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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

Introduction to Ratemaking Relativities

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

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?

- 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!

Considerations in setting rating distinctions

- Operational
- Social
- Legal
- Actuarial

Operational Considerations

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

Social Considerations

- Privacy
- Causality
- Controllability
- Affordability

Legal Considerations

- Constitutional
- Statutory
- Regulatory

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

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.

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

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

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

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

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

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

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

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.

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

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

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

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)

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

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.

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

Advanced Techniques

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

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

Examples

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

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

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

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

Minimum Bias Techniques

- Least Squares
- Baileys Minimum Bias

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

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

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

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

Baileys Minimum Bias

- Less sensitive to the experience of individual

cells than Least Squares Method - Widely used e.g.., ISO GL loss cost reviews

A Simple Baileys Example- Manufacturers

Contractors

SW 1.61

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

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

Baileys Example

- CGj ?i wij rij
- ?i wij TOPi

Class Group Baileys Output

Light Manuf .547

Medium Manuf .833

Heavy Manuf 1.389

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

Baileys Example

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

relativities

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

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

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

Generalized Linear Models

- ISO Applications
- Businessowners, Commercial Property (Variables

include Construction, Protection, Occupancy,

Amount of insurance) - GL, Homeowners, Personal Auto

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)

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