Credible Risk Classification CAS Ratemaking Journal 2004 Written by Ben Turner of Farmers Insurance.

Credible Risk Classification

CAS Ratemaking Journal 2004Written by Ben Turnerof Farmers Insurance

Skim the Cream vs. Adverse Selection

Segmentation

As book is sliced indications become erratic, unreliable, and disrupting to policyholders.

Hence actuaries opt to credibility weight

Roadmap

1 Simple Example--Overview 5 minutes2 Simple Example--Details 20 minutes3 Theoretic Underpinnings 5 minutes4 A Complex Example 5 minutes5 Alternate Configuration 3 minutes6 Conclusion 2 minutes

A Simple Example

ClassPolicy Counts Exposures Losses

Losses / Exposures

1 1,140 1,741 1,265,754 727 2 1,000 1,514 1,390,038 918 3 960 1,456 1,359,435 934 4 1,060 1,609 1,846,048 1,147

A Simple Example—Alternate Class Plan

GroupPolicy

Counts Exposures LossesLosses /

Exposures1 1,140 1,741 1,265,754 727

2-3 1,960 2,970 2,749,473 926 4 1,060 1,609 1,846,048 1,147

Potential Groupings of Four Levels

Class Plan 1 2 3 4

• 1-2-3-4 •1, 2-3-4 Group A

• 1-2, 3-4 • 1-2, 3, 4 Group B Group C

• 1, 2, 3-4 Group A Group B

• 1, 2-3, 4 Group A Group C

• 1-2-3, 4 Group B

• 1, 2, 3, 4 Group A Group B Group C Group D

Class

Group BGroup C

Group A

Group A

Group A

Group A

Group BGroup B

Segmentation vs. Credibility

1 2 3 4Variance of Means

Bühlmann Credibility

• 1-2-3-4 Minimum Maximum

•1, 2-3-4 • 1-2, 3-4 • 1, 2, 3-4 • 1-2, 3, 4 • 1, 2-3, 4 • 1-2-3, 4 • 1, 2, 3, 4 Maximum Minimum



Bühlmann Credibility Score

• 1-2-3-4 Minimum Maximum 0.00%

•1, 2-3-4 0.11%

• 1-2, 3-4 0.09%

• 1, 2, 3-4 0.10%

• 1-2, 3, 4 0.11%• 1, 2-3, 4 0.14%

• 1-2-3, 4 0.12%

• 1, 2, 3, 4 Maximum Minimum 0.12%

Simulation

Exposures Frequency E(Loss)Level P(Accident) Alpha Beta

1 1000 0.06 10 1000 6002 1000 0.08 10 1100 8803 1000 0.08 10 1100 8804 1000 0.12 10 1200 1440

Severity

Simulation Results

Did New Method Select the Underlying Simulation Assumptions?

Did Hypothesis Testing Select the Underlying Simulation Assumptions

Do the Two Methods Agree?

Yes 62 54 53No 38 46 47

Simulation Results

Count of AttemptTurner

Method 1-2-3-4 1-1,2-4 1-3,4-4 1-1,2-3,4-4 1-2,3-3,4-4 1,2,3,4 Grand Total

1-2-3-4 0

1-1,2-4 0

1-3,4-4 10 10

1-1,2-3,4-4 1 6 15 40 62

1-2,3-3,4-4 3 7 2 12

1,2,3,4 1 14 1 16

Grand Total 4 6 33 54 2 1 100

Hypothesis Method (Salam)

Roadmap


A Simple Example

ClassPolicy


Exposures"Losses

Squared"1 1,140 1,741 1,265,754 727 12,882,705,642

2 1,000 1,514 1,390,038 918 16,157,016,292

3 960 1,456 1,359,435 934 15,132,695,151

4 1,060 1,609 1,846,048 1,147 22,805,214,328

“Losses Squared”

• For EACH POLICY the losses are squared and then divided by the exposures of that policy.

• The results can then be summed up and the underlying detail does not need to be maintained.

• This allows the computation of variance without having to keep policy level detail.

A Simple Example

ClassPolicy


Exposures"Losses

Squared"1 1,140 1,741 1,265,754 727 12,882,705,642

2 1,000 1,514 1,390,038 918 16,157,016,292

3 960 1,456 1,359,435 934 15,132,695,151

4 1,060 1,609 1,846,048 1,147 22,805,214,328

Calculation of Credibility

Buhlmann-Empirical-Bayes • It assumes no underlying distribution.• It is relatively uncontroversial.• It supplies its own complement of credibility.• It does not require arbitrary selection of

parameters.See Loss Models, Klugman, et. al.


