On the Covariance Matrices Used in Value at Risk Models

Abstract

This article examines the covariance matrices that are often used for internal value at risk models. We first show how the large covariance matrices necessary for global risk management systems can be generated using orthogonalization procedures in conjunction with univariate volatility forecasting methods. We then examine the performance of three common volatility forecasting methods: the equally weighted average of squared returns; the exponentially weighted average; and generalized autortegressive conditional heteroscedasticity (GARCH). Standard statistical evaluation criteria using equity and foreign exchange data with 1996 as the test period give mixed results, although they generally favor the exponentially weighted moving average methodology for all but very short-term holding periods. But these criteria assess the ability to model the center of returns distributions, while value at risk models require accuracy in the tails. Operational evaluation takes the form of backtesting volatility forecasts following the Bank for International Settlements (BIS) guidelines. For almost all major equity markets and U.S. dollar exchange rates, both the equally weighted average and the GARCH models produce results falling within the acceptable "green zone." But on most of the test data, and particularly for foreign exchange, using predictions from exponentially weighted moving average models leads to an unacceptably high number of outliers. Thus value at risk measures calculated using this method would be understated.

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