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
1
Basic Ratemaking Equation and Its
Considerations:

Organization of Data

Premium Adjustments

Loss Adjustments

Expense Considerations

Other Considerations
ORGANIZATION OF DATA
I. CALENDAR YEAR DATA
(standard accounting year)
II. POLICY YEAR DATA
III. ACCIDENT YEAR DATA
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.
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.
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.
Basic Ratemaking Equation:
Future Premiums =
Future Losses +
Future Expenses +
Underwriting Profit and Contingency Provision
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
R
indicated rate per unit of exposure (R)
= [PP + FE] / [1-VER-Profit Ratio]
NOTE: THE TWO METHODS PRODUCE IDENTICAL RESULTS
WHEN IDENTICAL DATA AND ASSUMPTIONS ARE USED.
LOSS RATIO METHODOLOGY
Fixed Expense Approach
INDICATED (needed) RATE LEVEL CHANGE
=
Projected Experience Loss + Fixed Expense Ratio
Expected (Target) Loss + Fixed Expense Ratio
For Example:
90.3% - 1.0 = + 17.9%
76.6%
- 1.0
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

RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
A.
EXPERIENCE Loss + Fixed Expense Ratio = (9 + 10+ 12) / (4).. . . . . .
(1)
(2)
(3)
(4)
2006 Earned Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current Rate Level Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Premium Trend Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trended Premium @ 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) . . . . . . . . . . .
(10) Expected Catastrophe Loss & LAE for Projection Period. . . . . . . . .
(11) Fixed Expense Ratio (FER). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(12) Fixed Expenses = (1) * (11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. EXPECTED (Target) Loss + Fixed Expense Ratio. . . . . . . . . . . . . . . . . . . . .
C. INDICATED RATE LEVEL CHANGE = (A / B) - 1.0. . . . . . . . . . . . . . .
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
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.
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
CURRENT RATE LEVEL ADJUSTMENT
Extension of Exposures Method
2006 Earned Exposures
Class 1
1,500
1,995
2,700
Territory 1
Territory 2
Territory 3
Class 2
2,260
3,010
2,500
Current Rates
Class 1
$150
$175
$220
Territory 1
Territory 2
Territory 3
Class 2
$300
$350
$440
Premium @ Current Rates
Territory 1
Territory 2
Territory 3
Statewide total
Class 1
$225,000
$349,125
$594,000
Class 2
$678,000
$1,053,500
$1,100,000
$3,999,625
CURRENT RATE LEVEL ADJUSTMENT
Parallelogram Method
A
B
1/05
1/06
1/07
1/08
1/09
Rate Change History
Date
Change
Rate Index
From 1/1/05 to 6/30/06
None
1.000
A
7/1/06
+ 12%
1.12
(1 * 1.12)
B
CURRENT RATE LEVEL ADJUSTMENT
Calculation of On-Level Factor - Parallelogram Method
I. Rate Index for 2006:
Area
Percent
of 2006
Rate
Index
A
B
87.5
12.5
1.000
1.120
100.0
1.015
TOTAL
II. On-Level Factor for 2006:
(1) Current Index
(2) 2006 Index
(3) On-Level Factor (1) / (2)
1.120
1.015
1.103
(4) 2006 Earned Premium
$3,690,000
(5) 2006 Earned Premium @ Current Rate Level
$4,070,070
RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
A.
EXPERIENCE Loss + Fixed Expense Ratio = (9 + 10+ 12) / (4).. . . . . .
(1)
(2)
(3)
(4)
2006 Earned Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current Rate Level Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Premium Trend Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trended Premium @ 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) . . . . . . . . . . .
(10) Expected Catastrophe Loss & LAE for Projection Period. . . . . . . .
(11) Fixed Expense Ratio (FER). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(12) Fixed Expenses = (1) * (11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. EXPECTED (Target) Loss + Fixed Expense Ratio. . . . . . . . . . . . . . . . . . . .
C. INDICATED RATE LEVEL CHANGE = (A / B) - 1.0 . . . . . . . . . . . . . . .
3,690
1.103
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
Premium Adjustments
Premium Trend – Determination of Trend Period
•
•
Annual Policies. Rates to be revised as of JANUARY 1, 2008
EXPERIENCE PERIOD: ACCIDENT YEAR 2006
2006
Experience
Period
2007
2008
Policies
Effective
2009
<COVERAGE PROVIDED>
Avg. Earned
Date is 7/1/06
Avg. Earned Date under
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
RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
A.
EXPERIENCE Loss + Fixed Expense Ratio = (9 + 10+ 12) / (4).. . . . . .
(1)
(2)
(3)
(4)
2006 Earned Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current Rate Level Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Premium Trend Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trended Premium @ 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)] ** 2.5. . . . . . . . . . . . . . . . . . . .
(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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. EXPECTED (Target) Loss + Fixed Expense Ratio. . . . . . . . . . . . . . . . . . . . .
C. INDICATED RATE LEVEL CHANGE = (A / B) - 1.0 . . . . . . . . . . . . . . .
3,690
1.103
1.051
4,278
LOSS RATIO METHODOLOGY
Experience Loss + Fixed Expense Ratio Projection
Loss Adjustments


