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

Abstract

Systemic risk measures play a crucial role in analyzing individual losses conditional 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-based risk measures have 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 consider a multivariate loss system following a multivariate Sarmanov distribution with common marginal distributions exhibiting second-order regular variation. Third, within this setting, we provide second-order asymptotic results for both families of systemic risk measures. These results extend standard first-order asymptotics and allow for more accurate tail approximations. Through numerical and analytical examples, we demonstrate the superiority of second-order asymptotics in accurately assessing systemic risk. We further conduct a comprehensive comparison between expectile-based and VaR-based systemic risk measures. The results indicate that expectile-based measures often yield higher asymptotic accuracy 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 various risk measures. Finally, we extend and obtain the second-order asymptotic formulas for generalized-quantile-based systemic risk measures with power functions.