Why OLS Style Factor Portfolios Have Zero Exposure to Other Factors
Summary
The document derives the exposure properties of style factor portfolios formed from a cross-sectional ordinary least squares regression. Asset returns are regressed on a matrix whose columns contain the assets’ style loadings. The estimated factor returns are obtained by applying the regression operator to the return vector, and the rows of that operator can be interpreted as factor portfolios.
Multiplying those portfolios by the loading matrix yields the identity matrix, provided the regression design matrix has the required full rank. Each portfolio therefore has unit exposure to its own style and zero exposure to the other included styles. The response interprets this as reducing contamination between estimated factor returns and describes diversification as helping limit estimation error. The derivation concerns the factors included in the regression; it does not by itself guarantee neutrality to omitted exposures or explain every practical portfolio-construction choice.
Key ideas
- Cross-sectional OLS estimates factor returns by regressing asset returns on a matrix of factor loadings.
- Rows of the OLS regression operator can be viewed as portfolios that estimate factor returns.
- Multiplying the factor portfolios by the included loading matrix gives the identity matrix when the design matrix is full rank.
- Each portfolio has unit exposure to its target style and zero exposure to other included styles.
- The result does not guarantee neutrality to omitted factors or address all practical construction constraints.
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Full text
# characteristics of factor portfolios
# characteristics of factor portfolios
In the paper Characteristics of Factor Portfolios (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1601414), when it discusses pure factor portfolios, it says that simple style factor portfolios have zero exposure to all other style, country, and industry factors. Could someone help me understand the math for why the style factor portfolios have zero exposure to all other style, country, and industry factors?
## Answer by Tim Wilding (score 6)
https://quant.stackexchange.com/a/68646
Factor Portfolios are by-products of the cross-sectional regression techniques used to estimate style factor returns - the estimates of the return of a set of styles for a particular period. Style Factor Returns are estimated by regression of the returns for all assets in a single period on a matrix containing the styles for each asset in that period.
So, for example, if we are interested in the return of a P/E factor and a P/B factor, we would gather the P/E and P/B for all of our stocks into a matrix of loadings $B$. $B$ would have two columns – one containing P/E and one containing P/B for all assets. We then regress $R$ (a vector containing the returns of all assets) on $B$. OLS regression gives us $f= (B’B)^{-1} B’R$ = the returns of the style factors for this particular period. The rows of $(B’B)^{-1} B’$ are considered to be the factor portfolios.
So, let’s go one step further and look at the loadings of the portfolio on the individual styles by multiplying the factor portfolios with the matrix of loadings. This gives $(B’B)^{-1} B’B = I$ - an identity matrix. Hence, the loadings of each factor portfolio are 1 against the particular style and 0 against any other style.
Intuitively, this result makes sense. OLS is trying to find the best estimate of the returns to a particular style. So, OLS tries to find a set of portfolios that have the following characteristics – (1) a unit exposure to the style of interest, (2) zero exposure to any other style, and (3) minimal error by maximally diversifying the portfolio. We then look at the returns of that portfolio and say that that is the estimate of the factor return. Characteristic (1) ensures that the returns of the portfolio are approximately the returns of the style factor. Characteristic (2) ensures that the return of the factor portfolio is uncontaminated by other factors. Characteristic (3) ensures that there is minimal error on the estimate of the style factor return.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.