Covariance data

Construction of volatility and correlation estimates
At the heart of the variance/covariance methodology for computing VaR is a covariance
matrix for the relative returns. This is generally computed from historical time
series. For example, if ri (tj ), jó1, . . . , n, are the relative returns of the ith asset over
nò1 consecutive days, then the variance of relative returns can be estimated by the
sample variance
m2
i ó; n
jó1
r2
j (tj)/n
In this expression, we assume that relative returns have zero means. The rationale
for this is that sample errors for estimating the mean will often be as large as the
mean itself (see Morgan and Reuters, 1996). Volatility estimates provided in the
RiskMetrics data sets use a modification of this formula in which more recent data
are weighted more heavily. This is intended to make the estimates more responsive
to changes in volatility regimes. These estimates are updated daily.
Estimation of volatilities and correlations for financial time series is a complex
subject, which will not be treated in detail here. We will content ourselves with
pointing out that the production of good covariance estimates is a laborious task.

First, the raw data needs to be collected and cleaned. Second, since many of the risk
factors are not directly observed, financial abstractions, such as zero-coupon discount
factor curves, need to be constructed. Third, one needs to deal with a number
of thorny practical issues such as missing data due to holidays and proper treatment
of time-series data from different time zones. The trouble and expense of computing
good covariance matrices has made it attractive to resort to outside data providers,
such as RiskMetrics.