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An example diagram, taken from this paper, looks like follows:

enter image description here

What is its physical interpretation? The highest eigenvalue, the paper says, represents market mode. So, what does the difference in distance of the maximum eigenvalue from the next largest tell us? What does each (x,y) pair in the plot tell us?

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In a physical system the eigenvectors represent "modes" of random "vibration" (random movement) of a system and the eigenvalues represent the variance (i.e. amplitude) of each of these modes. The eigenvalues added together equal the variance of the overall (combined) movements (the Decomposition Property). In the stock market the main mode (biggest eigenvalue) is the common market movement (stocks' tendency to go up and down together) sometimes called the market factor or beta factor. There is no agreement on what the second is, but it might be, hypothetically, an upward movement in high book-to-market stocks on a day when the low book-to-market stocks go down. In any case, the eigenvalues 2, 3, ..., n involve some category of stock going down/up and some other category going up/down. In the 1970s it was thought that the second mode might be stocks that benefit from higher oil prices (oil companies) go up while stocks that are hurt by rising oil prices go down. Sensitivity to other macroeconomic factors were also considered as possible modes (interest rates, inflation, the USD exchange rate). Nowadays small cap vs large cap or Value vs growth are popular candidates. In any case the exact description of these factors is not easy (it probably changes over time) and is not really the objective of this type of article. They generally just try to describe how many modes would explain most (say 80 or 90%) of the variance, i.e. the "complexity" of the stock market. The main conclusion is that one mode (one eigenvalue) is not enough (therefore rejecting the CAPM as an adequate model of variance) and that there clearly other modes that are important. The size of the 2d eigenvalue compared to the first is a measure of the inadequacy of a single factor model to explain the stock market; all studies show that the second eigenvalue is clearly not negligible compared to the first.

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  • $\begingroup$ Thanks for your explanation! It's hard to find papers or articles describing these things well. I have a few questions, and would be glad if you address them. When you say, "modes of random vibration", what does that mean? Mode in statistics refers to the most commonly occurring element in a set. So, I am not able to put 2 and 2 together here. $\endgroup$ – Kristada673 Jun 2 '16 at 6:27
  • $\begingroup$ (2) Do eigenvalues denote variance among stocks, or their probability density does? Because if its the latter, that would mean in the figure there are lot of stocks with high variance, and only few with low variance (meaning they go up and down together), which is in conjunction with the discussions in the paper. Am I right? But if its the former, it means there are lot of stocks with low variance, and few stocks with high variance. But this is not in conjunction with the paper, as it says the high eigenvalues are interesting, and low ones are noise. Could you kindly elaborate a bit on that? $\endgroup$ – Kristada673 Jun 2 '16 at 6:39
  • $\begingroup$ (3) When you say, "They generally just try to describe how many modes would explain most (say 80 or 90%) of the variance", does it mean that the number of modes that explain 80-90% of the variance are the important ones (and the rest can be ignored when studying and analyzing the stock market) from the point of view of an investor looking to invest in a stock portfolio? If so, can the investor make an intelligent decision from this analysis, which stocks should they invest in to give the most returns? $\endgroup$ – Kristada673 Jun 2 '16 at 7:14
  • $\begingroup$ (1) Modes in physics is different from modes in statistics. Modes are different ways the system can move. For example an aircraft has at least three modes of rotation: yaw which points the aircraft to a different destination, roll which lifts one wing and drops the opposite wing, and pitch which lifts or drops the nose away or towards the ground. Motions take place according to a direction vector which is equivalent to an eigenvector. $\endgroup$ – noob2 Jun 2 '16 at 13:25
  • $\begingroup$ (2) This analysis does not consider expected returns, but only risks of stock portfolios (portfolio = linear combination of individual stocks). If only 3 eigeinvalues matter then we can accurately predict the variability (again: not necessarily the return) of any portfolio knowing only the exposure to three factors (three "betas") and the idiosyncratic risk of each stock. It would be very useful for risk measurement of portfolios. $\endgroup$ – noob2 Jun 2 '16 at 13:31

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