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I'm struggling to understand how Principal Component Analysis (PCA) is used in Factor Models of returns. For example, in the JPMorgan paper (p.19) the authors write:

In a multi asset portfolio, factor analysis will identify the main drivers such as momentum, value, carry, volatility, liquidity, etc. A very well-known method of factor analysis is Principal Component Analysis (PCA).

In my understanding of PCA, PCs are some factors that exist but are not known and cannot be described as momentum factor, volatility factor, etc. For example, I can show that 3 PCs explain 99% variation in returns, but I cannot label those components as momentum, volatility, etc. Clearly, I'm missing something here. Can somebody explain how PCA is used in identifying factors referred to in the quote above?

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  • $\begingroup$ Give a look at the eigenvectors. Those are the weights of the mutually uncorrelated "(eigen)portfolios". Take one of these portfolios, analyze the assets weights and try to give it an interpretation. Good luck. $\endgroup$ – Lisa Ann Nov 9 '20 at 19:12
  • $\begingroup$ "Can somebody explain how PCA is used in identifying factors referred to in the quote above?" - what I've seen (in FX and rates) has simply involved guesstimating what the 1st, 2nd (etc) PC might be correlated with and creating an overlay chart of the two. $\endgroup$ – user42108 Nov 9 '20 at 19:28
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You are right: the "factors" stemming from the literature of CAPM anomalies and the "components" of PCA are not of the same nature

  • as you underlined: factors are meant to have an economic sense (even if you have a factor like "betting against the beta", that are not that clear and have more a behavioral interpretation).
  • whereas PCA is a statistical procedure.

A first consequence is that Components are orthogonal while Factors are not. It means that you can think about thousands of Factors but you have only a limited of (statistically) significant Components (see Financial Applications of Random Matrix Theory: a short review, by Bouchaud and Potters for details about PCA in finance). Fundamentally any combination of characteristics of a set of financial instruments is a Factor. Some attempts are on-going to exhibit an objective and quantitative procedure to identify "real and meaningful Factors* (see A Protocol for Factor Identification, by Pukthuanthong, Richard Roll, and Subrahmanyam), but no consensus is reached by now.

Moreover, this proliferation of factors lead to an inflation of papers commenting them, until one of the editors of Journal of Finance, Campbell Harvey, published "…and the Cross-Section of Expected Returns" to try to counter this trend.

On the PCA side, note that a time scale is important: you can choose the one you want, corresponding to the time scale of your investment strategies. It is not really the case of Factors that have their own scale (for instance quarterly for factors exploiting fundamental / accounting characteristics).

These two concepts meet when you try to build a portfolio based on selecting investment instruments using factors. The risk of your portfolio will be driven by the covariance matrix between the instruments, while its expected returns will be driven by Factor: the two views are confronted there. I suggest the reading of Introduction to Risk Parity and Budgeting, by Thierry Roncalli, since it explores this kind of mixings.

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