1. CAS Seminar on Ratemaking
Development of an
Overall Indication and Calculation of Ratemaking Relativities
(INT-1)
March 8, 2007
Atlanta, GA
Presented by:
Gavin Lienemann, FCAS, MAAA &
Amy Juknelis, FCAS, MAAA
Welcome to “Basic Techniques for an Overall Rate Level Indication”.
Today we will be presenting some basic ratemaking concepts and applying them to the calculation of an overall rate level indication. We will be covering a great deal of material and will be moving fairly quickly. Please stop us if you have any questions or need some further explanation.

2. Basic Ratemaking Equation and Its Considerations:
Organization of Data
Premium Adjustments
Loss Adjustments
Expense Considerations
Other Considerations
Today we are going to discuss the basic ratemaking equations and its considerations, specifically:
The organization of the data
Premium adjustments
Loss Adjustments
Expenses
We will walk through a simplified ratemaking example to allow you to see how these concepts are applied.

3. ORGANIZATION OF DATA I. CALENDAR YEAR DATA II. POLICY YEAR DATA
(standard accounting year)
II. POLICY YEAR DATA
III. ACCIDENT YEAR DATA
These are the three main methods of organizing data. I will describe each method in the next few slides.

4. ORGANIZATION OF DATA I. CALENDAR YEAR DATA
Premium and Loss transactions that occur during the year.
Loss = Payments + change in reserves during year
Matches financial statements
Data available quickly, least time lag in development
Never changes after it is calculated at the end of a year.
Premium and Loss transactions DO NOT match
Reserve changes from prior years can distort the reliability of the data for ratemaking and management purposes.
Calendar Year data

5. ORGANIZATION OF DATA II. POLICY YEAR DATA
Premium and Loss transactions on policies with effective dates (new or renewal) during the year.
Loss = Payments + Reserves
Premium and Loss transactions DO match
Transactions from policies effective in prior years do not distort the data for ratemaking
Data with the greatest time lag (not available until one term after end of the year.)
Exact ultimate losses cannot be finalized until all losses settled.

6. ORGANIZATION OF DATA III. ACCIDENT YEAR DATA
Loss transactions for accidents occurring during the year. Premium transaction during the same 12 months.
Loss = Payments + Reserves
Premium and Loss transactions generally match
Transactions from accidents occurring in prior years do not distort the data for ratemaking
Data with slight time lag
Exact ultimate losses cannot be finalized until all losses settled.

7. Basic Ratemaking Equation:
Future Premiums =
Future Losses +
Future Expenses +
Underwriting Profit and Contingency Provision
The important element to remember here is that rates are estimates of FUTURE costs.
The estimates of the future premium, losses and expenses are based upon past observations of company experience, competitors, the economy, and the world in general. Ratemaking bridges the past to the future.
In the following slides we will discuss how the bridging occurs in more detail.

8. BASIC RATEMAKING METHODS
Loss Ratio Method
develops indicated rate change (A)
A = Experience LR / Target LR
Pure Premium (PP) Method
PP = Loss / Exposure Units
develops indicated rate per unit of exposure (R)
R = [PP + FE] / [1-VER-Profit Ratio]
There are two general pricing methodologies – LR and PP. When consistently applied to a common set of data, it can be shown that these 2 methodologies will result in identical rates.
In general, the PP method is used to set rates for a new line or product or when there are no existing rates. The loss ratio method is used to calculate the needed change in existing rates.
NOTE: THE TWO METHODS PRODUCE IDENTICAL RESULTS
WHEN IDENTICAL DATA AND ASSUMPTIONS ARE USED.

9. LOSS RATIO METHODOLOGY Fixed Expense Approach
INDICATED (needed) RATE LEVEL CHANGE =
Projected Experience Loss + Fixed Expense Ratio
Expected (Target) Loss + Fixed Expense Ratio
- 1.0
Today we will focus on the Loss Ratio methodology. This is a quick example of how the calculation works.
For Example: 90.3%
76.6%
- 1.0 = %

10. LOSS RATIO METHODOLOGY Experience Loss + Fixed Expense Ratio Projection
Premium Adjustments
Adjust to Current Rate Level
Premium Trend
Loss Adjustments
Loss Development
Loss Trend
Catastrophe Adjustments

11. RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
EXPERIENCE Loss + Fixed Expense Ratio = ( ) / (4)
(1) Earned Premium
(2) Current Rate Level Factor
(3) Premium Trend Factor
(4) Trended Current Rate Level = (1)*(2)*(3)
(5) Accident Year 2006 Ultimate Losses & ALAE
(6) Unallocated Loss Adjustment Expense (ULAE) Factor
(7) Annual Loss Trend ___% Trend Period:
(8) Exponential Trend Factor [1.0 + (7)] ** Trend Period
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8)
Expected Catastrophe Loss & LAE for Projection Period
(11) Fixed Expense Ratio (FER)
(12) Fixed Expenses = (1) * (11)
We will be coming back to this exhibit throughout the presentation. This details the specific elements and calculations that go into the calculation of the overall rate level indication. We will be filling in the various elements as we discuss them.
B. EXPECTED (Target) Loss + Fixed Expense Ratio
C. INDICATED RATE LEVEL CHANGE = (A / B)

12. Sample Rate Level Indication Assumptions
Annual Policies. Rates to be revised as of JANUARY 1, 2008
Loss Ratio Methodology
EXPERIENCE PERIOD: ACCIDENT YEAR 2006
2006 Earned Premium $3,690,000
Reported Incurred Losses as of 12/31/06: $1,900,000

13. Current Rate Level Adjustment
PREMIUM ADJUSTMENTS
Current Rate Level Adjustment
Loss Ratio Method analyzes the appropriateness of the CURRENT RATES for use in the future.
CRL adjustment reflects rate changes NOT already included in historical recorded premium.

14. Current Rate Level Adjustment - Common Techniques
PREMIUM ADJUSTMENTS
Current Rate Level Adjustment - Common Techniques
Extension of Exposures
Re-rate each exposure (policy)
Requires extensive detail and mechanization
Most accurate method
Parallelogram Method
Easier method
Specific policy information not required
Assumes even distribution of policies written throughout the year

15. CURRENT RATE LEVEL ADJUSTMENT Extension of Exposures Method
2006 Earned Exposures
Class Class 2
Territory , ,260
Territory , ,010
Territory , ,500
Current Rates
Class Class 2
Territory $ $300
Territory $ $350
Territory $ $440
Current Rates
Class Class 2
Territory $225, $678,000
Territory $349, $1,053,500
Territory $594, $1,100,000
Statewide total $3,999,625

16. CURRENT RATE LEVEL ADJUSTMENT
Parallelogram Method A B
1/ / / / /09
Rate Change History
Date Change Rate Index
From 1/1/05 to 6/30/06 None A
7/1/ % B
(1 * 1.12)

17. CURRENT RATE LEVEL ADJUSTMENT
Calculation of On-Level Factor - Parallelogram Method
I. Rate Index for 2006:
Percent Rate
Area of Index A B
TOTAL
II. On-Level Factor for 2006:
(1) Current Index
(2) Index
(3) On-Level Factor (1) / (2)
(4) Earned Premium $3,690,000
(5) Earned Current Rate Level $4,070,070

18. RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
EXPERIENCE Loss + Fixed Expense Ratio = ( ) / (4)
(1) Earned Premium
Current Rate Level Factor
(3) Premium Trend Factor
Trended Current Rate Level = (1)*(2)*(3)
(5) Accident Year 2006 Ultimate Losses & ALAE
(6) Unallocated Loss Adjustment Expense (ULAE) Factor
(7) Annual Loss Trend ___% Trend Period:
(8) Exponential Trend Factor [1.0 + (7)] ** Trend Period
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8)
Expected Catastrophe Loss & LAE for Projection Period
(11) Fixed Expense Ratio (FER)
(12) Fixed Expenses = (1) * (11)
3,690
1.103
B. EXPECTED (Target) Loss + Fixed Expense Ratio
C. INDICATED RATE LEVEL CHANGE = (A / B)

19. PREMIUM ADJUSTMENTS Premium Trend
To project the premium level which will exist during the period being priced. The premium trend accounts for shifts of business that will also impact the losses.
Must adjust for items such as:
Average car model year or price group
Average home value
Territorial distribution shift
Any item that would impact future premium or both premium and losses in the future except policy count

20. Premium Adjustments Premium Trend – Determination of Trend Period
Annual Policies. Rates to be revised as of JANUARY 1, 2008
EXPERIENCE PERIOD: ACCIDENT YEAR 2006
Experience Policies
Period Effective
<COVERAGE PROVIDED>
Avg. Earned Avg. Earned Date under
Date is 7/1/ Revised Rates is 1/1/2009
TREND PERIOD is 2.50 Years
Assuming an average annual trend of 2% for this example, the premium trend would be: (1.02) ^ 2.5 = 1.051

21. RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
EXPERIENCE Loss + Fixed Expense Ratio = ( ) / (4)
(1) Earned Premium ,690
(2) Current Rate Level Factor
(3) Premium Trend Factor
(4) Trended Current Rate Level = (1)*(2)*(3)
(5) Accident Year 2006 Ultimate Losses & ALAE
(6) Unallocated Loss Adjustment Expense (ULAE) Factor
(7) Annual Loss Trend ___% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] **
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8)
(10) Expected Catastrophe Loss & LAE for Projection Period
(11) Fixed Expense Ratio (FER)
(12) Fixed Expenses = (1) * (11)
1.051
4,278
B. EXPECTED (Target) Loss + Fixed Expense Ratio
C. INDICATED RATE LEVEL CHANGE = (A / B)

22. LOSS RATIO METHODOLOGY Experience Loss + Fixed Expense Ratio Projection
Loss Adjustments
Loss Development
Loss Adjustment Expenses
Allocated Loss Adjustment Expense (ALAE)
Generally included with loss
Unallocated Loss Adjustment Expense (ULAE)
Generally loaded to Loss & ALAE
Loss Trend
Catastrophe Adjustments

23. Loss Development Analysis
LOSS ADJUSTMENTS
Loss Development Analysis
Adjust historical losses to an expected ULTIMATE value
Reflects revisions to claim values as claims are settled
Used with policy and accident year data
Reflects IBNR reporting.
Reflects development on reported claims.
Key Factors for Consideration
Observation of historical patterns
Incurred and Paid developments
Development period

24. Accident Year Loss Development Analysis
INCURRED METHOD - Recognizes SYSTEMATIC inaccuracy of case reserves
INCURRED LOSSES & ALAE
Adjusted for Deductibles and Cats, (000’s)
ACCIDENT Reported as of:
YEAR 12 mos mos mos mos
, , , ,548
, , , ,843
, , , ,691
, , ,836
, ,968
,900
Age to Age Development Factor =
Incurred Later Report Period divided by Prior Report Period
AY mos TO 24 mos Factor = $1,800 / $1,500 = 1.20

25. Accident Year Loss Development Analysis
INCURRED AGE-TO-AGE FACTORS
ACCIDENT
YEAR mos mos mos
Average
Selected
x x
Cumulative
Age-to-Age Factors

26. LOSS DEVELOPMENT ANALYSIS
(1) (2) (3)
Cumulative Estimated
Accident Incurred Loss Age to Ultimate Ultimate Loss
Year & 12/06 Factor (1) * (2)
, ,691
, ,836
, ,017
, ,430

27. RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
EXPERIENCE Loss + Fixed Expense Ratio = ( ) / (4)
(1) Earned Premium ,690
(2) Current Rate Level Factor
(3) Premium Trend Factor
(4) Trended Current Rate Level = (1)*(2)*(3) ,278
(5) Accident Year 2006 Ultimate Losses & ALAE
(6) Unallocated Loss Adjustment Expense (ULAE) Factor
(7) Annual Loss Trend ___% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] **
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8)
(10) Expected Catastrophe Loss & LAE for Projection Period
(11) Fixed Expense Ratio (FER)
(12) Fixed Expenses = (1) * (11)
2,430
B. EXPECTED (Target) Loss + Fixed Expense Ratio
C. INDICATED RATE LEVEL CHANGE = (A / B)

28. Unallocated Loss Adjustment Expense
EXPENSE ANALYSIS
Unallocated Loss Adjustment Expense
Countrywide Figures
(in $ millions)
Unallocated
Loss ULAE to
Incurred Adjustment Losses & ALAE
Year Losses & ALAE Expenses Ratio
$61, $6, %
, , %
, , %
Estimated Future ULAE Percentage %
as a percentage of Incurred Losses & ALAE

29. RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
EXPERIENCE Loss + Fixed Expense Ratio = ( ) / (4)
(1) Earned Premium ,690
(2) Current Rate Level Factor
(3) Premium Trend Factor
(4) Trended Current Rate Level = (1)*(2)*(3) ,278
(5) Accident Year 2006 Ultimate Losses & ALAE ,430
(6) Unallocated Loss Adjustment Expense (ULAE) Factor
(7) Annual Loss Trend ___% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] **
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8)
(10) Expected Catastrophe Loss & LAE for Projection Period
(11) Fixed Expense Ratio (FER)
(12) Fixed Expenses = (1) * (11)
1.10
B. EXPECTED (Target) Loss + Fixed Expense Ratio
C. INDICATED RATE LEVEL CHANGE = (A / B)

