GLM III - The Matrix Reloaded
Claudine Modlin
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GLM III - The Matrix Reloaded
Claudine Modlin
© 2014 Towers Watson. All rights reserved.
Agenda
l
l
l
l
l
l
l
l
"Quadrant Saddles"
The Tweedie Distribution
"Emergent Interactions"
Dispersion Modeling
Modelling sparse claim types
Driver Averaging
Model Validation
Man (with GLM) vs machine
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Interactions
1.7
1.5
1.3
1.1
0.9
0.7
0.5
16
17
18
19
20
21
22
23
24
25
26
27 28 29 30 31
Policyholder Age
32
33
34
40-44
50-54
60-64
70+
35-39
45-49
55-59
65-69
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Interactions
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
16
17
18
19
20
21
22
23
24
25
26
27 28 29 30
Policyholder Age
31
32
33
34
40-44
35-39
45-49
70+
50-54
60-64
55-59
65-69
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Why are interactions present?
l
Because that's how the factors behave
l
Because the multiplicative model can go wrong at the edges
l
1.5 * 1.4 * 1.7 * 1.5 * 1.8 * 1.5 * 1.8 = 26!
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Interactions
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Interactions
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Interactions
Vehicle group
b b - b b b b b b b
Age
b
- - - - - - - - - -
b
- - - - - - - - - -
b
- - - - - - - - - -
-
- - - - - - - - - -
b
- - - - - - - - - -
b
- - - - - - - - - -
b
- - - - - - - - - -
b
b
b
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
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Interactions
Vehicle group
Vehicle group
b b - b b b b b b b
Age
b b - b b b b b b b
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - - - - - - -
b
b b - b b b b b b b
-
- - - - - - - - - -
-
- - - - - - - - - -
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - - - - - - -
b
b b - b b b b b b b
b
b
b
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
b
b
b
b b - b b b b b b b
b b - b b b b b b b
b b - b b b b b b b
Age
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Example
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Example
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Example
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Interactions
Vehicle group
b b - b b b b b b b
Age
Vehicle group
Vehicle group
b b - b b b b b b b
b b - b b b b b b b
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - - - - - - -
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - - - - - - -
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - - - - - - -
-
- - - - - - - - - -
-
- - - - - - - - - -
-
- - - - - - - - - -
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - - - - - - -
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - -
b
- - - - - - - - - -
b
b b - b b b b b b b
b
- - - - -
b
b
b
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
b
b
b
b b - b b b b b b b
b b - b b b b b b b
b b - b b b b b b b
b
b
b
- - - - - - - - - - - - -
Age
Age
b
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Saddles
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Saddles
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Saddles
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Saddles
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Saddles
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Saddles
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Saddles
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Saddles
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Saddles
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Saddles
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Saddles
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Saddles - model comparison
Motor frequency - out of sample
20%
400,000
18%
350,000
16%
300,000
14%
250,000
12%
200,000
10%
150,000
8%
100,000
6%
50,000
0
4%
50%
70%
90%
94%
98%
Exposure
102%
Saddle
106%
110%
130%
150%
Original
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Saddles - model comparison
Motor frequency - out of sample
20%
400,000
18%
350,000
16%
300,000
14%
250,000
12%
200,000
10%
150,000
8%
100,000
6%
50,000
0
4%
50%
70%
90%
94%
Exposure
98%
102%
Observed
106%
Saddle
110%
130%
150%
Original
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Saddles - model comparison
Motor frequency - out of sample
900000
2.6%
2.4%
800000
2.2%
700000
2.0%
600000
1.8%
500000
1.6%
400000
1.4%
300000
1.2%
200000
1.0%
100000
0.8%
0.6%
0
< 80%
80% 86%
83% 89%
86% 92%
