Factor Exposures and Factor Returns in Regression Models
Summary
The discussion clarifies the roles of variables in a factor return model. The question contrasts a familiar regression, where observed factor data are multiplied by estimated coefficients, with a formulation involving benchmark returns, pure factor portfolio returns, and exposure values. The answer explains that factor exposures are the independent-variable data, while the coefficients or factor premia describe how those exposures relate to asset or portfolio returns.
In the cited formulation, the quantities formed by subtracting the benchmark return from pure factor portfolio returns serve as the factor return contributions, and the b terms represent standardized exposures. Thus, the apparent reversal comes from notation and from distinguishing factor exposures from factor returns, rather than from a fundamentally different regression structure. The explanation is brief and points readers to a paper appendix for further detail; it does not derive the estimation procedure, address identification, or show empirical results.
Key ideas
- Factor exposures are input data in a cross-sectional factor return model.
- Regression coefficients and factor exposures play distinct roles.
- Pure factor portfolio returns relative to a benchmark represent factor return premia in the cited setup.
- Notation can obscure the familiar relationship between explanatory variables and coefficients.
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# Why do Factor Models set up their factors differently from regression?
# Why do Factor Models set up their factors differently from regression?
While this may be awkwardly-titled, I hope that my question becomes clearer upon reading.
So this is what I gather about Factor Models: they are statistical models set up to explain the returns, ${return}_i$ of a portfolio, based on certain security characteristics (or factors) plugged in as the independent variables, after which the resulting coefficients will be used as the factors' weightings in the realized portfolio.
However, what I do not understand is why the factors are not used in and of themselves, such as in this equation, from page 4 of this article (Clarke, DeSilva, Thorley: Pure Factor Portfolios and Multivariate Regression Analysis, JPM 2017) on factor investing:
$${return}_i = r_M + (r_1 - r_M)b1_i + (r_2 - r_M)b2_i + \ldots + \epsilon_i$$
($r_M$ stands for a "benchmark portfolio return", $r_i - r_M$ together stands for some "one-period return to a pure factor portfolio minus the benchmark return", and $b$ stands for some "factor exposure")
Why isn't this equation instead set up as like a normal linear model? $${return}_i = intercept + \beta_1{factor}_i + \beta_2{factor}_2 + \ldots + \epsilon_i \ \ ?$$
In general, I am quite confused about this set-up, and would extremely appreciate someone taking me step-by-step through the logic of this regression.
## Answer by develarist (score 3)
https://quant.stackexchange.com/a/46122
Thanks for editing your original post to show that betas are in front of the factors.
In factor models, $\beta$ are factor loadings (regression coefficients) while $X$ are factor exposures (independent variables/the data). The model in the paper uses $r_i-r_m$ as factor loadings (premia over some benchmark), while $b_i$ are standardized factor exposures (not to be confused with $\beta_i$), so the first formula is in the format of what you expect of factor models described in your second formula. In the pdf article, search the word 'factor exposure' to see how $b_i$, the data, should be incorporated in the model. It says there is a separate appendix that also explains it more.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.