Interpreting Eigenvalues of Stock Return Correlation Matrices
Summary
The document explains how to read eigenvalues from a correlation matrix of stock returns. Each eigenvalue measures the variance associated with a corresponding mode of co-movement, while its eigenvector describes the combination of stocks that moves together in that mode. The largest eigenvalue is commonly interpreted as the broad market factor, reflecting stocks’ tendency to move in concert.
The remaining eigenvalues represent additional patterns, which might correspond to contrasts such as value versus growth, small versus large companies, or sensitivities to macroeconomic conditions. Their precise interpretation can be difficult and may change over time. Comparing the second eigenvalue with the first indicates how much a single market factor misses, while the spectrum as a whole can indicate how many modes are needed to account for much of market variation. The discussion is conceptual: it gives no specific dataset or empirical procedure, and it cautions that smaller modes do not have universally agreed economic labels.
Key ideas
- An eigenvalue measures the variance associated with a mode of stock co-movement.
- The largest eigenvalue is commonly associated with the broad market factor.
- Other modes may reflect contrasts between stock groups or sensitivities to economic conditions.
- A substantial second eigenvalue suggests that a single-factor account misses important variation.
- The precise economic meaning of smaller modes can be uncertain and can change over time.
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# Answer by nbbo2 (score 1) # What does each bar in the empirical average eigenvalues spectrum of the correlation matrix of log-returns of stocks represent? An example diagram, taken from this paper, looks like follows: 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? ## Answer by nbbo2 (score 1) https://quant.stackexchange.com/a/26354 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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