Comparing Fama-French Three vs Five Factors

An explanation for the difference in the size factors of Fama and French 3 and 5 factor data
Data
Replications
R
Author
Affiliation

Christoph Scheuch

wikifolio Financial Technologies AG

Published

October 2, 2023

In our book chapters Replicating Fama and French Factors (R Version) and Replicating Fama and French Factors (Python Version), we show how to construct factor portfolios that are fairly close to the popular data from Prof. Kenneth French finance data library. In this blog post, I want to elaborate a bit more on the subtle difference between the size data in the Fama-French three (FF3)1 and five (FF5)2 factor data.

Analyzing monthly factor data

I first start by downloading the monthly Fama-French factors using the frenchdata package. The currently available data ranges from July 1926 to August 2023. 1926-07-01 to 2023-08-01.

library(tidyverse)
library(frenchdata)
library(fixest)

start_date <- "1926-07-01"
end_date <- "2023-08-01"

factors_ff3_monthly_raw <- download_french_data("Fama/French 3 Factors")
factors_ff3_monthly <- factors_ff3_monthly_raw$subsets$data[[1]] |>
  mutate(
    month = floor_date(ymd(str_c(date, "01")), "month"),
    across(c(RF, `Mkt-RF`, SMB, HML), ~as.numeric(.) / 100),
    .keep = "none"
  ) |>
  rename_with(str_to_lower) |>
  rename(mkt_excess = `mkt-rf`)

factors_ff5_monthly_raw <- download_french_data("Fama/French 5 Factors (2x3)")
factors_ff5_monthly <- factors_ff5_monthly_raw$subsets$data[[1]] |>
  mutate(
    month = floor_date(ymd(str_c(date, "01")), "month"),
    across(c(RF, `Mkt-RF`, SMB, HML, RMW, CMA), ~as.numeric(.) / 100),
    .keep = "none"
  ) |>
  rename_with(str_to_lower) |>
  rename(mkt_excess = `mkt-rf`) 

factors_ff_monthly <- factors_ff3_monthly |> 
  rename_with(.cols = c(mkt_excess, rf, smb, hml),
              ~str_c(.x, "_ff3")) |> 
  inner_join(
    factors_ff5_monthly |> 
      select(month, mkt_excess, rf, smb, hml) |> 
      rename_with(.cols = c(mkt_excess, rf, smb, hml),
              ~str_c(.x, "_ff5")), by = "month") |> 
  filter(month >= start_date & month <= end_date)

industries_ff_monthly_raw <- download_french_data("10 Industry Portfolios")
industries_ff_monthly <- industries_ff_monthly_raw$subsets$data[[1]] |>
  mutate(month = floor_date(ymd(str_c(date, "01")), "month")) |>
  mutate(across(where(is.numeric), ~ . / 100)) |>
  select(month, everything(), -date) |>
  rename_with(str_to_lower)

Let us first inspect the summary statistics of each factor.

factors_ff_monthly |> 
  pivot_longer(cols = - month) |> 
  select(name, value) |>
  drop_na() |>
  group_by(name) |>
  summarize(
    mean = mean(value),
    sd = sd(value),
    min = min(value),
    q05 = quantile(value, 0.05),
    q50 = quantile(value, 0.50),
    q95 = quantile(value, 0.95),
    max = max(value),
    n = n()
  ) |> 
  print(n = Inf)
# A tibble: 8 × 9
  name              mean      sd    min     q05     q50    q95    max     n
  <chr>            <dbl>   <dbl>  <dbl>   <dbl>   <dbl>  <dbl>  <dbl> <int>
1 hml_ff3        0.00286 0.0297  -0.138 -0.0414 0.00215 0.0538 0.129    722
2 hml_ff5        0.00286 0.0297  -0.138 -0.0414 0.00215 0.0538 0.129    722
3 mkt_excess_ff3 0.00567 0.0449  -0.232 -0.0725 0.00925 0.0703 0.161    722
4 mkt_excess_ff5 0.00567 0.0449  -0.232 -0.0725 0.0092  0.0704 0.161    722
5 rf_ff3         0.00362 0.00266  0      0      0.0038  0.0081 0.0135   722
6 rf_ff5         0.00362 0.00266  0      0      0.0038  0.0081 0.0135   722
7 smb_ff3        0.00174 0.0302  -0.174 -0.0418 0.0011  0.0501 0.212    722
8 smb_ff5        0.00209 0.0301  -0.155 -0.0429 0.0008  0.0493 0.185    722

