Mapping Risk Between Security Universes with Linear Regression

Summary

The document asks how to express risk in one set of securities in terms of another correlated set. It considers principal component analysis but identifies difficulty standardizing the inputs and concern about overfitting. The response recommends regressing the securities in the first universe on those in the second and using the fitted coefficients as a linear mapping between the two sets.

This approach represents the portion of the first universe's variation that can be explained by the second. Any remaining variation is residual risk and cannot be captured by the mapped exposures. The exchange provides no data, validation results, or specific guidance on choosing a sample, standardizing variables, or controlling overfitting, so the proposed mapping's reliability depends on those implementation choices.

Key ideas

• Linear regression can map exposures from one security universe into another correlated universe.
• The fitted regression coefficients form a linear transformation between the two sets of securities.
• The mapping leaves residual risk when the target securities do not explain all variation in the original set.
• The document does not establish how to prevent overfitting or select and validate the regression sample.

Tags

# Rebucketing Risk using PCA/other methods


# Rebucketing Risk using PCA/other methods












was working on a project and could use some help. New to the community and looking fwd to being an active part of it.

My question is, let's say we have a vector of securities V, and it trades with some correlation to another vector of securities W. I'm trying to figure out what the best way is to rebucket the risk in V to W equivalents. I tried running a PCA on V and W but am having trouble standardizing the variables. I'm also worried that I might be "overfitting" my dataset a little. Any guidance is appreciated.

## Answer by MattBecker82 (score 1)

https://quant.stackexchange.com/a/19481

Sounds like PCA is not the approach you're looking for. If you're looking to transform a risk vector in terms of securities V into a risk vector in terms of securities W, then the basic approach would be to perform a linear regression of V against W. The resulting regression coefficients will form a matrix B which will give a change of basis between V and W.

Note that there will in general be some residual "unexplained" risk, i.e. part of the variation in the securities V will not be explained by variation in the securities W.

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