Forecasting VaR and ES using a joint quantile regression and implications in portfolio allocation

In this paper we propose a multivariate quantile regression framework to forecast Value at Risk (VaR) and Expected Shortfall (ES) of multiple financial assets simultaneously, extending Taylor (2019). We generalize the Multivariate Asymmetric Laplace (MAL) joint quantile regression of Petrella and Raponi (2019) to a time-varying setting, which allows us to specify a dynamic process for the evolution of both VaR and ES of each asset. The proposed methodology accounts for the dependence structure among asset returns. By exploiting the properties of the MAL distribution, we then propose a new portfolio optimization method that minimizes the portfolio risk and controls for well-known characteristics of financial data. We evaluate the advantages of the proposed approach on both simulated and real data, using weekly returns on three major stock market indices. We show that our method outperforms other existing models and provides more accurate risk measure forecasts compared to univariate ones.