CAS Seminar on
Ratemaking
Introduction to Ratemaking Relativities
March 13-14, 2006
Salt Lake City Marriott
Salt Lake City, Utah
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
@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
0.61
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
– Example: $62 loss cost + $25 VE + $13 FE = $100

Multiplying both means fixed expense no longer
“fixed”
– 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
Loss Eliminated
L.E.R.
153,500 261,250 434,375
0.239
0.407
.677
Example: Expenses
Total
Variable
Fixed
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%
Commissions
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
– 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
A Simple Bailey’s ExampleManufacturers &
Contractors
Type of
Policy
Aggregate Loss Costs at
Current Level (ALCCL)
Experience Ratio
(ER)
Class Group
Class Group
Light
Manuf
Medium
Manuf
Heavy
Manuf
Light
Manuf
Medium
Manuf
Heavy
Manuf
Monoline
2000
250
1000
1.10
.80
.75
Multiline
4000
1500
6000
.70
1.50
2.60
SW = 1.61
Bailey’s Example
Experience Ratio Relativities
Class Group
Type of Policy
Statewide
Light
Manuf
Medium
Manuf
Heavy
Manuf
Monoline
.683
.497
.466
.602
Multiline
.435
.932
1.615
1.118
Bailey’s 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
Bailey’s Example
CGj = ∑i wij rij
∑i wij TOPi
Class Group
Bailey’s Output
Light Manuf
.547
Medium Manuf
.833
Heavy Manuf
1.389
Bailey’s 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
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
Heavy Manuf
Italic factors = newly calculated; continue until factors stop changing
Bailey’s Example



Apply Credibility
Balance to no overall change
Apply to current relativities to get new
relativities
Bailey’s

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