Value-at-Risk- and Expectile-based Systemic Risk Measures and Second-order Asymptotics: With Applications to Diversification

The systemic risk measure plays a crucial role in analyzing individual losses conditioned on extreme system-wide disasters. In this paper, we provide a unified asymptotic treatment for systemic risk measures. First, we classify them into two families of Value-at-Risk- (VaR-) and expectile-based systemic risk measures. While VaR has been extensively studied, in the latter family, we propose two new systemic risk measures named the Individual Conditional Expectile (ICE) and the Systemic Individual Conditional Expectile (SICE), as alternatives to Marginal Expected Shortfall (MES) and Systemic Expected Shortfall (SES). Second, to characterize general mutually dependent and heavy-tailed risks, we adopt a modeling framework where the system, represented by a vector of random loss variables, follows a multivariate Sarmanov distribution with a common marginal exhibiting second-order regular variation. Third, we provide second-order asymptotic results for both families of systemic risk measures. This analytical framework offers a more accurate estimate compared to traditional first-order asymptotics. Through numerical and analytical examples, we demonstrate the superiority of second-order asymptotics in accurately assessing systemic risk. Further, we conduct a comprehensive comparison between VaR-based and expectile-based systemic risk measures. Expectile-based measures output higher risk evaluation than VaR-based ones, emphasizing the former's potential advantages in reporting extreme events and tail risk. As a financial application, we use the asymptotic treatment to discuss the diversification benefits associated with systemic risk measures. The expectile-based diversification benefits consistently deduce an underestimation and suggest a conservative approximation, while the VaR-based diversification benefits consistently deduce an overestimation and suggest behaving optimistically.