89% 95%
92% 98%
95% 101%
98% 104%
Exposure
101% 107%
104% 110%
Observed
107% 113%
110% 116%
Saddle
113% 119%
116% 122%
119% 125%
122% 128%
125% 131%
128% 134%
> 130%
Original
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Saddles - model comparison
Motor frequency - out of time
1000000
0.026
900000
800000
700000
0.021
600000
500000
0.016
400000
300000
0.011
200000
100000
0.006
0
< 80%
80% 84%
82% 86%
84% 88%
86% 90%
88% 92%
90% 94%
92% 96%
94% 98%
96% 100%
98% 102%
100% - 102% - 104% - 106% 104% 106% 108% 110%
Exposure
Observed
108% 112%
Saddles
110% 114%
112% 116%
114% 118%
116% 120%
118% - > 120%
122%
Original
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Saddles - model comparison
Motor renewals - out of sample
0.4
14000
12000
0.35
10000
0.3
8000
6000
0.25
4000
0.2
2000
0.15
0
< 80%
80% - 86%
83% - 89%
86% - 92%
89% - 95%
92% - 98%
Exposure
95% - 101%
Observed
98% - 104% 101% - 107% 104% - 110% 107% - 113% 110% - 116% 113% - 119% 116% - 122%
Saddles
Original
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Agenda
l
l
l
l
l
l
l
l
"Quadrant Saddles"
The Tweedie Distribution
"Emergent Interactions"
Dispersion Modeling
Modelling sparse claim types
Driver Averaging
Model Validation
Man (with GLM) vs machine
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Tweedie GLMs
l
Consider the following empirical probability distribution function
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Tweedie GLMs
l
Raw pure premiums
l
Incurred losses have a point mass at
zero and then a continuous distribution
l
Poisson and gamma not suited to this
l
Tweedie distribution has
— point mass at zero
— a parameter which changes shape above
zero
¥
fY ( y;q , l,a ) = å
n=1
{(lw)
ka (-1/ y)}
exp{lw[q0 y - ka (q0 )]} for y > 0
G(-na)n! y
1-a
n
p (Y = 0) = exp{- lwk a (q 0 )}
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Formulization of GLMs
l
Generally accepted standards for link functions and error distribution
Most Appropriate
Link Function
Most Appropriate
Error Structure
Variance Function
--
Normal
µ0
Claim Frequency
Log
Poisson
µ1
Claim Severity
Log
Gamma
µ2
Claim Severity
Log
Inverse Gaussian
µ3
Raw Pure Premium
Log
Tweedie
µT
Retention Rate
Logit
Binomial
µ (1-µ)
Conversion Rate
Logit
Binomial
µ(1- µ)
Observed Response
--
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Formulization of GLMs
l
More formally:
φV( μˆ )
Var(Y) =
ω
Scale Parameter
l
Variance Function
Prior Weights
Tweedie’s Variance function: V(m) = mp
p=1
Poisson
l p=2
Gamma
l 1<p<2
Poisson/Gamma process
l Other concerns
l Need to estimate both j and p when fitting models
l Typically p»1.5 for incurred claims
l
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Example 1
Vehicle age - frequency
1.1
20
0.9
15
0.7
0.5
10
0.3
5
0.1
-0.1
0
0
1
2
3
4
Exposure
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5
6
7
8
9
10
11
12
Model Prediction at Base levels
13
14
15
16
17
18
19
20
21
22
23
24
Model Prediction +/- 2 Standard Errors
37
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Example 1
Vehicle age - amounts
1.2
1.1
20
1
0.9
15
0.8
0.7
10
0.6
0.5
5
0.4
0.3
0.2
0
0
1
2
Exposure
towerswatson.com
3
4
5
6
7
8
Model Prediction at Base levels
9
10
11
12
13
14
15
16
Model Prediction + 2 Standard Errors
17
18
19
20
21
22
23
24
Model Prediction - 2 Standard Errors
38
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Example 1
Vehicle age - pure premium
1.2
20
1
15
0.8
0.6
10
0.4
5
0.2
0
0
1
2
3
4
5
6
7
8
9
Exposure
towerswatson.com
10
11
12
13
14
15
16
Model Prediction +/- 2 Standard Errors
17
18
19
20
21
22
23
24
25
Traditional
39
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Example 1
Vehicle age - pure premium
1.2
20
1
15
0.8
0.6
10
0.4
5
0.2
0
0
0
towerswatson.com
1
2
3
4
5
6
Exposure
7
8
9
Tweedie
10 11
12 13 14 15 16 17
Model Prediction +/- 2 Standard Errors
18
19 20 21
Traditional
22
23
24
40
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Example 2
Gender - frequency
1.1
160
140
1.05
120
100
1
80
60
0.95
40
20
0.9
0
Male
Exposure
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Female
Model Prediction at Base levels
Model Prediction +/- 2 Standard Errors
41
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Example 2
Gender - frequency
500
450
0.065
400
350
0.06
300
250
0.055
200
150
0.05
100
50
0.045
0
Male
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Exposure (1998)
Exposure (1999)
Loss Year (1998)
Loss Year (1999)
Female
Exposure (2000)
Policyholder
Sex
Loss Year (2000)
Exposure (2001)
