Combining a Large Pool of Forecasts of Value-at-Risk and Expected Shortfall

Value-at-risk (VaR) and expected shortfall (ES) have become widely used measures of risk for daily portfolio returns. As a result, many methods now exist for forecasting the VaR and ES. These include GARCH-based modelling, approaches involving quantile-based autoregressive models, and methods incorporating measures of realised volatility. When multiple forecasting methods are available, an alternative to method selection is forecast combination. In this paper, we consider the combination of a large pool of VaR and ES forecasts. As there have been few studies in this area, we implement a variety of new combining methods. In terms of simplistic methods, in addition to the simple average, the large pool of forecasts leads us to use the median and mode. As a complement to the previously proposed performance-based weighted combinations, we use regularised estimation to limit the risk of overfitting due to the large number of weights. By viewing the forecasts of VaR and ES from each method as the bounds of an interval forecast, we are able to apply interval forecast combining methods from the decision analysis literature. These include different forms of trimmed mean, and a probability averaging method that involves a mixture of the probability distributions inferred from the VaR and ES forecasts. Among other methods, we consider smooth transition between two combining methods. Using six stock indices and a pool of 90 individual forecasting methods, we obtained particularly strong results for a trimmed mean approach, the probability averaging method, and performance-based weighting combining.