Appendix 2: Variance/covariance methodologies

In this appendix, we formalize a version of the RiskMetrics variance/covariance
methodology. The RiskMetrics Technical Document sketches a number of methodological
choices by example rather than a single rigid methodology. This has the
advantage that the user can tailor the methodology somewhat to meet particular
circumstances. When it comes time for software implementation, however, it is
advantageous to formalize a precise approach. One source of potential confusion is
in the description of the statistical model for portfolio value. In some places, it is
modeled as normally distributed (see Morgan and Reuters, 1996, §1.2). In other
places it is modeled as log-normally distributed (Morgan and Reuters, 1996, §1.1).
The explanation for this apparent inconsistency is that RiskMetrics depends on an
essential approximation. The most natural statistical model for changes in what we
earlier termed fundamental asset prices is a log-normal model. An additional attractive
feature of the log-normal model for changes in fundamental asset prices is that
it implies that primary asset prices are log-normal as well. However, the difficulty
with the log-normal model is that the distribution of a portfolio containing more than
one asset is analytically intractable. To get around this difficulty, the RiskMetrics
methodology uses an approximation in which relative changes in the primary assets
are equated to changes in the log prices of these assets. Since relative returns over