Exposure (2002)
Loss Year (2001)
Loss Year (2002)
42
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Example 2
Gender - amounts
160
1.1
140
1.05
120
1
100
0.95
80
0.9
60
0.85
40
0.8
20
0.75
0
Male
Exposure
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Female
Model Prediction at Base levels
Model Prediction +/- 2 Standard Errors
43
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Example 2
Gender - amounts
450
1450
400
1400
350
1350
300
1300
250
1250
200
150
1200
100
1150
50
1100
0
Male
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Female
Exposure (1998)
Exposure (1999)
Exposure Sex
(2000)
Policyholder
Loss Year (1998)
Loss Year (1999)
Loss Year (2000)
Exposure (2001)
Exposure (2002)
Loss Year (2001)
Loss Year (2002)
44
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Example 2
Gender – pure premium
1.2
160
1.15
140
1.1
120
1.05
100
1
80
0.95
60
0.9
40
0.85
20
0.8
0
Male
Model Prediction
at Base levels
Tweedie
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Female
Model Prediction + 2 Standard Errors
Model Prediction - 2 Standard Errors
Combined
45
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Tweedie GLMs
l
Helpful when it's important to fit to incurred costs directly
l
Similar results to frequency/severity traditional approach if frequency
and amounts effects are clearly weak or clearly strong
l
Distorted by large insignificant effects
l
Removes understanding of what is driving results
l
Smoothing harder
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Agenda
l
l
l
l
l
l
l
l
"Quadrant Saddles"
The Tweedie Distribution
"Emergent Interactions"
Dispersion Modeling
Modelling sparse claim types
Driver Averaging
Model Validation
Man (with GLM) vs machine
towerswatson.com
47
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Combining models
Collision
Frequency
x
Collision
Severity
Overall
rates
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Combining models
BI
Frequency
x
BI
Severity
+
PD
Frequency
x
PD
Severity
Overall
rates
+
Collision
Frequency
x
Collision
Severity
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Combining models
PI
Severity
TP
Frequency
x
PI
Propensity
x
PD
Severity
+
Collision
Frequency
x
Collision
Severity
Overall
rates
Apply e.g. trends, case
reserves adjustments,
target loss ratio etc.
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Combining models
BI
Frequency
x
BI
Severity
+
PD
Frequency
x
PD
Severity
Overall
rates
+
Collision
Frequency
x
Collision
Severity
Apply e.g. trends, case
reserves adjustments,
target loss ratio etc.
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Combining models
l
Take models
l
Take relevant mix of business
l
eg current in force policies
l
For each record calculate expected frequencies and severities
according to the models
l
For each record, calculate expected total cost of claims "C"
l
Fit a GLM to "C" using all available factors
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Combining models
Intercept
Sex
Area
Policy
…
82155654
82168746
82179481
82186845
…
Male
Female
Town
Country
Sex Area
…
M
F
M
F
…
…
T
T
C
C
…
PD
Numbers
32%
1.000
0.750
1.000
1.250
PD
Amounts
$1000
1.000
1.200
1.000
0.700
BI
Numbers
12%
1.000
0.667
1.000
0.750
BI
Amounts
$4860
1.000
0.900
1.000
0.833
NUM1
AMT1
NUM2
AMT2
CC1
CC2
RISKPREM
…
32%
24%
40%
30%
…
…
1000
1200
700
840
…
…
12%
8%
9%
6%
…
…
4860
4374
4050
3645
…
…
320
288
280
252
…
…
583.20
349.92
364.50
218.70
…
…
903.20
637.92
644.50
470.70
…
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Except…
Policy
…
82155654
82168746
82179481
82186845
…
Sex Area
…
M
F
M
F
…
…
T
T
C
C
…
NUM1
AMT1
NUM2
AMT2
CC1
CC2
RISKPREM
…
32%
24%
40%
30%
…
…
1000
1200
700
840
…
…
12%
8%
9%
6%
…
…
4860
4374
4050
3645
…
…
320
288
280
252
…
…
583.20
349.92
364.50
218.70
…
…
903.20
637.92
644.50
470.70
…
l
The global risk premium is not multiplicative
l
In the town, women have a modelled claim cost 29% lower than men
l
637.92/903.20=0.706
In the country, women have a modelled claim cost 27% lower than men
l
l
470.07/644.50=0.730
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To solve…
Policy
…
82155654
82168746
82179481
82186845
…
l
Sex Area
…
M
F
M
F
…
…
T
T
C
C
…
NUM1
AMT1
NUM2
AMT2
CC1
CC2
RISKPREM
…
32%
24%
40%
30%
…
…
1000
1200
700
840
…
…
12%
8%
9%
6%
…
…
4860
4374
4050
3645
…
…
320
288
280
252
…
…
583.20
349.92
364.50
218.70
…
…
903.20
637.92
644.50
470.70
…
We can capture this result exactly with an interaction
Thi
si
magecannotcur
r
ent
l
ybedi
s
pl
ayed.
