2001 CAS Seminar on Ratemaking - Las Vegas, Nevada Considerations in Estimating Loss Cost Trends Kurt Dickmann & Jim Merz

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

Considerations in Estimating Loss Cost Trends

Kurt Dickmann& Jim Merz


Overview of the PaperImportance of Trends in RatemakingOrdinary Least Squares RegressionCharacteristics of Insured LossesIdentifying OutliersAdjusting for OutliersComparison of ResultsSimulation Results


Importance of Trends in RatemakingHigh Leverage on Estimated LossesRequires JudgmentScarcity of Actuarial Literature on

Estimation


RegressionUnderstood by ManagementCommonly UsedWidely UnderstoodAccepted by RegulatorsIncluded in Spreadsheet Software


Regression Basics

General Form

Linear

Exponential

Exponential to Linear Transformation

XYLn

eY

XY

XfY

i

Xi

iii

iiii

i

10

10

10

)(

),(

2


Properties of Least Squares EstimatorsUnbiasedEfficientMaximum Likelihood Estimators

when error term is normal IID


Regression Statistics

R2 – Coefficient of Determination“Explained Variance”

Durbin - WatsonSerial Correlation


Insured Loss Data

TrendSeasonalCyclicalRandom


Insured Loss DataHigh Frequency Loss EventsHigh Severity LossesPositively Skewed Error


Insured Loss Data

Seasonal Nature of Losses New York Collision

Qtrly Paid Frequency

6.0

7.0

8.0

9.0

10.0

11.0

94/1

95/1

96/1

97/1

98/1

99/1

Year/Qtr


Treatment of Seasonality

tii DDDXYLn 44332210)(

12-Month-Moving Data

Residual Plot

-0.06

0.00

0.06

Seasonal Regression Model

(creates serial correlation)


Kurt’s Favorite SlideSimulation Results

Underlying Trend = 0%

0.00

0.20

0.40

0.60

0.80

1.00

-3.5% -2.5% -1.5% -0.5% 0.5% 1.5% 2.5% 3.5%

Estimated Trend

R-S

quar

ed


Comments on R2

“by itself, is a poor measure of goodness-of-fit” – Lee Barclay

Low Trend implies Low R2

High Error Variance implies Low R2


Identifying Outliers

Visual MethodStudentized ResidualStudentized Deleted ResidualDFITSCooks D


Visual MethodOregon

Homeowners4QE Paid Frequency

5.0

7.0

9.0

11.0

13.0

94/1

95/1

96/1

97/1

98/1

99/1

Year/Qtr


QPV for SeasonalityIndicator Variables for each “Season”

tii DDDXYLn 44332210)(

Single Trend Parameter

Parameters for Seasonal Adjustments


QPV for Seasonality

First Quarter

Second Quarter

Third Quarter

Fourth Quarter t

t

t

t

Xt

Xt

Xt

Xt

eeeY

eeeY

eeeY

eeY

104

103

102

10

][

][

][

][


QPV verses Multiple Regression

QPV Single TrendMultiple Regression Multiple

Trends


Comparison of Results

Results using five year dataOregon Homeowner

FrequencyNew York PPA Collision

FrequencyNevada PPA Bodily

Injury Severity

Trend R2 Trend R2 Trend R2

12 MM -6.9% .17 -1.9% .58 3.1% .78

Quarterly -3.9% .03 -1.7% .10 2.7% .25

Annual -10.1% .34 -1.2% .37 3.7% .85

ManualAdjustment

-2.6% .20 -0.8% .84 1.4% .41

IndicatorVariables

-2.6% .27 -1.2% .74 2.7% .27


Simulation – Personal Auto BI SeverityTrendQuarterly Seasonal AdjustmentRandom Error (Normally Distributed)Shock Variable

ˆˆ][)( 44332210 iii WDDDXYLn


Simulation in ExcelUsing RAND, NORMSDIST, LN, EXPMacro Loop to

generate simulated data remove outliers calculate regressions record results


Simulation Results

Shocked EarlyAll Methods except Manual Adjustment

underestimate the trendShocked Late

All Methods except Manual Adjustment overestimate the trend


Conclusion

Familiar IdeasAdditional ToolsAreas for Further Research

Documents

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada Considerations in Estimating Loss Cost Trends Kurt Dickmann & Jim Merz