Loss Development
Loss Adjustment Expenses

Allocated Loss Adjustment Expense (ALAE)


Unallocated Loss Adjustment Expense (ULAE)



Generally included with loss
Generally loaded to Loss & ALAE
Loss Trend
Catastrophe Adjustments
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
Accident Year Loss Development Analysis
INCURRED METHOD - Recognizes SYSTEMATIC inaccuracy of case reserves
INCURRED LOSSES & ALAE
Adjusted for Deductibles and Cats, (000’s)
ACCIDENT
YEAR
2001
2002
2003
2004
2005
2006
12 mos
1,200
1,300
1,400
1,500
1,600
1,900
Reported as of:
24 mos
36 mos
1,488
1,548
1,755
1,843
1,708
1,691
1,800
1,836
1,968
48 mos
1,548
1,843
1,691
Age to Age Development Factor =
Incurred Loss @ Later Report Period divided by Loss @ Prior Report Period
AY 2004 12 mos TO 24 mos Factor = $1,800 / $1,500 = 1.20
Accident Year Loss Development Analysis
INCURRED AGE-TO-AGE FACTORS
ACCIDENT
YEAR
12-24 mos
24-36 mos
36-48 mos
2001
2002
2003
2004
2005
1.24
1.35
1.22
1.20
1.23
1.04
1.05
0.99
1.02
1.00
1.00
1.00
Average
1.248
1.025
1.000
Selected
1.248
1.025
1.000
x
Cumulative
Age-to-Age Factors
1.279
x
1.025
1.000
LOSS DEVELOPMENT ANALYSIS
(1)
Accident
Year
2003
Incurred Loss
& ALAE @ 12/06
1,691
(2)
Cumulative
Age to Ultimate
Factor
1.000
(3)
Estimated
Ultimate Loss
(1) * (2)
1,691
2004
1,836
1.000
1,836
2005
1,968
1.025
2,017
2006
1,900
1.279
2,430
RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
A.
EXPERIENCE Loss + Fixed Expense Ratio = (9 + 10+ 12) / (4).. . . . . .
(1)
(2)
(3)
(4)
2006 Earned Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current Rate Level Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Premium Trend Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trended Premium @ Current Rate Level = (1)*(2)*(3) . . . . . . . . . .
3,690
1.103
1.051
4,278
(5) Accident Year 2006 Ultimate Losses & ALAE . . . . . . . . . . . . . . . . . . 2,430
(6) Unallocated Loss Adjustment Expense (ULAE) Factor. . . . . . . . . . .
(7) Annual Loss Trend ___% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] ** 2.5. . . . . . . . . . . . . . . . . . . .
(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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. EXPECTED (Target) Loss + Fixed Expense Ratio. . . . . . . . . . . . . . . . . . . . .
C. INDICATED RATE LEVEL CHANGE = (A / B) - 1.0 . . . . . . . . . . . . . . .
EXPENSE ANALYSIS
Unallocated Loss Adjustment Expense
Countrywide Figures
(in $ millions)
Year
Incurred
Losses & ALAE
Unallocated
Loss
Adjustment
Expenses
ULAE to
Losses & ALAE
Ratio
2004
2005
2006
$61,200
79,000
82,300
$6,500
7,800
8,300
10.6%
9.9%
10.1%
Estimated Future ULAE Percentage
as a percentage of Incurred Losses & ALAE
10.0%
RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
A.
EXPERIENCE Loss + Fixed Expense Ratio = (9 + 10+ 12) / (4).. . . . . .
(1)
(2)
(3)
(4)
2006 Earned Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current Rate Level Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Premium Trend Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trended Premium @ Current Rate Level = (1)*(2)*(3) . . . . . . . . . .
3,690
1.103
1.051
4,278
(5) Accident Year 2006 Ultimate Losses & ALAE . . . . . . . . . . . . . . . . . . 2,430
(6) Unallocated Loss Adjustment Expense (ULAE) Factor. . . . . . . . . . . 1.10
(7) Annual Loss Trend ___% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] ** 2.5. . . . . . . . . . . . . . . . . . . .
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8) . . . . . . . . . . .
(10) Expected Catastrophe Loss & LAE for Projection Period. . . . . . . . .
(11) Fixed Expense Ratio (FER). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(12) Fixed Expenses = (1) * (11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. EXPECTED (Target) Loss + Fixed Expense Ratio. . . . . . . . . . . . . . . . . . . . .
C. INDICATED RATE LEVEL CHANGE = (A / B) - 1.0. . . . . . . . . . . . . . .
29
LOSS ADJUSTMENTS
Loss Trend Analysis
 Project to the loss level predicted to exist during pricing period

