Insurance Demand under Government Interventions and Distorted Probabilities

In this article, we investigate the optimal insurance demand for an individual under distorted probabilities, considering the participation of government interventions, such as premium subsidies and disaster relief. We model the premium subsidy as a non-decreasing function ranging from 0 to 1, representing the percentage of government support, whereas the relief assistance is characterized by a 1-Lipschitz relief scheme function, reflecting the government’s effort in post-disaster recovery. When the expected-value premium principle is employed, the general form of the optimal retained loss function for the policyholder is derived under a concave government relief scheme. We demonstrate that the optimal retained loss function takes a layered form, shaped by the trade-off between government premium subsidies and relief assistance, and can be further characterized by an ordinary integro-differential equation. In particular, explicit solutions are obtained for VaR and general convex distortion risk measures. To provide further insights, we explore two extensions: one investigates the design of the optimal safety loading from the insurer’s perspective, while the other examines the impact of the government’s budget constraint. Finally, we present numerical examples to illustrate and validate the main findings of the paper.