In this paper, we propose a new stochastic volatility model based on a generalized skew-Student-t distribution for stock returns. This new model allows a parsimonious and flexible treatment of the skewness and heavy tails in the conditional distribution of the returns. An efficient Markov chain Monte Carlo (MCMC) sampling algorithm is developed for computing the posterior estimates of the model parameters. Value-at-Risk (VaR) and Expected Shortfall (ES) forecasting via a computational Bayesian framework are considered. The MCMC-based method exploits a skewnormal mixture representation of the error distribution. The proposed methodology is applied to the Shenzhen Stock Exchange Component Index (SZSE-CI) daily returns. Bayesian model selection criteria reveal that there is a significant improvement in model fit to the SZSE-CI returns data by using the SV model based on a generalized skew-Student-t distribution over the usual normal and Student-t models. Empirical results show that the skewness can improve VaR and ES forecasting in comparison with the normal and Student-t models. We demonstrate that the generalized skew-Studentt tail behavior is important in modeling stock returns data.