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Example "emergent" interaction
Thi
si
magecannotcur
r
ent
l
ybedi
s
pl
ayed.
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"Emergent" interactions
l
In the above examples the interaction "emerged" from the risk premium
step
l
Emergent interactions are not risk insights, there is no subtle risk effect
we have just discovered
l
The different behaviour is by peril, and the rating factors are just bad proxies
for the peril effects
l
Emergent interactions are corrections to fix problems we have
introduced
l
Best solution is by peril pricing
l
l
Reflects true behaviour
l
Underlying models simple to understand and implement
If not, check for emergent interactions in the risk premium
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Agenda
l
l
l
l
l
l
l
l
"Quadrant Saddles"
The Tweedie Distribution
"Emergent Interactions"
Dispersion Modeling
Modelling sparse claim types
Driver Averaging
Model Validation
Man (with GLM) vs machine
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Modeling the Insurance Risk
ISSUE:
Heterogeneous exposure bases
l
l
Different policies within the same line can cover entirely different structures
(i.e. commercial property)
Goal of a predictive model
l
Ideally would like to separate the heterogenous exposure bases
l
Joint-modeling techniques and quasi-likelihood functions allow for analysis of
heterogeneous environment without separation
3.5
> .2
> .5
> .7
3.0
> 1 .4
> 2 .2
Studentized Standardized Deviance R esiduals
> 2 .9
2.5
> 3 .6
> 4 .3
> 5 .0
2.0
> 5 .7
> 6 .5
> 7 .2
1.5
> 7 .9
> 8 .6
> 9 .3
1.0
> 1 0. 1
> 1 0. 8
> 1 1. 5
> 1 2. 2
0.5
> 1 2. 9
> 1 3. 6
> 1 4. 4
0.0
> 1 5. 1
> 1 5. 8
> 1 6. 5
-0.5
> 1 7. 2
> 1 7. 9
> 1 8. 7
-1.0
> 1 9. 4
> 2 0. 1
> 2 0. 8
-1.5
> 2 1. 5
-2.0
> 2 3. 7
> 2 2. 3
> 2 3. 0
> 2 4. 4
> 2 5. 1
> 2 5. 8
-2.5
> 2 6. 6
> 2 7. 3
-3.0
-3.5
-1, 000
0
1 ,00 0
2,00 0
3, 000
4,00 0
5, 000
6,0 00
7,0 00
8 ,00 0
9, 000
F it te d V alu e
Two concentrations suggests two perils:
split or use joint modeling
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Heterogeneous Exposure Bases
l
l
If possible should be modeled separately
l
If model together, exposures with high variability may mask patterns of less
random risks
l
If loss trends vary by exposure class, the proportion each represents of the
total will change and may mask important trends
l
Independent predictors can have different effects on different perils
If cannot, use joint modeling techniques to improve overall fit
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Generalized Linear Models
l
Formulation of deviance – logarithm of a ratio of likelihoods
Where:
æ Act ö
D
÷÷
= lnçç
a(j)
è Expø
2
(
)
()
(
)
()
~
~
Act = f Y y;θ ,j ' E(Y) = y = b' q
Exp = f Y y;θˆ,j ' E(Y) = mˆ = b' qˆ
Then:
( ) ()
2
~
~
é y~
æ
ö
D
fY (y;θ,j)
θ - yθˆ - b θ + b θˆ ù
÷ = 2´ ê
= lnçç
ú
ˆ ,j) ÷
(
)
a(j)
a
j
f
(y;
θ
è Y
ø
û
ë
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Generalized Linear Models
l
Analyzing the scale parameter
l
When modeling homogeneous data
a (j ) =
l
j
D
Þj=
ω
dof
Heterogeneous data requires a more rigorous definition of the scale
function
— Scale parameter could vary in a systematic way with other predictors
— Construct and fit formal models for the dependence of both the mean and
the scale
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Dispersion Model Form
l
Double generalized linear models
l
Response model
Y~fY (y;θ;j)
E(Y) = b'(θ)
jb''(θ)
Var(Y)=
ω
l
Dispersion model
D~fD (d;x,t)
E(D) = b'(x)
tb''(x)
Var(D) =
ω
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Where
2
Y - m)
(
d=
V(m)
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Dispersion Model Form
l
Dispersion adjustments
l
l
Pearson residual has excess variability (deviance residual has bias)
Distribution
Adjustment
Normal
0
Poisson
f/(2m)
Gamma
3f
Parameter in the adjustment term is the scale parameter from the original
response model
