Minimum Variance Portfolios as Factor Mimicking Portfolios
Summary
The document addresses why minimum variance weights can be used to estimate a factor return, even though factor models are often associated with explaining common variation. Its explanation distinguishes common factor exposure from diversifiable idiosyncratic risk. When assets are assumed to share equal exposure to the market factor, combining their returns can estimate the common market return; reducing noise from asset-specific residuals can make that estimate more useful.
The stated minimum variance weights are proportional to the inverse covariance matrix applied to a vector of ones, then normalized. The response interprets this as precision-weighting asset returns and notes the estimation challenge that factor loadings and factor returns depend on one another. It also presents inverse variances as a practical workaround when residual variances are unknown. The discussion is an intuitive sketch, not a complete derivation; its equal-loading assumption and simplified treatment of residual risk limit how broadly it applies.
Key ideas
- Minimum variance weights can reduce idiosyncratic noise in an estimate of a shared factor return.
- The illustration assumes each asset has equal exposure to the market factor.
- The weights use the inverse covariance matrix and are normalized to form a portfolio.
- Estimating factor returns and factor loadings involves a mutual dependence between the two quantities.
- Using asset variances as a proxy for residual risk is presented as a workaround, not an exact model.
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# Why minimum variance portfolio is used to construct factor models
# Why minimum variance portfolio is used to construct factor models
I am reading Tsay's classic "Analysis of Financial Time Series" and I have seen him using minimum variance portfolio
Relevant passage on the minimum variance portfolio here (Chapter 9, Page 411）
And "factor mimicking portfolio" here. (Chapter 9, Page 419）
Question: I'm a bit confused why we need to find the weights that minimize the portfolio variance as the weights to compute the factor returns. I'd think that if we want to define one factor, we want it to explain as much variance as possible, rather than minimizing the variance?
## Answer by krkeane (score 0, accepted)
https://quant.stackexchange.com/a/80270
> I'm a bit confused why we need to find the weights that minimize the portfolio variance as the weights to compute the factor returns. I'd think that if we want to define one factor, we want it to explain as much variance as possible, rather than minimizing the variance?
As a casual answer, idiosyncratic risk is diversifiable; common factor risk is non-diversifiable.
In the equation $$ w = \frac{\Sigma^{-1} \boldsymbol{1}}{\boldsymbol{1}^{\textrm{T}}\Sigma^{-1} \boldsymbol{1}} ~, $$
the numerator $\Sigma^{-1} \boldsymbol{1}$ is a precision weighted combination of the assets and the denominator $ \boldsymbol{1}^{\textrm{T}}\Sigma^{-1} \boldsymbol{1} $ is a scaling factor for portfolio weights $w$.
If all assets are equally impacted by the market return, $$ r_i = r_{\textrm{market}} + \ldots + \epsilon_{i}~, $$ a reasonable attempt at an estimate of $r_{\textrm{market}}$ is $$ {\hat{~r~}_\textrm{market}} \propto \Sigma^{-1}r ~ . $$
Not considering $\textrm{Var}\left(\epsilon_i\right)$ results in noisier estimates of $r_{\textrm{market}}$ . We don't know $\textrm{Var}\left(\epsilon_i\right)$, so $\sigma_{i,i}^{-1}$ is a work-around.
Edit
You need factor loadings to estimate factor returns, and you need factor returns to estimate factor loadings.
Here, $\boldsymbol{1}$ is a vector of loadings on market return. $$ \begin{aligned} r_i &= l_{i,\textrm{market}} &r_{\textrm{market}} + \ldots + \epsilon_{i} \\ r_i &= 1 &r_{\textrm{market}} + \ldots + \epsilon_{i} \end{aligned} $$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.