CAS Seminar on Ratemaking

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CAS Seminar on Ratemaking
Introduction to Ratemaking Relativities March
10-11, 2005 New Orleans Marriott New Orleans,
Louisiana
Presented by Brian M. Donlan, FCAS Theresa A.
Turnacioglu, FCAS
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Introduction to Ratemaking Relativities

• Why are there rate relativities?
• Considerations in determining rating distinctions
• Basic methods and examples
• Advanced methods

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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

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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 150
• Subgroup A costs 100
• Subgroup 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!

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Considerations in setting rating distinctions

• Operational
• Social
• Legal
• Actuarial

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Operational Considerations

• Objective definition - clear who is in group
• Administrative expense
• Verifiability

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Social Considerations

• Privacy
• Causality
• Controllability
• Affordability

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Legal Considerations

• Constitutional
• Statutory
• Regulatory

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Actuarial 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

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Basic Methods for Determining 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.

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Data and Data Adjustments

• Policy Year or Accident Year data
• Premium 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

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Loss Ratio Relativity Method
Class Premium _at_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
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Pure Premium Relativity Method
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
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Incorporating 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

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Properties of Credibility

• 0 Z 1
• at Z 1 data is fully credible (given full
  weight)
• ? Z / ? E gt 0
• credibility increases as experience increases
• ? (Z/E)/ ? Elt0
• percentage change in credibility should decrease
  as volume of experience increases

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Methods to Estimate Credibility

• Judgmental
• Bayesian
• Z E/(EK)
• 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

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Loss Ratio Method, Continued
Class Loss Ratio Credibility Credibility Weighted Loss Ratio Loss Ratio Relativity Current Relativity New Relativity
1 0.65 0.50 0.61 1.00 1.00 1.00
2 0.52 0.90 0.52 0.85 2.00 1.70
Total 0.56
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Off-Balance Adjustment
Class Premium _at_CRL Current Relativity Premium _at_ 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 1.70 2,406,775
Total 3,999,625 3,574,900
Off-balance of 11.9 must be covered in base
rates.
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Expense 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 (622513) 1.70 170
• Should charge (621.70 13)/(1-.25) 158
• Flattening relativities accounts for fixed
  expense
• Flattened factor (1-.25-.13)1.70 .13 1.58
  1 - .25

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Deductible Credits

• Insurance policy pays for losses left to be paid
  over a fixed deductible
• Deductible credit is a function of the losses
  remaining
• Since expenses of selling policy and non claims
  expenses remain same, need to consider these
  expenses which are fixed

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Deductible Credits, Continued

• Deductibles relativities are based on Loss
  Elimination Ratios (LERs)
• The LER gives the percentage of losses removed by
  the deductible
• Losses lower than deductible
• Amount of deductible for losses over deductible
• LER (Losseslt D)(D of ClmsgtD)
• Total Losses

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Deductible Credits, Continued

• F Fixed expense ratio
• V Variable expense ratio
• L Expected loss ratio
• LER Loss Elimination Ratio
• Deductible credit L(1-LER) F
  (1 - V)

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Example Loss Elimination Ratio
Loss Size of Claims Total Losses Average Loss Losses Net of Deductible Losses Net of Deductible Losses Net of Deductible
Loss Size of Claims Total Losses Average Loss 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
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Example Expenses
Total 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.
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Example 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
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Advanced Techniques

• Multivariate techniques
• Why use multivariate techniques
• Minimum Bias techniques
• Example
• Generalized Linear Models

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Why Use Multivariate Techniques?

• One-way analyses
• Based on assumption that effects of single rating
  variables are independent of all other rating
  variables
• Dont consider the correlation or interaction
  between rating variables

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Examples

• Correlation
• Car value model year
• Interaction
• Driving record age
• Type of construction fire protection

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Multivariate Techniques

• Removes potential double-counting of the same
  underlying effects
• Accounts for differing percentages of each rating
  variable within the other rating variables
• Arrive at a set of relativities for each rating
  variable that best represent the experience

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Minimum Bias Techniques

• Multivariate procedure to optimize the
  relativities for 2 or more rating variables
• Calculate relativities which are as close to the
  actual relativities as possible
• Close measured by some bias function
• Bias function determines a set of equations
  relating the observed data rating variables
• Use iterative technique to solve the equations
  and converge to the optimal solution

