Post on 04-Nov-2020
29/11/2011
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© 2011 Towers Watson - used with permission by The Actuarial Profession
Redefining the deviance objective for generalised linear models Tony Lovick, Pete Lee
Relieving the pressure from tight
fitting models
October 2011
Agenda
• Motivation for method
• Concepts behind
Case Deleted Deviance
• Simple Example
• Concepts behind
Noise Reduction Method
• Real Examples
• Other applications
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© 2011 Towers Watson - used with permission by The Actuarial Profession
29/11/2011
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Redefining the deviance objective for generalised linear models
Motivation for method
© 2011 Towers Watson - used with permission by The Actuarial Profession
Higher Competitiveness Lower
– Unintended competitiveness through unwitting under-pricing is punished severely
Aggregator
policies written
Aggregator
conversion rate
Internet
conversion rate
Internet
policies
written
Internet quotes
Motivation for method
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100+ candidate variables
~25 factors – more when looking at
e.g. external data for postcoding
Up to ~12 interactions
Multi-dimension effects via “scores”
Reduced scope for competitor
benchmarking
Dilemma of over-fitting vs. danger of
anti-selection Old cars
New cars
Motivation for methods
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0
2
4
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12
3%
5%
7%
9%
11%
13%
15%
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 71 73 75+
Policyholder Age - Female
Exposure Observed Average Fitted Average
Motivation for methods
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0
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-5%
0%
5%
10%
15%
20%
25%
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 71 73 75+
Policyholder Age * Gender
Male Exposure Female Exposure Male Female
Motivation for methods
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0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.7%
0.8%
0 2 4 6 8 10 12 14 16
Chi-squared & F-tests
Wald p-values
Akaike information criteria
Motivation for methods
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Motivation for methods
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Motivation for methods
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Sample
Motivation for methods
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Sample Linear (Sample)
Motivation for methods
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Sample Linear (Sample)
Motivation for methods
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Sample Hold-Out Linear (Sample)
Motivation for methods
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Sample
Model Data Input
Solve for
Parameters
Sample Hold-Out Linear (Sample)
Motivation for methods
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Sample
Model
Hold Out
Hold Out
Test
Sample Hold-Out Linear (Sample)
Motivation for methods
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Sample
Model
Hold Out
Sample Hold-Out Linear (Sample)
Circular
Reference
Motivation for methods
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Redefining the deviance objective for generalised linear models
Concepts behind Case Deleted
Deviance
© 2011 Towers Watson - used with permission by The Actuarial Profession
29/11/2011
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Standard Deviance
– SD ( yi, μi)
“Case Deleted” Deviance
– CDD (yi, μ(i))
“Pattern”
– Pattern1,2 = CDD1 - CDD2
“Noise”
– Noise1,2 = SD1 - SD2 - Pattern1,2
“Value”
– Value1,2 = Pattern1,2 – 5 * Noise1,2
Concepts behind the Case Deleted Deviance
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Concepts behind the Case Deleted Deviance
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The approximate method to calculate μi is
– 99.9% accurate
– “n” times faster
Concepts behind the Case Deleted Deviance
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Redefining the deviance objective for generalised linear models
Simple Example
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29/11/2011
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0
5
10
15
20
25
30
35
40
45
-0.01
0
0.01
0.02
0.03
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0.08
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Unknown
Vehicle Group
Predicted Values
Original Fitted
0
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35
40
45
-0.01
0
0.01
0.02
0.03
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Unknown
Vehicle Group
Predicted Values
Original Fitted
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40
