Backtesting expected shortfall: a simple recipe?

Felix Moldenhauer; Marcin Pitera

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Need to know

We develop a new backtesting framework for Expected Shortfall by building on the standard Value-at-Risk procedure. Instead of counting the number of VaR breaches (i.e. exceedances of projected risk quantile by subsequent losses), our method consists of identifying the number of worst realisations for the secured position (i.e. sum of projected risk and realised PnL) that still add up to a negative total.

Our framework has strong mathematical foundations in the duality between risk measures and performance measures (the latter also referred to as acceptability indices). We show that the same relationship provides the basis for the standard VaR breach backtest.

In contrast to many existing frameworks, our approach is model-free and does neither require the joint estimation of Value-at-Risk and Expected Shortfall, nor a reference estimation procedure. Also, the proposed test is easy to implement in any programming language.

The simplicity and effectiveness of our framework, combined with the clear financial context, make our proposal a suitable candidate for regulatory backtesting of Expected Shortfall. Moreover, we show in empirical studies that our test statistic produces results that are consistent with the current regulatory framework, allowing for transferability of existing model features into the new risk model landscape.

Abstract

We propose a new backtesting framework for expected shortfall (ES) that can be used by regulators. Instead of looking at estimated capital reserve and realized cashflow separately, one can bind them into a secured position, for which risk measurement is much easier. Using this simple concept combined with monotonicity of ES with respect to its target confidence level, we introduce a natural and efficient backtesting framework. Our test statistics is given by the biggest number of worst realizations for the secured position that adds up to a negative total. Surprisingly, this simple quantity can be used to construct an efficient backtesting framework for unconditional coverage of ES in a natural extension of the regulatory traffic-light approach for value-at-risk. While being easy to calculate, the test statistic is based on the underlying duality between coherent risk measures and scale-invariant performance measures.

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