پیش بینی های چند مرحله ای برای مدل های گارچ چند متغیری: راه حل فرم بسته و ارزش برای مدیریت پرتفولیو

This paper derives the closed form solution for multistep predictions of the conditional means and covariances for multivariate ARMA-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 cumulated higher frequency returns are required as inputs in the mean-variance portfolio problem. The empirical value of the result is evaluated by comparing the performance of quarterly and monthly rebalanced portfolios using monthly MSCI index data across a large set of GARCH models. Using correct multistep predictions generally results in lower risk and higher returns.

This paper derives the closed form solution for multistep predictions of the conditional means and covariances from multivariate GARCH models. These predictions are useful in mean-variance portfolio analysis, when the rebalancing frequency is lower than the data frequency. In the application the empirical value of this result is evaluated in the performance of quarterly rebalanced portfolios based on correct three-step predictions. We compare their performance with that of quarterly rebalanced portfolios incorrectly based on one-step predictions and with the performance of monthly rebalanced portfolios. We use monthly Morgan Stanley Capital International (MSCI) index data for six regions. Multistep prediction in GARCH models has been considered previously in e.g. Baillie and Bollerslev (1992). They derive the minimum mean squared error forecasts for the conditional mean and the conditional variance of univariate GARCH processes. We extend their results to the multivariate case and derive closed form representations for the conditional mean and the conditional covariances h-steps ahead. In addition we derive the explicit formula for the conditional covariance of the sum of the conditional means up to h-steps ahead. This corresponds to the conditional variance of the cumulative returns over an h-period horizon, when modelling asset returns. In our empirical application portfolios are adjusted quarterly based on GARCH models estimated with monthly data. This implies that the conditional variances of monthly returns cumulated over three months have to be computed. The empirical part of our study is related to Ledoit, Santa-Clara and Wolf (2003), who apply one-step predictions from multivariate GARCH models for portfolio selection using — as we do — MSCI regional indices. However, our study is based on multistep predictions and the results are based on a larger set of GARCH models. In particular, the value of the derived multistep predictions for portfolio management is evaluated on monthly data for six regional MSCI indices during the evaluation period January 1992 to December 2003. The minimum variance portfolios are tracked for 48 different GARCH models, both for monthly and quarterly rebalancing. In the latter case the quarterly rebalanced portfolios correctly based on multistep predictions and those incorrectly based on one-step predictions are evaluated. We find that 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 paper is organized as follows: In Section 2 the multistep prediction problem is discussed. Section 3 contains the empirical application in portfolio management. Section 4 briefly summarizes and provides conclusions.

In this paper we have derived the closed form solution for multistep predictions of the conditional means and covariances for multivariate GARCH models and have illustrated their value for portfolio management. Multistep predictions of the conditional means and covariances are needed for mean-variance portfolio analysis when the rebalancing frequency is lower than the data frequency. In order to deal with this problem we have also derived the explicit formula for the conditional covariance matrix of the corresponding cumulative higher frequency returns. The closed form solution for the general ARMA (p, q)-GARCH (k, l) case is provided in Section 2 along with a convenient recursive representation. The practical relevance of the theoretical results is assessed empirically with an application to six regional MSCI indices using a large variety of GARCH models. Based on monthly data, the portfolio performance of monthly and quarterly rebalanced portfolios is investigated and compared to the naive portfolio, which is based on the sample mean and covariance. The quarterly rebalancing decision is either correctly based on three-step predictions or incorrectly on one-step predictions. The evaluation period is January 1992 to December 2003. The following main results are obtained: First, portfolios based on GARCH models — labelled GARCH portfolios — have on average higher return, lower risk and higher Sharpe ratio than the naive portfolio. Second, almost all quarterly rebalanced GARCH portfolios based on correct multistep predictions exhibit both higher returns and higher risk than the monthly rebalanced GARCH portfolios even in the absence of transaction costs. The higher returns are surprising because monthly adjusted portfolios incorporate new information faster and should therefore outperform quarterly adjusted portfolios. Third, portfolios based on correct predictions show on average lower risk than the corresponding portfolios based on incorrect predictions. Furthermore, all GARCH portfolios based on correct predictions result in higher returns and Sharpe ratios than those based on incorrect predictions.