Estimating Portfolio Risk with a Covariance Matrix
Summary
The document introduces covariance as a measure of how two assets move together, then outlines a matrix-based approach to portfolio risk. It describes collecting observations for several stocks, calculating each series’ mean, subtracting that mean, and using the centered data to form a symmetric covariance matrix. The diagonal entries represent individual variances, while the off-diagonal entries capture co-movement.
Given portfolio weights that sum to one, the matrix can be combined with the weights to estimate portfolio variance; its square root gives standard deviation. Varying the weights produces portfolios that can be compared by risk and plotted to illustrate an efficient frontier, including minimum-risk and risk-return choices. The article is conceptual and provides no worked numerical example or empirical results. It describes centering closing prices, whereas practical portfolio risk estimation generally uses returns; the observation window, frequency, and estimation choices can materially affect the result.
Key ideas
- Covariance indicates whether two asset series tend to move together or in opposite directions.
- A covariance matrix summarizes individual variances and pairwise co-movements.
- Portfolio variance depends on both the covariance matrix and the vector of asset weights.
- Portfolio weights are treated as capital shares that sum to one.
- Changing weights creates portfolios with different risk and return characteristics.
- The article centers price levels, so its method needs care when applied to practical risk estimates.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.