Condition Numbers of Covariance and Correlation Matrices
Summary
The document asks whether a correlation matrix is necessarily better conditioned than the covariance matrix from the same multivariate return data. One answer reports observing that correlation matrices were more well-conditioned in randomly generated examples, then raises the concern that covariance-based financial models may be numerically unstable. This observation is anecdotal and does not establish a general ordering or proof.
A second answer says condition numbers can be compared when they describe the same computational problem, such as matrix inversion, and points to numerical-analysis material for further characterization. The exchange does not derive a theorem relating the two matrices. Correlation standardizes each variable by its volatility, while covariance retains scale and risk information, so substituting one for the other changes the model input and interpretation. Conditioning alone therefore does not show that a correlation matrix is universally preferable; matrix conditioning must be considered alongside the task and the information the model needs.
Key ideas
- The reported examples suggest correlation matrices can be better conditioned, but they do not prove a universal rule.
- Condition numbers are meaningful to compare when they assess the same computational problem.
- Covariance matrices retain asset variance information, while correlation matrices standardize by volatility.
- Changing from covariance to correlation changes model inputs and can omit scale and risk information.
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Full text
# Which is more ill-conditioned, the asset correlation matrix or covariance matrix? # Which is more ill-conditioned, the asset correlation matrix or covariance matrix? If i have a matrix of multivariate asset returns for $N$ stocks, and i compute from it the covariance matrix and then the correlation matrix, can I always know which of the two will have the higher condition number (higher to infinity means more ill-conditioned, as opposed to near 1 for well-conditioned)? or is the condition number of two different (types of) matrices completely incomparable? If one is always more well-conditioned than the other, is there a mathematical proof for this? other criteria besides the condition number are welcome ## Answer by develarist (score 1, accepted) https://quant.stackexchange.com/a/57431 After having tried this with randomly generated vectors, I am consistently seeing the correlation matrix of randomly generated numbers, regardless of which distribution they are sampled from, are always more well-conditioned than the covariance matrix. Which is strange because the covariance matrix exists before the correlation matrix: the correlation matrix must be computed from the covariance matrix, and the other way around cannot be done. In other words, the covariance matrix, being more ill-conditioned, in fact is transformed into a more well-conditioned, stable, matrix when it is converted to the correlation matrix. which makes me wonder if all the financial models that rely on the covariance matrix would be better of using the correlation matrix as an input instead, given all the animosity towards the instability and ill-conditioning of the covariance. I know that the covariance possess variance, or risk, so slanting models to strictly interpret correlations instead would result in missing out on the more relevant measure, which is risk, not correlation, so it seems that we are putting interpretability first compared to other highly-related options, which comes at the price of numerical instability and estimation error ## Answer by Quantoisseur (score 1) https://quant.stackexchange.com/a/57415 Yes, you can compare matrix condition numbers if evaluating them for the same problem, for example taking the matrix's inverse. For L2: For the additional mathematical characterization of conditioning and its impact, check out the first half of these lecture notes from a class I took: https://github.com/mandli/intro-numerical-methods/blob/master/12_LA_conditioning_stability.ipynb
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.