CAS Seminar on RatemakingCAS Seminar on Ratemaking

Introduction to Ratemaking Relativities(INT - 3)

March 11, 2004

Wyndham Franklin Plaza Hotel

Philadelphia, Pennsylvania

Presented by:

Francis X. Gribbon, FCAS & Julie A. Jordan, FCAS

IntroductionIntroduction to Ratemaking to Ratemaking RelativitiesRelativities

Why are there rate relativities? Considerations in determining rating

distinctionsBasic methods and examplesAdvanced methods

Why are there rate relativities?Why are there rate relativities?

Individual Insureds differ in . . .– Risk Potential– Amount of Insurance Coverage Purchased

With Rate Relativities . . . – Each group pays its share of losses – We achieve equity among insureds (“fair

discrimination”)– We 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 setting Considerations in setting rating distinctionsrating distinctions

OperationalSocialLegalActuarial

Operational ConsiderationsOperational Considerations

Objective definition - clear who is in groupAdministrative expenseVerifiability

Social ConsiderationsSocial Considerations

PrivacyCausalityControllabilityAffordability

Legal ConsiderationsLegal Considerations

ConstitutionalStatutoryRegulatory

Actuarial ConsiderationsActuarial Considerations

Accuracy - the variable should measure cost differences

Homogeneity - all members of class should have same expected cost

Reliability - should have stable mean value over time

Credibility - groups should be large enough to permit measuring costs

Basic Methods for Basic Methods for Determining Rate RelativitiesDetermining 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.

Data and Data AdjustmentsData and Data Adjustments

Policy Year or Accident Year dataPremium Adjustments

– Current Rate Level– Premium Trend/Coverage Drift – generally not necessary

Loss Adjustments– Loss Development – if different by group (e.g., increased limits)– Loss Trend – if different by group– Deductible Adjustments– Catastrophe Adjustments

Loss Ratio Relativity MethodLoss Ratio Relativity Method

Class Premium @CRL

Losses Loss Ratio

Loss Ratio

Relativity

Current Relativity

New Relativity

1 $1,168,125 $759,281 0.65 1.00 1.00 1.00

2 $2,831,500 $1,472,719 0.52 0.80 2.00 1.60

Pure Premium Relativity Pure Premium Relativity MethodMethod

Class Exposures Losses Pure Premium

Pure Premium Relativity

1 6,195 $759,281 $123 1.00

2 7,770 $1,472,719 $190 1.55

Incorporating CredibilityIncorporating Credibility

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

Credibility is usually designated by Z Credibility weighted Loss Ratio is

LR= (Z)LRclass i + (1-Z) LRstate

Properties of CredibilityProperties of Credibility

0 – at Z = 1 data is fully credible (given full weight)

Z / E > 0– credibility increases as experience increases

(Z/E)/ E<0– percentage change in credibility should decrease

as volume of experience increases

Methods to Estimate CredibilityMethods to Estimate Credibility Judgmental Bayesian

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

variance between classes

Classical / Limited Fluctuation– Z = (n/k).5

– n = observed number of claims– k = full credibility standard

Loss Ratio Method, ContinuedLoss Ratio Method, Continued

Class Loss Ratio

Credibility Credibility Weighted Loss Ratio

Loss Ratio Relativity

Current Relativity

New Relativity

1 0.65 0.50 0.60 1.00 1.00 1.00

2 0.52 0.90 0.52 0.87 2.00 1.74

Total 0.56

Off-Balance AdjustmentOff-Balance Adjustment

Class Premium @CRL

Current Relativity

Premium @ Base Class Rates

Proposed Relativity

Proposed Premium

1 $1,168,125 1.00 $1,168,125 1.00 $1,168,125

2 $2,831,500 2.00 $1,415,750 1.74 $2,463,405

Total $3,999,625 $3,631,530

Off-balance of 9.2% must be covered in base rates.

Expense FlatteningExpense Flattening

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”– Example: (62+25+13) * 1.74 = $174– Should charge: (62*1.74 + 13)/(1-.25) = $161

“Flattening” relativities accounts for fixed expense – Flattened factor = (1-.25-.13)*1.74 + .13 = 1.61

1 - .25

Deductible CreditsDeductible CreditsInsurance policy pays for losses left to be

paid over a fixed deductibleDeductible credit is a function of the losses

remainingSince expenses of selling policy and non

claims expenses remain same, need to consider these expenses which are “fixed”

Deductible Credits, ContinuedDeductible Credits, Continued

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

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) + (D * # of Claims >D) Total Losses

Deductible Credits, ContinuedDeductible Credits, Continued

F = Fixed expense ratio V = Variable expense ratioL = Expected loss ratioLER = Loss Elimination Ratio

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

Example: Loss Elimination RatioExample: Loss Elimination Ratio

Loss Size # of Claims

Total Losses

Average Loss

Losses Net of Deductible

$100 $200 $500

0 to 100 500 30,000 60 0 0 0

101 to 200 350 54,250 155 19,250 0 0

201 to 500 550 182,625 332 127,625 72,625 0

501 + 335 375,125 1120 341,625 308,125 207,625

Total 1,735 642,000 370 488,500 380,750 207,625

Loss Eliminated 153,500 261,250 434,375

L.E.R. 0.239 0.407 .677

Example: ExpensesExample: ExpensesTotal Variable Fixed

Commissions 15.5% 15.5% 0.0%

Other Acquisition 3.8% 1.9% 1.9%

Administrative 5.4% 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%

Total 38.6% 25.3% 13.3%

Use same expense allocation as overall indications.

Example: Deductible CreditExample: Deductible Credit

Deductible Calculation Factor

$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

Advanced TechniquesAdvanced Techniques

Multivariate techniques– Bailey’s Minimum Bias– Generalized Linear Models

Curve fitting

Why Use Multivariate Why Use Multivariate Techniques?Techniques?

