Considerations in
Estimating Loss Cost Trends
Kurt Dickmann
& Jim Merz
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Overview of the Paper
Importance of Trends in Ratemaking
Ordinary Least Squares Regression
Characteristics of Insured Losses
Identifying Outliers
Adjusting for Outliers
Comparison of Results
Simulation Results
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Importance of Trends in
Ratemaking
High Leverage on Estimated Losses
Requires Judgment
Scarcity of Actuarial Literature on
Estimation
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Regression
Understood by Management
Commonly Used
Widely Understood
Accepted by Regulators
Included in Spreadsheet Software
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Regression Basics
General Form
 
Yi  f ( X i ,  i )   i
Linear
Yi   0  1 X i   i
Exponential
Yi   0   1e 2 X i  
Exponential to Linear Transformation
Ln(Yi )   0  1 X  
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Properties of Least
Squares Estimators
Unbiased
Efficient
Maximum Likelihood Estimators when
error term is normal IID
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Regression Statistics
R2 – Coefficient of Determination
“Explained Variance”
Durbin - Watson
Serial Correlation
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Insured Loss Data
Trend
Seasonal
Cyclical
Random
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Insured Loss Data
High Frequency Loss Events
High Severity Losses
Positively Skewed Error
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Insured Loss Data
Seasonal Nature of Losses
New York Collision
Qtrly Paid Frequency
11.0
10.0
9.0
8.0
7.0
Year/Qtr
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
99/1
98/1
97/1
96/1
95/1
94/1
6.0
Treatment of Seasonality
12-Month-Moving Data
(creates serial correlation)
Residual Plot
0.06
0.00
-0.06
Seasonal Regression Model
Ln(Yi )   0  1 X i   2 D2   3 D3   4 D4   t
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Kurt’s Favorite Slide
Simulation Results
Underlying Trend = 0%
1.00
R-Squared
0.80
0.60
0.40
0.20
0.00
-3.5%
-2.5%
-1.5%
-0.5%
0.5%
1.5%
Estimated Trend
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
2.5%
3.5%
Comments on R2
“by itself, is a poor measure of goodnessof-fit” – Lee Barclay
Low Trend implies Low R2
High Error Variance implies Low R2
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Identifying Outliers
Visual Method
Studentized Residual
Studentized Deleted Residual
DFITS
Cooks D
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Visual Method
Oregon Homeowners
4QE Paid Frequency
13.0
11.0
9.0
7.0
Year/Qtr
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
99/1
98/1
97/1
96/1
95/1
94/1
5.0
QPV for Seasonality
Indicator Variables for each “Season”
Ln(Yi )   0  1 X i   2 D2   3 D3   4 D4   t
 Single Trend Parameter
 Parameters for Seasonal Adjustments
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
QPV for Seasonality
First Quarter
Yt  [e  0 ]  e 1 X t
Second Quarter
Yt  e  2  [e  0 ]  e 1 X t
Third Quarter
Yt  e  3  [e  0 ]  e 1 X t
Fourth Quarter
Yt  e  4  [e  0 ]  e 1 X t
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
QPV verses Multiple
Regression
QPV  Single Trend
Multiple Regression  Multiple Trends
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Comparison of Results
Results using five year data
Oregon Homeowner
Frequency
New York PPA Collision
Frequency
Nevada PPA Bodily
Injury Severity
12 MM
Trend
-6.9%
R2
.17
Trend
-1.9%
R2
.58
Trend
3.1%
R2
.78
Quarterly
-3.9%
.03
-1.7%
.10
2.7%
.25
Annual
-10.1%
.34
-1.2%
.37
3.7%
.85
Manual
Adjustment
-2.6%
.20
-0.8%
.84
1.4%
.41
Indicator
Variables
-2.6%
.27
-1.2%
.74
2.7%
.27
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Simulation – Personal Auto
BI Severity
Trend
Quarterly Seasonal Adjustment
Random Error (Normally Distributed)
Shock Variable
Ln(Yi )  [  0  1 X i   2 D2  3 D3   4 D4 ] Wˆi  ˆ
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Simulation in Excel
Using RAND, NORMSDIST, LN, EXP
Macro Loop to
generate simulated data
remove outliers
calculate regressions
record results
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Simulation Results
Shocked Early
All Methods except Manual Adjustment
underestimate the trend
Shocked Late
All Methods except Manual Adjustment
overestimate the trend
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada
Conclusion
Familiar Ideas
Additional Tools
Areas for Further Research
2001 CAS Seminar on
Ratemaking - Las Vegas, Nevada