Interpreting PCA Eigenvectors and Weights in Long-Short Portfolios

# How to extract normalised portfolio weights from PCA, when the eigenvector has negative elements?


# How to extract normalised portfolio weights from PCA, when the eigenvector has negative elements?












Most of the examples of using PCA of asset returns to construct an eigen portfolio seem to tend to focus on equities, which tend to all be positively correlated. As such I usually see normalised (such that they sum to 1) asset weights along the lines of:

```
weights = eigenvector[0, :] / sum(eigenvector[0, :])
```

Where eigenvector zero is associated with the eigenvalue of the largest variance.

Occasionally negative values will appear in that PC0 (for example if there is a negatively correlated asset introduced), which will be explained as a weight towards a short sale. But I am confused about how these weights should be correctly normalised in the presence of negative value(s).

Following the same procedure as above, the weights will of course sum to one, but the net and gross exposures now will of course be different. Is this just accepted?

Follow up question, should weights here actually be the loadings (eigenvector * sq.root(eigenvalues))?

## Answer by Carson McKee (score 3)

https://quant.stackexchange.com/a/65631

I'm not entirely sure what you are carrying out the PCA on, are you using a correlation matrix or covariance matrix? As for the negative eigenvalue issue, the sign associated with an eigenvalue is not important as eigenvectors are only unique up to a constant. If you have a square matrix, $X$, an Eigenvalue $\lambda_i$ and eigenvector $v_i$ for $X$ satisfy, \begin{equation} Xv_i = \lambda_i v_i \end{equation} If you flip the sign of $\lambda_i$ to $-\lambda_i$ then it implies that we have $-v_i$ as an eigenvector of $X$ and the above equation is still valid. These eigenvalues represent the variance explained by the corresponding Eigenvector in relation to the original axis.

For your question on weights, I'm not fully sure what you are weighting. You could allocate a 'weight' to each Eigenvector as $\frac{\sum_{j=1}^{k}\lambda_j}{\sum_{i=1}^{p}\lambda_i}$ which would represent the the proportion of variance explained by including the first k components. You may then use this to drop or retain components that contribute little to the overall variance of the portfolio.

If by weights you are instead interested in how to map your original data (an equities returns portfolio??) onto the new axis (principal components). Then for this you would use the loadings matrix. For a given column in the loadings matrix you have an eigenvector. Each element in the vector represents a 'weight' corresponding to one of your original variables which maps that variable onto the new axis (principal component) for this eigenvector. It then follows that by multiplying your original data matrix by the loadings matrix, you arrive at the score matrix (your original data mapped to the new axis of principal components). I believe the loadings matrix is the matrix of weights that you are looking for. Please add a comment if I've gone in the wrong direction here.