Risk Modeling of Multi-year, Multi-line Reinsurance Using Copulas

Relationships between loss variables covered by multi-year, multi-line reinsurance are complicated in the sense that correlation may be present in two dimensions: from year to year and across business lines. While reinsurers may be able to choose and accept risk exposures from different lines of business to achieve risk diversification, it is unlikely that they will be able to completely eliminate the temporal dependence of annual loss experience throughout the period of coverage. Assuming temporal independence may likely result in an underestimation of the risk embedded in the contract. Using two independent lines of business, this paper applies copulas for the purpose of modeling the year-to-year dependence of annual loss experience of a multi-year, multi-line reinsurance agreement. We then simulate annual losses for separate lines of business by making random draws from different multivariate loss distributions and then analyze the characteristics of the resulting Total Loss variable upon which the contract payoff is determined. As an illustrative example, we study the performance of a stop-loss reinsurance contract covering a primary insurer’s losses over a three-year period arising from workers compensation and commercial multiple perils coverage. This approach can be easily extended to address, simultaneously, dependencies among lines of business.