Risk Aggregation and Allocation in the Presence of Systematic Risk via Stable Laws

Abstract

In order to properly manage risk, practitioners must understand the aggregate risks they are exposed to. Additionally, to properly price policies and calculate bonuses the relative riskiness of individual business units must be well understood. Certainly, Insurers and Financiers are interested in the properties of the sums of the risks they are exposed to and the dependence of risks therein. Realistic risk models however must account for a variety of phenomena: ill-defined moments, lack of elliptical dependence structures, excess kurtosis and highly heterogeneous marginals. Equally important is the concern over industry-wide systematic risks that can affect multiple business lines at once. Many techniques of varying sophistication have been developed with all or some of these problems in mind. We propose a modification to the classical individual risk model that allows us to model company-wide losses via the class of Multivariate Stable Distributions. Stable Distributions incorporate many of the unpleasant features required for a realistic risk model while maintaining tractable aggregation and dependence results. We additionally compute the Tail Conditional Expectation of aggregate risks within the model and the corresponding allocations.