 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 Schmid Senior Economist NCCI

 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.

 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 3 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. 4 Forecasting As Signal Extraction And Extrapolation (NOT CURVE FITTING TO NOISE!) R 2 = 100% R 2 = 55%

 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 5 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

 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 6 STS + UC Local Linear Model 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.

 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 7 The Kalman Filter Uses estimates for σ ε, σ ν, σ η, and σ ζ, together with actual observations of y t to filter out measurement noise ε t and produce a piecewise least squares estimate θ t, similar to Bϋhlmann credibility. Since the likelihood function for the observations has arguments ε t and σ ε, the values of σ ε, σ ν, σ η, and σ ζ, can be MLE estimated from the Kalman filter estimates for θ t.

 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 8 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

 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 9

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

 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 11

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 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 14 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 , and RMSE (root mean squared error) of For the last observed rates of growth, the absolute forecast error is and the RMSE is

 2007 National Council on Compensation Insurance, Inc. All Rights Reserved. 15 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