Average Premium ModelActuarial Research Conference

Brant Wipperman, FCAS FCIA MSc (SFU)

July 27, 2010

Motivation

• Revenue Requirements

• Monitoring our book of business

1

2

Basic On-Level Average Premium

600

605

610

615

620

625

630

2002 2003 2004 2005 2006

Avg.

Premium

(

$)

Year

3

Personal TPL On-Level Avg. Premium

600

605

610

615

620

625

630

2005 2006 2007 2008 2009

Avg.

Premium

(

$)

Year

Overview

4

ExposureForecasts

AveragePremiumForecasts

PCA

Exposure Model

• Historical exposure data– Split into Personal and Commercial– Further split into vehicle use, location, and

bonus-malus groups

• An econometric regression model is fit to each group– Demographic– Economy

5

Vehicle Use Groups

• Personal

– Pleasure– Commute– Business– Senior– Motorcycle– Motor home– Collector

6

Location Groups

• Lower Mainland• Ridge Meadows• Fraser Valley• Squamish/Whistler• Pemberton/Hope• Okanagan• Kootenays• Cariboo• Prince George• Peace River• North Coast• South Island• Mid Island• North Island

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Bonus-Malus Groups

• Claim Rated Scale

– Roadstar (43% discount)

– 25% to 40% discount

– 5% to 20% discount

– Base or surcharge

8

Overview

9

HistoricalExposure

Data

ExternalFactorsEconometric

Models FactorForecasts

ExposureForecasts

AveragePremiumForecasts

PCAExposure Model

Historical Exposure Data

• Too many groups for the average premium model

• Need a dimension reduction technique

• Want to keep all of the groups

• Linear dependencies exist

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What is PCA?

• It transforms a number of correlated variables into a smaller number of uncorrelated variables

• Uses linear algebra

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PCA Notation

12

)(1 ZZA Tn ⋅=

VVA λ=⋅

21−⋅= LVB

BZP ⋅= LBLVS ⋅=⋅= 2

1

TTTC ⋅=

Eigen Decomposition

• Linear algebra problem• Done on correlation matrix of

explanatory variables• Eigenvectors are new explanatory

variables (i.e. principal components)• Each associated eigenvalue represents

variability of eigenvector (or PC)

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PCA Resolves the Issues

• Number of dimensions reduced• All groups ‘retained’• Linear dependencies eliminated

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PCA Process

• Step 1: Create new set of explanatory variables

• Step 2: Determine how many new explanatory variables to retain

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PCA

How many components?

0%

5%

10%

15%

20%

25%

30%

35%

1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132

%

of

Variance

Principal Component

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How many components?

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132

%

of

Variance

Principal Component

17

88% 94%

Overview

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Chosen PCs

HistoricalExposure

Data

ExternalFactorsEconometric

Models FactorForecasts

ExposureForecasts

Exposure ModelPCA

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HistoricalExposure

Data30 correlated variables

Chosen PCs

Ortho-normaltransformation

PrincipalComponents 30 uncorrelated variables

ScreeProportion of varianceOther

6 uncorrelated variables

Overview

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Chosen PCs

HistoricalExposure

Data

LinearRegression

Models

HistoricalAveragePremium

ExternalFactorsEconometric

Models FactorForecasts

ExposureForecasts

Chosen PCForecasts

AveragePremiumForecasts

PCAExposure Model

Average Premium Model

Modeled vs. Actual – Personal TPL

580

600

620

640

660

Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10 Jan-11

Avg.

Premium

(

$)

MonthModeled Actual

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Modeled vs. Actual – Personal TPL

600

605

610

615

620

625

630

2004 2005 2006 2007 2008 2009 2010 2011

Avg.

Premium

YearActual 6 PCs 4 PCs 8 PCs

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Recap - Advantages

• PCs uncorrelated

• PCs organized to reduce dimensionality

• Keeps most of original information

• Determine contribution of each variable

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Recap - Disadvantages

• PCA process not familiar

• PCs can be hard to interpret

• PC weights may change upon updating

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Is PCA Right For You?

• Does multi-collinearity roll off your tongue too easily?

• Are you confident in the set of explanatory variables?

• Do you want to reduce dimensionality without throwing away information?

• Have you been modeling for more than 4 consecutive hours?

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For More Information

• CAS Discussion Paper

– PCA and Partial Least Squares: Two Dimension Reduction Techniques for Regression

• http://www.casact.org/pubs/dpp/dpp08/08dpp76.pdf