Measuring market risk under the Basel Accords: Var, stressed var, and expected shortfall

Each of the most recent accords of the Basel Committee on Banking Regulation (known as Basel II, 2.5, and III), 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 paper evaluates how well the reforms embraced by Basel 2.5 and III — stressed VaR and expected shortfall — have addressed longstanding regulatory concerns with conventional VaR.

For illustrative purposes, this paper describes the calculation of conventional, parametric VaR according to a Gaussian distribution. Known weaknesses of conventional VaR include from model and estimation risk, readily illustrated by performing parametric VaR with more robust statistical distributions that assume higher levels of excess kurtosis and greater tail risks. VaR not only performs poorly under extreme economic stress; it also fails to satisfy theoretical constraints on “coherent” measurements of risk.

Expected shortfall can be calculated as an extension of conditional VaR. Unlike conventional VaR, expected shortfall provides a coherent measure of risk. On the other hand, VaR outperforms expected shortfall (and, for that matter, nearly every other general spectral measure of risk) by satisfying the mathematical demands of “elicitability.”

Under any of the Basel Accords, the choice of a measure of market risk poses an irreconcilable conflict between coherence and elicitability. Mathematical limitations on measures of risk therefore force regulators and bankers to choose between the theoretically sound consolidation of diverse risks and reliable backtesting of risk forecasts against historical observations.

Journal of Economic Literature (JEL) categories:

F30 International finance
G10 General financial markets
G15 International financial markets
G20 Financial institutions and services
K10 Law and economics