45
-0.01
0
0.01
0.02
0.03
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Unknown
Vehicle Group
Predicted Values
Original Fitted
0
5
10
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40
45
-0.01
0
0.01
0.02
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Unknown
Vehicle Group
Predicted Values
Original Fitted
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-0.01
0
0.01
0.02
0.03
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0.08
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Unknown
Vehicle Group
Predicted Values
Original Fitted
Simple Example
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Simple Example
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29/11/2011
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Redefining the deviance objective for generalised linear models
Concepts behind Noise Reduction
Method
© 2011 Towers Watson - used with permission by The Actuarial Profession
Sample Hold-Out Linear (Sample)
Sample
Model
Hold Out
Noise
Reduced
Model
Solve for
Parameters
Motivation for methods
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Model 1
– SD ( yi, μi) = CDD ( yi, μ(i))
Concepts behind the Noise Reduction Method
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Model 2
– SD ( yi, μi)
Concepts behind the Noise Reduction Method
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Model 2
– CDD ( yi, μ(i))
Concepts behind the Noise Reduction Method
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Parameter
NumberName Value Standard Error
Standard
Error (%)
Alias
Indicator (%)Weight Weight (%) Exp(Value)
1 Mean - 2.743 0.031 1.1 236,207 100.0 0.064
2 Business Use Indicator (Y) - 0.060 0.020 34.1 100,320 42.5 0.942
- Business Use Indicator (N) 135,888 57.5
- Garage Indicator (N) 200,845 85.0
3 Garage Indicator (Y) 0.114 0.027 23.6 35,362 15.0 1.120
13 New Rating Area (OPoly(1)) 0.117 0.010 8.9 236,207 100.0 1.124
14 VA Curve 1 (OPoly(1)) - 0.265 0.016 6.0 219,928 93.1 0.768
15 VA Curve 1 (OPoly(2)) - 0.076 0.017 22.5 219,928 93.1 0.926
16 PA Curve 1 spline 1 (OPoly(1)) 0.212 0.039 18.5 8,194 3.5 1.236
17 PA Curve 1 spline 3 (OPoly(1)) 0.041 0.009 23.1 229,373 97.1 1.042
18 PA Curve 1 spline 4 (OPoly(1)) - 0.064 0.009 13.9 229,373 97.1 0.938
19 YADA Curve 1 (OPoly(1)) - 0.176 0.014 8.2 236,207 100.0 0.838
20 YADA Curve 1 (OPoly(2)) 0.062 0.012 19.0 236,207 100.0 1.064
21 VG Curve 1 spline 1 (OPoly(1)) - 0.242 0.119 49.4 203,278 86.1 0.785
22 VG Curve 1 spline 2 (OPoly(1)) - 0.116 0.070 60.0 235,863 99.9 0.890
23 VG Curve 1 spline 3 (OPoly(1)) - 0.050 0.055 109.6 235,863 99.9 0.951
24 VG Curve 1 spline 4 (OPoly(1)) - 0.177 0.092 52.3 235,863 99.9 0.838
Find the “best” scalars in the “Case Deleted
Deviance” sense
Higher Variance parameters get scaled back
most
Take account of parameter correlations
Concepts behind the Noise Reduction Method
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29/11/2011
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-14%
-12%
-10%
-8%
-6%
-4%
-2%
0%
2%
4%
6%
0% 20% 40% 60% 80% 100%
Parameter Inc. Variance
Cjk Is the Variance Covariance Matrix
iii
i
iii yg
h
h
1
jk
iikjkiji WXCXh
kjCovCov
kjVarVarC
kjkjkkjj
jjjj
jk,,,
,2
* 1
p
pkp
m
pjjk XWXC
iii
i
iii yg
h
h*
***
1
jk
iikjkiji WXCXh **
-14%
-12%
-10%
-8%
-6%
-4%
-2%
0%
2%
4%
6%
0% 20% 40% 60% 80% 100%
Parameter Inc. Variance
-14%
-12%
-10%
-8%
-6%
-4%
-2%
0%
2%
4%
6%
0% 20% 40% 60% 80% 100%
Parameter Inc. Variance
Concepts behind the Noise Reduction Method
30
Find the “best” scalars in the “Case Deleted
Deviance” sense
Higher Variance parameters get scaled back
most
Take account of parameter correlations
© 2011 Towers Watson - used with permission by The Actuarial Profession
Redefining the deviance objective for generalised linear models
Real Examples
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29/11/2011
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Profit Margin Value 0.57%
Real Examples – Log Poisson
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Profit Margin Value 0.69%
Real Examples – Log Gamma
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Profit Margin Value 3.4%
Real Examples – Logit Binomial
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Redefining the deviance objective for generalised linear models
Other applications
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29/11/2011
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Other Applications
• “Case Deleted” Deviance
– Completely generic concept, that can be applied to any model type
– For models with quadratic parameter convergence, a similar
approximation should exist
• “Value” Measure
– An absolute mechanism to compare disparate model options.
