Why Fama–French SMB Returns Differ Across Three- and Five-Factor Models

# SMB data for 3-factor and 5-factor are different on French's website


# SMB data for 3-factor and 5-factor are different on French's website












Does anyone know why the SMB data published in the 3-factor and 5-factor data files on French's website are different? Which one should be used then?

## Answer by skoestlmeier (score 3, accepted)

https://quant.stackexchange.com/a/45038

The difference arises because of the underlying procedure to calculate factor returns. Each SMB-return is an appropriate measurement of the size-effect, but you should be aware to use them appropriate (i.e. use the 5-factor SMB when applying the 5-factor model or when analyzing the SMB-return with regards to a companies profitability or investment behavior).

#### Construction of the portfolios

As explained on Kenneth French's website, the Fama/French 3-factor model is based on

> the 6 value-weight portfolios formed on size and book-to-market. (See the description of the 6 size/book-to-market portfolios.)

and further:

> The 6 value-weight portfolios formed on size and book-to-market. (See the description of the 6 size/book-to-market portfolios.)

The 5-factor model however is based on

> using the 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. (See the description of the 6 size/book-to-market, size/operating profitability, size/investment portfolios.)

so

> SMB (Small Minus Big) is the average return on the nine small stock portfolios minus the average return on the nine big stock portfolios.

Each single portfolio return is calculated as a value-weighted return. These returns are finally equal weighted to obtain the SMB-return. As stocks are sorted into different portfolios, this procedure is exactly where differences in the SMB-return arise.

#### Analysis of the SMB-returns

The difference of 3-factor SMB and 5-factor SMB is very little on average. An analysis of their difference from July 1963 - Feb. 2019 gives the following results:

```
# 3-factor SMB return
> summary(FF3$SMB)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-16.8700  -1.5025   0.0950   0.2129   2.0150  21.7100

# 5-factor SMB return
> summary(FF5$SMB)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-14.9100  -1.4825   0.0850   0.2458   2.0700  18.3100

> diff <- FF3$SMB - FF5$SMB
> summary(diff)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-3.40000 -0.21000  0.00000 -0.03292  0.16250  3.46000
```

The following plot shows the difference of 3-factor SMB returns and 5-factor SMB returns:

So in fact, the 3-factor SMB return (on average) is just 0.03% per month (or about 0.36% per year) lower than the 5-factor SMB return, which is also less extreme distributed. The most differences arise in times of the "Dotcom-crisis", where the technology sector collapsed. However, this difference is not significant different from zero on average, based on a Newey/West corrected regression using a lag of six:

```
> library(sandwich)
> library(lmtest)
> reg <- lm(diff~1)
> coeftest(reg, NeweyWest(reg, lag = 6))

> t test of coefficients:

             Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.032919   0.023100 -1.4251   0.1546
```

In conclusion, you are fine to use both SMB-returns for any analysis of the size-effect. It is especially their differences, which arise from different underlying portfolio-sorts, which give us more insight on the underlying economic reasons for anomalies like the size-effect.