30. LOSS ADJUSTMENTS Loss Trend Analysis
Project to the loss level predicted to exist during pricing period
Data Issues
Separate Claim frequency and Severity Trends?
Internal Vs. External Data ?
Paid, Incurred, Reported data ?
Calendar Vs. Accident year ?
Length of Historical period ?
Credibility ?
Extrapolations of Historical Data? (Least Squares Regression, Time Series, Econometric Models)

31. Other Possible Trend Sources
LOSS TREND ANALYSIS
Calendar Paid Losses Earned Exposures Pure
Year ($ 000’s) (000’s) Premium
, $
, $102.73
, $112.48
, $128.81
, $127.21
, $134.23
, $143.75
, $150.57
Annual Trend based on Least Squares (exponential ) %
Most Recent Annual Change ( / ) %
Other Possible Trend Sources
C.P.I. Medical Care Index %
C.P.I. Auto Body Work Index %
C.P.I. Home Maintenance & Repair Index %

32. RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
EXPERIENCE Loss + Fixed Expense Ratio = ( ) / (4)
(1) Earned Premium ,690
(2) Current Rate Level Factor
(3) Premium Trend Factor
(4) Trended Current Rate Level = (1)*(2)*(3) ,278
(5) Accident Year 2006 Ultimate Losses & ALAE ,430
(6) Unallocated Loss Adjustment Expense (ULAE) Factor
(7) Annual Loss Trend _5.0__% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] **
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8)
(10) Expected Catastrophe Loss & LAE for Projection Period
(11) Fixed Expense Ratio (FER)
(12) Fixed Expenses = (1) * (11)
1.13
3,020
B. EXPECTED (Target) Loss + Fixed Expense Ratio
C. INDICATED RATE LEVEL CHANGE = (A / B)

33. LOSS ADJUSTMENTS CATASTROPHES Example:
Catastrophes should be eliminated from losses
Average provision should be used as a loss loading
Example:
Expected Annual Catastrophe Loss & ALAE
for Projection Period 394
(2) Projected Premium 4,278
(3) Catastrophe Load (1) / (2) 9.21%

34. RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
EXPERIENCE Loss + Fixed Expense Ratio = ( ) / (4)
(1) Earned Premium ,690
(2) Current Rate Level Factor
(3) Premium Trend Factor
(4) Trended Current Rate Level = (1)*(2)*(3) ,278
(5) Accident Year 2006 Ultimate Losses & ALAE ,430
Unallocated Loss Adjustment Expense (ULAE) Factor
Annual Loss Trend _5.0__% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] **
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8) ,020
(10) Expected Catastrophe Loss & LAE for Projection Period
(11) Fixed Expense Ratio (FER)
(12) Fixed Expenses = (1) * (11)
394
B. EXPECTED (Target) Loss + Fixed Expense Ratio
C. INDICATED RATE LEVEL CHANGE = (A / B)

35. UNDERWRITING EXPENSE ANALYSIS
Direct Expenses Other Than Loss Adjustment
Countrywide Figures (In $ Millions)
Selected
$ % $ % $ % %
Written Premium , , ,
Commissions , , ,
Other
Acquisition , , ,
Administrative , , ,
Taxes, Licenses
& Fees , , ,
Commissions and Premium Taxes vary directly with premiums
Other acquisition and general expenses are “fixed” expenses
Not really fixed - vary with inflation

36. DEVELOPMENT of EXPECTED LOSS RATIO & FIXED EXPENSE RATIO
Total Variable Fixed
Commissions % % %
Other Acquisition
General
Taxes, Licenses & Fees
Profit & Contingency
Other Costs *
TOTAL % % %
TARGET Loss, LAE & Fixed Expense Ratio = % % = 76.6%
* Policyholder Dividends, Involuntary Market Costs, Guaranty Fund Assessments, Etc. (if allowable)

37. RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
EXPERIENCE Loss + Fixed Expense Ratio = ( ) / (4)
(1) Earned Premium ,690
(2) Current Rate Level Factor
(3) Premium Trend Factor
(4) Trended Current Rate Level = (1)*(2)*(3) ,278
(5) Accident Year 2006 Ultimate Losses & ALAE ,430
(6) Unallocated Loss Adjustment Expense (ULAE) Factor
(7) Annual Loss Trend _5.0__% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] **
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8) ,020
(10) Expected Catastrophe Loss & LAE for Projection Period
(11) Fixed Expense Ratio (FER)
(12) Fixed Expenses = (1) * (11)
90.3%
12.2%
450
B. EXPECTED (Target) Loss + Fixed Expense Ratio
76.6%
C. INDICATED RATE LEVEL CHANGE = (A / B)
+17.9%

38. Introduction to Ratemaking Relativities
Why are there rate relativities?
Considerations in determining rating distinctions
Basic methods and examples
Advanced methods

39. 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

40. 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!

