Mapping Whitened Factor Transformations to Asset Portfolios

Summary

The document poses a portfolio mapping problem that arises when factor returns are whitened and then reduced or transformed into secondary factors. It distinguishes the time series of original factor returns from the matrix of asset portfolios associated with those factors, and asks how to construct secondary portfolios whose returns match the transformed factor representation at every time step.

The setup gives the relevant matrix dimensions and defines secondary factor returns as the centered original returns multiplied by a whitening matrix and a transformation matrix. However, it does not include an answer or derive the corresponding portfolio transformation. A solution would need to account for the centering of factor returns as well as the order and orientation of the matrices, but those derivation details are outside the supplied text. The document is therefore useful as a clearly specified linear algebra question, rather than as a complete method or evidence-backed result.

Key ideas

• Whitening and dimensionality reduction transform factor return series through matrix multiplication.
• Each original factor is associated with a portfolio of underlying assets.
• The secondary portfolio matrix should reproduce the transformed factor returns at each time step.
• The supplied discussion specifies the problem dimensions but does not derive the required portfolio mapping.

Tags

# Transforming portfolios when transforming factors


# Transforming portfolios when transforming factors












This should be a relatively simple, mostly linear algebra question. It starts with factors, let's say 50 of them. Each factor has a time series of returns, let's say 250, and an associated portfolio of assets, let's say 1000. The factor returns get whitened and then transformed into a fewer number of secondary factors, let's say 10, using a transformation matrix.

Here are relevant matrices:

```
X (250, 50) - factor returns
P (1000, 50) - factor portfolios (unit norm for each factor)
xmeans (50) - a vector of means for each time series (to center them)
W (50, 50) - the whitening matrix
T (50, 10) - the transformation matrix
Z (250, 10) - secondary factor returns
R (1000, 10) - secondary factor portfolios <--- to compute
```

so that `Z = (X - xmeans) * W * T`

The question is, how to transform the original portfolio matrix into a corresponding secondary portfolio matrix (so that the portfolio returns from both representations - X with P, Z with R - are the same for any given time step)?

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.