The above table shows that risk free rates rf_*, market excess returns mkt_excess_*, and value factors hml_* show de facto identical value across all statistics for FF3 and FF5. However, the size factors smb_* seem to be different between the data sets. Another way to show the difference is running regressions, as we do in our replication chapters:

model_smb <- lm(smb_ff3 ~ smb_ff5, data = factors_ff_monthly)
summary(model_smb)

Call:
lm(formula = smb_ff3 ~ smb_ff5, data = factors_ff_monthly)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.034411 -0.001966  0.000286  0.002143  0.033894 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.0003225  0.0002174  -1.483    0.138    
smb_ff5      0.9837614  0.0072071 136.499   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.005827 on 720 degrees of freedom
Multiple R-squared:  0.9628,    Adjusted R-squared:  0.9627 
F-statistic: 1.863e+04 on 1 and 720 DF,  p-value: < 2.2e-16

Regressing the FF3 size factor on its FF5 counterpart yields a coefficient of 0.98 and an R-squared around 96%, so definitely no perfect co-movement.

Is this difference just an artifact in the data, limited to a certain time period? Figure Figure 1 shows that there are differences throughout the whole sample.

factors_ff_monthly |> 
  mutate(difference = smb_ff3 - smb_ff5) |> 
  ggplot(aes(x = month, y = difference, fill = difference > 0)) +
  geom_col() +
  scale_x_date(date_breaks = "10 years", date_labels = "%Y") +
  theme(legend.position = "none") + 
  labs(
    x = NULL, y = "smb_ff3 - smb_ff5",
    title = "Difference between monthly size factors from FF3 and FF5 data"
  )
Title: Difference between monthly size factors from FF3 and FF5 data. The figure shows a bar chart of end-of-month differences between monthly size factors Fama-French three (FF3) and five (FF5) data. The figure demonstrates that the differences exist throughout the full period from 1926 to 2023.
Figure 1: End-of-month difference between monthly size factors from Fama-French three (FF3) and five (FF5) factor data.

Where does this difference come from? In my opinion, there is one likelyexplanation for the differences: the size portfolios portfolio_size and value portfolios portfolio_bm are constructed as independent sorts in FF3, while portfolio_bm, portfolio_op, and portfolio_inv are the result of dependent sorts in FF5 (depending on portfolio_size). In FF5, portfolio_size is then calculated on averages based on portfolio_bm, portfolio_op, and portfolio_inv portfolios. As all portfolios are the result of value-weighted return aggregation, it is hence very likely that these aggregations lead to different values.

These subtle differences might even impact your statistical tests. As an example, let us consider one of the industry portfolios from industry_ff_monthly. We use the ‘other’ portfolio, which contains sectors such as mines, construction, entertainment, finance, etc. We run a gression of the corresponding industry portfolios against the market, size, and value factors of FF3 and FF5, respectively.

industry_returns <- industries_ff_monthly |> 
  select(month, ret_other = other) |> 
  inner_join(factors_ff_monthly, by = "month")

model_ff3 <- feols(
  ret_other ~ mkt_excess_ff3 + smb_ff3 + hml_ff3, 
  industry_returns
)

model_ff5 <- feols(
  ret_other ~ mkt_excess_ff5 + smb_ff5 + hml_ff5, 
  industry_returns
)

etable(model_ff3, model_ff5, coefstat = "tstat")
                        model_ff3         model_ff5
Dependent Var.:         ret_other         ret_other
                                                   
Constant         0.0017** (2.602)  0.0016** (2.598)
mkt_excess_ff3   1.139*** (76.81)                  
smb_ff3           0.0516* (2.364)                  
hml_ff3         0.3950*** (18.23)                  
mkt_excess_ff5                     1.135*** (76.54)
smb_ff5                            0.0692** (3.201)
hml_ff5                           0.3861*** (17.95)
_______________ _________________ _________________
VCOV type                     IID               IID
Observations                  722               722
R2                        0.90001           0.90070
Adj. R2                   0.89959           0.90029
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The results show that the size factor is only significant at the 5% level for the FF3 data, but it is significant at the 1% level for the FF5 version!

