Dynamic expected shortfall: A spectral decomposition of tail risk across time horizons

The tail risk of financial institutions is traditionally measured by Expected Shortfall (ES) that does not characterize risk changes over investment horizons. Using wavelet analysis, we propose a new method to capture the dynamics of ES across time horizons. The new method decomposes the stock return of financial institutions into different frequency (e.g., short-, mid-, and long-run) components, and then, models the dynamics of these components separately to produce an aggregated ES forecast. We provide numerical and empirical examples to illustrate the new method. We also study the relevance of each frequency component to out-of-sample ES forecasts over different predictive horizons. Our empirical results confirm that the different frequency components of stock returns exhibit different persistence. Explicitly considering this distinction when modeling ES significantly improves the out-of-sample forecasting performance. In addition, excluding the long-run (e.g., yearly) return component can largely reduce short-run (e.g., weekly or monthly) ES forecasts without impacting the regulatory quality of the risk assessment.