Choosing Weights When Repairing a Non-Positive-Definite Correlation Matrix

Summary

The document asks how to choose weights in a weighted Frobenius norm when using alternating projections to adjust an equity correlation matrix that is not positive definite. It raises the possibility of weighting entries according to portfolio exposures, so correlations among heavily held assets are changed less than correlations among lightly held assets. No definitive weighting rule is provided.

The response points to work on a variant of the method that accounts for factor structure before applying an unweighted norm. It also notes that equal weighting has often been adequate as a quick correction when a matrix is only slightly non-positive-definite. The suggested caution is that this kind of repair may be a stopgap: a fuller treatment should represent the matrix’s factor structure. The exchange gives no comparison of exposure-based weights, nor implementation details or performance evidence, so it leaves the portfolio-specific choice unresolved.

Key ideas

• Alternating projections can adjust a correlation matrix toward positive definiteness by minimizing a weighted Frobenius distance.
• Portfolio exposure weights are proposed as a possible way to limit changes to correlations in heavily held assets, but no answer validates that choice.
• An unweighted norm can serve as a quick correction when the matrix is only slightly non-positive-definite.
• Accounting for factor structure before matrix adjustment may provide a more complete approach.

Tags

# What weights should be used when adjusting a correlation matrix to be positive definite?


# What weights should be used when adjusting a correlation matrix to be positive definite?












I have a correlation matrix $A$ for an equity market that is not positive definite. Higham (2002) proposes the Alternating Projections Method, minimising the weighted Frobenius norm $||A-X||_W$ where $X$ is the resulting positive definite matrix.

How should one choose the weight matrix $W$?

The easy alternative is to weigh them equally (W is an identity matrix), but if one has exposures to a portfolio, wouldn't it be natural to weigh the correlations according to your weights of exposure in the different assets, in order to alter their historical correlation less than for those assets you have little exposure in? Or is there a more natural choice?

## Answer by Tal Fishman (score 1)

https://quant.stackexchange.com/a/2431

You may want to have a look at a later paper by Borsdorf, Higham, and Raydan (2010). I believe a variant of the same method may apply in your case. That is, you may want to account for some of the factor structure of your correlation matrix before you apply an unweighted Frobenius norm. Otherwise, using unweighted norms has often given me fine results anyhow, and this is often used only as a quick fix to slightly adjust matrices that are just barely not positive definite. A full approach should definitely be applying some factor structure (see a previous question of mine, as well as others on the site).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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