Impact of common shocks on the optimal upper limits of insurance contracts

In many insurance contracts that include multiple coverage lines, actuaries often assume that claims occur independently across those lines. However, common shocks – events that simultaneously impact multiple coverages – are frequently overlooked, despite their significant influence. This study explores how such shocks affect key elements of insurance design. We focus on contracts with multiple coverage lines, capped claim payments, and premiums calculated using the expected value principle. Cash flow processes are analyzed over a defined time horizon. Optimal upper limits for coverage are derived under two scenarios: (1) independent claim processes, and (2) dependent claim processes arising from common shocks. In the independent case, the optimal limits are determined by solving a single equation. Under dependence, a system of equations with unique solutions is required. Mathematically, we show that accounting for common shocks results in lower optimal upper limits. These findings are illustrated through two numerical examples and a real-world case study involving short-term travel health insurance, which covers policyholders' healthcare costs for a limited period.