We examine equilibria in competitive insurance markets with adverse selection when wealth differences arise endogenously from unobservable savings or labor supply decisions. The endogeneity of wealth implies that high risk individuals may ceteris paribus exhibit the lower marginal willingness to pay for insurance than low risks, a phenomenon that we refer to as irregular-crossing preferences. In our model, both risk and patience (or productivity) are privately observable. In contrast to the models in the existing literature, where wealth heterogeneity is exogenously assumed, equilibria in our model no longer exhibit a monotone relation between risk and coverage. Individuals who purchase larger coverage are no longer higher risks, a phenomenon frequently observed in empirical studies.