  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.
