RDP 1999-04: Value at Risk: On the Stability and Forecasting of the Variance-Covariance Matrix 1. Introduction

Over the past decade value at risk (VaR) has become the most widely used technique for the quantification of market-risk exposure. From January 1998 banks in Australia have been permitted (subject to a range of conditions) to use their VaR models as the basis for determining the capital that is required to cover market-risk exposure. VaR is a measure of the potential loss that may occur from adverse moves in market prices (interest rates, exchange rates, equity prices and so forth). Specifically it is the dollar amount that portfolio losses are not expected to exceed, with a specified degree of statistical confidence, over a pre-specified period of time. There are a number of different methodologies used to calculate VaR (Cassidy and Gizycki (1997) provide a discussion of these models). The most widely used method is that known as the variance-covariance approach. This approach is based on the simplifying assumption that financial-asset returns are normally distributed and hence, the statistical distribution of these returns can be completely described by the mean of the market returns, the variance of market returns and the correlations between the various market rates (that is, the variance-covariance matrix).

The capacity for a variance-covariance VaR measure to accurately predict future risk exposures depends upon the quality of forecasts of the variance-covariance matrix incorporated into the VaR model. In this paper we first present the results of tests of the stability of the variances, covariances and correlations for exchange rates and Australian interest rates. Next we assess the performance of several time-series models that may be used to forecast the variance-covariance matrix, in particular three models for the variance-covariance matrix are considered: simple historical variances, exponentially weighted averages of historical variances (where the weights are progressively smaller for observations further in the past) and generalised autoregressive conditional heteroskedasticity (GARCH) models. While most Australian banks use the simpler models (past, observed variances or exponentially weighted moving averages) there are some banks that use the more complex GARCH approaches within their risk measurement models. This paper investigates the benefits of the more sophisticated approaches.