Interpreting Covariance-Based Portfolio Risk as a Forecast

Summary

The document discusses portfolio variance calculated as the weight vector multiplied by a historical variance–covariance matrix and by the transposed weight vector. With weights held fixed, this calculation represents the variance of portfolio returns over the historical sample when those weights are applied to the asset returns. Taking its square root gives the corresponding volatility estimate. It is not automatically a measure of future risk.

To use the historical result as a forecast, one must assume that future return behavior will resemble the period used to estimate the covariance matrix. The answer notes GARCH as another approach to estimating future volatility, but does not compare forecasting methods or describe how to choose a sample window. If portfolio weights changed during the historical period, applying today's weights throughout that period may not match the risk actually realized by the portfolio as it was held.

Key ideas

• A covariance matrix and fixed portfolio weights produce historical portfolio variance when applied to historical asset returns.
• The square root of that variance is the corresponding portfolio volatility.
• Using historical volatility as a forecast assumes future behavior resembles the estimation period.
• GARCH is mentioned as an alternative future-risk estimation approach.
• Applying today's weights to historical returns may differ from realized risk under changing historical holdings.

Tags

# Portfolio risk estimation through variance covariance matrix


# Portfolio risk estimation through variance covariance matrix












Is the portfolio risk calculated through variance covariance matrix an estimate of the current risk of the portfolio? Suppose I am using the weights as of today, and I have estimated the variance covariance matrix from historical returns of the assets and I calculate portfolio risk as wVw'. Is it an estimate of the future risk in the portfolio?

How does this estimate relate to the realized portfolio risk i.e. the standard deviation of the portfolio returns?

## Answer by AlRacoon (score 1)

https://quant.stackexchange.com/a/38374

The calculation you provide is the realized historical variance of the portfolio. The volatility would be the square root of this calculation and would be equal to the realized portfolio risk.

If one views that the future will look like the historical period which was used to calculate the volatility you describe, one can use this as an estimate of future risk. There are other methods to estimate future risk such as GARCH.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.