2007 national council on compensation insurance, inc. all rights reserved. 1 “forecasting workers...

2007 National Council on Compensation Insurance, Inc. All

Rights Reserved.

1

“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 SchmidSenior EconomistNCCI

2007 National Council on Compensation Insurance, Inc.

All Rights Reserved.2

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.



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



Forecasting As Signal Extraction And Extrapolation (NOT CURVE FITTING TO NOISE!)

R2 = 100%

R2 = 55%



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



STS + UC Local Linear Model

2, ~ (0, ) t t t t t ty x N

t t

21 , ~ (0, ) t t t t N

21 1 , ~ (0, ) t t t t t N

21 , ~ (0, ) t t t t N

Observation (measurement)

Signal

Exogenous Regression Parameter

Level

Slope

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.



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 .

ˆ ˆ ˆ ˆ ˆ ˆ( , , , , )t t t ty y y

2

21

ˆ1exp

ˆ2ˆ 2

nt

t

L



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



Logarithmic Growth Rates of Indemnity and Medical Severities, State-Level Data, Accident

Years 1987-2004

1985 1990 1995 2000 2005 2010

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Accident Year

Lo

garith

mic R

ate of G

row

th

Indemnity Severity Medical Severity



Note: The Rate of Unemployment was measured in percent; for scaling purposes, the first difference was divided by 10 (in this exhibition only).

Logarithmic Growth Rate of Frequency and First Difference in Rate of Unemployment,

State-Level Data, Accident Years 1987-2004

1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

Accident Year

Lo

g R

ate of G

row

th (F

requ

ency

) and

First D

ifference (U

nem

plo

ym

ent)

Frequency Rate of Unemployment



Regression Diagnostics (Local Level UC Model) for the Log Growth Rate of Medical

Severity

1 2 3 4 5

0

1Correlogram

-1.0 -0.5 0.0 0.5 1.0 1.5

-1

0

1

2 QQ Plot (Versus Normal)

1990 1995 2000 2005

-10

0

10

Lag Length

Accident Year

Cumulative Sum of Residuals

1990 1995 2000 2005

0.5

1.0

Accident Year

Cumulative Sum of Squared Residuals



Holdout-Window Forecasts (Local Level STS Model) for the Growth Rate of Frequency

2001 2002 2003 2004 2005

-0.175

-0.150

-0.125

-0.100

-0.075

-0.050

-0.025

0.000

0.025

0.050

0.075

Accident Year

Logarith

mic R

ate of G

row

th

Actual Forecasts



Forecasts (Local Level UC Model) for the Log Growth Rate of Medical Severity

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

0.05

0.10

Accident Year

Logarith

mic R

ate of G

row

thL

ogarith

mic R

ate of G

row

th

Actual Forecasts

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

0.04

0.06

0.08

Accident Year

Level (Trend Log Growth Rate)



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



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

2007 national council on compensation insurance, inc. all rights reserved. 1 “forecasting workers...

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