MAF Fall Meeting
September 26, 2002
GLMs in Personal
Lines Pricing
Claudine Modlin, FCAS
Watson Wyatt Insurance &
Financial Services Inc.
www.watsonwyatt.com/pretium
WWW.WATSONWYATT.COM
Agenda

Overview of GLMs in the rating process

GLMs in practice
–
data
–
diagnostics
–
interactions

Territory analysis

How to get started
Agenda

Overview of GLMs in the rating process

GLMs in practice
–
data
–
diagnostics
–
interactions

Territory analysis

How to get started
Objective
Age
Sex
Vehicle
Rate Scheme
Area
Claim
Limit
Premium
Modeling the cost of claims
Age
Sex
Vehicle
Model
Area
Claim
Limit
Expected
cost of
claims
Modeling the cost of claims
BI
Freq
x
Amt
= Cost 1
PD
Freq
x
Amt
= Cost 2
MED Freq
x
Amt
= Cost 3
COL Freq
x
Amt
= Cost 4
OTC Freq
x
Amt
= Cost 5
Modeling the cost of claims

Rating factors

Statistical techniques
Example auto rating factors

Standard factors:
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
Age
Sex
Marital status
Number years licensed
Claim experience
Territory
Usage
Mileage
Limits
Deductibles
Make/Model of vehicle
Violations
Credit
Multi-line
Multi-car
Safety devices
Theft devices

External data:
– geodemographic data
– geophysical data

Data from other
products:
– banking data
– other insurance data
The failings of one way analysis
M
F
T
40%
20%
Claims
*2
True risk
C
20%
10%
M
F
Total
M
F
Total
C
100
200
300
C
20
20
40
Total
100
40
140
One-way
Exposure
T
200
100
300
T
80
20
100
Total
300
300
600
*
2.5
M
F
Exp
300
300
Claims
100
40
Ratio
33.3%
13.3%
T
C
300
300
100
40
33.3%
13.3%
Example correlation
Number of policies
35
30
25
20
15
10
5
Old
High
Vehicle Age
Vehicle Value
New
Low
Generalized linear models
E[Y] = m =
-1
g (X.b
+ x)
Var[Y] = f.V(m) / w

Consider all factors simultaneously

Allow for nature of random process

Robust and transparent

EU industry standard
Why GLMs over other methods

One-way and two-way analyses
–

Iteratively standardized one-ways
–

Not transparent, hard to interpret, can be unstable with new
types of policy, easy to over/under fit
Cluster analyses / "segmenting"
–

No diagnostics, no faster than GLMs, less flexibility for allowance
of random process, not always tractable solution
Neural networks
–

