A Nonlinear Regression Model of Incurred But Not Reported Losses

The process of loss development has been studied by casualty actuaries for many years. When an accident period is closed, the ultimate claim liabilities are unknown because many of the claims are still unreported and some that are reported remain unsettled. The difference between ultimate losses and reported losses is known as "Incurred But Not Reported" loss or IBNR. The reserve for IBNR losses is the largest liability on an insurer's balance sheet. Quantifying the uncertainty is estimates of IBNR is of great importance to the financial health of casualty insurance companies. Most of the current methods for estimating ultimate losses focus on estimation of loss development factors which relate the emergence of losses to the amount of losses already reported. This paper presents a model for predicting incremental losses as a function of exposures, calendar period and development age. A nonlinear regression model is used for estimating the 95% confidence interval of IBNR for an accident period. The model predicts the incremental pure premium for a development interval as a function of development age, calendar quarter and exposure. The estimated IBNR is the sum of forecasted incremental pure premiums. The regression model produces confidence interval estimates for the model parameters and for IBNR. The regression model is applied to trended losses. We assume that the trend has been estimated by some reasonable time series method that produces confidence interval estimates of trend factors. Many good methods are available. We use the confidence interval estimate of the trend factors to adjust the IBNR estimates for uncertainty in loss trend. The model presented here assumes normally distributed residuals. Although the underlying loss severities are probably not normal, the central limit theorem implies that this assumption would be appropriate if the number of claims is large. Thus, the model will most likely work well for high frequency lines of business such as personal auto. We will present methods for estimating parameters, confidence intervals for the parameters, and the distribution of IBNR. These methods will be illustrated using simulated automobile bodily injury liability data. Model predictions will be compared to actual emerged losses. Based on a comparison of predicted IBNR to the "actual" IBNR from the simulated data, the model appears to produce unbiased predictions and reasonable confidence interval estimates of IBNR. We conclude that the distribution of incremental pure premiums is close to normal and there is not a significant correlation between development age intervals. Thus, traditional regression methods can be used to estimate the distribution of forecasted incremental pure premiums and consequently, IBNR.