Decomposing Portfolio Variance into Factor Contributions
Summary
The document asks how to attribute a portfolio’s factor-model variance to individual factors. It starts from a covariance decomposition into factor risk and specific risk, then forms position-weighted factor exposures by multiplying asset weights by the exposure matrix. The resulting factor variance is a quadratic form in those exposures and the factor covariance matrix.
The answer outlines the relevant matrix dimensions and suggests expanding that quadratic form as a double sum. This gives a starting point for understanding factor risk, but it does not complete the decomposition or explain how to assign covariance terms to individual factors. When factors are correlated, a factor’s total contribution depends on how cross-factor terms are allocated, so a convention must be specified. The discussion is therefore an incomplete pointer rather than a full calculation method.
Key ideas
- Factor variance can be calculated separately from specific variance in a factor covariance model.
- Portfolio weights and asset factor exposures combine to produce position-weighted factor exposures.
- The resulting factor variance is a quadratic form using the factor covariance matrix.
- Correlated factors create cross terms that require an explicit attribution convention.
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# Decompose portfolio in factor risk
# Decompose portfolio in factor risk
I am reading the risk chapter of Grinold Active Portfolio Managment. I understand how to calculated specific and factor risk of my portfolio, what I don't understand is how to calculate how much risk I have in each factor.
For example, given the following:
$\sigma^2_p = w \textbf{S} w = w \textbf{X} \textbf{F} \textbf{X}^T w' + w \textbf{O} w'$
where: S is portoflio covariance matrix X is the factor exposure w is my holding position F is my factor covariance matrix O is diagonal matrix of specific variance
how do I calculate how much risk I have in each factor? Basically, I am trying to decompose my portoflio in each direction of risk.
## Answer by KaiSqDist (score 1)
https://quant.stackexchange.com/a/79751
I am also reading up on this recently.
Since you are not looking at the specific variance, you should ignore $w^TOw$, where I treat $w$ are column vectors (different notation from you).
If we look at the factor variance, we have $w^TX^TFXw$. Assuming we have $f$ number of factors and $n$ number of assets, the matrices are of the following dimensions:
- $w$ is $n \times 1$ and $w^T$ is $1 \times n$
- $X$ is $f \times n$ and $X^T$ is $n \times f$
- $F$ is $f \times f$
We can perform the matrix multiplication of $w^TX^T = W^T$ and $Xw = W$ first, obtaining the position-weighted factor exposures of dimensions of $1 \times f$ and $f \times 1$, respectively.
Once you move onto $W^TFW$, it becomes a double summation if I recall correctly. You need to organize the double summation such that they are separated into their factor risks. This should work!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.