CAS Seminar on Ratemaking
Introduction to Ratemaking Relativities
(INT - 3)
March 11, 2004
Wyndham Franklin Plaza Hotel
Philadelphia, Pennsylvania
Presented by:
Francis X. Gribbon, FCAS & Julie A. Jordan, 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
@CRL
1
$1,168,125
2
Losses
Loss
Ratio
Loss
Ratio
Relativity
Current
Relativity
New
Relativity
$759,281
0.65
1.00
1.00
1.00
$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
Properties of Credibility

0  
– at Z = 1 data is fully credible (given full weight)

Z/E>0
– credibility increases as experience increases

 (Z/E)/  E<0
– percentage change in credibility should decrease
as volume of experience increases
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
0.60
1.00
1.00
1.00
2
0.52
0.90
0.52
0.87
2.00
1.74
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
1.74
$2,463,405
Total
$3,999,625
$3,631,530
Off-balance of 9.2% 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: (62+25+13) * 1.74 = $174
– Should charge: (62*1.74 + 13)/(1-.25) = $161

“Flattening” relativities accounts for fixed expense
– Flattened factor = (1-.25-.13)*1.74 + .13 = 1.61
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 Claims >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
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%
38.6%
25.3%
13.3%
Total
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
– Bailey’s Minimum Bias
– Generalized Linear Models

Curve fitting
Why Use Multivariate
Techniques?

Many rating variables are correlated
 Different variables, when viewed one at a
time, may be “double counting” the same
underlying effect
 Using a multivariate approach removes
potential double-counting and can account
for interaction effects
A Simple Example
Exposures
Pure Premium
Car Size
Car Size
Age
Group
Large
Medium
Small
Large
Medium
Small
1
100
1200
500
100
310
840
2
300
500
400
470
1460
2530
One-Way Relativities
Class
Exposures
Pure
Premium
Relativity
Large car
400
380
1.00
Medium car
1700
650
1.70
Small car
900
1590
4.20
Age Group 1
1800
450
1.00
Age Group 2
1200
1570
3.50
Multi-way vs. One-way
Multi-Way Relativities
One-way Relativities
Car Size
Car Size
Age
Group
Large
Medium
Small
Large
Medium
Small
1
1.00
3.10
8.40
1.00
1.70
4.20
2
4.70
14.60
25.30
3.50
6.00
14.60
When to use Multivariate?

Can use Multivariate techniques for entire
rating plan, or for particular variables that
are correlated or have interaction effects
 Example of correlation
– Value of car and Model Year

Examples of interaction effects
– Driving record and Age
– Type of construction and Fire protection
Bailey’s Minimum Bias

To get toward multivariate but still have
simple method to calculate premiums
 Can have credibility issues with many cells
 Can use either Loss Ratio or Pure Premium
methods
 Can assume multiplicative and/or additive
relationships of rating variables and
dependent variable
Bailey’s Example

Start with initial guess at factors for one
variable
Class
Pure Premium
Relativity
Age group 1
$450
1.00
Age group 2
$1570
3.50
Bailey’s Example: Step 1A

What would the premiums be, assuming
base rate = $100 and this rating plan?
Exposures
Theoretical Premium
Car Size
Car Size
Age
Group
Large
Medium
Small
Large
Medium
Small
1
100
1200
500
10000
120000
50000
2
300
500
400
105000
175000
140000
Bailey’s Example: Step 1B

What should the factors for car size be,
given the rating factors for age group?
Car size
Theoretical
Premium
Theoretical Loss
Ratio
Loss Ratio
Relativity
Large
115000
1.30
1.00
Medium
295000
3.70
2.80
Small
190000
7.50
5.70
Bailey’s Example: Step 2A

What would the premiums be, assuming
base rate = $100 and this rating plan?
Exposures
Theoretical Premium
Car Size
Car Size
Age
Group
Large
Medium
Small
Large
Medium
Small
1
100
1200
500
10000
336000
285000
2
300
500
400
30000
140000
228000
Bailey’s Example: Step 2B

What should the factors for age group be,
given the rating factors for car size?
Age group
Theoretical
Premium
Theoretical Loss
Ratio
Loss Ratio
Relativity
Age group 1
631000
1.30
1.00
Age group 2
398000
4.70
3.70
Bailey’s Example: Steps 3-6

What if we continued iterating this way?
Class
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Large Car
1.00
1.00
1.00
1.00
1.00
1.00
Medium Car
2.80
2.80
2.90
2.90
2.90
2.90
Small Car
5.70
5.70
5.80
5.80
5.80
5.80
Age Group 1
1.00
1.00
1.00
1.00
1.00
1.00
Age Group 2
3.50
3.70
3.70
3.60
3.60
3.60
Italic factors = newly calculated; continue until factors stop changing
Bailey’s Example: Results
Multi-Way Relativities
Bailey Relativities
Car Size
Car Size
Age
Group
Large
Medium
Small
Large
Medium
Small
1
1.00
3.10
8.40
1.00
2.90
5.80
2
4.70
14.60
25.30
3.60
10.40
20.10
Bailey’s Minimum Bias

Bailey Relativities get much closer to multiway relativities than univariate approach
 Premium calculation by multiplying factors
vs. table lookup for multi-way
 This example assumed two multiplicative
factors, but approach can be modified for
more variables and/or additive rating plans
Generalized Linear Models

Generalized Linear Models (GLM) is a
generalized framework for fitting
multivariate linear models
 Bailey’s method is a specific case of GLM
 Factors can be estimated with SAS or other
statistical software packages
Curve Fitting

Can calculate certain type of relativities
using smooth curves
 Fit exposure data to a curve
 Determine a functional relationship of loss
data and exposure data
 Taking derivative of this function and
relating the value at any given point to a
base point produces relativity
Curve Fitting

HO Policy Size Relativities
 Assume the distribution of exposures by
amount of insurance is log normal
 Assume the cumulative loss distribution has
a functional relationship to the cumulative
exposure distribution
Curve Fitting

Let r = amount of insurance
 f (r) is density of exposures at r
 = exposures at r / total exposures
 g (r) is density of losses at r
 = losses at r / total losses
 F(A) and G (A) are the cumulative
functions of f and g
Curve Fitting

F (A) and G (A) are cumulative functions of
f and g
 G (A) = H[ F (A)]
 Then dG (A)/dF (A) = g(a)/f(a)
=
(losses at A / total losses)
(exposures at A / total exposures)
 = pure premium at A/ total pure premium
Suggested Readings





ASB Standard of Practice No. 9
ASB Standard of Practice No. 12
Foundations of Casualty Actuarial Science,
Chapters 2 and 5
Insurance Rates with Minimum Bias, Bailey
(1963)
Something Old, Something New in Classification
Ratemaking with a Novel Use of GLMs for Credit
Insurance, Holler, Sommer, and Trahair (1999)