Distorted by correlations, no diagnostics
Suitable for marketing but less appropriate for assessing
continuous risk; does not fit with rating structures
Data mining
–
General term for all of the above but can often be merely oneway or two-way analyses on subsets of data
Example of GLM output
(real UK data)
0.25
180
0.2
160
0.15
0.1
140
0.05
0%
Log of multiplier
-5%
-4%
-0.05
100
-0.1
-15%
-17%
-0.15
80
-19%
-20%
-0.2
60
-0.25
40
-0.3
-0.35
20
-0.4
-0.45
0
1
2
3
4
5
Factor
Exposure
Approx 2 SE from estimate
GLM estimate
6
7
Exposure (policy years)
120
0
Example of GLM output
0.25
(real UK data)
22%
180
0.2
160
0.15
10%
0.1
7%
6%
140
0.05
0%
Log of multiplier
-4%
-5%
-0.05
100
-0.1
-15%
-16%
-0.15
-19%
-17%
80
-19%
-20%
-0.2
60
-0.25
40
-0.3
-0.35
20
-0.4
-0.45
0
1
2
3
4
5
6
Factor
Exposure
Oneway relativities
Approx 2 SE from estimate
GLM estimate
7
Exposure (policy years)
120
0
Modeling the cost of claims
BI
Freq
x
Amt
= Cost 1
PD
Freq
x
Amt
= Cost 2
MED Freq
x
Amt
= Cost 3
COL Freq
x
Amt
= Cost 4
OTC Freq
x
Amt
= Cost 5
The premium rating process
BI
Freq
x
Amt
= Cost 1
PD
Freq
x
Amt
= Cost 2
MED Freq
x
Amt
= Cost 3
COL
Freq
x
Amt
= Cost 4
OTC
Freq
x
Amt
= Cost 5
Rate level adjustments
Profit loadings
Risk model
The premium rating process
BI
Freq
x
Amt
= Cost 1
PD
Freq
x
Amt
= Cost 2
MED Freq
x
Amt
= Cost 3
COL
Freq
x
Amt
= Cost 4
OTC
Freq
x
Amt
= Cost 5
Current Rates
Rate level adjustments
Profit loadings
Risk
Model
Compare
Factor effect analysis
Demonstration job
Run 10 Model 2 - Third party material, standard risk premium run - Unsmoothed standard risk premium model
0.8
250000
89%
0.6
68%
67%
200000
30%
150000
22%
0.2
22%
22%
0%
11%
0%
1%
100000
0
0%
0%
0%
0%
-22%
-0.2
-10%
50000
-18%
-0.4
0
17-21
22-24
25-29
30-34
35-39
40-49
50-59
60-69
70+
MAGE - Age of driver
Approx 2 SEs from unsmoothed estimate
Unsmoothed unrestricted estimate
Unsmoothed restricted estimate
Current rating structure
Exposure
Log of multiplier
0.4
Factor effect analysis
Demonstration job
Run 10 Model 2 - Third party material, standard risk premium run - Unsmoothed standard risk premium model
0.8
200000
0.6
82%
150000
49%
0.2
22%
8%
31%
11%
100000
0%
0
-7%
0%
-16%
-15%
50000
-10%
-0.2
-18%
-0.4
0
2-33%
to 7
8
9
10
11
12
13 to 17
MGROUP - Group of vehicle
Approx 2 SEs from unsmoothed estimate
Unsmoothed unrestricted estimate
Unsmoothed restricted estimate
Current rating structure
Exposure
Log of multiplier
0.4
Factor effect analysis
Demonstration job
Run 10 Model 2 - Third party material, standard risk premium run - Unsmoothed standard risk premium model
0.35
0.3
500000
28%
0.25
0.2
300000
0.15
12%
0.1
200000
0.05
0%
5%
5%
100000
0
0%
-0.05
0
Yearly
Half-yearly
Quarterly
MPFREQ - Payment frequency
Approx 2 SEs from unsmoothed estimate
Unsmoothed unrestricted estimate
Unsmoothed restricted estimate
Current rating structure
Exposure
Log of multiplier
400000
Impact analysis
Example job
Currently
profitable
business
7000
6000
Count of records
5000
4000
3000
Currently
unprofitable
business
2000
1000
0
0.450 - 0.550 0.500
0.600
0.650 0.700
0.750 0.800
0.850 0.900
0.950 1.000
1.050 - 1.150 1.100
1.200
1.250 1.300
1.350 1.400
1.450 1.500
1.550 - 1.650 1.600
1.700
1.750 1.800
Ratio: Risk Premium / Current tariff
1.850 1.900
1.950 2.000
2.050 2.100
2.150 - 2.250 2.200
2.300
2.350 2.400
2.450 2.500
Impact analysis
Example job
7000
180%
170%
160%
6000
150%
140%
5000
120%
4000
110%
100%
3000
90%
80%
2000
70%
60%
1000
50%
40%
0
30%
0.450 0.500
0.600 0.650
0.750 0.800
0.900 0.950
1.050 1.100
1.200 1.250
1.350 1.400
1.500 1.550
1.650 1.700
Ratio: Risk Premium / Current tariff
Yearly
Claims / Earnedprem
1.800 1.850
1.950 2.000
2.100 2.150
2.250 2.300
2.400 2.450
Loss ratio
Count of records
130%
Impact analysis
Example job
Age of driver
7000
180%
170%
160%
6000
150%
140%
5000
120%
4000
110%
100%
3000
90%
80%
2000
70%
60%
1000
50%
40%
0
30%
0.450 0.500
0.600 0.650
0.750 0.800
0.900 0.950
1.050 1.100
1.200 1.250
1.350 1.400
1.500 1.550
1.650 1.700
1.800 1.850
1.950 2.000
2.100 2.150
2.250 2.300
2.400 2.450
Ratio: Risk Premium / Current tariff
17-21
22-24
25-29
30-34
35-39
40-49
50-59
60-69
70+
Claims / Earnedprem
Loss ratio
Count of records
130%
Impact analysis
Example job
Area of garage
7000
180%
170%
160%
6000
150%
140%
5000
120%
4000
110%
100%
3000
90%
80%
2000
70%
60%
1000
50%
40%
0
30%
0.450 0.500
0.600 0.650
0.750 0.800
0.900 0.950
1.050 1.100
1.200 1.250
1.350 1.400
1.500 1.550
1.650 1.700
1.800 1.850
1.950 2.000
2.100 2.150
Ratio: Risk Premium / Current tariff
A
B
C
D
E
F
G
H
Claims / Earnedprem
2.250 2.300
2.400 2.450
Loss ratio
Count of records
130%
Impact analysis
Example job
Payment frequency
7000
180%
170%
160%
6000
150%
140%
5000
120%
4000
110%
100%
3000
90%
80%
2000
70%
60%
1000
50%
40%
0
30%
0.450 0.500
0.600 0.650
0.750 0.800
0.900 0.950
1.050 1.100
1.200 1.250
1.350 1.400
1.500 1.550
1.650 1.700
1.800 1.850
1.950 2.000
Ratio: Risk Premium / Current tariff
Yearly
Half-yearly
Quaterly
Claims / Earnedprem
2.100 2.150
2.250 2.300
2.400 2.450
Loss ratio
Count of records
130%
The premium rating process
Freq
x
Amt
= Cost 1
TPPD Freq
x
Amt
= Cost 2
AD
Freq
x
Amt
= Cost 3
FT
Freq
x
Amt
= Cost 4
WS
Freq
x
Amt
= Cost 5
TPBI
Current Rates
Expense loadings
Profit loadings
Risk
Model
Compare
New
Rates
Competitor
Model
Competitive position
Survey market
–
–
–
–

