Description: Under Yaari's dual theory of risk, this paper determines the equilibrium separating contracts for high and low risks in a competitive insurance market, in which risks are defined only by their expected losses, that is, a high risk is a risk that has a greater expected loss than a low risk. The authors also determine the pooling equilibrium contract when insurers are assumed non-myopic. Utility theory generally predicts that optimal insurance indemnity payments are nonlinear functions of the underlying loss due to the nonlinearity of agents' utility functions. Under Yaari's dual theory, the authors show that under mild technical conditions the indemnity payment is a piecewise linear function of the loss, a common property of insurance coverages.