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Dispersion Model Results
l
Dispersion model is integrated with original response model
Response
Weight
Initial Response
Model
Loss /
Exposure
Exposure
Dispersion
Model
Squared
Pearson
Residual
Exposure/
(Exposure + Adjustment)
Final Response
Model
Loss /
Exposure
Exposure/ Squared Pearson
Residual
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Agenda
l
l
l
l
l
l
l
l
"Quadrant Saddles"
The Tweedie Distribution
"Emergent Interactions"
Dispersion Modeling
Modelling sparse claim types
Driver Averaging
Model Validation
Man (with GLM) vs machine
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Amplification of the BI signal using PD experience
l
Fit straight to BI
l
Use PD model as a guide in free fitting BI
l
Use PD model structure
l
Offset PD relativities onto BI data as starting point
l
BI/PD proportion model:
l
BI frequency = BI/PD proportion * PD frequency
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More
Data
Less
Data
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Reference models
BI
Frequency
x
BI
Severity
PD
Frequency
x
PD
Severity
BI
Severity
TP
Frequency
x
BI
Propensity
x
PD
Severity
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Reference models
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
A
B
C
D
E
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Reference models
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
A
B
C
D
E
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Reference models
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
A
B
C
D
E
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Reference models
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
A
B
C
D
E
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Reference models
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
A
B
C
D
E
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Reference models
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
A
B
C
D
E
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Reference models
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
A
B
C
D
E
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Reference models
E[Yi] = mi = g-1(SXij.bj+xi)
Offset term
l
When modeling BI set PD fitted values to be offset term
l
GLM will seek effects over and above assumed PD effect
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Experiment
(1) GLM on BI claims on all the data - the "correct" answer
Real large company
10% random sample
Pretend
small
company
(2) Traditional GLM on BI claims on the "small company"
(3) Propensity reference model on BI claims cf PD claims
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Example result
1.2
20,000
1.15
18,000
1.1
16,000
1.05
14,000
1
12,000
0.95
10,000
0.9
8,000
0.85
6,000
0.8
4,000
0.75
2,000
0.7
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Vehicle Group
Exposure
"Correct"
Traditional
Reference
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Example result
200,000
1.9
180,000
160,000
1.7
140,000
120,000
1.5
100,000
80,000
1.3
60,000
40,000
1.1
20,000
0.9
0
Provisional
0
1
2
3
Licence Age
Exposure
"Correct"
Traditional
Reference
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Agenda
l
l
l
l
l
l
l
l
"Quadrant Saddles"
The Tweedie Distribution
"Emergent Interactions"
Dispersion Modeling
Modelling sparse claim types
Driver Averaging
Model Validation
Man (with GLM) vs machine
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Household Averaging
l
l
l
Historically companies assigned operators to vehicles for the purpose of
rating
More recently driver averaging strategies are deployed to capture
household
Average may consider all drivers or a subset
l
l
l
This choice may affect other household composition factors
Types of averages
l
Straight vs. geometric average
l
Weighted average
l
Modified
l
Average/assignment hybrid
Modeling data needs to mimic the transaction
