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Given a set of asset classes (specifically their log-returns), I would like to decompose their returns in to the principle components, in a manner that will preserve the scale of the returns. I'm not sure if I'm explaining properly so I'll try using an example: if I look at US large-cap equities, small-cap equities and treasuries, I expect the first PC to be the "equities returns" (both in shape and magnitude), the 2nd to be the small-vs-large premium and the 3rd the bonds returns (or something very similar in nature).

The problem I come across is that PCA requires mean centering, so the mean returns of all the PCs is zero. To overcome this, I've tried performing the transformation on the non-centered returns, using the eigenvectors calculated on the centered returns. There is a similar question here (PCA scores in a for portfolio replication task: stumble over mean-centering question) but it doesn't address the same problem nor provides a solution. The transformation I described distorts the scale of the transformed returns, so that "equities returns" are almost twice as large compared to large-cap or small-cap alone (which have very similar returns overall). So my question boils down to - how do I perform the transformation while preserving the scale of the returns?

I'm attaching a visualization of my results in hopes that it will make my questions clearer. The graph is log-scaled. It easy to see that PC-1 matches equity-returns, PC-2 matches the small-cap premium and PC-3 matches bonds returns, but that the scales of the PC returns are off.

Sample PCs and corresponding assets

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