Data Issues
• Separate Claim frequency and Severity Trends?
• Internal Vs. External Data ?
• Paid, Incurred, Reported data ?
• Calendar Vs. Accident year ?
• Length of Historical period ?
• Credibility ?
• Extrapolations of Historical Data? (Least Squares Regression,
Time Series, Econometric Models)
LOSS TREND ANALYSIS
Calendar
Year
1999
2000
2001
2002
2003
2004
2005
2006
Paid Losses Earned Exposures
($ 000’s) (000’s)
Premium
1,212
13.0
1,356
13.2
1,496
13.3
1,726
13.4
1,730
13.6
1,839
13.7
1,984
13.8
2,108
14.0
Pure
$ 93.23
$102.73
$112.48
$128.81
$127.21
$134.23
$143.75
$150.57
Annual Trend based on Least Squares (exponential )
6.6%
Most Recent Annual Change (150.57 / 143.75)
4.7%
Other Possible Trend Sources
C.P.I. Medical Care Index
C.P.I. Auto Body Work Index
C.P.I. Home Maintenance & Repair Index
3 - 4%
4 - 5%
3 - 4%
RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
A.
EXPERIENCE Loss + Fixed Expense Ratio = (9 + 10+ 12) / (4).. . . . . .
(1)
(2)
(3)
(4)
2006 Earned Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current Rate Level Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Premium Trend Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trended Premium @ Current Rate Level = (1)*(2)*(3) . . . . . . . . . .
3,690
1.103
1.051
4,278
(5) Accident Year 2006 Ultimate Losses & ALAE . . . . . . . . . . . . . . . . . . 2,430
(6) Unallocated Loss Adjustment Expense (ULAE) Factor. . . . . . . . . . . 1.10
(7) Annual Loss Trend _5.0__% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] ** 2.5. . . . . . . . . . . . . . . . . . . . 1.13
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8) . . . . . . . . . . . . 3,020
(10) Expected Catastrophe Loss & LAE for Projection Period. . . . . . . . .
(11) Fixed Expense Ratio (FER). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(12) Fixed Expenses = (1) * (11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. EXPECTED (Target) Loss + Fixed Expense Ratio. . . . . . . . . . . . . . . . . . . . .
C. INDICATED RATE LEVEL CHANGE = (A / B) - 1.0. . . . . . . . . . . . . . .
LOSS ADJUSTMENTS
CATASTROPHES
 Catastrophes should be eliminated from losses
 Average provision should be used as a loss loading
Example:
(1) Expected Annual Catastrophe Loss & ALAE
for Projection Period
394
(2) Projected Premium
4,278
(3) Catastrophe Load (1) / (2)
9.21%
RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
A.
EXPERIENCE Loss + Fixed Expense Ratio = (9 + 10+ 12) / (4).. . . . . .
(1)
(2)
(3)
(4)
2006 Earned Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current Rate Level Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Premium Trend Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trended Premium @ Current Rate Level = (1)*(2)*(3) . . . . . . . . . .
3,690
1.103
1.051
4,278
(5) Accident Year 2006 Ultimate Losses & ALAE . . . . . . . . . . . . . . . . . . 2,430
(6) Unallocated Loss Adjustment Expense (ULAE) Factor. . . . . . . . . . . 1.10
(7) Annual Loss Trend _5.0__% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] ** 2.5. . . . . . . . . . . . . . . . . . . . 1.13
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8) . . . . . . . . . . . . 3,020
(10) Expected Catastrophe Loss & LAE for Projection Period. . . . . . . . . 394
(11) Fixed Expense Ratio (FER). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(12) Fixed Expenses = (1) * (11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. EXPECTED (Target) Loss + Fixed Expense Ratio. . . . . . . . . . . . . . . . . . . . .
C. INDICATED RATE LEVEL CHANGE = (A / B) - 1.0. . . . . . . . . . . . . . .
UNDERWRITING EXPENSE ANALYSIS
Direct Expenses Other Than Loss Adjustment
Countrywide Figures (In $ Millions)
2004
Written Premium
Commissions
2005
2006
Selected
$
%
$
%
$
%
107,400
100
121,600
100
142,400
100
16,647 15.5
%
18,850
15.5
22,100
15.5
15.5
Other
Acquisition
6,703
6.2
7,250
6.0
8,235
5.8
5.8
Administrative
7,332
6.8
7,977
6.6
9,101
6.4
6.4
3,652
3.4
4,100
3.4
4,900
3.4
3.4
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
DEVELOPMENT of
EXPECTED LOSS RATIO &
FIXED EXPENSE RATIO
Total
Commissions
Variable
Fixed
15.5%
15.5%
Other Acquisition
5.8
0.0
5.8
General
6.4
0.0
6.4
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
35.6%
23.4%
12.2%
TOTAL
TARGET Loss, LAE & Fixed Expense Ratio = 100.0% - 23.4% = 76.6%
* Policyholder Dividends, Involuntary Market Costs, Guaranty Fund Assessments, Etc. (if allowable)
0.0%
RATE INDICATION WORKSHEET
Loss Ratio Methodology - Fixed Expense Approach
A.
EXPERIENCE Loss + Fixed Expense Ratio = (9 + 10+ 12) / (4).. . . . . . 90.3%
(1)
(2)
(3)
(4)
2006 Earned Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current Rate Level Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Premium Trend Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trended Premium @ Current Rate Level = (1)*(2)*(3) . . . . . . . . . .
3,690
1.103
1.051
4,278
(5) Accident Year 2006 Ultimate Losses & ALAE . . . . . . . . . . . . . . . . . . 2,430
(6) Unallocated Loss Adjustment Expense (ULAE) Factor. . . . . . . . . . . 1.10
(7) Annual Loss Trend _5.0__% Trend Period: 2.5 years
(8) Exponential Trend Factor [1.0 + (7)] ** 2.5. . . . . . . . . . . . . . . . . . . . 1.13
(9) Trended Ultimate Losses and LAE = (5) * (6) * (8) . . . . . . . . . . . . 3,020
(10) Expected Catastrophe Loss & LAE for Projection Period. . . . . . . . . 394
(11) Fixed Expense Ratio (FER). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2%
(12) Fixed Expenses = (1) * (11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
B. EXPECTED (Target) Loss + Fixed Expense Ratio. . . . . . . . . . . . . . . . . . . . . 76.6%
C. INDICATED RATE LEVEL CHANGE = (A / B) - 1.0. . . . . . . . . . . . . . . +17.9%
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
 Objective definition
 Administrative expense
 Verifiability
LEGAL
 Constitutional
 Statutory
 Regulatory
SOCIAL
 Privacy
 Causality
 Controllability
 Affordability
ACTUARIAL
 Accuracy
 Homogeneity
 Reliability
 Credibility
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.
Loss Ratio Relativity Method
Class
Premium
@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
1
6,195
2
7,770
Losses
Pure
Premium
Pure
Premium
Relativity
$759,281
$123
1.00
$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
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
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 @CRL
Current
Relativity
Premium @
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


