The principle of a single big jump from the perspective of tail moment risk measure

Abstract

Consider a financial or insurance system with a finite number of individual components. The famous principle of a single big jump (PSBJ) says that a system crisis occurs mainly due to a single but unusually large loss from some individual component. Most of literatures modeled the PSBJ through the tail probabilities of the largest risk and the total risk of the system. Different from the existing works, this paper is devoted to explore the PSBJ from a new perspective. We aim to establish the PSBJ based on a kind of risk measure defined via the tail moments of the related risks. Our study is mainly conducted under a widely used framework, in which the individual risks are pairwise asymptotically independent and have the distributions from the Fréchet or Gumbel max-domain of attraction. The asymptotic behavior of the tail mixed moments is also discussed in detail. The results obtained are applied to an optimal capital allocation problem based on a tail mean-variance model. A numerical study is given to illustrate the accuracy of our main asymptotic results. We also give a thorough discussion on some interesting theoretical properties regarding the PSBJ.