Asymptotics of the loss-based tail risk measures in the presence of extreme risks

Evaluating the risk exposure of a financial/insurance company when some extreme scenario occurs is one of the fundamental aspects of risk management. Well-known tail risk measures, such as the Conditional Tail Expectation and the Marginal Expected Shortfall, are used for measuring a massive downside in adverse scenarios. Most of these risk measures are based on the conditional expectation of a specific loss function, subject to the assumption of an extreme loss event. In this paper, we study the loss-based tail risk measures under the condition that an extreme loss event has occurred. Asymptotic approximations are derived under a regularly varying loss function for individual and multivariate heavy-tailed risks. Some further examples with applications are given to show how our asymptotic approximations can be used to approximate many other loss-based tail risk measures.