Replicating Fama-French Factors

Felix Dimmerling

2025-07-03

Hintergründe des Empirical Assets Pricing

Bisherige Portfolio Bildung

  • Capital Asset Pricing Modell (CAPM) → hat Limits
    • Bewertung beruht nur auf der Beziehung von systematischem Risiko zu den Assets Returns
    • „Alle Firmen gleich“ (nur Beta im Bezug auf Marktüberrendite)
    • Ohne Auflösung, da Firmeneigenschaften vollkommen ignoriert werden
      → mangelnde Belastbarkeit in Detailbetrachtungen
    • Return-Muster, die nicht in dem Modell erklärt werden können

Firmeneigenschaften zur verbesserten Vorhersage

CAPM FF3 FF5
Faktoren 1 3 5
Granularität (Portfolios) 1 6 (independent sort) ~18 (dependent sort)
Theoretische Eleganz Hoch Solide Redundanzdiskussion (HML vs. RMW/CMA)
Erklärungskraft Schwach Gut (Size & Value) Sehr gut (Profitabilität & Investment)
Komplexität Niedrig Mittel Hoch
Empirische Robustheit Niedrig Langfristig bestätigt Noch nicht über alle Zeiträume stabil
Praktische Relvanz Eingeschränkt Hoch Sehr hoch

Faktoren im Detail

CAPM FF3 FF5
Marktüberrendite ✅ ✅ ✅
Size SMB ✅ ✅
Value HML ✅ ✅
Profitabilität RMW ✅
Investition CMA ✅
  • Firmeneigenschaften werden systematisch genutzt
  • Renditeunterschiede werden erstmal nur korrelativ und nicht kausal erklärt

Portfolio Bildung bei Fama-French

  • Firmen werden anhand der Portfolio Gruppen sortiert
  • Granularität bei FF3 weniger als bei FF5
  • Renditeerwartungen werden mit einer Long/Short-Strategie gesteigert
    • Höher performende Long-Portfolios werden durch den Leerverkauf von Short-Portfolios finanziert
  • Renditemuster werden systematisch durch Firmeneigenschaften bergründet

Pakete & Datenimport

library(tidyverse)
library(nanoparquet)

crsp_monthly <- read_parquet("data/crsp_monthly.parquet") |>
  select(permno, gvkey, date, ret_excess, mktcap, mktcap_lag, exchange)

compustat <- read_parquet("data/compustat.parquet") |>
  select(gvkey, datadate, be, op, inv)

factors_ff3_monthly <- read_parquet("data/factors_ff3_monthly.parquet") |>
  select(date, smb, hml)

