returns - Fama French & Solving for Alpha

This is a question about comparing results from the Fama french 3 factor model.

I have not physically done this, but let's assume a Fama French 3 factor regression was performed for Coca-Cola (KO) and Pepsi (PEP). The model used was: $$r_{it}-r_{ft1}=\alpha_i+\beta_{im}(r_{mt}-r_{ft2})+\beta_{is}SMB+\beta_{ih}HML$$

$r_{it}$: return of Asset (either KO or PEP)

$r_{ft1}$: Risk Free Rate (3-month T-bill or equivalent investor uses), also the benchmark in this example

$\alpha_i$: what we are solving for aka output from regression aka intercept, Portfolio (Asset) Return minus Benchmark Return

$(r_{mt}-r_{ft2})$:Market Return minus the Risk Free Rate (3-month T-bill or equivalent investor uses)

Assume the rest of the variables are their regular assumptions as found in textbooks

Now I distinguish between $r_{ft1}$ and $r_{ft2}$ because I have read on this site found here that the $r_{ft}$ is the benchmark, the risk-free market return. Now in their original model, they did not distinguish the 1 and 2 on the risk free rate as I did. This leads me to think that $r_{ft1}$ is interchangeable with a benchmark such as the S&P 500 for example.

My question is, this alpha value solved assumes the benchmark is risk-free rate universally across all assets. Although this is a way to compare all assets, that doesn't mean in theory you can compare two different funds/portfolios/assets this way when they are comprised of different items. You should use the other definition of alpha=Asset minus Benchmark return. So can I change the $r_{ft1}$ to be the benchmark of my asset.

Is this an acceptable/practiced method of thinking about $r_{ft1}$?

Lastly, it seems like two different definitions of alpha are being used. Define alpha as Portfolio (Asset) Return minus Benchmark, how we tend to think of alpha. But re-arranging the model would make alpha equal to the Portfolio Return minus Benchmark plus other factors. So now alpha = alpha + stuff. As you can see I am lost and need some clarification about alpha in Fama French.

Answer

It's fine to put any excess return on the left hand side of the regression.

Definition of excess return

The difference between two returns is called an excess return.

An excess return is the result of going long one portfolio return and short another (such as risk risk free rate). An excess return is a payoff that can be achieved at zero cost (in some idealized, somewhat unrealistic world).

Let $R^f$ be the 1 month risk free rate, let $R^A$ be the return of Apple, and let $R^G$ be the return of Google. Examples of excess returns:

• $R^A_t - R^f_t$ is an excess return


• $R^A_t - R^G_t$ is an excess return


• $2 \left( R^A_t - R^G_t \right)$ is an excess return

(For the mathematically inclined, the space of excess returns is a vector space.)

Any excess return can go on the left hand side of a regression in factor models

In the Fama-French five factor model and other factor models, what you place on the left hand side of the regression is an excess return.

$$ R^x_t = \alpha + \beta_1 \mathit{RMRF}_{t} + \beta_2 \mathit{SMB}_{t} + \beta_3 \mathit{HML}_{t} + \beta_4 \mathit{RMW}_{t} + \beta_5 \mathit{CMA}_{t} + \epsilon_t$$

It's fine to put any excess return on the left hand side. You could put the return of Apple minus the 1 month risk free rate on the left hand side, but you could also put the return of Apple minus the return of the Dominos pizza.

A simple argument to justify this

If model A (below) is well specified:

$$ R^A_t - R^f_t = \alpha_A + \beta_{A,1} \mathit{RMRF}_{t} + \beta_{A,2} \mathit{SMB}_{t} + \beta_{A,3} \mathit{HML}_{t} + \epsilon_{A,t}$$

And model B is well specified:

$$ R^B_t - R^f_t = \alpha_B + \beta_{B,1} \mathit{RMRF}_{t} + \beta_{B,2} \mathit{SMB}_{t} + \beta_{B,3} \mathit{HML}_{t} + \epsilon_{B,t}$$

Then you can take the difference of the two equations and you get the a well specified regression model: $$ R^B_t - R^A_t = \alpha + \beta_{1} \mathit{RMRF}_{t} + \beta_{2} \mathit{SMB}_{t} + \beta_{3} \mathit{HML}_{t} + \epsilon_{t}$$

Where $\alpha = \alpha_A - \alpha_B$, $\beta_1 = \beta_{A,1} - \beta_{B,1}$ etc...