Multinomial VaR Backtests: A simple implicit approach to backtesting expected shortfall

TL;DR

This paper introduces a multinomial backtesting approach for expected shortfall risk measures, demonstrating its superior power over traditional methods through simulations and real-data application, enhancing risk model validation.

Contribution

It proposes a novel multinomial test framework for backtesting expected shortfall, improving detection of model misspecifications compared to standard binomial tests.

Findings

Multinomial tests with N≥4 outperform binomial tests in detecting model misspecifications.

Pearson test is simple; Nass test is robust; LRT is highly powerful but computationally intensive.

Application to 2008 financial crisis data illustrates practical effectiveness.

Abstract

Under the Fundamental Review of the Trading Book (FRTB) capital charges for the trading book are based on the coherent expected shortfall (ES) risk measure, which show greater sensitivity to tail risk. In this paper it is argued that backtesting of expected shortfall - or the trading book model from which it is calculated - can be based on a simultaneous multinomial test of value-at-risk (VaR) exceptions at different levels, an idea supported by an approximation of ES in terms of multiple quantiles of a distribution proposed in Emmer et al. (2015). By comparing Pearson, Nass and likelihood-ratio tests (LRTs) for different numbers of VaR levels $N$ it is shown in a series of simulation experiments that multinomial tests with $N\geq 4$ are much more powerful at detecting misspecifications of trading book loss models than standard binomial exception tests corresponding to the case $N=1$. Each test has its merits: Pearson offers simplicity; Nass is robust in its size properties to the choice of $N$; the LRT is very powerful though slightly over-sized in small samples and more computationally burdensome. A traffic-light system for trading book models based on the multinomial test is proposed and the recommended procedure is applied to a real-data example spanning the 2008 financial crisis.