Interpreting Cointegration Eigenvectors as Portfolio Hedge Weights
Summary
The question asks why eigenvectors from the Johansen cointegration procedure can serve as hedge ratios for a portfolio intended to be stationary. The answer offers a geometric account of eigenvectors and relates the weights to historical dependence among asset prices. It suggests that coefficients derived from the data can scale the instruments so their combined movements produce a portfolio with a stable relationship.
The explanation compares this idea with adjusting one instrument by a simple price-scale factor, then says eigenvector-based coefficients also reflect covariance and average volatility. However, it conflates eigenvectors and eigenvalues in its discussion and does not accurately derive the Johansen result. In the cointegration framework, the relevant cointegrating vector defines a linear combination of nonstationary series that is stationary under the model assumptions; the answer's account of volatility equalization is not a sufficient explanation. It provides intuition but omits model selection, testing details, and validation of estimated hedge ratios.
Key ideas
- A cointegrating vector specifies weights for a linear combination of price series that is stationary under the fitted model.
- The vector's coefficients can be interpreted as relative hedge weights for constructing a spread.
- Data-derived weights reflect relationships estimated across the observed series rather than simple price scaling alone.
- The answer's geometric explanation is imprecise and confuses eigenvectors with eigenvalues.
- Estimated cointegration relationships require validation and may not remain stable out of sample.
Tags
Full text
# What is the logic of the eigenvectors of the Johanson cointegration test determining hedge ratios? # What is the logic of the eigenvectors of the Johanson cointegration test determining hedge ratios? Reading Algorithmic Trading: Winning Strategies and Their Rationale, Ernie Chan and there is a short section about the Johanson test for cointegration where it is mentioned that > the eigenvectors resulting from this test can be used as a vector of hedge ratios for the instruments in question to form a stationary portfolio. My question is: what is the logic in doing this / how does this make sense (Ie. what is the logic in taking the resulting eigenvector values and using them as the hedge ratios for the portfolio)? What property about using these values then makes the portfolio stationary? ## Answer by Anonymous (score 3, accepted) https://quant.stackexchange.com/a/57800 Necropost. I know, but this article helped me some time ago. https://georgemdallas.wordpress.com/2013/10/30/principal-component-analysis-4-dummies-eigenvectors-eigenvalues-and-dimension-reduction/ The geometric meaning If you have multiple vectors in space, e.g. matrix of prices for several stocks, then eigenvalue is an angle measuring of how much each vector needs to be rotated to align it with other vectors in the matrix. You can get a clearer picture if you check algorithms for Jacobi, Givens, or just a plain rotation. The main idea is that trigonometric `cos` and `sin` functions can define an angle between vectors. So, if you iteratively multiply elements on the main diagonal of the matrix by `cos(X)` and other by `sin(X)` and keep the value of angle X between iterations, then eventually you'll find a combination of X values that make all vectors in the matrix to be aligned along the main diagonal and all values outside main diagonal will be -> 0, which means that these vectors (stocks) are now heading in the same direction. https://en.wikipedia.org/wiki/Plane_of_rotation Physical meaning Using eigenvalues as weights in the portfolio means that you equalize the volatility of these stocks to make them move together. The primitive solution is to compare prices of the stocks in the portfolio and multiply them by missing volatility factor, e.g. ``` SPX is 3000 SPY is 300 x 10 = 3000 ``` So, both of them are equally heavy now. The advantage of eigenvalues is that you're using more precise method to find coefficients based on a list of historical prices and their covariance, i.e. level of dependency between them, which takes into account not only the current difference in prices but average volatility, as well.
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