Backtesting Expected Shortfall: Accounting for both duration and severity with bivariate orthogonal polynomials

TL;DR

This paper introduces a novel backtesting framework for Expected Shortfall that separately tests the frequency and severity of violations using bivariate orthogonal polynomials, improving model diagnostics.

Contribution

It develops a new two-part duration-severity backtesting method for ES that allows separate testing of violation frequency and severity, addressing limitations of existing PIT-based tests.

Findings

The proposed test shows good finite sample properties in simulations.

Application to stock indices reveals insights into ES model mis-specifications.

The framework can extend to other systemic risk measures like Marginal Expected Shortfall.

Abstract

We propose an original two-part, duration-severity approach for backtesting Expected Shortfall (ES). While Probability Integral Transform (PIT) based ES backtests have gained popularity, they have yet to allow for separate testing of the frequency and severity of Value-at-Risk (VaR) violations. This is a crucial aspect, as ES measures the average loss in the event of such violations. To overcome this limitation, we introduce a backtesting framework that relies on the sequence of inter-violation durations and the sequence of severities in case of violations. By leveraging the theory of (bivariate) orthogonal polynomials, we derive orthogonal moment conditions satisfied by these two sequences. Our approach includes a straightforward, model-free Wald test, which encompasses various unconditional and conditional coverage backtests for both VaR and ES. This test aids in identifying any mis-specified components of the internal model used by banks to forecast ES. Moreover, it can be extended to analyze other systemic risk measures such as Marginal Expected Shortfall. Simulation experiments indicate that our test exhibits good finite sample properties for realistic sample sizes. Through application to two stock indices, we demonstrate how our methodology provides insights into the reasons for rejections in testing ES validity.