factors_ff5_monthly <- read_parquet("data/factors_ff5_monthly.parquet") |>
  select(date, smb, hml, rmw, cma)
[1] "crsp_monthly"
# A data frame: 3,378,303 × 7
   permno gvkey  date       ret_excess mktcap mktcap_lag exchange
    <dbl> <chr>  <date>          <dbl>  <dbl>      <dbl> <chr>   
 1  10000 013007 1986-02-01    -0.262   12.0       16.1  NASDAQ  
 2  10000 013007 1986-03-01     0.359   16.3       12.0  NASDAQ  
 3  10000 013007 1986-04-01    -0.104   15.2       16.3  NASDAQ  
 4  10000 013007 1986-05-01    -0.228   11.8       15.2  NASDAQ  
 5  10000 013007 1986-06-01    -0.0102  11.7       11.8  NASDAQ  
 6  10000 013007 1986-07-01    -0.0860  10.8       11.7  NASDAQ  
 7  10000 013007 1986-08-01    -0.620    4.15      10.8  NASDAQ  
 8  10000 013007 1986-09-01    -0.0616   3.91       4.15 NASDAQ  
 9  10000 013007 1986-10-01    -0.247    3.00       3.91 NASDAQ  
10  10000 013007 1986-11-01     0.0561   3.18       3.00 NASDAQ  
# ℹ 3,378,293 more rows
[1] "compustat"
# A data frame: 495,358 × 5
   gvkey  datadate       be      op     inv
   <chr>  <date>      <dbl>   <dbl>   <dbl>
 1 001000 1961-12-31 NA     NA      NA     
 2 001000 1962-12-31  0.552  2.88   NA     
 3 001000 1963-12-31  0.561  0.0463 NA     
 4 001000 1964-12-31  0.627  0.150  NA     
 5 001000 1965-12-31  0.491 -0.452   0.631 
 6 001000 1966-12-31  0.834  0.516   0.0519
 7 001000 1967-12-31  0.744  0.140   0.0107
 8 001000 1968-12-31  2.57   0.370   1.41  
 9 001000 1969-12-31 10.2    0.435   3.85  
10 001000 1970-12-31 10.5    0.430   0.165 
# ℹ 495,348 more rows
[1] "factors_ff3_monthly"
# A data frame: 768 × 3
   date           smb     hml
   <date>       <dbl>   <dbl>
 1 1960-01-01  0.0209  0.0278
 2 1960-02-01  0.0051 -0.0193
 3 1960-03-01 -0.0049 -0.0294
 4 1960-04-01  0.0032 -0.0228
 5 1960-05-01  0.0121 -0.037 
 6 1960-06-01 -0.0021 -0.0034
 7 1960-07-01 -0.0051  0.0198
 8 1960-08-01  0.0087 -0.0018
 9 1960-09-01 -0.0111  0.0162
10 1960-10-01 -0.0408  0.0283
# ℹ 758 more rows
[1] "factors_ff5_monthly"
# A data frame: 726 × 5
   date           smb     hml     rmw     cma
   <date>       <dbl>   <dbl>   <dbl>   <dbl>
 1 1963-07-01 -0.0041 -0.0097  0.0068 -0.0118
 2 1963-08-01 -0.008   0.018   0.0036 -0.0035
 3 1963-09-01 -0.0052  0.0013 -0.0071  0.0029
 4 1963-10-01 -0.0139 -0.001   0.028  -0.0201
 5 1963-11-01 -0.0088  0.0175 -0.0051  0.0224
 6 1963-12-01 -0.021  -0.0002  0.0003 -0.0007
 7 1964-01-01  0.0013  0.0148  0.0017  0.0147
 8 1964-02-01  0.0028  0.0281 -0.0005  0.0091
 9 1964-03-01  0.0123  0.034  -0.0221  0.0322
10 1964-04-01 -0.0152 -0.0067 -0.0127 -0.0108
# ℹ 716 more rows

Fama-French Datums-Anpassungen

size <- crsp_monthly |>
  filter(month(date) == 6) |>
  mutate(sorting_date = date %m+% months(1)) |>
  select(permno, exchange, sorting_date, size = mktcap)

market_equity <- crsp_monthly |>
  filter(month(date) == 12) |>
  mutate(sorting_date = ymd(str_c(year(date) + 1, "0701)"))) |>
  select(permno, gvkey, sorting_date, me = mktcap)

book_to_market <- compustat |>
  mutate(sorting_date = ymd(str_c(year(datadate) + 1, "0701"))) |>
  select(gvkey, sorting_date, be) |>
  inner_join(market_equity, join_by(gvkey, sorting_date)) |>
  mutate(bm = be / me) |>
  select(permno, sorting_date, me, bm)

