average premium model - soa

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

7

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

10

What is PCA?

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

• Uses linear algebra

11

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)

13

PCA Resolves the Issues

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

14

PCA Process

• Step 1: Create new set of explanatory variables

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

15

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

16

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

18

Chosen PCs

HistoricalExposure

Data

ExternalFactorsEconometric

Models FactorForecasts

ExposureForecasts

Exposure ModelPCA

19

HistoricalExposure

Data30 correlated variables

Chosen PCs

Ortho-normaltransformation

PrincipalComponents 30 uncorrelated variables

ScreeProportion of varianceOther

6 uncorrelated variables

Overview

20

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

21

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

22

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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

25

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?

26

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