Implicit Multivariate Backtesting Expected Shortfall

For risk management, implementing risk measures and backtesting them are essential tasks. Since the expected shortfall (ES) possesses coherence and tail sensitivity, the Basel Committee has raised the option to replace the classical risk measure value-at-risk (VaR) with ES. However, the techniques for the assessment of ES are still scarce. Meanwhile, practitioners are increasingly interested in the behavior of a financial system rather than a single financial institution. Hence we attach significance to the multivariate backtesting, namely evaluating multiple portfolios simultaneously. This paper presents a comprehensive framework for multivariate ES backtesting. We design three types of methods with different focuses on unconditional coverage (UC), spatial temporal independence (STI), and conditional coverage (CC) properties. The proposed framework is illustrated through well-designed simulations that shed light on its practical application. An empirical study demonstrates that the proposed framework can acutely reflect the violations of certain properties in a complex financial system