A quick look at daily factors

Let us take a quick look at the daily factors to check whether the difference in size premia exists there as well. If my explanation for the difference is correct, then there should be differences. We can download the daily factor data in a similar fashion as the monthly data.

factors_ff3_daily_raw <- download_french_data("Fama/French 3 Factors [Daily]")
factors_ff3_daily <- factors_ff3_daily_raw$subsets$data[[1]] |>
  mutate(
    date = ymd(date),
    across(c(RF, `Mkt-RF`, SMB, HML), ~as.numeric(.) / 100),
    .keep = "none"
  ) |>
  rename_with(str_to_lower) |>
  rename(mkt_excess = `mkt-rf`)

factors_ff5_daily_raw <- download_french_data("Fama/French 5 Factors (2x3) [Daily]")
factors_ff5_daily <- factors_ff5_daily_raw$subsets$data[[1]] |>
  mutate(
    date = ymd(date),
    across(c(RF, `Mkt-RF`, SMB, HML, RMW, CMA), ~as.numeric(.) / 100),
    .keep = "none"
  ) |>
  rename_with(str_to_lower) |>
  rename(mkt_excess = `mkt-rf`) 

factors_ff_daily <- factors_ff3_daily |> 
  rename_with(.cols = c(mkt_excess, rf, smb, hml),
              ~str_c(.x, "_ff3")) |> 
  inner_join(
    factors_ff5_daily |> 
      select(date, mkt_excess, rf, smb, hml) |> 
      rename_with(.cols = c(mkt_excess, rf, smb, hml),
              ~str_c(.x, "_ff5")), by = "date")  |> 
  filter(date >= start_date & date <= end_date)

model_smb <- lm(smb_ff3 ~ smb_ff5, data = factors_ff_daily)
summary(model_smb)

Call:
lm(formula = smb_ff3 ~ smb_ff5, data = factors_ff_daily)

Residuals:
       Min         1Q     Median         3Q        Max 
-0.0193259 -0.0003965  0.0000245  0.0004195  0.0138805 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.568e-05  1.019e-05  -1.539    0.124    
smb_ff5      9.695e-01  1.881e-03 515.316   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.001253 on 15122 degrees of freedom
Multiple R-squared:  0.9461,    Adjusted R-squared:  0.9461 
F-statistic: 2.656e+05 on 1 and 15122 DF,  p-value: < 2.2e-16

Regressing the FF3 size factor on its FF5 counterpart yields a coefficient of 0.97 and an R-squared around 95%, so again no perfect co-movement. Unreported results of the distributions and differences over time confirm the regression results and are in line with differences among monthly factor data.

Conclusion

Are there any implications for empirical applications? In my opinion, you should be careful when you want to test your portfolios against FF3 and FF5 factors. It is strictly speaking not correct to just use a subsample of factors from FF5 if you want to test against the FF3 factors. I rather recommend downloading both FF3 and FF5 and run tests with each data set separately.

Footnotes

  1. Fama, E. F.; French, K. R. (1993). “Common risk factors in the returns on stocks and bonds”. Journal of Financial Economics. 33: 3–56. https://doi.org/10.1016/0304-405X(93)90023-5↩︎

  2. Fama, E. F.; French, K. R. (2015). “A Five-Factor Asset Pricing Model”. Journal of Financial Economics. 116: 1–22. CiteSeerX 10.1.1.645.3745. https://doi.org/10.1016/j.jfineco.2014.10.010↩︎

© Christoph Frey, Christoph Scheuch, Stefan Voigt & Patrick Weiss
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