
1 - .25 1 - .25
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, Deductible Credits, ContinuedContinued
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, Deductible Credits, ContinuedContinued
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)
Example: Loss Example: Loss Elimination RatioElimination Ratio
Loss SizeLoss Size # of # of ClaimsClaims
Total Total LossesLosses
Average Average LossLoss
Losses Net of DeductibleLosses Net of Deductible
$100$100 $200$200 $500$500
0 to 1000 to 100 500500 30,00030,000 6060 00 00 00
101 to 200101 to 200 350350 54,25054,250 155155 19,25019,250 00 00
201 to 500201 to 500 550550 182,625182,625 332332 127,62127,6255
72,62572,625 00
501 +501 + 335335 375,125375,125 11201120 341,62341,6255
308,12308,1255
207,62207,6255
TotalTotal 1,7351,735 642,000642,000 370370 488,50488,5000
380,75380,7500
207,62207,6255
Loss Loss EliminatedEliminated
153,50153,5000
261,25261,2500
434,37434,3755
L.E.R.L.E.R. 0.2390.239 0.4070.407 .677.677
Example: ExpensesExample: ExpensesTotalTotal 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.
Example: Deductible Example: Deductible CreditCredit
DeductibleDeductible CalculationCalculation FactorFactor
$100$100(.614)*(1-.239) (.614)*(1-.239)
+ .133+ .133 (1-.253) (1-.253)
0.8040.804
$200$200(.614)*(1-.407) (.614)*(1-.407)
+ .133+ .133 (1-.253) (1-.253)
0.6650.665
$500$500(.614)*(1-.677) (.614)*(1-.677)
+ .133+ .133 (1-.253) (1-.253)
0.4440.444
Advanced TechniquesAdvanced Techniques
Multivariate techniquesMultivariate techniques– Why use multivariate Why use multivariate
techniquestechniques– Minimum Bias techniquesMinimum Bias techniques– ExampleExample
Generalized Linear Generalized Linear Models Models
Why Use Multivariate Why Use Multivariate Techniques?Techniques? One-way analyses:One-way analyses:
– Based on assumption that effects of Based on assumption that effects of single rating variables are single rating variables are independent of all other rating independent of all other rating variablesvariables
– Don’t consider the correlation or Don’t consider the correlation or interaction between rating variablesinteraction between rating variables
ExamplesExamples
Correlation:Correlation:– Car value & model yearCar value & model year
InteractionInteraction– Driving record & ageDriving record & age– Type of construction & fire Type of construction & fire
protectionprotection
Multivariate Multivariate TechniquesTechniques Removes potential double-counting Removes potential double-counting
of the same underlying effects of the same underlying effects Accounts for differing percentages Accounts for differing percentages
of each rating variable within the of each rating variable within the other rating variablesother rating variables
Arrive at a set of relativities for Arrive at a set of relativities for each rating variable that best each rating variable that best represent the experiencerepresent the experience
Minimum Bias Minimum Bias TechniquesTechniques Multivariate procedure to optimize the Multivariate procedure to optimize the
relativities for 2 or more rating variablesrelativities for 2 or more rating variables Calculate relativities which are as close to Calculate relativities which are as close to
the actual relativities as possiblethe actual relativities as possible ““Close” measured by some bias functionClose” measured by some bias function Bias function determines a set of equations Bias function determines a set of equations
relating the observed data & rating relating the observed data & rating variablesvariables
Use iterative technique to solve the Use iterative technique to solve the equations and converge to the optimal equations and converge to the optimal solutionsolution
Minimum Bias Minimum Bias TechniquesTechniques 2 rating variables with relativities 2 rating variables with relativities
XXii and Y and 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 newly calculated YUse newly calculated Yjjs to solve s to solve for each Xfor each Xii
Process continues until solutions at Process continues until solutions at each interval converge each interval converge
Minimum Bias Minimum Bias TechniquesTechniques Least SquaresLeast Squares
Bailey’s Minimum BiasBailey’s Minimum Bias
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 = relativities for rating variables i and jj
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 and j = relativities for rating variables i and j
wwijij = weights = weights
rrijij = observed relativity = observed relativity
∑∑j j w wij ij ( ( YYjj))22