41. Considerations in Setting Rating Distinctions
OPERATIONAL
Objective definition
Administrative expense
Verifiability
SOCIAL
Privacy
Causality
Controllability
Affordability
LEGAL
Constitutional
Statutory
Regulatory
ACTUARIAL
Accuracy
Homogeneity
Reliability
Credibility
OPERATIONAL
Objective definition – clear who is in the group
ACTUARIAL
Accuracy – variable should measure cost differences
Homogeneity – all members of the class should have the same expected cost
Reliability – should have stable mean value over time
Credibility – groups should be large enough to permit measuring costs

42. 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.

43. Loss Ratio Relativity Method
Class
Losses
Loss Ratio
Loss Ratio Relativity
Current Relativity
New Relativity 1
$1,168,125
$759,281
0.65
1.00 2
$2,831,500
$1,472,719
0.52
0.80
2.00
1.60

44. 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

45. 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

46. Methods 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

47. 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 2
0.52
0.90
0.85
2.00
1.70
Total
0.56

48. Off-Balance Adjustment
Class
Current Relativity
Base Class Rates
Proposed Relativity
Proposed Premium 1
$1,168,125
1.00 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.

49. 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: ( ) * 1.70 = $170
Should charge: (62* )/(1-.25) = $158
“Flattening” relativities accounts for fixed expense
Flattened factor = ( )* =

50. 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”

51. Deductible 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 Clms>D)
Total Losses

52. 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)

53. Example: 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
101 to 200
350
54,250
155
19,250
201 to 500
550
182,625
332
127,625
72,625
501 +
335
375,125
1120
341,625
308,125
207,625
Total
1,735
642,000
370
488,500
380,750
Loss Eliminated
153,500
261,250
434,375
L.E.R.
0.239
0.407
.677

54. Use same expense allocation as overall indications.
Example: Expenses
Total
Variable
Fixed
Commissions
15.5%
0.0%
Other Acquisition
5.8%
General
6.4%
Unallocated Loss Expenses
6.0%
Taxes, Licenses & Fees
3.4%
Profit & Contingency
4.0%
Other Costs
0.5%
41.6%
23.4%
18.2%
Use same expense allocation as overall indications.

55. Example: Deductible Credit
Calculation
Factor
$100
(.614)*(1-.239) (1-.234)
0.848
$200
(.614)*(1-.407) (1-.234)
0.713
$500
(.614)*(1-.677) (1-.234)
0.497

56. Advanced Techniques Multivariate techniques Generalized Linear Models
Why use multivariate techniques
Minimum Bias techniques
Example
Generalized Linear Models

57. Why Use Multivariate Techniques?
One-way analyses:
Based on assumption that effects of single rating variables are independent of all other rating variables
Don’t consider the correlation or interaction between rating variables

58. Examples Correlation: Interaction Car value & model year
Driving record & age
Type of construction & fire protection

59. 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

60. 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

61. 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

62. Minimum Bias Techniques
Least Squares
Bailey’s Minimum Bias

63. 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

64. Least Squares Method ∑j wij ( Yj)2 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

65. Bailey’s 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

66. Bailey’s Minimum Bias ∑j wij Yj Formula: Xi = ∑j wij rij where
Xi and Yj = relativities for rating variables i and j
wij = weights
rij = observed relativity
∑j wij Yj

67. Bailey’s Minimum Bias
Less sensitive to the experience of individual cells than Least Squares Method
Widely used; e.g.., ISO GL loss cost reviews
Can be multiplicative or additive
Can be used for many dimensions (convergence may be difficult)
Easily coded in spreadsheets

68. 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

69. 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

70. Generalized Linear Models
ISO Applications:
Businessowners, Commercial Property (Variables include Construction, Protection, Occupancy, Amount of insurance)
GL, Homeowners, Personal Auto

71. 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)
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)

72. Suggested Readings (Continued)
A Practitioners Guide to Generalized Linear Models, Anderson, et al (2004)
Statement of Principles Regarding P&C Insurance Ratemaking
Insurance Operations, Webb et al (CPCU) Chapters 10 & 11
Introduction to Ratemaking and Loss Reserving for P&C Insurance, Robert L. Brown Chapter 3
Trend and Loss Development Factors, Cook (1970)