Standard diversification models have a fundamental weakness: during periods of large market fluctuations and crises, historical relationships stop working. This happens because correlations between assets tend to increase, often sharply.
To address this problem, practitioners use dynamic covariance models, such as EWMA or GARCH. These models allow us to capture the impact of recent shocks on the covariance of asset returns.
While a static model treats historical correlation as given, dynamic models continuously adjust both the variance of asset returns and the covariance matrix in response to new market information. As a result, correlation itself becomes time-varying:
As an example, let us consider an upgrade of the linear VaR (Value at Risk) model using a dynamic covariance matrix.
Assume the following portfolio:
Next, we collect historical data:
Then we compute return dynamics (assuming zero dividends):
After that, for the standard linear VaR model, we construct a covariance matrix based on historical data:
Finally, we calculate the risk measure (see details here: VaR Linear Model).
Now let us see how the model is updated using the EWMA approach.
First, we construct the variance dynamics:
For more details on lambda and EWMA, see: EWMA vs MA.
Note that in the first row we use the last available observation from the dataset — the model requires the most recent (previous day’s) information.
Next, we compute covariances between asset returns, for example for AAPL:
In the following step, we obtain the corresponding dynamic correlation measures. After processing all asset pairs, we construct the full dynamic covariance matrix:
Finally, we compute the updated risk measures:
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As the results from the two models show, the difference in risk estimates is not always large under normal conditions. However, the key point is that this difference becomes significant during crisis periods, when correlations increase and diversification weakens.
Excel file: Dynamic Covariance
Adapted from:
Options, Futures, and Other Derivatives — John C. Hull