Updating Correlations While Preserving a Valid Correlation Matrix

Summary

The document asks how to revise a historical correlation matrix when one pairwise correlation receives new information, while retaining as much of the original matrix as possible. Its example considers changing the equity–Treasury relationship and asks how that change should affect correlations involving corporate bonds. The central constraint is positive definiteness, which limits the values that remaining correlations can take.

The author describes an iterative workaround: replace the target correlation, check the eigenvalues, adjust negative eigenvalues to small positive values, reconstruct the matrix, and repeat while restoring the desired correlation and diagonal. No validated algorithm, comparison, or empirical result is supplied, and the author acknowledges flaws in this procedure. The document therefore frames a covariance regularization and constrained estimation problem, but leaves open how to choose a principled point estimate that incorporates the new information without distorting the rest of the matrix.

Key ideas

• Changing one pairwise correlation can make a correlation matrix fail positive definiteness.
• Positive definiteness constrains the feasible values of the other correlations.
• The author proposes iterative eigenvalue adjustment and reconstruction as a workaround.
• The iterative procedure is acknowledged as flawed and is not supported by comparative evidence.
• A principled update would need to balance the revised correlation against preservation of historical structure.

Tags

# Adjusting a correlation matrix based on one changed correlation


# Adjusting a correlation matrix based on one changed correlation












I have a correlation matrix that is created by historical asset returns, but I want to see how changing one of those correlations would affect the rest of the correlation matrix. How would I go through changing the correlation matrix to account for the new information?

Let's say I have 3 asset indices - Equity, US Treasury, and US Corporate Bonds. Historically, Equity Treasury correlation is 0.2, Equity Corporate is 0.4, and Treasury Corporate is 0.4 as well (I'm making these up). But, I think that in the future, Equity Treasury will be -0.2. How do I use that new information to update what the impact to the Corporate related correlations will be?

Are there any articles that discuss this that the Stack Exchange community is aware of?

I know that given two correlations a range exists around the possible values the third in order to maintain positive definite-ness. But is there a way to arrive at a point estimate given as much information as possible is retained from the initial correlation matrix.

Right now, I'm just changing the correlation, checking eigenvalues, and if any are negative, I'm making them slightly positive, and reconstructing. Then, I iterate by resetting the changed correlation and diagonals and test again. I'm using ~10 assets for my purpose and the correlation matrix doesn't look too funky. I'd still love to know what other options people think is better. ... mine definitely has flaws.

Thanks!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.