Long-range contagion in automobile insurance data: estimation and implications for experience rating

The purpose of the paper is to use the date of claims in the predictionof risks. This issue has already been addressed by actuarial literature.Solutions are obtained from credibility models which can be updated (Ger-ber, Jones (1975)), and from credibility estimators with geometric weights(Sundt (1988)).The rating models presented in this paper are obtained after statisticalinference on longitudinal count data. Let us clarify …rst the motivationswhich led us to question the assumption of time-independence for the ran-dom e¤ects.Hiddenfeaturesofriskdistributionsvarywithtime,asdoratingfactors.The random e¤ects added in an a priori rating model on longitudinal datashould then be dynamic. Variations of rating factors between two datesshould increase with the related lag, and the same result is expected forhiddenfeatures in risk distributions. Hence the predictive ability of a claimshoulddecreasewiththelagbetweenthedateofriskpredictionandthedateof the claim. Besides, economic analysis suggests that optimal insurancecontracts with moral hazard should penalize recent claims more than olderones (Henriet, Rochet (1986)). A stationary process for the random e¤ectswill relate the predictive power of claims to the aforementioned lag. If thepreceding intuitionis veri…ed, the estimated autocorrelation function of therandom e¤ects should be decreasing, a point already mentioned by Sundt(1988).In Section 2, we present di¤erent Poisson models with random e¤ects.The variance of a time-independent random e¤ect can be estimatedfromdisaggregateddataorfromnumbersofclaimsandfrequencypremiumswhich are summed across the periods. If the estimated variance obtainedfrom disaggregated data is greater than the second one, the estimation ofdistributions for dynamic random e¤ects can be thought of. This conditionis veri…ed on our data set, which is drawn from the portfolio of a majorSpanish insurance company.An unconstrained autocorrelation function for the dynamic random ef-fects is then estimated from a Poisson model with regression components.For each lag, the corresponding autocorrelation is estimated from pairedo¤ products of lagged number-residuals and frequency-premiums. In theempirical study, we do …nd a decreasing autocorrelation function, with adecreasing shape which is slower than a geometric one.5