Credible Risk Classification CAS Ratemaking Journal 2004 Written by Ben Turner of Farmers Insurance.
-
Upload
joella-thornton -
Category
Documents
-
view
217 -
download
0
description
Transcript of 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
Segmentation vs. Credibility
1 2 3 4Variance of Means
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
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 LossesLosses /
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
Counts Exposures LossesLosses /
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.
Calculation of Credibility
Required Calculations• V = Process Variance• A = Variance of Hypothetical Means• K = V/A• Credibility = Exposures / (Exposures + K)
Calculation of Credibility
Calculation of V, the Process Variance
}{}{ )()(i
ii
i Exposures^2LossesredLossesSqua*
1)nts(PolicyCou1
V
Calculation of Credibility
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 Credibility
Calculation of K and Credibility• K = V/A• Credibility = Exposures / (Exposures + K)
Calculation of Credibility
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
Calculation of Score
Variance Total Variance Credible Explained
Score
)[BookMean](Exposures)redLossesSqua(
lassMeanCred.Wtd.CBookMeanExposures
2
ii
i
2ii
Score
0.12%Score ,74461,541,785
75,017,324
Segmentation vs. Credibility
1 2 3 4Variance of Means
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%
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
Segmentation vs. Credibility
1 2 3 4Variance of Means
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%
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
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
Calculation of Score
Variance Total Variance Credible Explained
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
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
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
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
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
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
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