“Forecasting Workers Compensation Severities
And Frequency Using The Kalman Filter”
Frank Schmid
and Jonathan Evans
presented by
Jonathan Evans, FCAS, MAAA
Actuary
CAS Seminar on Ratemaking
Atlanta, GA
March 8, 2007
Dr. Frank Schmid
Senior Economist
NCCI
 2007 National Council on Compensation Insurance, Inc. All
Rights Reserved.
1
Frank Schmid, director and senior economist in Actuarial and Economic Services at the National
Council on Compensation Insurance, recently accepted a Hicks-Tinbergen Medal from the European Economic Association
(EEA). The award was presented for the research paper, "Capital, Labor, and the Firm: A Study of German
Codetermination," which he coauthored with Gary Gorton of the University of Pennsylvania prior to joining NCCI. The
EEA recognized the research paper as the best paper published in the Journal of the European Economic Association in
2004 and 2005.
 2007 National Council on Compensation Insurance, Inc.
All Rights Reserved.
2
Forecasting Frequency And Severity Is Crucial
To Workers Compensation Ratemaking
• Prospective loss costs are very sensitive to trends in
frequency and severity
• Trend rates change over time
• Forecasting changes in trend rates, or even turning
points, greatly enhances rate adequacy
 2007 National Council on Compensation Insurance, Inc.
All Rights Reserved.
3
Forecasting As Signal Extraction And Extrapolation
(NOT CURVE FITTING TO NOISE!)
R2 = 55%
R2 = 100%
 2007 National Council on Compensation Insurance, Inc.
All Rights Reserved.
4
Time Series Models
• ARIMA - Auto Regressive Integrated Moving
Average: focused on patterns of serial
autocorrelation coefficients in observed data
• UC – Unobserved Components: data assumed to
be observed with white noise on top of signal
• STS – Structural Time Series: combines UC with
linear regression on exogenous explanatory time
series
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5
STS + UC Local Linear Model
Observation (measurement)
y t  t  t  xt  t,
 t ~ N (0, 2 )
Signal
t t
Exogenous Regression Parameter
 t   t 1   t ,
 t ~ N (0, 2 )
Level
 t   t 1   t 1   t ,
 t ~ N (0, 2 )
Slope
 t   t 1   t ,
 t ~ N (0, 2 )
The Local Level Model is the special case where the slope and exogenous
regression parameter is set to constant 0. The Local Level STS Model is
the special case where the slope is set to constant 0.
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6
The Kalman Filter
Uses estimates for σε, σν, ση, and σζ, together with actual observations
of yt to filter out measurement noise εt and produce a piecewise least
squares estimate θt , similar to Bϋhlmann credibility. Since the
likelihood function for the observations has arguments εt and σε, the
values of σε, σν, ση, and σζ, can be MLE estimated from the Kalman filter
estimates for θt .
ˆt  yt  yˆt ( yt , ˆ  , ˆ , ˆ , ˆ )
n
L
t 1
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1
ˆ
 ˆt2 
exp   2 
2
 2ˆ 
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NCCI Frequency And Severity Applications
• Objective to forecast the 3 year growth factor for
the indemnity and medical severities, and
frequency of claims (per on-leveled and wage
adjusted premium)
• 18 observed log growth rates for accident years
1986 through 2004
• Severity data on a paid basis
• Models applied to log growth rates of data points
– Local Level model used for severity log growth rates
– STS Local Level model used for frequency with the
change in unemployment as the exogenous
explanatory series
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8
Logarithmic Growth Rates of Indemnity and Medical Severities, State-Level Data, Accident
Years 1987-2004
0.12
Indemnity Severity
Medical Severity
0.10
L o g arith m ic R ate o f G ro w th
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
1985
1990
1995
2000
Accident Year
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9
2005
2010
L o g R ate o f G ro w th (F req u en cy ) an d F irst D ifferen ce (U n em p lo y m en t)
Logarithmic Growth Rate of Frequency and First Difference in Rate of Unemployment,
State-Level Data, Accident Years 1987-2004
0.20
Frequency
Rate of Unemployment
0.15
0.10
0.05
0.00
-0.05
-0.10
1985
1987
1989
1991
1993
1995
1997
Accident Year
1999
2001
2003
Note: The Rate of Unemployment was measured in percent; for scaling purposes, the first
difference was divided by 10 (in this exhibition only).
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10
2005
Regression Diagnostics (Local Level UC Model) for the Log Growth Rate of Medical
Severity
1
Correlogram
2
QQ Plot (Versus Normal)
1
0
0
-1
1
10
2
3
Lag Length
4
5
-1.0
1.0
Cumulative Sum of Residuals
0
-0.5
0.0
0.5
1.0
1.5
Cumulative Sum of Squared Residuals
0.5
-10
1990
1995
2000
Accident Year
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2005
11
1990
1995
2000
Accident Year
2005
Holdout-Window Forecasts (Local Level STS Model) for the Growth Rate of Frequency
0.075
Actual
Forecasts
0.050
Logarithmic Rate of Growth
0.025
0.000
-0.025
-0.050
-0.075
-0.100
-0.125
-0.150
-0.175
2001
2002
2003
Accident Year
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12
2004
2005
Logarithmic Rate of Growth
Forecasts (Local Level UC Model) for the Log Growth Rate of Medical Severity
Actual
Forecasts
0.10
0.05
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2002
2003
2004
2005
2006
2007
2008
Logarithmic Rate of Growth
Accident Year
Level (Trend Log Growth Rate)
0.08
0.06
0.04
1995
1996
1997
1998
1999
2000
2001
Accident Year
 2007 National Council on Compensation Insurance, Inc.
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13
Kalman Filtered Forecasts Versus Forecasts
Disregarding Measurement Noise
For the holdout forecast for medical severity presented:
• Kalman filtered forecasts of the annual log rates of growth have a
sum of absolute forecast error (for periods T+1, T+2, and T+3)
equal to 0.0387, and RMSE (root mean squared error) of 0.0090
• For the last observed rates of growth, the absolute forecast error
is 0.1154 and the RMSE is 0.0234
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14
Conclusion
• The experience of NCCI with Kalman filtered
estimation of trend rates during the policy year
2006 rate filing season was encouraging
• Current research at NCCI has shifted from Kalman
Filter+MLE estimation to Bayesian estimation
(Gibbs sampling using WinBUGS) of underlying
models similar to the UC and STS models in the
paper
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All Rights Reserved.
15