rate filings
quotation systems
question policyholder
mystery shopping
Investigate competitors'
structures
Apply "cheapest" tariff to own
portfolio

20
18
16
Percentage of contracts

14
Zone A
12
Zone B
10
Zone C
8
6
4
2
0
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
60
70
80
90
100
Percentage change in premium

Use in retention / new
business model
110
120
130
140
150
160
The premium rating process
TPBI
Freq
x
Amt
= Cost 1
TPPD Freq
x
Amt
= Cost 2
AD
Freq
x
Amt
= Cost 3
FT
Freq
x
Amt
= Cost 4
WS
Freq
x
Amt
= Cost 5
Current Rates
Competitor
Model
Compare
Lapse/take-up
Model
Expense loadings
Profit loadings
Risk
Model
New
Rates
Modeling retention
Age
Sex
Vehicle age
D Premium
Model
Probability
of lapsing
Claims
Premium /
Competitors' premium

Model
- rating factors
- payment method
- discount expectation
- source
- claims history
- other products held
- change in coverage
plus…
- change in premium
- competitiveness
Log of multiplier
Retention model - Policyholder age
20
25
30
35
40
45
50
Age of policyholder
Approx 2 SEs from estimate
Unsmoothed estimate
55
60
65
70
Retention model - Change in premium
1
Log of multiplier
0.7
0.4
0.1
-0.2
-0.5
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
Change in premium on renewal
Approx 2 SEs from estimate
Unsmoothed estimate
40
50
60
70
80
90
New business model
Log of multiplier of p/(1-p)
Competitiveness of premium
-47
0.6
0.7
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.3
1.4
1.5
1.6
Quote/Average of the three cheapest quotes on the market
Approx 2 SD from estimate
Smoothed estimate
1.7
1.8
1.9
2
Customer lifetime value
Profitability
Current Rates
Lapse model
Low
Retention
High
Low
-
Risk
Model
High
Target marketing at
these
Increase premiums
Actively target at
renewal (discount
vouchers / phone
calls)
Price elasticity
10
5
0
-80
-5
-70
-60
-50
-40 -30
-20
-10
0
10
20
30
40
50
Premium change ($)
Number of policies
Profit
60
70
80
90
100 110 120
The premium rating process
TPBI
Freq
x
Amt
= Cost 1
TPPD Freq
x
Amt
= Cost 2
AD
Freq
x
Amt
= Cost 3
FT
Freq
x
Amt
= Cost 4
WS
Freq
x
Amt
= Cost 5
Current Rates
Competitor
Model
Compare
Lapse/take-up
Model
Expense loadings
Profit loadings
Risk
Model
New
Rates
Model
office
Agenda

Overview of GLMs in the rating process

GLMs in practice
–
data
–
diagnostics
–
interactions

Territory analysis

How to get started
Data required

Linked policy + claims data

Record: one insured risk (eg car) for one policy
period or portion of policy period for which risk
has not changed

