Optimal insurance with adverse selection and comonotonic background risk

In this note, we consider an adverse selection problem involving an insurance market à la Rothschild-Stiglitz. We assume that part of the loss is uninsurable as in the case with health care or environmental risk. We characterize sufficient conditions such that adverse selection by itself does not distort competitive insurance contracts. A sufficiently large uninsurable loss provides an incentive to high-risk policy holders not to mimic low-risk policy holders without distorting the optimal coverage