Quantile-based modeling of scale dynamics in financial returns for Value-at-Risk and Expected Shortfall forecasting

Abstract: We introduce a semiparametric approach for forecasting Value-at-Risk (VaR) and Expected Shortfall (ES) by modeling the conditional scale of financial returns, defined as the difference between two specified quantiles, via restricted quantile regression. Focusing on downside risk, VaR is derived from the left-tail quantile of rescaled returns, and ES is approximated by averaging quantiles below the VaR level. The method delivers robust, distribution-free estimates of extreme losses and captures skewness, heavy tails, and leverage effects. Simulation experiments and empirical analysis show that it often outperforms established models, including GARCH and joint VaR-ES conditional-quantile approaches. An application to daily returns on major international stock indices, spanning the COVID-19 period, highlights its effectiveness in capturing risk dynamics.