Multiple-Period Market Risk Prediction under Long Memory: When VaR is Higher than Expected∗

Several authors, including Andersen and Bollerslev (1998), stress the importance of long-term volatility dependence for value-at-risk (VaR) prediction. The present paper addresses multiple-period market risk forecasts under long memory persistence in market volatility. To this aim, we propose volatility forecasts based on a combination of the GARCH(1,1)-model with potentially fat-tailed and skewed innovations and a long memory specification of the slowly declining influence of past volatility shocks. As the square-root-of-time rule is known to be mis-specified, we use the GARCH setting of Drost and Nijman (1993) as the benchmark forecasting model. Our empirical study of equity market risk is based on daily index returns during the period January 1975 to December 2010. We study the out-of-sample accuracy of VaR predictions for five, ten, 20 and 60 trading days and document that our approach remarkably improves VaR forecasts for the longer horizons. The result is only in part due to higher predicted risk levels. Ex-post calibration to equal unconditional risk levels illustrates that our approach also enhances efficiency in allocating VaR capital through time.