sorting_variables <- size |>
  inner_join(
    book_to_market, join_by(permno, sorting_date)
    ) |>
  drop_na() |>
  distinct(permno, sorting_date, .keep_all = TRUE)
[1] "size"
# A data frame: 281,110 × 4
   permno exchange sorting_date  size
    <dbl> <chr>    <date>       <dbl>
 1  10000 NASDAQ   1986-07-01   11.7 
 2  10001 NASDAQ   1986-07-01    6.03
 3  10001 NASDAQ   1987-07-01    5.82
 4  10001 NASDAQ   1988-07-01    6.2 
 5  10001 NASDAQ   1989-07-01    7.01
 6  10001 NASDAQ   1990-07-01   10.1 
 7  10001 NASDAQ   1991-07-01   11.3 
 8  10001 NASDAQ   1992-07-01   12.6 
 9  10001 NASDAQ   1993-07-01   18.0 
10  10001 NASDAQ   1994-07-01   18.9 
# ℹ 281,100 more rows
[1] "market_equity"
# A data frame: 281,726 × 4
   permno gvkey  sorting_date    me
    <dbl> <chr>  <date>       <dbl>
 1  10000 013007 1987-07-01    1.98
 2  10001 012994 1987-07-01    6.94
 3  10001 012994 1988-07-01    5.83
 4  10001 012994 1989-07-01    6.36
 5  10001 012994 1990-07-01   10.3 
 6  10001 012994 1991-07-01   10.0 
 7  10001 012994 1992-07-01   15.6 
 8  10001 012994 1993-07-01   15.1 
 9  10001 012994 1994-07-01   20.0 
10  10001 012994 1995-07-01   17.8 
# ℹ 281,716 more rows
[1] "book_to_market"
# A data frame: 256,864 × 4
   permno sorting_date    me    bm
    <dbl> <date>       <dbl> <dbl>
 1  25881 1971-07-01   26.6  0.397
 2  25881 1972-07-01   15.3  0.549
 3  25881 1973-07-01   13.6  0.537
 4  25881 1974-07-01    4.65 1.89 
 5  25881 1975-07-01    4.71 1.76 
 6  25881 1976-07-01    9.24 1.19 
 7  25881 1977-07-01   12.7  1.22 
 8  25881 1978-07-01   20.6  0.858
 9  10015 1984-07-01   26.8  0.292
10  10015 1985-07-01   13.6  0.685
# ℹ 256,854 more rows
[1] "sorting_variables"
# A data frame: 233,216 × 6
   permno exchange sorting_date  size    me    bm
    <dbl> <chr>    <date>       <dbl> <dbl> <dbl>
 1  10001 NASDAQ   1987-07-01    5.82  6.94 1.01 
 2  10001 NASDAQ   1988-07-01    6.2   5.83 1.21 
 3  10001 NASDAQ   1989-07-01    7.01  6.36 1.15 
 4  10001 NASDAQ   1990-07-01   10.1  10.3  0.818
 5  10001 NASDAQ   1991-07-01   11.3  10.0  0.943
 6  10001 NASDAQ   1992-07-01   12.6  15.6  0.668
 7  10001 NASDAQ   1993-07-01   18.0  15.1  0.710
 8  10001 NASDAQ   1994-07-01   18.9  20.0  0.573
 9  10001 NASDAQ   1995-07-01   18.6  17.8  0.705
10  10001 NASDAQ   1996-07-01   18.6  21.4  0.642
# ℹ 233,206 more rows

Sorting für FF3

assign_portfolio <- function(data, 
                             sorting_variable, 
                             percentiles) {
  breakpoints <- data |>
    filter(exchange == "NYSE") |>
    pull({{ sorting_variable }}) |>
    quantile(
      probs = percentiles,
      na.rm = TRUE,
      names = FALSE
    )

  assigned_portfolios <- data |>
    mutate(portfolio = findInterval(
      pick(everything()) |>
        pull({{ sorting_variable }}),
      breakpoints,
      all.inside = TRUE
    )) |>
    pull(portfolio)
  
