Alexander-Leigh (1997) On the Covariance Matrices Used in Value-at-Risk Models

Summary

Alexander and Leigh examine the covariance matrices used inside internal value-at-risk (VaR) models. They show how the large covariance matrices required by firm-wide risk systems can be generated by combining univariate volatility forecasts with orthogonalization (principal-component) procedures, and then compare three common volatility-forecasting methods — the equally weighted average, the exponentially weighted moving average (EWMA), and GARCH — on equity and foreign-exchange data with 1996 as the test period. The central message is that a method's fit to the center of the returns distribution can mislead: VaR requires accuracy in the tails, and by that standard EWMA fares poorly.

Key Claims

• Building large covariance matrices. For global risk management the full covariance matrix is generated via orthogonalization of the risk factors (PCA) combined with univariate volatility forecasts — a dimension-reducing route that became orthogonal / principal-component GARCH.
• Statistical vs. operational evaluation diverge. Standard statistical criteria (which assess the middle of the returns distribution) generally favour EWMA for all but very short holding periods.
• Backtesting reverses the ranking. Operational evaluation — backtesting volatility forecasts under the Bank for International Settlements (BIS) traffic-light guidelines — places the equally weighted average and GARCH in the "green zone" for almost all major equity markets and USD exchange rates, while EWMA predicts an unacceptably high number of outliers, especially for foreign exchange.
• Implication for VaR. Choosing a volatility model by central-fit criteria is unsafe for VaR; tail/exception performance is what matters for capital adequacy.

Concepts Introduced or Extended

• Value at Risk
• Covariance Matrix Estimation

Entities Mentioned

• Carol Alexander

Quotes

"Standard statistical evaluation criteria … generally favour the exponentially weighted moving average methodology for all but very short term holding periods. But these criteria assess the ability to model the centre of returns distributions, whereas value-at-risk models require accuracy in the tails … on most of the test data, and particularly for foreign exchange, exponentially weighted moving average models predict an unacceptably high number of outliers."

My Take

A useful early corrective to the RiskMetrics-era default of EWMA volatility: the paper's design — pairing a statistical evaluation (center-of-distribution fit) against an operational one (BIS exception backtesting) — is exactly the right way to expose that the two disagree, and that only the second is decision-relevant for VaR. The finding that EWMA looks best statistically yet fails in the tails, while the humble equally-weighted average and GARCH pass the green-zone test, is a durable caution against optimizing volatility models for the wrong loss function. The orthogonalization half of the paper is the more influential seed: it is the germ of Carol Alexander's principal-component GARCH programme (Alexander 2002) for large-dimensional covariance matrices. The dataset is dated (1996 test period, pre-crisis), so the specific green-zone verdicts should be read as illustrative rather than current.