Relieving the pressure from tight fitting models · 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

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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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Redefining the deviance objective for generalised linear models

Motivation for method


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

© 2011 Towers Watson - used with permission by The Actuarial Profession 3

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http://www.carinsurancemanual.co.uk/images/pic-logo-confused.gif

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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

4 © 2011 Towers Watson - used with permission by The Actuarial Profession

0

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3%

5%

7%

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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



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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



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0.1%

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Chi-squared & F-tests

Wald p-values

Akaike information criteria



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Sample



Sample Linear (Sample)



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Sample Linear (Sample)



Sample Hold-Out Linear (Sample)



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Sample

Model Data Input

Solve for

Parameters




Sample

Model

Hold Out

Hold Out

Test




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Sample

Model

Hold Out


Circular

Reference




Concepts behind Case Deleted

Deviance


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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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The approximate method to calculate μi is

– 99.9% accurate

– “n” times faster




Simple Example


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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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Vehicle Group

Predicted Values

Original Fitted

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Vehicle Group

Predicted Values

Original Fitted

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0

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Vehicle Group

Predicted Values

Original Fitted

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Vehicle Group

Predicted Values

Original Fitted

Simple Example


Simple Example


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Concepts behind Noise Reduction

Method



Sample

Model

Hold Out

Noise

Reduced

Model

Solve for

Parameters



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Model 1

– SD ( yi, μi) = CDD ( yi, μ(i))

Concepts behind the Noise Reduction Method


Model 2

– SD ( yi, μi)



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Model 2

– CDD ( yi, μ(i))



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




Find the “best” scalars in the “Case Deleted

Deviance” sense

Higher Variance parameters get scaled back

most

Take account of parameter correlations



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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%

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-4%

-2%

0%

2%

4%

6%

0% 20% 40% 60% 80% 100%


-14%

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-8%

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0%

2%

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6%

0% 20% 40% 60% 80% 100%



30

Find the “best” scalars in the “Case Deleted

Deviance” sense

Higher Variance parameters get scaled back

most

Take account of parameter correlations



Real Examples


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Profit Margin Value 0.57%

Real Examples – Log Poisson



Real Examples – Log Gamma


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Real Examples – Logit Binomial



Other applications


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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”.


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.


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


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

29/11/2011

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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.

•


41

Contact Details

• Tony Lovick

– Senior Consultant

– Saddlers Court, 64-74 East Street, Epsom, KT17 1HB

– +44 1372 751060

– [email protected]

• Pete Lee

– Director

– Saddlers Court, 64-74 East Street, Epsom, KT17 1HB

– +44 1372 751060

– [email protected]


Documents

Relieving the pressure from tight fitting models · 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