Constructing SMB and Momentum Factors in Fama–French Models

# Carhart 4 factor model and six factor model


# Carhart 4 factor model and six factor model












The value of SMB of Fama French 3 factor model is calculated as follows: $$ \frac{1}{3} (Small Value + Small Neutral + Small Growth) - \frac{1}{3} (Big Value + Big Neutral + Big Growth). $$

However, in Fama French 5 factor model the value of SMB is calculated as follows: $$ \frac{1}{3} ( SMB(B/M) + SMB(OP) + SMB(INV) ). $$

My questions are:

- Shall I add the SMB (Momentum) to the to SMB(B/M) value and divide them when I want to estimate Carhart model (as in 5 factor)?

- Shall I add it to the other SMB values when I want to estimate the six factor model (Fama French+ momentum)?

## Answer by Konstantinos (score 3)

https://quant.stackexchange.com/a/65970

Fama and French (1992, JFE, "Common risk factors in the returns on stocks and bonds") "use portfolios formed on size and BE/ME because [they] seek to determine whether the mimicking portfolios SMB and HML capture common factors in stock returns related to size and book-to-market equity". They used 2x3 independent sorts of stocks for this. 2 portfolios based on size (Market Equity) and 3 based on the ratio of Book equity to Market Equity.

Fama and French (2016, RFS, Dissecting Anomalies with a Five-Factor Model), add the profitability (RMW) and investment (CMA) risk factor. Their "RMW and CMA produce two additional Size factors, $SMB_{OP}$ and $SMB_{Inv}$. The size factor SMB used in the tests is the average of the returns on the nine small stock portfolios of the three 2x3 sorts minus the average of the returns on the nine big stock portfolios". To construct the five risk factors they use a total of 18 different portfolio definitions!: "6 value-weight portfolios formed on size and book-to-market, the 6 value-weight portfolios formed on size and operating profitability, and the 6 value-weight portfolios formed on size and investment" as it's described in French's data library.

Every step they take is clearly motivated (explained why) in their research papers. My general answer is that you need to precisely motivate whatever decision you want to make first. Then you can ask more focused questions. I mention this in good will because I am missing your motivation in your questions and your questions are vague and imprecise.

Now, constructing the portfolios/factors can become a complicated and long process. I suggest you use the factors/portfolios directly from the French's data library, but if you want to learn (or torture) yourself you can calculate them on your own.

Also, mind that in the case you want to explain some portfolio returns (dependent variable) by different factors (independent), your results will defer when you consider different portfolios. For instance, SMB might be significant when you have portfolios formed on Size and Investment as Dependent and insignificant when they are formed on Operating Profitability and Investment. So in your work (I assume something like essay/small research exercise) you need to also motivate why you use the portfolios you use.

To your specific questions, respectively:

- There is no SMB(Momentum). This appears to be a confusion. Also, I do not understand what you mean when you want to divide (what is the nominator? denominator?) The Carhart is specified as $r_i = r_f + \beta_1 Mkt + \beta_2 HML + \beta_3 SMB + \beta_4MOM + e$ so you need not to add any SMB or MOM values. They are different independent variables.

- Based on Fama and French (2018, JFE, "Choosing factors") their momentum factor, $UMD$, is irrelevant to the $SMB_{??}$ values.

Last, when you ask questions, as my best mentor of my life said, take the reader by the hand and don't assume you talk to a second "Sima". :) In this specific question and your comment I literally see like 6+ different questions (with different branches) and I find this confusing. (personal opinion) If you want to ask something specific from now on it's a good idea to also provide proper mathematical notation.