  return(assigned_portfolios)
}
portfolios <- sorting_variables |>
  group_by(sorting_date) |>
  mutate(
    portfolio_size = assign_portfolio(
      data = pick(everything()),
      sorting_variable = size,
      percentiles = c(0, 0.5, 1)
    ),
    portfolio_bm = assign_portfolio(
      data = pick(everything()),
      sorting_variable = bm,
      percentiles = c(0, 0.3, 0.7, 1)
    )
  ) |>
  ungroup() |> 
  select(permno, sorting_date, 
         portfolio_size, portfolio_bm)
portfolios <- crsp_monthly |>
  mutate(sorting_date = case_when(
    month(date) <= 6 ~ ymd(str_c(year(date) - 1, "0701")),
    month(date) >= 7 ~ ymd(str_c(year(date), "0701"))
  )) |>
  inner_join(portfolios, join_by(permno, sorting_date))
# A data frame: 2,666,586 × 6
    permno date       sorting_date mktcap portfolio_size portfolio_bm
     <dbl> <date>     <date>        <dbl>          <int>        <int>
  1  10001 1987-07-01 1987-07-01     5.95              1            3
  2  10001 1987-08-01 1987-07-01     6.44              1            3
  3  10001 1987-09-01 1987-07-01     6.2               1            3
  4  10001 1987-10-01 1987-07-01     6.32              1            3
  5  10001 1987-11-01 1987-07-01     6.14              1            3
  6  10001 1987-12-01 1987-07-01     5.83              1            3
  7  10001 1988-01-01 1987-07-01     6.2               1            3
  8  10001 1988-02-01 1987-07-01     6.70              1            3
  9  10001 1988-03-01 1987-07-01     6.08              1            3
 10  10001 1988-04-01 1987-07-01     6.26              1            3
 11  10001 1988-05-01 1987-07-01     6.39              1            3
 12  10001 1988-06-01 1987-07-01     6.2               1            3
 13  10001 1988-07-01 1988-07-01     6.39              1            3
 14  10001 1988-08-01 1988-07-01     6.57              1            3
 15  10001 1988-09-01 1988-07-01     6.36              1            3
 16  10001 1988-10-01 1988-07-01     6.61              1            3
 17  10001 1988-11-01 1988-07-01     6.61              1            3
 18  10001 1988-12-01 1988-07-01     6.36              1            3
 19  10001 1989-01-01 1988-07-01     6.49              1            3
 20  10001 1989-02-01 1988-07-01     6.74              1            3
 21  10001 1989-03-01 1988-07-01     6.74              1            3
 22  10001 1989-04-01 1988-07-01     7.24              1            3
 23  10001 1989-05-01 1988-07-01     6.99              1            3
 24  10001 1989-06-01 1988-07-01     7.01              1            3
 25  10001 1989-07-01 1989-07-01     7.26              1            3
 26  10001 1989-08-01 1989-07-01     9.26              1            3
 27  10001 1989-09-01 1989-07-01     9.04              1            3
 28  10001 1989-10-01 1989-07-01     9.68              1            3
 29  10001 1989-11-01 1989-07-01    10.1               1            3
 30  10001 1989-12-01 1989-07-01    10.3               1            3
 31  10001 1990-01-01 1989-07-01    10.2               1            3
 32  10001 1990-02-01 1989-07-01    10.1               1            3
 33  10001 1990-03-01 1989-07-01    10.1               1            3
 34  10001 1990-04-01 1989-07-01    10.1               1            3
 35  10001 1990-05-01 1989-07-01    10.0               1            3
 36  10001 1990-06-01 1989-07-01    10.1               1            3
 37  10001 1990-07-01 1990-07-01    10.3               1            2
 38  10001 1990-08-01 1990-07-01     9.79              1            2
 39  10001 1990-09-01 1990-07-01    10.2               1            2
 40  10001 1990-10-01 1990-07-01    10.0               1            2
 41  10001 1990-11-01 1990-07-01    10.0               1            2
 42  10001 1990-12-01 1990-07-01    10.0               1            2
 43  10001 1991-01-01 1990-07-01    10.1               1            2
 44  10001 1991-02-01 1990-07-01    10.3               1            2
