Multistep Predictions for Multivariate GARCH Models: Closed Form Solution and the Value for Portfolio Management

Abstract

In this paper we derive the closed form solution for multistep predictions of the conditional means and covariances for multivariate GARCH models. These predictions are useful e.g. in mean variance portfolio analysis when the rebalancing frequency is lower than the data frequency. In this situation the conditional mean and the conditional covariance matrix of the cumulative higher frequency returns until the next rebalancing period are required as inputs in the mean variance portfolio problem. The closed form solution for this quantity is derived as well. We assess the empirical value of the result by evaluating and comparing the performance of quarterly and monthly rebalanced portfolios using monthly MSCI index data across a large set of GARCH models. The value of using correct multistep predictions is assessed by comparing the performance of the quarterly rebalanced portfolios based on the correct multistep predictions with the quarterly rebalanced portfolios incorrectly based on 1-step predictions and the monthly rebalanced portfolios. Using correct multistep predictions generally results in lower risk and higher returns. Furthermore the correctly computed quarterly rebalanced portfolios exhibit higher returns than monthly rebalanced portfolios. The empirical results thus forcefully demonstrate the substantial value of multistep predictions for portfolio management.

Keywords: Multivariate GARCH models, volatility forecasts, portfolio optimization, minimum variance portfolio

JEL Classification: C32, C61, G11