Development of the Spatial Credibility Model

Classical credibility theory can be used to develop a best estimate for a territory by weighting the territorial average loss cost with the overall statewide loss cost. Generally, credibility is applied to the indicated changes in loss costs instead. However, the classical credibility formulas are developed under two assumptions which are not valid for MPCI. The first is that the true territorial expected loss costs are independent of one another, i.e., that there is no spatial correlation. The second is that the intertemporal random fluctuations in one territory are independent of the random fluctuations in other territories.

For MPCI, the spatial correlation between counties as a function of distance can be described using a variogram. The first step in the preparation of the variogram is to calculate the statistic Yij as ½ of the squared difference in pure premiums for each pair of counties (i,j). The distance between each pair of counties is also required. Given this information, distances are grouped into ranges and the average of all Yij within each range is determined. This produces a variogram similar to that shown in Chart 24 for yields. The variogram is an estimator for E[(Xi-Xj)2/2 | counties i,j in distance range k], where X represents pure premium. The variogram is also a proxy for the spatial covariance. Notice that E[(Xi-Xj)2/2] = ½ VXi + ½ VXj – Cov(Xi,Xj) + ½ (?i – ?j)2 = ?2 - Cov(Xi,Xj), where all Xk are assumed to be from the same distribution. That is, a small value for the variogram implies a high value for the spatial covariance. The chart shows that the variogram is low for nearby counties and gradually increases as the distance increases, within a certain range.

Similarly, the intertemporal correlation of a county's experience to that of its nearest neighbors is also greater than its correlation to more distant counties, as shown in Chart 25 for Iowa corn in Adams county. The average intertemporal correlation across all counties as a function of distance is shown in Chart 26. Each type of correlation needs to be taken into account in a spatial credibility formula.