This paper defines asymptotic marginal expected shortfall (AMES) for banks within a financial system and provides corresponding estimation method based on multivariate extreme value theory. The estimation method does not assume a specific dependence structure among bank equity returns. Both theoretical AMES and the estimator possess additive property and thus can serve as a tool to allocate system-wide risk to individual institutions. We apply the AMES to 30 global systemically important financial institutions (G-SIFIs). We show that the AMES outperforms the MES in predicting extreme losses during extreme systemic events. By taking the AMES as the reference point for allocating systemic risk to individual institutions, we show that an allocation according to simple bank characteristics such as size and individual risk can be imperfect. The allocation unfairness of individual risk or size across all the G-SIFIs has increased since 2008.