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.

• First could you explain, are $(r_1 - r_M)$ supposed to be the regression coefficients and $b1_i$ the independent variables? And in your own formula, is $factor_i$ a regression coefficient or an independent variable? Jun 14, 2019 at 19:18
• This is possibly where I am confused. Based on the article on page 3, I cannot tell if $(r_1 - r_M)$ is supposed to be the "data" representing the "pure factor return" - which sounds like an accounting of how much a factor is worth, having taken into account some base market return - or if it is being estimated as well. $r_1$ refers to an individual factor, whereas $r_M$ refers to a market rate of sorts, which leaves me confused about what $b1_i, b2_i, \ldots$ represents. Jun 14, 2019 at 21:09
• The reason behind this is the difference between a statistical model and a factor model. While statistical ones try to explain past returns, factor ones try to extract risk premia. For this reason, they look at the excess return over the benchmark. Jun 15, 2019 at 16:12

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.