# Why does the 2nd Principle Component of a portfolio explain lots of variance in the data but has a low R^2 to portfolio?

I have used QQQ (nasday etf)holdings and weights data in a PCA model to see what components drive the daily returns of QQQ. What I found was PCA 1 explains 52% of the variance and PCA 2 explains 19% of the variance in the data set. Here is an example of what the data looks like (only 3 stocks shown of 102):

      MSFT      AAPL       NVDA
1 -0.00103    -0.00118  -0.00124
2 -0.000144   -0.00320   0.000188
3  0.00193     0.00125  -0.000655
4  0.00000505  0.000803  0.00101


Here is a summary of the PCA results:

What i am not understanding is, when i regress PCA 1 onto QQQ returns, I get an R^2 of 89% and when i regress PCA 2 onto QQQ returns I get an R^2 of 3%. How can this be the case if PCA 2 explains 19% of the variance in the data set?

• If a component makes some stocks go up and some stocks go down, then it could explain a lot of the movements of individual stocks but relatively little of the QQQ index (the effects would largely cancel when you take the average of all stocks). Usually the 1st component affects all stocks together, but the 2d, 3d affect different groups of stocks differently. (Put another way the 2d, 3d etc eigenvectors usually have both positive and negative entries). Nov 19, 2021 at 11:51