Most automobile insurance databases contain a large number of policy holders with zero claims. This high frequency of zeros may reflect the fact that some insureds make little use of their vehicle, or that they do not wish to make a claim for small accidents in order to avoid an increase in their premium, but it might also be because of good driving. We analyse information on exposure to risk and driving habits using telematics data from a Pay-as-you-Drive sample of insureds. We include distance travelled per year as part of an offset in a zero- inflated Poisson model to predict the excess of zeros. We show the existence of a learning effect for large values of distance travelled, so that longer driving should result in higher premium, but there should be a discount for drivers that accumulate longer distances over time due to the increased proportion of zero claims. We confirm that speed limit violations and driving in urban areas increase the expected number of accident claims. We discuss how telematics information can be used to design better insurance and to improve traffic safety.