Fields:
– explanatory variables - rating, underwriting,
marketing, external
– stats - earned exposure, incurred claim count,
incurred loss, earned premium (optional)

Minimum of 100,000 earned exposures
Data considerations

Reflect cancellation/endorsement

Include time lag to reduce effect of IBNR

Include dummy variables to standardize for
geography (if countrywide study) and time

Display rating factors applicable at time of
exposure, categorized on current basis
Model iteration diagnostics
Factor 3
Factor 4
Factor 5
Factor 2
Factor 6
Factor 1

Standard errors of
parameter estimates

F-tests / c2 tests on
deviances (with ranks)

Consistency over time

Common sense
Factor 7
Factor 3
Factor 5
Factor 6
Standard errors of
parameter estimates
0.8
101%
82%
65%
0.5
49%
35%
Log of multiplier
22%
0.2
11%
0%
-10%
-0.1
-18%
-26%
-33%
-0.4
-39%
-45%
-50%
-0.7
-1
1
2
3
4
5
6
7
8
9
10
11%
16%
11
12
13
14
-10%
-5%
13
14
15
1.9
1.6
1.3
1
Log of multiplier
0.7
57%
28%
0.4
0.1
-18%
22%
0%
-10%
-18%
-26%
-0.2
28%
-18%
-39%
-0.5
-0.8
-1.1
-1.4
-1.7
-2
1
2
3
4
5
6
7
8
9
10
11
12
15
Deviances
Age
Sex
Vehicle
Model A
Zone
Fitted
value
Deviance = 9585
df = 109954
Multi-car
?
Claims
Age
Sex
Vehicle
Model B
Multi-car
Claims
Fitted
value
Deviance = 9604
df = 109965
Consistency over time
A
B
C
1996
A
1997
B
1998
C
1996
1997
D
1998
D
Common sense

Does it make sense given correlations?

Are ordered categorical variables well behaved?

Can you believe it?

Can underwriters believe it?

Consider results for frequency and amounts at
the same time

Consider results for each claim type at the same
time
Interactions
Example job
Run 12 Model 3 - Small interaction - Third party material damage, Numbers
1
250000
122%
0.8
105%
200000
0.6
150000
0.4
26%
24%
0.2
Exposure
Log of multiplier
56%
100000
0%
-6%
0
-11%
50000
-20%
-0.2
-0.4
0
17-21
22-24
25-29
30-34
35-39
40-49
50-59
60-69
70+
Age of driver
Approx 2 SEs from estimate
Unsmoothed estimate
Smoothed estimate
P level = 0.0%
Rank 7/7
Interactions
Example job
Run 12 Model 3 - Small interaction - Third party material damage, Numbers
0.03
0%
0
400000
-0.06
-0.09
300000
-0.12
200000
Exposure
500000
-0.03
Log of multiplier
600000
-13%
-0.15
100000
-0.18
0
Female
Male
Sex of driver
Approx 2 SEs from estimate
Unsmoothed estimate
Smoothed estimate
P level = 0.0%
Rank 2/7
Interactions
Example job
Run 5 Model 3 - Small interaction - Third party material damage, Numbers
1
155%
138%
300000
0.8
250000
63%
63%
46%
0.4
200000
40%
28%
19%
24%
20%
150000
0.2
13%
Exposure
Log of multiplier
0.6
6%
0%
-2%
100000
-6%
0
-11%
-18%
-19%
-0.2
50000
-0.4
0
17-21
22-24
25-29
30-34
35-39
40-49
50-59
60-69
70+
P level = 0.0%
Rank 6/6
Age of driver.Sex of driver
Approx 2 SEs from estimate, Sex of driver: Female
Approx 2 SEs from estimate, Sex of driver: Male
Unsmoothed estimate, Sex of driver: Female
Unsmoothed estimate, Sex of driver: Male
Smoothed estimate, Sex of driver: Female
Smoothed estimate, Sex of driver: Male
Interactions
4
3
2
5
6
Company
7 8 9 10 11 12 13 14 15 16 17 18
Interaction
1
Age / Sex
Area / garaged
   
        
     
     
   
     


   

Age / occupation

Age / vehicle group
Area / vehicle group


Age / marital status / sex




Occupation / use
Group >
Age v
17
18
19
20
21-23
24-26
27-30
31-35
36-40
41-45
46-50
51-60
60+




