CAS Seminar on Ratemaking
Introduction to Ratemaking Relativities
March 17-18, 2008
Royal Sonesta Hotel
Boston, Mass.
Presented by:
Michael J. Miller, FCAS
Introduction to Ratemaking
Relativities




What is the purpose of rate
relativities?
Considerations in determining rating
distinctions
Basic methods and examples
Advanced methods
The Purpose of Rate Relativities
Example – Personal Auto:
Overall Indicated Change for State = +10%
Should everyone’s rate be increased by 10%?
Same for youthful drivers vs. adults?
Same for urban vs. rural?
Same for all policy limits? Deductibles?
The Purpose of Rate Relativities
Example:
Base Rate = $100 (Adult, Terr 1, No Deductible)
Insured
Age
Territory
Deductible
Premium
Male Age 40
Terr 1
$0 Ded
1.00
1.00
1.00
$100
Male 18
Terr 2
$500 Ded
3.50
1.50
0.60
$315
Female 18
Terr 2
$100 Ded
2.00
1.50
0.85
$255
Considerations in Selecting
Rate Relativities




Actuarial (Statistical)
Operational
Social
Legal
Actuarial Considerations

Accuracy




Rating variable closely related to cost differences
Provides “fairest” price (fair discrimination)
Reduces adverse selection (discussed later)
Homogeneity


Members of a class have similar expected cost
Variability within class always exists – grouping is
necessary since individual lacks credibility
Actuarial Considerations (cont.)

Credibility



Class groups should be large enough to measure
costs with sufficient accuracy
There is a trade-off between the need to estimate
costs accurately for an individual and the need for
enough data to do it.
Reliability


Estimated cost differences between groups should be
relatively stable over time
This does not mean they will be the same over time
Adverse Selection
Adverse selection can result when a group can be
accurately separated into 2 or more distinct groups, but
has not been.
Consider the following scenario:
Group
A expected costs = $100
Group B expected costs = $200
Your company charges $150 for both
Competitor charges $100 for A, and $200 to B
Adverse Selection (cont.)
At the outset, your company is collecting enough to
cover expected costs for both groups. Life is good.
All of your insureds in Group A learn about your
competitor’s lower rate and switch.
Your company is left with all of Group B at a $150 rate.
You have been selected against!
Typically this process happens gradually
Operational Considerations


Objective - Age & Marital Status vs “Maturity”
Administrative expense – Actual mileage driven
may be better predictor of accident potential than where
an insured lives, but one is much cheaper to obtain

Verifiability –
The amount of sleep a person has
gotten in the previous 24 hours may be a significant
predictor of auto accident potential. Aside from the
expense of trying to get this information, how could it be
verified?
Social Considerations

Privacy – certain personal info is off limits
Causality – as opposed to correlation
Controllability – something the insured can

Affordability –


impact (e.g. install sprinklers in commercial
property, non-smoker in health insurance)
balance with availability (e.g.
hospitals closing ER and OB due to high cost of med
mal insurance for these classes)
Legal Considerations
Choice of rating variable may be
prohibited by law at many levels (e.g.
Federal, State). Some examples:
Race
Gender (always in Health ins,
sometimes in other lines – even auto)
Income
Basic Methods for Determining
Rate Relativities

Loss ratio relativity method


Compare “actual” LR to expected LR to
produce an indicated change in relativity
Pure premium relativity method

Develop expected cost per unit of exposure to
produce 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 (LR method)



Current Rate Level
Premium Trend/Coverage Drift (not typical)
Loss Adjustments




Loss Development (project to ultimate)
Loss Trend (project to same time period)
Coverage Adjustments (diff Ded’s, Limits?)
Catastrophe Adjustments (“Shock Losses”)
Loss Ratio Relativity Method
Class
Premium
@CRL
1
$1,168,125
2
Trended &
Developed
Losses
Loss
Ratio
Loss
Ratio
Adjustmt
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
Trended &
Developed
Losses
Pure
Premium
Pure
Premium
Relativity
$759,281
$123
1.00
$1,472,719
$190
1.55
Incorporating Credibility



Credibility: how much predictive weight
do you assign to a given body of data?
Credibility is usually designated by Z
Credibility Weighted Loss Ratio:
LR= (Z) * LRclass + (1-Z) * LRcomplement
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
Adjustmt
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
Impact on Current Premium (“Off-Balance”)
$3,574,900
-11.9%
Off-balance of 11.9% must be covered in base rates. (How?)
Multivariate Techniques

Univariate (One-Way) Analyses:


Based on assumption that effects of single
rating variables are independent of all other
rating variables
Multivariate Analyses:

Give consideration to the correlation or
interaction between rating variables
Multivariate Techniques (cont.)



