Optimal reimbursement limitation for a health plan

The goal of this paper is the assessment of an optimal reimbursement strategy for employer-based health insurance plans (HP), that cover several categories of medical services. Indeed, a health plan may offer several cost-sharing provisions for these categories, and the “Percent Expense Paid” by the plan also known as “Actuarial Value” represents a summary measure of the protection provided. Starting from a wide industrial applied model in insurance for rate-making, the generalized linear models (GLM), we estimate the expected value and variance of the health expenditure for each category. Moreover, different reimbursement rules (e.g. deductibles, co-payments, policy limits, etc.) involve a change of the “Actuarial Value” calculated as the ratio between the benefits paid by the plan (i.e. reimbursement amounts) and the expenses paid by policyholders (i.e. expenditures). The latter is the percentage of expenditure reimbursed by the plan and is sometimes defined in actuarial literature as Indicated Deductible Relativity (IDR); an IDR can be calculated for each category covered by the Health Plan or per policyholder and is a commonly used method for scoring the benefits of health insurance. Hence, we calculate the optimal IDR for each category, using the optimization problem proposed by de by Finetti (Il problema dei pieni, Giornale dell’istituto italiano degli attuari, 1940) in the context of proportional reinsurance. The goal is the minimization of the variance of the total reimbursement of the Health Plan by fixing the total gain. Furthermore, we propose a numerical application to a real dataset, containing observed expenditures of an Italian HP.