  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.