Bailey’s Method
Least Squares Method
Generalized Linear Model (GLM) Method
Example: Bailey’s Method





2 rating variables with relativities Xi and Yj
Select initial value for each Xi
Use model to solve for each Yj
Use new Yjs to solve for each Xi
Process continues until solutions at each
interval converge
Bailey’s Minimum Bias

“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
Aggregate Loss Costs at
Experience Ratio
Current Level (wij)
Class Group
Type of
Class Group (CGj)
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
Policy
SW = 1.61
Bailey’s Example
Experience Ratio Relativities
Class Group (CGj)
Type of Policy
(TOPi)
Statewide
Light
Manuf
Medium
Manuf
Heavy
Manuf
Monoline
.683
.497
.466
.602
Multiline
.435
.932
1.615
1.118
Bailey’s Example

Initial guess for relativities of one variable
(e.g., TOP: Mono = .602; Multi = 1.118)

Use TOP relativities and Bailey’s Minimum
Bias formulas to determine the Class
Group (CG) relativities
Bailey’s Example
CGj = ∑i wij rij
Class Group
Bailey’s Output
Light Manuf
.547
Medium Manuf
.833
Heavy Manuf
1.389
∑i wij TOPi
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
circled factors = newly calculated; continue until factors stop changing
Bailey’s


Can be used for multiplicative or additive
rating relativities
Can be used for many dimensions
(convergence may be difficult)

Easily coded in spreadsheets
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 =
where
∑j wij rij Yj
∑j wij ( Yj)2
Xi and Yj = relativities for rating variables i and j
wij = weights
rij = observed relativity
Generalized Linear Models



Generalized Linear Models (GLM) provide a
generalized framework for fitting multivariate
linear models
User-friendly software allows for ease of
changing assumptions, adding variables, testing
results, visual depiction of actual and expected
results
Methodology based on Maximum Likelihood
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 3 (Ratemaking) & 6 (Risk
Classification)


Insurance Rates with Minimum Bias,
Bailey (1963)
The Minimum Bias Procedure – A
Practitioners Guide, Feldblum et al (2002)
Suggested Readings



Something Old, Something New in
Classification Ratemaking with a Novel Use
of GLMs for Credit Insurance, Holler, et al
(1999)
A Practitioners Guide to Generalized Linear
Models, Anderson, et al
A Systematic Relationship Between
Minimum Bias and Generalized Linear
Models, Mildenhall (1999)
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 (cont.)


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 (cont.)





F = Fixed expense ratio
V = Variable expense ratio
L = Expected loss ratio
LER = Loss Elimination Ratio
Deductible credit = L*(1-LER) + F
(1 - V)
Deductible Credits (cont.)
Example: Loss Elimination Ratio
Loss Size
# of
Claims
Total
Losses
Average
Loss
0 to 100
500
30,000
101 to 200
350
201 to 500
Loss Eliminated by Deductible
$100
$200
$500
60
30,000
30,000
30,000
54,250
155
35,000
54,250
54,250
550
182,625
332
55,000
110,000
182,625
501 +
335
375,125
1120
33,500
67,000
167,500
Total
1,735
642,000
370
153,500
261,250
434,375
0.239
0.407
.677
L.E.R.
Deductible Credits (cont.)
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.
Deductible Credits (cont.)
Calculation of Deductible Factor
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
Deductible Credits (cont.)



Calculations above are at one point in time
Need to perform calculation for deductible
credits with each periodic review
Compare results of most recent review to
factors in place to decide if deductible
factors need to change