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The covariance matrix tabulates pair-wise interactions between variables (assets) one-at-a-time into a grid, which can quickly become large as the number of assets included in a portfolio, for example, is increased to the hundreds or thousands, contributing to the curse of dimensionality. Elongating the covariance matrix like this also often just merely bumps up the number of corresponding rows and columns that are deemed redundant due to eventually numerous collinearities. If the covariance matrix is fundamental to many multivariate financial models, it is more of a necessity, due to lack of better alternative measures, and far from an ideal.

Like how total correlation does for correlation, is there a measure that represents the entire covariance structure as a scalar or something similarly small? If so, would the pair-wise mentality found in finance prevail regardless because its intuitiveness overrides its issues mentioned above?

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    $\begingroup$ Usually, I have a look at the first couple of largest eigenvalues and their proportions of sum total in order to get a first feeling for the dependence... $\endgroup$
    – Kermittfrog
    Commented Oct 20, 2020 at 10:01
  • $\begingroup$ eigenvalues are abstract and don't correspond to individual assets though $\endgroup$
    – develarist
    Commented Oct 20, 2020 at 10:19
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    $\begingroup$ Eigenvalues give you information about how much assets comove due common factor(s). Also, it is probably easier quantified and communicated. Though it offers information on the linear relationship, only. $\endgroup$
    – Kermittfrog
    Commented Oct 20, 2020 at 15:35
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    $\begingroup$ Specifically, I would calculate $\frac{\lambda_1-1}{N-1}$. If the assets share many common sources of (linear) variation, this number is close to 1. The more orthogonal risk sources there are, the faster this measure goes to 0. $\endgroup$
    – Kermittfrog
    Commented Oct 22, 2020 at 8:56
  • $\begingroup$ Is there a source article or textbook that talks about this approach in more detail $\endgroup$
    – develarist
    Commented Oct 22, 2020 at 14:03

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