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I have N stocks, and a covariance matrix that indicates the covariance of these N random variables. Now, if I run PCA on the covariance matrix, what can you tell about the principle component?

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    $\begingroup$ If you have a covariance matrix of the returns, the first principle component is comparable to the CAPM or the Single Index Model, ie, there is one common factor that explains a good part of the individual returns $\endgroup$ – David Duarte Nov 29 '20 at 12:23
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PCA is nothing more than a special change of basis, such that most of the variance in the data is concentrated in the first eigendirections. So the first eigendirection will explain more variance than any other eigendirection. The princal components are then the original data transformed into this new basis.

If $N$ is large, then the first eigendirection or pc is the main factor that moves these assets. So one might say that this "common single factor among all stocks" is the general market risk of the market portfolio ($\beta$ in CAPM), i.e. a factor not attributed to individual stocks, rather global macro aspects.

While the first factor is generally considered the market risk/systemic risk, the other factors are more fuzzy. To my knowledge, there is no clear interpretation of these, as it also depends on what your stock universe consists of. Potential interpreration for these other factors could be equity sectors (healthcare, financial, industrial), equity stock fundamentals (revenue, balance sheets), etc...

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Your first PC will be the "true" beta of the stock, compared to the other stocks in your sample. As opposed to the beta to the usual market index that people choose to measure this against.

If it isn't, then you have a very strange sample of stocks ;-)

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