Required Calculations• V = Process Variance• A = Variance of Hypothetical Means• K = V/A• Credibility = Exposures / (Exposures + K)


Calculation of V, the Process Variance

}{}{ )()(i

ii

i Exposures^2LossesredLossesSqua*

1)nts(PolicyCou1

V


Calculation of A, the Variance of the Hypothetical Means

2^* )(i

i

i

iii Exposures

LossesExposuresLossesExposuresW

i

ii

i

Exposures^2Exposures)Exposures(

1) - Classes of(Number *V)W(

A


Calculation of K and Credibility• K = V/A• Credibility = Exposures / (Exposures + K)


ClassPolicy Counts Exposures Losses

Losses / Exposures

"Losses Squared" Credibility

1 1,140 1,741 1,265,754 727 12,882,705,642 0.7212 1,000 1,514 1,390,038 918 16,157,016,292 0.6923 960 1,456 1,359,435 934 15,132,695,151 0.6834 1,060 1,609 1,846,048 1,147 22,805,214,328 0.705

Credibility-Weighted Class Mean

Class ExposuresBook Mean

Losses / Exposures Credibility

Credibility-Weighted

Class Mean1 1,741 927 727 0.721 783

2 1,514 927 918 0.692 921

3 1,456 927 934 0.683 932

4 1,609 927 1,147 0.705 1,082

Calculation of Score

Variance Total Variance Credible Explained

Score

)[BookMean](Exposures)redLossesSqua(

lassMeanCred.Wtd.CBookMeanExposures

2

ii

i

2ii

Score

Score: Calculation of Numerator

Class ExposuresBook Mean

Credibility-Weighted

Class Mean Difference SquaredMultiplied by

Exposures1 1,741 927 783 -144 20,855 36,307,970

2 1,514 927 921 -6 41 62,568

3 1,456 927 932 4 18 26,647

4 1,609 927 1,082 155 24,003 38,620,138

Sum 75,017,324

i

2ii lassMeanCred.Wtd.CBookMeanExposuresNumerator

Score: Calculation of Denominator

)[BookMean](Exposures)redLossesSqua( 2

ii rDenominato

Class"Losses

Squared"Book Mean Exposures

Book Mean Squared

Exposure * Book Mean Squared

1 12,882,705,642 927 1,741 859,329 1,496,091,789

2 16,157,016,292 927 1,514 859,329 1,301,024,106

3 15,132,695,151 927 1,456 859,329 1,251,183,024

4 22,805,214,328 927 1,609 859,329 1,382,660,361 Sum 66,977,631,413 5,430,959,280

Denominator = 66,977,631,413-5,430,959,280 = 61,546,672,133



Score

)[BookMean](Exposures)redLossesSqua(

lassMeanCred.Wtd.CBookMeanExposures

2

ii

i

2ii

Score

0.12%Score ,74461,541,785

75,017,324




• 1-2-3-4 Minimum Maximum 0.00%•1, 2-3-4 0.11%• 1-2, 3-4 0.09%• 1, 2, 3-4 0.10%• 1-2, 3, 4 0.11%• 1, 2-3, 4 0.14%• 1-2-3, 4 0.12%• 1, 2, 3, 4 Maximum Minimum 0.12%

A Simple Example—Alternate Class Plan

GroupPolicy Counts Exposures Losses

Losses / Exposures

"Losses Squared"

1 1,140 1,741 1,265,754 727 12,882,705,642

2-3 1,960 2,970 2,749,473 926 31,289,711,443

4 1,060 1,609 1,846,048 1,147 22,805,214,328




• 1-2-3-4 Minimum Maximum 0.00%•1, 2-3-4 0.11%• 1-2, 3-4 0.09%• 1, 2, 3-4 0.10%• 1-2, 3, 4 0.11%• 1, 2-3, 4 0.14%• 1-2-3, 4 0.12%• 1, 2, 3, 4 Maximum Minimum 0.12%

Roadmap


Score’s Factors

An increase in any of the following, will raise Score, ceterus paribus:

• The difference between the class means • The credibility of each class • The number of classes



Score

Constant

^2)ClassMean -BookMean ( *^2 )ty(CredibiliExposuresi

iii Score

Factors: 1) Difference between means, 2) Credibility, 3) Number of Classes

Segmentation vs. CredibilityCredibility-weighted Class Means

750

850

950

1050

1150

1-2,3-4 .092%

1,2,3-4 .104%

1,2-4 .107%

1-2,3,4 .110%

1-3,4 .118%

1,2,3,4 .122%

1,2-3,4 .144%

1-4 .000%

PlanScore

Score’s Theory

Score is theoretically correct because it: • Will tend to occur inadvertently via the free

markets• Is designed explicitly for this actuarial issue• Uses the correct standard of proof

Roadmap


Complex Hypothetical Example

Company introduced specialty line and tracked: • Location• Radius of operation• Whether the business is owner-operatedIt now seeks to create a class plan, and is

willing to have the plan be nonlinear.

