A parametric approach to estimating and forecasting Value-at-Risk (VaR) and expected shortfall (ES) for a heteroscedastic financial return series is proposed. The well-known GJR-GARCH form models the volatility process, capturing the leverage effect. To capture potential skewness and heavy tails, the model assumes an asymmetric Laplace form as the conditional distribution of the series. Furthermore, dynamics in higher moments are modeled by allowing the shape parameter in this distribution to be time-varying. Estimation is via an adaptive Markov chain Monte Carlo (MCMC) sampling scheme, employing the Metropolis-Hastings (MH) algorithm with a mixture of Gaussian proposal distributions. A simulation study highlights accurate estimation and improved inference compared to a single-Gaussian-proposal MH method. The model is illustrated by applying it to four international stock market indices and two exchange rates, generating one-step-ahead forecasts of VaR and ES. Standard and non-standard tests are applied to these forecasts, and the finding is that the proposed model performs favourably compared to some popular competitors: in particular it is the only conservative model of risk over the period studied, which includes the recent global financial crisis. © 2010 Elsevier B.V. All rights reserved.
Chen, Q., Gerlach, R., & Lu, Z. (2012). Bayesian Value-at-Risk and expected shortfall forecasting via the asymmetric Laplace distribution. Computational Statistics and Data Analysis, 56(11), 3498–3516. https://doi.org/10.1016/j.csda.2010.06.018