Tail Moment for Gamma‐like Risks With Arbitrary Scale Parameters

Consider a financial or insurance system with a finite number of individual components. The Tail Moment (TM) risk measure is defined as the conditional moment of some individual risk given that a system crisis occurs, and it is one of the systemic risk measures that have garnered significant scholarly attention in recent years. This paper is devoted to study the asymptotic behavior of the TM under a general framework, in which the individual risks are modelled by real‐valued random variables and they follow gamma‐like distributions with scale parameters not necessarily the same. Some precise asymptotic estimates of the TM are obtained when the individual risks are mutually independent or have a dependence structure of the Farlie–Gumbel–Morgenstern (FGM) type. When the system contains two individual risks with the FGM joint distribution, we discuss in detail the critical case of extremely negative dependence, which is rarely considered in the existing works. Moreover, a numerical study is given to illustrate the approximation performance of our main asymptotic results. We also provide an application in capital allocation based on real insurance data.