A Sample from the Database

Policy ID Location RadiusOwner

Operated Counts Exposures LossesLosses

Squared77854A5 Rural Over 10 miles Yes 1 1 0 0

77943A5 SuburbanLess than 10

miles Yes 1 2 3,000 4,500,00078949A6 Urban Over 10 miles No 1 1 0 0

78951A6 UrbanLess than 10

miles Yes 1 1 4,000 16,000,000*** *** *** *** *** *** *** ***

Summarized DataIndex Location Radius

Owner Operated Counts Exposures Losses Losses Squared

Loss / Exposure

1Suburban Less than 10

milesYes 1,535 1,919 733,418 8,147,186,304 382

2 Rural Over 10 Miles Yes 810 1,013 427,466 4,777,629,538 422

3Rural Less than 10

milesYes 1,014 1,268 593,773 6,724,286,005 468

4 Suburban Over 10 Miles Yes 1,272 1,591 886,662 10,558,821,934 557

5City Less than 10

milesYes 1,250 1,563 1,092,799 13,345,632,591 699

6Suburban Less than 10

milesNo 1,232 1,541 1,687,092 19,948,652,768 1,095

7City Less than 10

milesNo 2,412 3,016 3,443,920 39,503,485,208 1,142

8 City Over 10 Miles Yes 207 259 295,748 3,479,777,226 1,142

9Rural Less than 10

milesNo 1,405 1,757 2,174,428 26,350,261,440 1,238

10 Suburban Over 10 Miles No 1,890 2,363 3,088,961 35,940,106,619 1,30711 Rural Over 10 Miles No 430 538 871,625 11,051,938,151 1,62012 City Over 10 Miles No 2,105 2,632 10,316,789 135,036,132,799 3,920

2,048 Potential Class PlansClass Plan Score

1-4, 5-5,6-8,9-10,11-11,12-12 8.10%1-3, 4-4,5-5,6-8,9-10,11-11,12-12 8.10%1-4, 5-5,6-8,9-9,10-10,11-11,12-12 8.10%1-3, 4-4,5-5,6-8,9-9,10-10,11-11,12-12 8.10%1-3, 4-5,6-8,9-10,11-11,12-12 8.10%*** ***The intermediate 2,038 Class Plans are omitted. *** ***1-1, 2-2,3-3,4-12 1.49%1-2, 3-12 1.02%1-1, 2-2,3-12 1.00%1-1, 2-12 0.64%

Selected Class PlanIndex

Underwriting Class Location Radius

Owner Operated

1 A Less than 10 miles Suburban Yes2 A Over 10 Miles Rural Yes3 A Less than 10 miles Rural Yes4 A Over 10 Miles Suburban Yes5 B Less than 10 miles City Yes6 C Less than 10 miles Suburban No7 C Less than 10 miles City No8 C Over 10 Miles City Yes9 D Less than 10 miles Rural No10 D Over 10 Miles Suburban No11 E Over 10 Miles Rural No12 F Over 10 Miles City No

Underwriting Guidelines

Underwriting Group Guidelines

A Owner Operated-Suburban or RuralB Owner Operated-City-Less than 10 MilesC Owner Operated-City-More than 10 Miles; or, Not

Owner Operated-City or Suburban-Less than 10 MilesD Not Owner Operated-Rural-Less than 10 Miles or

Suburban-Over 10 MilesE Not Owner Operated-Rural-Over 10 Miles

Roadmap


Complex Example—Linear Class Plan

Owner Operated Yes

Losses / Exposures LocationRadius City Suburban RuralLess than 10 miles 699 382 468 Over 10 Miles 1,142 557 422

Owner Operated No

Losses / Exposures LocationRadius City Suburban RuralLess than 10 miles 1,142 1,095 1,238 Over 10 Miles 3,920 1,307 1,620

Complex Example—Linear Class Plan—Alternate A

Owner Operated Yes

Losses / Exposures Location2Radius City Suburban-RuralLess than 10 miles 699 425 Over 10 Miles 1,142 490

Owner Operated No

Losses / Exposures Location2Radius City Suburban-RuralLess than 10 miles 1,142 1,167 Over 10 Miles 3,920 1,464

Complex Example—Linear Class Plan—Alternate B

Owner Operated Yes

Losses / Exposures Location2Radius City-Suburban RuralLess than 10 miles 541 468 Over 10 Miles 850 422

Owner Operated No

Losses / Exposures Location2Radius City-Suburban RuralLess than 10 miles 1,119 1,238 Over 10 Miles 2,614 1,620

Roadmap


ConclusionWe’ve seen: • Score is a theoretically correct method• Score can be done in a spreadsheet• Score can be iterated over all possible plans

via a computer program• Score can be used on just the class plans that

are of interest• Score can help you design superior class plans

Credible Risk Classification CAS Ratemaking Journal 2004 Written by Ben Turner of Farmers Insurance.

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Transcript of Credible Risk Classification CAS Ratemaking Journal 2004 Written by Ben Turner of Farmers Insurance.