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Average Prem ium Model Actuarial Research Conference
Brant Wipperman, FCAS FCIA MSc (SFU) July 27, 2010
Motivation
- Revenue Requirements
- Monitoring our book of business
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Average Prem ium Model Actuarial Research Conference Brant - - PowerPoint PPT Presentation
Average Prem ium Model Actuarial Research Conference Brant - - PowerPoint PPT Presentation
Average Prem ium Model Actuarial Research Conference Brant Wipperman, FCAS FCIA MSc (SFU) July 27, 2010 Motivation Revenue Requirements Monitoring our book of business 1 Basic On-Level Average Premium 630 A 625 v g 620 . P
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Basic On-Level Average Premium
600 605 610 615 620 625 630 2002 2003 2004 2005 2006 A v g . P r e m i u m ( $ ) Year
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Personal TPL On-Level Avg. Premium
600 605 610 615 620 625 630 2005 2006 2007 2008 2009 A v g . P r e m i u m ( $ ) Year
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Overview
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Exposure Forecasts Average Premium Forecasts PCA
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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
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Vehicle Use Groups
- Personal
– Pleasure – Commute – Business – Senior – Motorcycle – Motor home – Collector
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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
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Overview
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Historical Exposure Data External Factors Econometric Models Factor Forecasts Exposure Forecasts Average Premium Forecasts PCA Exposure Model
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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
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) (
1
Z Z A
T n
⋅ =
V V A λ = ⋅
2 1
−
⋅ = L V B
B Z P ⋅ =
L B L V S ⋅ = ⋅ =
2 1
T
T T C ⋅ =
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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
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How many components?
0% 5% 10% 15% 20% 25% 30% 35% 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132 %
- f
V a r i a n c e Principal Com ponent
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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 %
- f
V a r i a n c e Principal Com ponent
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88% 94%
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Overview
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Chosen PCs Historical Exposure Data External Factors Econometric Models Factor Forecasts Exposure Forecasts Exposure Model PCA
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Historical Exposure Data 30 correlated variables Chosen PCs Ortho-normal transformation Principal Components 30 uncorrelated variables Scree Proportion of variance Other 6 uncorrelated variables
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Overview
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Chosen PCs Historical Exposure Data Linear Regression Models Historical Average Premium External Factors Econometric Models Factor Forecasts Exposure Forecasts Chosen PC Forecasts Average Premium Forecasts PCA Exposure Model Average Premium Model
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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 A v g . P r e m i u m ( $ ) Month Modeled 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 A v g . P r e m i u m Year Actual 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