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Optimal design of reinsurance contracts with a continuum of risk assessments

In this article, we employ a principal-agent model to analyze optimal contract design in a monopolistic reinsurance market under adverse selection with a continuum of insurer types. Instead of using the classical expected utility framework, we model each insurer's risk preference through their VaR at their chosen risk tolerance level. Under informational asymmetry, the reinsurer (principal) seeks to maximize expected profit by offering an optimal menu of reinsurance contracts to a continuum of insurers (agents) with hidden characteristics. To avoid the complexity of the traditional duality approach, which yields indirect multivariate utility functions, we introduce a change of variables that reduces the problem to a univariate one. We show that the optimal indirect utility for both stop-loss and quota-share reinsurance is in stop-loss form, implying that the reinsurer will classify agents into two risk groups-high and low-even in the continuum setting. Utilizing this new class of indirect utility functions, we fully solve the problem for three common reinsurance structures: stop-loss, quota-share, and change-loss. Numerical examples are also provided for illustrating the main findings.