Measuring Market Risk Under the Basel Accords: VaR, Stressed VaR, and Expected Shortfall

Abstract

Each of the most recent accords of the Basel Committee on Banking Regulation, known as Basel II, 2.5, and II, has embraced a different primary measure of market risk in global banking regulation: traditional value-at-risk (VaR), stressed VaR, and expected shortfall. After introducing the mathematics of VaR and expected shortfall, this note will evaluate how well the reforms embraced by Basel 2.5 and III - stressed VaR and expected shortfall - have addressed longstanding regulatory concerns with traditional VaR.

Part I describes the calculation of VaR in its conventional form. For illustrative purposes, Part I will describe parametric VaR on a Gaussian distribution. Part II summarizes known weaknesses in VaR, from inherent model and estimation risk to VaR’s failure to perform under extreme economic stress and VaR’s failure to satisfy the theoretical constraints on “coherent” measurements of risk. Part III describes how to calculate expected shortfall as an extension of conditional VaR. It further describes how expected shortfall, but not VaR, provides a coherent measure of risk. Part III then reverses field. It explains how VaR, but not expected shortfall (or, for that matter, nearly every other general spectral measure of risk), satisfies the mathematical requirement of “elicitability.” Mathematical limitations on measures of risk therefore force regulators and bankers to choose between coherence and elicitability, between theoretically sound consolidation of diverse risks (on one hand) and reliable backtesting of risk forecasts against historical observations.

Keywords: Value-at-risk, VaR, expected shortfall, coherence, elicitability, Basel

JEL Classification: G15, G28, G32