PELVE: Probability Equivalent Level of VaR and ES

Abstract

In the recent Fundamental Review of the Trading Book (FRTB), the Basel Committee on Banking Supervision proposed the shift from the 99% Value-at-Risk (VaR) to the 97.5% Expected Shortfall (ES) for internal models in market risk assessment. Inspired by the above transition, we introduce a new distributional index, the probability equivalence level of VaR and ES (PELVE), which identifies the balancing point for the equivalence between VaR and ES. PELVE enjoys many desirable theoretical properties and it distinguishes empirically heavy-tailed distributions from light-tailed ones via a threshold of 2.72. Convergence properties and asymptotic normality of the empirical PELVE estimators are established. Applying PELVE to financial asset and portfolio data leads to interesting observations that are not captured by VaR or ES alone. We find that, in general, the transition from VaR to ES in the FRTB yields an increase in risk capital for single-asset portfolios, but for well-diversified portfolios, the capital requirement remains almost unchanged. This leads to both a theoretical justification and an empirical evidence for the conclusion that the use of ES rewards portfolio diversification more than the use of VaR.