Quantitative Finance Asked by lampShadesDrifter on October 9, 2020

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?

Necropost. I know, but this article helped me some time ago.

**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.

Correct answer by Anonymous on October 9, 2020

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