Multiplying both means fixed expense no longer “fixed”



Example: $62 loss cost + $25 VE + $13 FE = $100
Example: (62+25+13) * 1.70 = $170
Should charge: (62*1.70 + 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 (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
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
$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
153,500
261,250
434,375
0.239
0.407
.677
Loss Eliminated
L.E.R.
Example: Expenses
Total
Variable
Fixed
15.5%
15.5%
0.0%
Other Acquisition
5.8%
0.0%
5.8%
General
6.4%
0.0%
6.4%
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%
41.6%
23.4%
18.2%
Commissions
Total
Use same expense allocation as overall indications.
Example: Deductible Credit
Deductible
Calculation
Factor
$100
(.614)*(1-.239) + .182
(1-.234)
0.848
$200
(.614)*(1-.407) + .182
(1-.234)
0.713
$500
(.614)*(1-.677) + .182
(1-.234)
0.497
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
 Don’t 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

Bailey’s 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
∑j wij ( Yj)2
where
Xi and Yj = relativities for rating variables i and j
wij = weights
rij = observed relativity
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
Bailey’s Minimum Bias
Formula:
Xi =
∑j wij rij
∑j wij Yj
where
Xi and Yj = relativities for rating variables i and j
wij = weights
rij = observed relativity
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
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)
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)
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)
