Bayesian forecasting of Value at Risk and Expected Shortfall using adaptive importance sampling

Abstract

An efficient and accurate approach is proposed for forecasting the Value at Risk (VaR) and Expected Shortfall (ES) measures in a Bayesian framework. This consists of a new adaptive importance sampling method for the Quick Evaluation of Risk using 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 alternative approaches, in the sense that it produces more accurate VaR and ES estimates given the same amount of computing time, or, equivalently, that it requires less computing time for the same numerical accuracy.