In this paper, we address the identification and estimation of insurance models where insurees have private information about their risk and risk aversion. The model includes random damages and allows for several claims, while insurees choose from a finite number of coverages. We show that the joint distribution of risk and risk aversion is nonparametrically identified despite bunching due to multidimensional types and a finite number of coverages. Our identification strategy exploits the observed number of claims as well as an exclusion restriction, and a full support assumption. Furthermore, our results apply to any form of competition. We propose a novel estimation procedure combining nonparametric estimators and GMM estimation that we illustrate in a Monte Carlo study.