A note on insurance coverage in incomplete markets

Tn this paper, "full insurance coverage on average" is defined as the coinsurance rate that eliminates all insurable risk when the uninsurable risk is evaluated at its mean. The regressibility assumption is used to derive the conditions on the correlation between background and insurable risks and the actuarial unfairness of insurance under which full, over-, or underinsurance on average is optimal. These conditions are compared to those for the case of default risk. Together, they explain intuitively the different results under the cases of background risk and default risk obtained in the literature.