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

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.
Comment: 30 pages; 7 tables; 5 figures