An efficient and accurate approach is proposed for forecasting Value at Risk [VaR] and Expected Shortfall [ES] measures in a Bayesian framework. This consists of a new adaptive importance sampling method for Quantile Estimation via Rapid Mixture of t approximations [QERMit]. As a first step the optimal importance density is approximated, after which multi-step `high loss' scenarios are efficiently generated. Numerical standard errors are compared in simple illustrations and in an empirical GARCH model with Student-t errors for daily S&P 500 returns. The results indicate that the proposed QERMit approach outperforms several alternative approaches in the sense of more accurate VaR and ES estimates given the same amount of computing time, or equivalently requiring less computing time for the same numerical accuracy.

expected shortfall, importance sampling, mixture of Student-t distributions, numerical accuracy, numerical standard error, value at risk, variance reduction technique
Bayesian Analysis (jel C11), Simulation Methods; Monte Carlo Methods; Bootstrap Methods (jel C15), Forecasting and Other Model Applications (jel C53), Criteria for Decision-Making under Risk and Uncertainty (jel D81)
Tinbergen Institute
Tinbergen Institute Discussion Paper Series
Discussion paper / Tinbergen Institute
Tinbergen Institute

Hoogerheide, L.F, & van Dijk, H.K. (2008). Bayesian Forecasting of Value at Risk and Expected Shortfall using Adaptive Importance Sampling (No. TI 2008-092/4). Discussion paper / Tinbergen Institute. Tinbergen Institute. Retrieved from