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A Simple Traffic Light Approach To Backtesting Expected Shortfall

Abstract. We propose a Traffic Light approach to backtesting Expected Shortfall which is completely consistent with, and analogous to, the Traffic Light approach to backtesting VaR initially proposed by the Basel Committee on Banking Supervision in their 1996 consultative document [5]. The approach relies on the generalised coverage test for Expected Shortfall developed in [6].

Nick Costanzino and Michael Curran 25 January 2018

Introduction

Even before the initial Basel Committee consultative document [10] there had been a push by both risk managers and academics to replace VaR with another risk measure that addresses VaR's deficiencies. In particular, coherent risk measures [2, 3, 13] satisfy the basic desired properties required by a risk measure as outlined in [4]. Expected Shortfall is the natural choice among all coherent risk measures and therefore no surprise that it has been chosen by the Basel Committee as the risk measure to replace VaR. However, unlike the case of VaR, there is no well-established backtesting framework for Expected Shortfall. Indeed the current Base proposal to backtest ES at the 97.5 quantile is to backtest the related VaR estimate at the 97.5 and 99 quantiles which is a grossly insufficient test. Nevertheless some recent backtesting methods have been proposed including, but not limited to, [1, 6, 8, 9, 11, 15].

The main result of this paper is the development of a Traffic Light backtest for Expected Shortfall which extends the Traffic Light backtest for VaR. The test relies on the computation of critical values derived from the finite-sample distribution of the ES test statistic (3.5) first introduced in [6].

The note is organised as follows. In Section 2 we briefly review the VaR Coverage Test to provide context for out ES Traffic Light test. In Section 3 we define the Traffic Light test for ES and compute the distribution of the finite-sample test statistic from which we calculate the critical values using a root-finding algorithm. Finally in Section 4 we discuss the test and some implications.

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