Extremal Risk Management: Expected Shortfall Value Verification Using the Bootstrap Method

In this paper, we refer to the axiomatic theory of risk and investigate the problem of formal verification of the expected shortfall (ES) model based on a sample ES. Recognizing the infeasibility of parametric methods, we explore the bootstrap technique, which, unlike the current value-at-risk model-based (VaR model-based) Basel III testing framework, permits the creation of more powerful sample ES-based procedures. Our contribution to the debate on the possibilities of sample ES-based testing is twofold. First, we introduce a bootstrap test based on the idea of ES prediction corrected variables. In this way, we obtain a procedure that makes no distributional assumptions about the underlying returns process, and whose p-value computation does not assume any asymptotic convergence. Second, we provide a unifying framework for ES value verification, in which we compare alternative sample ES-based approaches: the residual-based procedures versus the ES prediction corrected tests as well as the VaR model-dependent approach versus the fixed failure rate tests. By examining its statistical properties and practical applicability, we find evidence that the proposed bootstrap procedure, based on ES prediction corrected variables, is superior to other methods. This provides important guidance for developing international standards of market risk management.