• Noise Reduction Method
– GLMs are convenient in that they generate the Variance Covariance
matrix as part of the solution.
– But could be applied to any model where this can be estimated.
– Result is a set of scaled parameters which are “most predictive”.
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Other Applications
• Can be applied to
– Neural Networks,
– Genetic Algorithms,
– Decision Trees, etc.
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Questions or comments?
Expressions of individual views by
members of The Actuarial Profession
and its staff are encouraged.
The views expressed in this presentation
are those of the presenter.
38 © 2011 Towers Watson - used with permission by The Actuarial Profession
Tony Lovick, MA FIA
Pricing Actuary
• Tony graduated in Mathematics from Oxford University in 1987, and qualified as a Fellow of the Institute of
Actuaries in 1994. He spent twenty one years with Aviva Group, before joining EMB as a Senior
Consultant.
• Tony undertook a number of roles within Aviva, most recently as Price Optimisation Actuary, “Pay as you
drive” Actuary and Head of Statistics and Development, in the Personal Lines Pricing Division of Norwich
Union.*
• Tony is interested in innovative actuarial research and its delivery through pragmatic systems development.
As Price Optimisation Actuary he undertook the client side pricing and architecture design, concluding in a
successful Motor Renewal pilot.
• As the actuary leading the research for Pay as you drive, he helped inspire the analysis, build of the data
warehouse systems**, and launch of the product to market. As part of this project Aviva prepared two
patents with Tony listed as the inventor, one of which is now granted***.
• As Head of Statistics he led the implementation of full postcode risk cost models for motor and home
insurance, pioneering the introduction of external data to Aviva rating systems.
* http://www.linkedin.com/in/anthonylovick
** http://www.silicon.com/financialservices/0,3800010322,39169285,00.htm
*** http://v3.espacenet.com/textdoc?DB=EPODOC&IDX=GB2436880&F=0
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Peter Lee FIA
Director
• Peter Lee is a Director at Towers Watson and global lead in pricing innovation with over twenty years
experience in non-life insurance. Prior to joining EMB Peter worked at Allianz UK as the Personal Lines
Actuary.
• Whilst at EMB, Peter worked for a large number of insurers throughout the world in different regulatory
regimes, advising over a broad spectrum of areas and products ranging from claims reserving to pricing and
the design of management information. Throughout his career Peter has been at the forefront of innovation,
being one of the pioneers of the application of statistical modelling to personal lines pricing and then
extending these techniques to commercial lines.
• More recently Peter developed EMB’s price optimisation solution which has now been implemented in many
of the largest general insurers in the world. Much of Peter’s work involves embedding technical analysis and
demand-based pricing into a wider pricing process, allowing these enhanced capabilities to be more
effectively leveraged. Peter is now working with clients to link pricing and marketing to provide an enhanced
framework for managing customer value.
•
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Contact Details
• Tony Lovick
– Senior Consultant
– Saddlers Court, 64-74 East Street, Epsom, KT17 1HB
– +44 1372 751060
– Tony.Lovick@TowersWatson.com
• Pete Lee
– Director
– Saddlers Court, 64-74 East Street, Epsom, KT17 1HB
– +44 1372 751060
– Pete.Lee@TowersWatson.com
© 2011 Towers Watson - used with permission by The Actuarial Profession