Second-order properties of risk concentrations without the condition of asymptotic smoothness

Abstract

For the purpose of risk management, the quantification of diversification benefits due to risk aggregation has received more attention in the recent literature. Consider a portfolio of n independent and identically distributed loss random variables with a common survival function \(\overline {F}\) possessing the property of second-order regular variation. Under the additional assumption that \(\overline {F}\) is asymptotically smooth, Degen et al. (Insur Math Econ 46:541–546, 2010) and Mao et al. (Insur Math Econ 51:449–456, 2012) derived second-order approximations of the risk concentrations based on the risk measures of Value-at-Risk and conditional tail expectation, respectively. In this paper, we remove the assumption of the asymptotic smoothness, and reestablish the second-order approximations of these two risk concentrations.