Historical variance–covariance matrix: problems

In risk management, the monitoring of changes in the variance and covariance of
the assets that comprises a portfolio is an extensive process of distinguishing shifts
over a period of time. Overseeing changes in the variance (or risk) of each asset and
the relationship between the assets in a portfolio through the variance is achieved
using the variance–covariance matrix. For the variance and covariance estimates in
the matrix to be reliable it is necessary that the joint distributions of security returns
are multivariate normal and stationary. However, as discussed earlier, investigations
have found that the distribution of speculative price changes are not normally
distributed and exhibit fat tails. Consequently, if the distribution of return for individual series fails to satisfy the i.i.d. assumptions, it is inconceivable to expect
the joint distribution to do so. Therefore, forecasts based on past extrapolations of
historical estimates must be viewed with skepticism. Hence, the usefulness of
conditional heteroskedastic models in multivariate setting to be discussed next.