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Model Design
l
In all modeling projects, it is imperative that the data set up mimic the
rating
l
Consider the following example…
l
Operator
Class Factor
Vehicle
Operator
Vehicle Rate
V1
Dad
$500
Dad
0.80
V2
Mom
$450
Mom
0.85
Junior
2.80
Assume Mom had a $1000 claim in Dad’s car
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Assignment
l
Driver assignment methodology each record represents a single
vehicle with one assigned operator
Veh
Op
Sym
MYR
Age
Sex
Type
Yths
Drvrs
Vehs
Exp
Clm
Losses
Prem
V1
Junior
17
2006
16
M
OO
1
3
3
1
1
1,000
1,400.00
V2
Mom
17
2005
43
F
PO
1
3
3
1
0
0
382.50
l
Operator characteristics based on assigned operator
l
Vehicle characteristics based on vehicle
l
Policy characteristics “catch” other drivers
l
Losses assigned to vehicle
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Straight Average
l
Straight average methodology:
ℎ
l
×
1
+
2
3
+
3
Which can be deconstructed:
towerswatson.com
ℎ
×
ℎ
×
ℎ
×
1
3
2
3
3
3
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Straight Average
l
Straight average methodology each record represents a single vehicle
and operator combination
Veh
Op
Sym
MYR
Age
Sex
Yths
Drvrs
Vehs
Exp
Clm
Losses
Prem
V1
Dad
17
2006
45
M
1
3
3
1/3
0
0
133.33
V1
Mom
17
2006
43
F
1
3
3
1/3
1
1,000
141.67
V1
Junior
17
2006
16
M
1
3
3
1/3
0
0
466.67
V2
Dad
17
2005
45
M
1
3
3
1/3
0
0
120.00
V2
Mom
17
2005
43
F
1
3
3
1/3
0
0
127.50
V2
Junior
17
2005
16
M
1
3
3
1/3
0
0
420.00
l
Policy characteristics are same, but less predictive
l
Exposure split amongst the vehicle
l
Losses assigned to vehicle/operator combination
l
iid is a major concern
l
What about Comprehensive?
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Geometric Average
l
Geometric average methodology:
ℎ
l
×
1
+
2
+
3
/
No direct decomposition
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Geometric Average
l
Geometric methodology each record represents a single vehicle
Veh
Sym
MYR
# of
Dads
# of
Moms
# of
Juniors
Exp
Clm
Losses
V1
17
2006
1/3
1/3
1/3
1
1
1,000
619.72
V2
17
2005
1/3
1/3
1/3
1
0
0
557.74
Prem
l
Policy characteristics are same, but less predictive
l
Predictors are translated to counts
l
Losses assigned to vehicle
l
More challenging to add operator interactions or variates
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Weighted Average
l
Weighted average methodology for a straight average approach
Veh
Op
Sym
MYR
Age
Sex
Type
Yths
Drvrs
Vehs
Exp
Clm
Losses
Prem
V1
Dad
17
2006
45
M
PO
1
3
3
1/3
0
0
133.33
V1
Mom
17
2006
43
F
OC
1
3
3
1/3
1
1,000
141.67
V1
Junior
17
2006
16
M
OC
1
3
3
1/3
0
0
466.67
V2
Dad
17
2005
45
M
OC
1
3
3
1/3
0
0
120.00
V2
Mom
17
2005
43
F
PO
1
3
3
1/3
0
0
127.50
V2
Junior
17
2005
16
M
OC
1
3
3
1/3
0
0
420.00
l
Creates a relationship between the vehicle and the operator
l
Uses the model to determine the weights
l
More accurate as it requires more information
ℎ
towerswatson.com
1
×
1
∗
+
2
∗
+
3
∗
3
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Agenda
l
l
l
l
l
l
l
l
"Quadrant Saddles"
The Tweedie Distribution
"Emergent Interactions"
Dispersion Modeling
Modelling sparse claim types
Driver Averaging
Model Validation
Man (with GLM) vs machine
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Validate Models
Holdout samples
l
Holdout samples are effective at validating model
l
Determine estimates based on part of data set
l
Uses estimates to predict other part of data set
Full Test/Training for Large Data Sets
Train
Data
Data
Partial Test/Training for Smaller Data Sets
All
Data
Build
Models
Build
Models
Data
Split Data
Test
Data
Compare
Predictions
to Actuals
Train
Refit
Data Parameters
Split Data
Test
Data
Compare
Predictions
to Actuals
Predictions should be close to actuals for heavily populated cells
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Validate Models
Gains curves
— Compare predictiveness of models
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Validate Models
Lift curves