Many rating variables are correlatedDifferent variables, when viewed one at a

time, may be “double counting” the same underlying effect

Using a multivariate approach removes potential double-counting and can account for interaction effects

A Simple ExampleA Simple Example

Age Group

Exposures Pure Premium

Car Size Car Size

Large Medium Small Large Medium Small

1 100 1200 500 100 310 840

2 300 500 400 470 1460 2530

One-Way RelativitiesOne-Way RelativitiesClass Exposures Pure

PremiumRelativity

Large car 400 380 1.00

Medium car 1700 650 1.70

Small car 900 1590 4.20

Age Group 1 1800 450 1.00

Age Group 2 1200 1570 3.50

Multi-way vs. One-wayMulti-way vs. One-way

Age Group

Multi-Way Relativities One-way Relativities

Car Size Car Size

Large Medium Small Large Medium Small

1 1.00 3.10 8.40 1.00 1.70 4.20

2 4.70 14.60 25.30 3.50 6.00 14.60

When to use Multivariate?When to use Multivariate?

Can use Multivariate techniques for entire rating plan, or for particular variables that are correlated or have interaction effects

Example of correlation– Value of car and Model Year

Examples of interaction effects– Driving record and Age– Type of construction and Fire protection

Bailey’s Minimum BiasBailey’s Minimum Bias

To get toward multivariate but still have simple method to calculate premiums

Can have credibility issues with many cellsCan use either Loss Ratio or Pure Premium

methodsCan assume multiplicative and/or additive

relationships of rating variables and dependent variable

Bailey’s ExampleBailey’s Example

Start with initial guess at factors for one variable

Class Pure Premium Relativity

Age group 1 $450 1.00

Age group 2 $1570 3.50

Bailey’s Example: Step 1ABailey’s Example: Step 1A

What would the premiums be, assuming base rate = $100 and this rating plan?

Age Group

Exposures Theoretical Premium

Car Size Car Size

Large Medium Small Large Medium Small

1 100 1200 500 10000 120000 50000

2 300 500 400 105000 175000 140000

Bailey’s Example: Step 1BBailey’s Example: Step 1B

What should the factors for car size be, given the rating factors for age group?

Car size Theoretical Premium

Theoretical Loss Ratio

Loss Ratio Relativity

Large 115000 1.30 1.00

Medium 295000 3.70 2.80

Small 190000 7.50 5.70

Bailey’s Example: Step 2ABailey’s Example: Step 2A

What would the premiums be, assuming base rate = $100 and this rating plan?

Age Group

Exposures Theoretical Premium

Car Size Car Size

Large Medium Small Large Medium Small

1 100 1200 500 10000 336000 285000

2 300 500 400 30000 140000 228000

Bailey’s Example: Step 2BBailey’s Example: Step 2B

What should the factors for age group be, given the rating factors for car size?

Age group Theoretical Premium

Theoretical Loss Ratio

Loss Ratio Relativity

Age group 1 631000 1.30 1.00

Age group 2 398000 4.70 3.70

Bailey’s Example: Steps 3-6Bailey’s Example: Steps 3-6

What if we continued iterating this way?

Class Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

Large Car 1.00 1.00 1.00 1.00 1.00 1.00

Medium Car 2.80 2.80 2.90 2.90 2.90 2.90

Small Car 5.70 5.70 5.80 5.80 5.80 5.80

Age Group 1 1.00 1.00 1.00 1.00 1.00 1.00

Age Group 2 3.50 3.70 3.70 3.60 3.60 3.60

Italic factors = newly calculated; continue until factors stop changing

Bailey’s Example: ResultsBailey’s Example: Results

Age Group

Multi-Way Relativities Bailey Relativities

Car Size Car Size

Large Medium Small Large Medium Small

1 1.00 3.10 8.40 1.00 2.90 5.80

2 4.70 14.60 25.30 3.60 10.40 20.10

Bailey’s Minimum BiasBailey’s Minimum Bias

Bailey Relativities get much closer to multi-way relativities than univariate approach

Premium calculation by multiplying factors vs. table lookup for multi-way

This example assumed two multiplicative factors, but approach can be modified for more variables and/or additive rating plans

Generalized Linear ModelsGeneralized Linear Models

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

Bailey’s method is a specific case of GLMFactors can be estimated with SAS or other

statistical software packages

Curve FittingCurve Fitting

Can calculate certain type of relativities using smooth curves

Fit exposure data to a curve Determine a functional relationship of loss

data and exposure dataTaking derivative of this function and

relating the value at any given point to a base point produces relativity

Curve FittingCurve Fitting

HO Policy Size Relativities Assume the distribution of exposures by

amount of insurance is log normalAssume the cumulative loss distribution has

a functional relationship to the cumulative exposure distribution

Curve FittingCurve Fitting

Let r = amount of insurancef (r) is density of exposures at r= exposures at r / total exposuresg (r) is density of losses at r = losses at r / total lossesF(A) and G (A) are the cumulative

functions of f and g

Curve Fitting Curve Fitting

F (A) and G (A) are cumulative functions of f and g

G (A) = H[ F (A)]Then dG (A)/dF (A) = g(a)/f(a)

= (losses at A / total losses)

(exposures at A / total exposures) = pure premium at A/ total pure premium

Suggested ReadingsSuggested Readings

ASB Standard of Practice No. 9 ASB Standard of Practice No. 12 Foundations of Casualty Actuarial Science,

Chapters 2 and 5 Insurance Rates with Minimum Bias, Bailey

(1963) Something Old, Something New in Classification

Ratemaking with a Novel Use of GLMs for Credit Insurance, Holler, Sommer, and Trahair (1999)