 45  10001 1991-03-01 1990-07-01    10.0               1            2
 46  10001 1991-04-01 1990-07-01    10.4               1            2
 47  10001 1991-05-01 1990-07-01    10.4               1            2
 48  10001 1991-06-01 1990-07-01    11.3               1            2
 49  10001 1991-07-01 1991-07-01    10.9               1            2
 50  10001 1991-08-01 1991-07-01    12.3               1            2
 51  10001 1991-09-01 1991-07-01    12.3               1            2
 52  10001 1991-10-01 1991-07-01    13.9               1            2
 53  10001 1991-11-01 1991-07-01    15.8               1            2
 54  10001 1991-12-01 1991-07-01    15.6               1            2
 55  10001 1992-01-01 1991-07-01    14.8               1            2
 56  10001 1992-02-01 1991-07-01    11.8               1            2
 57  10001 1992-03-01 1991-07-01    12.6               1            2
 58  10001 1992-04-01 1991-07-01    12.8               1            2
 59  10001 1992-05-01 1991-07-01    12.9               1            2
 60  10001 1992-06-01 1991-07-01    12.6               1            2
 61  10001 1992-07-01 1992-07-01    13.4               1            2
 62  10001 1992-08-01 1992-07-01    14.0               1            2
 63  10001 1992-09-01 1992-07-01    16.1               1            2
 64  10001 1992-10-01 1992-07-01    15.7               1            2
 65  10001 1992-11-01 1992-07-01    15.5               1            2
 66  10001 1992-12-01 1992-07-01    15.1               1            2
 67  10001 1993-01-01 1992-07-01    15.1               1            2
 68  10001 1993-02-01 1992-07-01    15.4               1            2
 69  10001 1993-03-01 1992-07-01    15.3               1            2
 70  10001 1993-04-01 1992-07-01    16.4               1            2
 71  10001 1993-05-01 1992-07-01    16.3               1            2
 72  10001 1993-06-01 1992-07-01    18.0               1            2
 73  10001 1993-07-01 1993-07-01    17.8               1            2
 74  10001 1993-08-01 1993-07-01    17.3               1            2
 75  10001 1993-09-01 1993-07-01    18.3               1            2
 76  10001 1993-10-01 1993-07-01    18.8               1            2
 77  10001 1993-11-01 1993-07-01    18.5               1            2
 78  10001 1993-12-01 1993-07-01    20.0               1            2
 79  10001 1994-01-01 1993-07-01    19.1               1            2
 80  10001 1994-02-01 1993-07-01    19.1               1            2
 81  10001 1994-03-01 1993-07-01    18.8               1            2
 82  10001 1994-04-01 1993-07-01    16.1               1            2
 83  10001 1994-05-01 1993-07-01    17.2               1            2
 84  10001 1994-06-01 1993-07-01    18.9               1            2
 85  10001 1994-07-01 1994-07-01    20.3               1            2
 86  10001 1994-08-01 1994-07-01    19.7               1            2
 87  10001 1994-09-01 1994-07-01    20.5               1            2
 88  10001 1994-10-01 1994-07-01    19.3               1            2
 89  10001 1994-11-01 1994-07-01    18.5               1            2
 90  10001 1994-12-01 1994-07-01    17.8               1            2
 91  10001 1995-01-01 1994-07-01    17.2               1            2
 92  10001 1995-02-01 1994-07-01    16.8               1            2
 93  10001 1995-03-01 1994-07-01    16.8               1            2
 94  10001 1995-04-01 1994-07-01    16.8               1            2
 95  10001 1995-05-01 1994-07-01    17.7               1            2
 96  10001 1995-06-01 1994-07-01    18.6               1            2
 97  10001 1995-07-01 1995-07-01    18.6               1            2
 98  10001 1995-08-01 1995-07-01    18.0               1            2
 99  10001 1995-09-01 1995-07-01    18.8               1            2
100  10001 1995-10-01 1995-07-01    18.2               1            2