Bailey’s Minimum BiasBailey’s Minimum Bias
Minimize bias along the dimensions of Minimize bias along the dimensions of the class systemthe class system
““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 variables i = relativities for rating variables i
and jand 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 Less sensitive to the experience of individual cells than Least of individual cells than Least Squares MethodSquares Method
Widely used; e.g.., ISO GL loss Widely used; e.g.., ISO GL loss cost reviewscost reviews
A Simple Bailey’s A Simple Bailey’s Example- Example- Manufacturers & Manufacturers & ContractorsContractors
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
SW = 1.61SW = 1.61
Bailey’s ExampleBailey’s Example
Experience Ratio RelativitiesExperience Ratio Relativities
Class GroupClass Group StatewideStatewide
Type of Type of PolicyPolicy
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
• Start with an initial guess for Start with an initial guess for relativities for one variable relativities for one variable
• e.g.., TOP: Mono = .602; Multi = e.g.., TOP: Mono = .602; Multi = 1.1181.118
• Use TOP relativities and Baileys Use TOP relativities and Baileys Minimum Bias formulas to determine Minimum Bias formulas to determine the Class Group relativities the Class Group relativities
Bailey’s ExampleBailey’s Example
CGCGjj = = ∑∑ii w wij ij rrijij
∑ ∑ii w wij ij TOPTOPii
Class GroupClass Group Bailey’s Bailey’s OutputOutput
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
Italic factors = newly calculated; continue until factors stop changing
Bailey’s ExampleBailey’s Example
Apply Credibility Apply Credibility Balance to no overall changeBalance to no overall change Apply to current relativities to get Apply to current relativities to get
new relativitiesnew relativities
Bailey’sBailey’s
Can use multiplicative or additiveCan use multiplicative or additive– All formulas shown were All formulas shown were
MultiplicativeMultiplicative Can be used for many dimensionsCan be used for many dimensions
– Convergence may be difficultConvergence may be difficult Easily coded in spreadsheetsEasily coded in spreadsheets
Generalized Linear Generalized Linear ModelsModels Generalized Linear Models (GLM) Generalized Linear Models (GLM)
provide a generalized framework for provide a generalized framework for fitting multivariate linear modelsfitting multivariate linear models
Statistical models which start with Statistical models which start with assumptions regarding the distribution assumptions regarding the distribution of the dataof the data– Assumptions are explicit and testableAssumptions are explicit and testable– Model provides statistical framework to Model provides statistical framework to
allow actuary to assess resultsallow actuary to assess results
Generalized Linear Generalized Linear ModelsModels Can be done in SAS or other Can be done in SAS or other
statistical software packagesstatistical software packages Can run many variablesCan run many variables Many Minimum bias models, are Many Minimum bias models, are
specific cases of GLMspecific cases of GLM– e.g., Baileys Minimum Bias can also be e.g., Baileys Minimum Bias can also be
derived using the Poisson distribution derived using the Poisson distribution and maximum likelihood estimation and maximum likelihood estimation
Generalized Linear Generalized Linear ModelsModels ISO Applications:ISO Applications:
– Businessowners, Commercial Businessowners, Commercial Property (Variables include Property (Variables include Construction, Protection, Occupancy, Construction, Protection, Occupancy, Amount of insurance) Amount of insurance)
– GL, Homeowners, Personal AutoGL, Homeowners, Personal Auto
Suggested ReadingsSuggested Readings
ASB Standards of Practice No. 9 ASB Standards of Practice No. 9 and 12and 12
Foundations of Casualty Actuarial Foundations of Casualty Actuarial Science, Science, Chapters 2 & 5Chapters 2 & 5
Insurance Rates with Minimum Insurance Rates with Minimum Bias, Bailey (1963)Bias, Bailey (1963)
A Systematic Relationship A Systematic Relationship Between Minimum Bias and Between Minimum Bias and Generalized Linear Models, Generalized Linear Models, Mildenhall (1999)Mildenhall (1999)
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)
The Minimum Bias Procedure – A The Minimum Bias Procedure – A Practitioners Guide, Feldblum et al Practitioners Guide, Feldblum et al (2002)(2002)
A Practitioners Guide to Generalized A Practitioners Guide to Generalized Linear Models, Anderson, et alLinear Models, Anderson, et al