— Compare predictiveness of models
Model Validation
Model Validation
0.18
26,000
24,000
0.16
26,000
24,000
0.16
Data
22,000
20,000
20,000
0.12
14,000
0.06
12,000
10,000
0.04
Weights
We i g h te d a v er a g e
0.10
0.08
Current
model
18,000
16,000
W ei gh t s
We i g h te d a v er a g e
0.12
Current
model
18,000
0.10
Data
22,000
0.14
0.14
16,000
0.08
14,000
0.06
12,000
W ei gh t s
0.18
Weights
10,000
0.04
8,000
8,000
0.02
0.02
6,000
6,000
0.00
0.00
4,000
4,000
-0.02
2,000
-0.04
0
>5.36262778826825E-02,
<=5.52043444104771E-02
>7.60324645215353E-02,
<=7.87366371098482E-02
Absol ute value: Current model
-0.02
2,000
-0.04
0
>5.40121604953223E-02,
<=5.55854196128298E-02
>7.57104580487717E-02,
<=7.83821667344998E-02
Absol ute value: Current model
— More intuitive but difficult to assess performance
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Validate Models
X-Graphs
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Validate Models
Residual analysis
l
Recheck residuals to ensure appropriate shape
Studentized Standardized Deviance Residuals by Policyholder Age
10
8
6
4
2
0
-2
-4
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81
+
36
-4
0
41
-4
5
46
-5
0
51
-5
5
56
-6
0
61
-6
5
66
-7
0
71
-7
5
75
-8
0
31
-3
2
33
-3
5
29
30
28
26
27
24
25
22
23
20
21
18
19
17
lt
Is the contour plot symmetric?
17
-6
Does the Box-Whisker show
symmetry across levels?
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Agenda
l
l
l
l
l
l
l
l
"Quadrant Saddles"
The Tweedie Distribution
"Emergent Interactions"
Dispersion Modeling
Modelling sparse claim types
Driver Averaging
Model Validation
Man (with GLM) vs machine
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Machine vs man
vs
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Machine vs man
vs
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Machine vs man
vs
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Machine vs man
What is the
underlying
process?
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Machine vs man
Underwriting
Claims
Cover Level
Fraud Likelihood
1.25
50
1.25
50
40
1.15
40
1.15
30
1.05
30
1.05
20
20
0.95
0.95
10
0.85
<= 4. 5
> 4. 5, <= 5
> 5, <= 5.5
> 5. 5, <= 6
> 6, <= 6. 5
New Business Historic
> 6. 5, <= 7
> 7, <= 7. 5
>7. 5, <= 8
> 8, <= 8. 5
>8. 5, <= 9
New Business Recent
> 9, <= 9. 5
> 9. 5
0
Relativity
10
0.85
1
2
3
4
5
6
New Business Historic
7
8
9
10
11
12
13
New Business Recent
14
15
0
Relativity
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Machine vs man
What are the
underlying
drivers?
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Drivers of elasticity
Need
Affordability
Elasticity
Alternatives
Shopping
Preferences
Brand
Affinity
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Machine vs man
I reckon that lots of recent bodily
injury changes are down to new types
of claims
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Claims management companies
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105
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BI models - "insurance" and "compensation" risk
Modelled Relativities - Accident Year
1.20
50
45
40
1.10
35
30
1.00
25
20
15
0.90
10
5
0.80
0
2004
2005
Exposure
2006
Compensation Risk
2007
2008
Insurance Risk
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Modelled Relativities - ABI20 Vehicle Group
1.20
30
25
1.10
20
1.00
15
Proportion TPD with BI - Rated Driver Age
10
0.90
10
5
0.80
0
8
Exposure
Compensation Risk
Insurance Risk
6
Proportion TPD with BI - Car Age
4
20
2
15
0
17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
10
Exposure
Compensation Risk
Insurance Risk
5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25+
Unknown
0
1
0
towerswatson.com
Exposure
Compensation Risk
Insurance Risk
© 2014 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.
GLM III - The Matrix Reloaded
Duncan Anderson, Serhat Guven
© 2012 Towers Watson. All rights reserved.
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