101  10001 1995-11-01 1995-07-01    20.0               1            2
102  10001 1995-12-01 1995-07-01    21.4               1            2
103  10001 1996-01-01 1995-07-01    20.8               1            2
104  10001 1996-02-01 1995-07-01    21.1               1            2
105  10001 1996-03-01 1995-07-01    21.9               1            2
106  10001 1996-04-01 1995-07-01    20.3               1            2
107  10001 1996-05-01 1995-07-01    19.9               1            2
108  10001 1996-06-01 1995-07-01    18.6               1            2
109  10001 1996-07-01 1996-07-01    19.0               1            2
110  10001 1996-08-01 1996-07-01    19.7               1            2
111  10001 1996-09-01 1996-07-01    20.5               1            2
112  10001 1996-10-01 1996-07-01    19.9               1            2
113  10001 1996-11-01 1996-07-01    20.5               1            2
114  10001 1996-12-01 1996-07-01    19.2               1            2
115  10001 1997-01-01 1996-07-01    20.3               1            2
116  10001 1997-02-01 1996-07-01    20.3               1            2
117  10001 1997-03-01 1996-07-01    20.3               1            2
118  10001 1997-04-01 1996-07-01    20.3               1            2
119  10001 1997-05-01 1996-07-01    20.3               1            2
120  10001 1997-06-01 1996-07-01    19.4               1            2
121  10001 1997-07-01 1997-07-01    20.3               1            3
122  10001 1997-08-01 1997-07-01    20.6               1            3
123  10001 1997-09-01 1997-07-01    21.1               1            3
124  10001 1997-10-01 1997-07-01    21.1               1            3
125  10001 1997-11-01 1997-07-01    20.8               1            3
126  10001 1997-12-01 1997-07-01    21.6               1            3
127  10001 1998-01-01 1997-07-01    21.6               1            3
128  10001 1998-02-01 1997-07-01    21.4               1            3
129  10001 1998-03-01 1997-07-01    21.0               1            3
130  10001 1998-04-01 1997-07-01    21.2               1            3
131  10001 1998-05-01 1997-07-01    20.9               1            3
132  10001 1998-06-01 1997-07-01    20.7               1            3
133  10001 1998-07-01 1998-07-01    21.0               1            3
134  10001 1998-08-01 1998-07-01    21.0               1            3
135  10001 1998-09-01 1998-07-01    22.3               1            3
136  10001 1998-10-01 1998-07-01    22.3               1            3
137  10001 1998-11-01 1998-07-01    22.7               1            3
138  10001 1998-12-01 1998-07-01    23.3               1            3
139  10001 1999-01-01 1998-07-01    23.3               1            3
140  10001 1999-02-01 1998-07-01    21.2               1            3
141  10001 1999-03-01 1998-07-01    21.2               1            3
142  10001 1999-04-01 1998-07-01    21.5               1            3
143  10001 1999-05-01 1998-07-01    21.0               1            3
144  10001 1999-06-01 1998-07-01    21.1               1            3
145  10001 1999-07-01 1999-07-01    21.6               1            2
146  10001 1999-08-01 1999-07-01    21.4               1            2
147  10001 1999-09-01 1999-07-01    19.6               1            2
148  10001 1999-10-01 1999-07-01    21.3               1            2
149  10001 1999-11-01 1999-07-01    21.2               1            2
150  10001 1999-12-01 1999-07-01    20.8               1            2
151  10001 2000-01-01 1999-07-01    19.9               1            2
152  10001 2000-02-01 1999-07-01    20.2               1            2
153  10001 2000-03-01 1999-07-01    19.7               1            2
154  10001 2000-04-01 1999-07-01    19.9               1            2
155  10001 2000-05-01 1999-07-01    19.5               1            2
156  10001 2000-06-01 1999-07-01    19.8               1            2
157  10001 2000-07-01 2000-07-01    19.5               1            2