Use / mileage

1
2
3
4
5
6
7
8
9
10
11
12
13
1.36
1.12
1.08
0.98
0.96
0.82
0.78
0.63
0.55
0.51
0.46
0.40
0.43
1.64
1.31
1.30
1.18
1.13
0.99
0.90
0.78
0.64
0.61
0.55
0.49
0.52
1.79
1.47
1.46
1.36
1.24
1.10
1.07
0.86
0.71
0.66
0.61
0.56
0.55
2.09
1.76
1.63
1.54
1.51
1.31
1.19
0.99
0.85
0.79
0.70
0.64
0.67
2.27
1.84
1.82
1.68
1.65
1.43
1.32
1.09
0.91
0.88
0.76
0.68
0.72
2.42
2.00
1.91
1.79
1.64
1.52
1.39
1.17
0.93
0.88
0.81
0.71
0.73
2.56
2.11
2.02
1.83
1.80
1.51
1.41
1.22
0.99
0.94
0.84
0.78
0.78
2.65
2.19
2.11
1.97
1.85
1.64
1.51
1.32
1.07
0.99
0.92
0.82
0.83
3.27
2.43
2.53
2.19
2.04
1.81
1.65
1.42
1.18
1.09
1.02
0.90
0.93
3.71
2.97
2.88
2.66
2.26
1.93
1.77
1.54
1.29
1.15
1.07
0.99
0.98
4.08
3.29
3.30
3.02
2.55
2.13
1.91
1.66
1.40
1.29
1.12
1.02
1.04
4.36
3.55
3.35
3.20
2.53
2.22
2.01
1.71
1.41
1.31
1.18
1.12
1.11
4.84
3.90
3.63
3.38
2.89
2.47
2.24
1.88
1.53
1.42
1.31
1.20
1.25
Group
1
2
3
4
5
6
7
8
9 10 11 12 13
Factor 0.54 0.65 0.73 0.85 0.92 0.96 1.00 1.08 1.19 1.26 1.36 1.43 1.56
Age

17
18
19
20
21-23
24-26
27-30
31-35
36-40
41-45
46-50
51-60
60+
Factor
2.52
2.05
1.97
1.85
1.75
1.54
1.42
1.20
1.00
0.93
0.84
0.76
0.78
1.00
1.17
1.00
1.00
Agenda

Overview of GLMs in the rating process

GLMs in practice
–
data
–
diagnostics
–
interactions

Territory analysis

How to get started
Geographic rating

Territory is one of the main drivers of cost

Considerable variety in how insurers rate for
territory

One insurer will have limited exposure in any
one area
Spatial smoothing

Fit GLM (excluding current territories)

Map "residual" risk by "region"

Make this residual risk more predictive

Categorize into territories to derive
appropriate loadings
Residual risk
High residual
Low residual
A model form
ri* = Z.ri + ( 1 - Z ) . neighboring experience
where
ri*= smoothed residual risk
ri = unsmoothed residual risk
Definitions of "neighboring"
Example results
Unsmoothed residuals
Smoothed residuals
Finding the parameters
Effect of smoothed vs unsmoothed residual zone
Zone based on smoothed residuals
2.5
400
650%
350
2
345%
300
287%
1.5
Zone based on
smoothed residuals
Log of multiplier
189%
250
139%
1
101%
200
74%
0.5
26%
150
15%
0%
0
-16%
-18%
-22%
-35%
-18%
-22%
100
Exposure (thousand policy years)
418%
464%
-34%
-37%
-0.5
50
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Zone
2 S.E from GLM estimate
GLM estimate
Zone based on unsmoothed residuals
2.5
400
300
Zone based on
unsmoothed residuals
Log of multiplier
1.5
250
1
91%
0.5
29%
41%
99%
79%
64%
200
74%
49%
49%
30%
24%
19%
16%
2%
-3%
63%
150
6%
0%
-4%
-1%
0
100
-0.5
50
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Zone
2 S.E from GLM estimate
GLM estimate
14
15
16
17
18
19
20
Exposure (thousand policy years)
350
2
Agenda

Overview of GLMs in the rating process

GLMs in practice
–
data
–
diagnostics
–
interactions

Territory analysis

How to get started
How can I start?

Programming from scratch

Software applications
– tailored to personal lines
– easy to navigate
– fast, even on PC
– clear output
– cost is often less than the annual
compensation of one actuary
MAF Fall Meeting
September 26, 2002
GLMs in Personal
Lines Pricing
Claudine Modlin, FCAS
Watson Wyatt Insurance &
Financial Services Inc.
www.watsonwyatt.com/pretium
WWW.WATSONWYATT.COM