A heavy traffic approach to modeling large life insurance portfolios

We explore a new framework to approximate life insurance risk processes in the scenario of plentiful policyholders, via a bottom-up approach. Given the insurance contract structure, we aggregate the balance of individual policy accounts, and derive an approximating Gaussian process with computable correlation structure. The methodology is borrowed from heavy traffic theory in the literature of many-server queues, and involves the so-called fluid and diffusion approximations. Our framework is different from individual risk model by taking into account the time dimension and the specific policy structure including the premium payments. It is also different from classical risk theory by building the risk process from micro-level contracts and parameters instead of assuming claim and premium processes outright. As a result, our approximating process behaves differently depending on the issued contract structure. We also illustrate the flexibility of our approach by formulating a finite-horizon ruin problem that incorporates actuarial reserve in the consideration. The study of risk processes is a central topic in actuarial science. Among the literature, the majority focuses on the calculation of ruin probability, as well as the optimal control of premiums, reinsurance levels and investment allocation. These questions have been studied under a variety of stochastic settings, from the classical Cramer-Lundberg approximation to diffusion processes. The central theme is that random-walk-type models, with a negatively drifted premium process and a jump process of claims, provide a rich framework to allow plenty of extensions, modifications and problem formulations (see, for example, Asmussen (2000) for survey on ruin probability calculations, and Schmidli (2008) for the counterpart in stochastic control problems). In this paper we take a different view from the existing literature. Rather than focusing on the computation of risk-related quantities, we explore the question of the construction of risk process itself. The approach we use is bottom-up: Given the structure and parameters of the individual insurance contracts, how does the risk process of the insurer look like on an aggregate scale? Naturally, the risk process under this framework is the sum of all the individual accounts i.e. the balances of policyholders who entered contract with the insurer over time. For actuaries, this points to the standard one-period individual and collective risk models. However, these standard models do not consider the time dimension. This in turn also restrains the power of such models to capture the specific contract structure involved e.g. the premium payments. In this regard, our work can be seen as a generalization of the standard risk models to a process-level approximation. Of course, mere summation of all individual accounts might end up getting an unpleasant process that is hardly computable. To tackle this issue, we borrow techniques in so-called heavy traffic theory in the literature on many-server queues. The basic idea is that under the assumption of large number of policyholders, one can approximate the functionals of these policyholders’ statuses using fluid and diffusion approximations. In the statistics literature, these correspond to stochastic-process version of Law of Large Numbers and Central