Solved – Principal Component Analysis Stock Returns

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I am new to PCA and am having trouble understanding some parts of the methodology. In multi-factor models you can run regressions like:

Stock Return = A(1)*Size + A(2)*Market Return + …

So in this case, you have a clear understanding of what is driving the stocks return. Now, when you have a system of say 100 stocks and extract the 3 most important principal components, is there any way of actually understanding where the risk is coming from? For example, if you ran a regression like:

Stock Return = A(1)*PC1 + A(2)*PC2

While that would tell you the effect of each component, how can that actually help an investor manage their portfolio? Because the PC doesn't really tell you what the actual risk factor is right? Or are you supposed to then run something like:

PC1 = A(1)*Size +…+

To determine what underlies each component?

Please help me with an intuitive answer.

Best Answer

Stock returns is not the good place to start with learning PCA, because it will not produce you interpretable factors. If your objective is to understan PCA then it's better to start with interest rates, yield curves. There you often get first three factors roughly corresponding to the level, slope and the curvature of the yield curves.

With the stock returns, maybe the first factor could be mapped to the market factors, i.e. similar to CAPM. Otherwise, you'll get a bunch of PCs that are hard to assign to any intuitive or usual risks.

Related Solutions

Solved – Clustering of stock market returns

This example from the scikit-learn project might give you some ideas on how to combine sparse covariance graph estimation with traditional clustering so as to identify some of the underlying structure of a market from daily price data.

Disclaimer: I contribute to the scikit-learn project even though I am not the one who wrote this example.

Solved – Principal component analysis in R

In R you do not need to compute the covariance matrix etc. yourself. Better use the prcomp() function.

If data is your 524x50 matrix you need to do

# scale. = TRUE if you want to scale variables to same variance.
pca <- prcomp(data, scale. = TRUE)
plot(pca)

The latter plot is a screeplot, with the amount of explained variance per principal component. To get the loadings, you simply do

pca$rotation
pca$rotation[,1]

It is fine to have only positive or negative loadings on the first component. If all your variables are very correlated that can be the case.

Not knowing what you want to do with this PCA, you can do a biplot.

biplot(pca)

This shows the individuals projected on the first 2 principal components, and the variables with an identical projection so that you can compare them.

You can also do it manually on each principal component.

data %*% pca$rotation[,1]
# Get projecions on both axes.
data %*% pca$rotation[,1:2]

Regarding Q4, it means that you can simplify your 50-dimensional space in a 20-dimensional space while keeping 80% of your variance. The score you are proposing is a 1D space. If you want to keep the first 20 PC, you need to project on the first 20 PC as follows:

projects <- data %*% pca$rotation[,1:20]

But 20-dimensional can barely be called a simplification. Most of the time the aim is to have 2-3 because you can still represent them in our space. Keeping 20 might make sense though. Depends on how you want to follow up.