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Minimum Bias Techniques

• 2 rating variables with relativities Xi and Yj
• Select initial value for each Xi
• Use model to solve for each Yj
• Use newly calculated Yjs to solve for each Xi
• Process continues until solutions at each
  interval converge

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Minimum Bias Techniques

• Least Squares
• Baileys Minimum Bias

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Least Squares Method

• Minimize weighted squared error between the
  indicated and the observed relativities
• i.e., Min xy ?ij wij (rij xiyj)2
• where
• Xi and Yj relativities for rating
  variables i and j
• wij weights
• rij observed relativity

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Least Squares Method

• Formula
• Xi ?j wij rij Yj
• where
• Xi and Yj relativities for rating
  variables i and j
• wij weights
• rij observed relativity

?j wij ( Yj)2
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Baileys Minimum Bias

• Minimize bias along the dimensions of the class
  system
• Balance Principle
• ? observed relativity ? indicated relativity
• i.e., ?j wijrij ?j wijxiyj
• where
• Xi and Yj relativities for rating
  variables i and j
• wij weights
• rij observed relativity

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Baileys Minimum Bias

• Formula
• Xi ?j wij rij
• where
• Xi and Yj relativities for rating
  variables i and j
• wij weights
• rij observed relativity

?j wij Yj
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Baileys Minimum Bias

• Less sensitive to the experience of individual
  cells than Least Squares Method
• Widely used e.g.., ISO GL loss cost reviews

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A Simple Baileys Example- Manufacturers
Contractors
SW 1.61
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Baileys Example
Experience Ratio Relativities Experience Ratio Relativities Experience Ratio Relativities Experience Ratio Relativities Experience Ratio Relativities
Class Group Class Group Class Group Statewide
Type of Policy Light Manuf Medium Manuf Heavy Manuf
Monoline .683 .497 .466 .602
Multiline .435 .932 1.615 1.118
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Baileys Example

• Start with an initial guess for relativities for
  one variable
• e.g.., TOP Mono .602 Multi 1.118
• Use TOP relativities and Baileys Minimum Bias
  formulas to determine the Class Group
  relativities

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Baileys Example

• CGj ?i wij rij
• ?i wij TOPi

Class Group Baileys Output
Light Manuf .547
Medium Manuf .833
Heavy Manuf 1.389

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Baileys Example

• What if we continued iterating?

Step 1 Step 2 Step 3 Step 4 Step 5
Light Manuf .547 .547 .534 .534 .533
Medium Manuf .833 .833 .837 .837 .837
Heavy Manuf 1.389 1.389 1.397 1.397 1.397
Monoline .602 .727 .727 .731 .731
Multiline 1.118 1.090 1.090 1.090 1.090
Italic factors newly calculated continue until
factors stop changing
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Baileys Example

• Apply Credibility
• Balance to no overall change
• Apply to current relativities to get new
  relativities

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Baileys

• Can use multiplicative or additive
• All formulas shown were Multiplicative
• Can be used for many dimensions
• Convergence may be difficult
• Easily coded in spreadsheets

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Generalized Linear Models

• Generalized Linear Models (GLM) provide a
  generalized framework for fitting multivariate
  linear models
• Statistical models which start with assumptions
  regarding the distribution of the data
• Assumptions are explicit and testable
• Model provides statistical framework to allow
  actuary to assess results

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Generalized Linear Models

• Can be done in SAS or other statistical software
  packages
• Can run many variables
• Many Minimum bias models, are specific cases of
  GLM
• e.g., Baileys Minimum Bias can also be derived
  using the Poisson distribution and maximum
  likelihood estimation

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Generalized Linear Models

• ISO Applications
• Businessowners, Commercial Property (Variables
  include Construction, Protection, Occupancy,
  Amount of insurance)
• GL, Homeowners, Personal Auto

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Suggested Readings

• ASB Standards of Practice No. 9 and 12
• Foundations of Casualty Actuarial Science,
  Chapters 2 5
• Insurance Rates with Minimum Bias, Bailey (1963)
• A Systematic Relationship Between Minimum Bias
  and Generalized Linear Models, Mildenhall (1999)

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Suggested Readings

• Something Old, Something New in Classification
  Ratemaking with a Novel Use of GLMs for Credit
  Insurance, Holler, et al (1999)
• The Minimum Bias Procedure A Practitioners
  Guide, Feldblum et al (2002)
• A Practitioners Guide to Generalized Linear
  Models, Anderson, et al