158  10001 2000-08-01 2000-07-01    20.4               1            2
159  10001 2000-09-01 2000-07-01    21.8               1            2
160  10001 2000-10-01 2000-07-01    22.5               1            2
161  10001 2000-11-01 2000-07-01    23.9               1            2
162  10001 2000-12-01 2000-07-01    24.4               1            2
163  10001 2001-01-01 2000-07-01    24.7               1            2
164  10001 2001-02-01 2000-07-01    24.4               1            2
165  10001 2001-03-01 2000-07-01    25.1               1            2
166  10001 2001-04-01 2000-07-01    24.5               1            2
167  10001 2001-05-01 2000-07-01    26.8               1            2
168  10001 2001-06-01 2000-07-01    29.7               1            2
169  10001 2001-07-01 2001-07-01    30.4               1            2
170  10001 2001-08-01 2001-07-01    31.0               1            2
171  10001 2001-09-01 2001-07-01    30.1               1            2
172  10001 2001-10-01 2001-07-01    29.1               1            2
173  10001 2001-11-01 2001-07-01    29.3               1            2
174  10001 2001-12-01 2001-07-01    29.4               1            2
175  10001 2002-01-01 2001-07-01    29.0               1            2
176  10001 2002-02-01 2001-07-01    27.5               1            2
177  10001 2002-03-01 2001-07-01    26.7               1            2
178  10001 2002-04-01 2001-07-01    25.6               1            2
179  10001 2002-05-01 2001-07-01    26.0               1            2
180  10001 2002-06-01 2001-07-01    25.0               1            2
181  10001 2002-07-01 2002-07-01    22.5               1            2
182  10001 2002-08-01 2002-07-01    22.9               1            2
183  10001 2002-09-01 2002-07-01    22.5               1            2
184  10001 2002-10-01 2002-07-01    22.2               1            2
185  10001 2002-11-01 2002-07-01    21.7               1            2
186  10001 2002-12-01 2002-07-01    19.0               1            2
187  10001 2003-01-01 2002-07-01    21.9               1            2
188  10001 2003-02-01 2002-07-01    22.6               1            2
189  10001 2003-03-01 2002-07-01    19.8               1            2
190  10001 2003-04-01 2002-07-01    13.4               1            2
191  10001 2003-05-01 2002-07-01    21.9               1            2
192  10001 2003-06-01 2002-07-01    15.6               1            2
193  10001 2003-07-01 2003-07-01    15.9               1            3
194  10001 2003-08-01 2003-07-01    17.3               1            3
195  10001 2003-09-01 2003-07-01    17.9               1            3
196  10001 2003-10-01 2003-07-01    15.6               1            3
197  10001 2003-11-01 2003-07-01    15.5               1            3
198  10001 2003-12-01 2003-07-01    15.4               1            3
199  10001 2004-01-01 2003-07-01    15.6               1            3
200  10001 2004-02-01 2003-07-01    16.8               1            3
# ℹ 2,666,386 more rows

Fama-French Three-Factor Model

factors_replicated <- portfolios |>
  group_by(portfolio_size, portfolio_bm, date) |>
  summarize(
    ret = weighted.mean(ret_excess, mktcap_lag), .groups = "drop"
  ) |>
  group_by(date) |>
  summarize(
    smb_replicated = mean(ret[portfolio_size == 1]) -
      mean(ret[portfolio_size == 2]),
    hml_replicated = mean(ret[portfolio_bm == 3]) -
      mean(ret[portfolio_bm == 1])
  )
# A tibble: 750 × 3
   date       smb_replicated hml_replicated
   <date>              <dbl>          <dbl>
 1 1961-07-01      -0.0179         -0.00284
 2 1961-08-01       0.000446       -0.0220 
 3 1961-09-01      -0.00352        -0.0215 
 4 1961-10-01      -0.0124          0.00571
 5 1961-11-01      -0.000233        0.00495
 6 1961-12-01      -0.0211          0.0427 
 7 1962-01-01       0.00893         0.0423 
 8 1962-02-01      -0.00768         0.0165 
 9 1962-03-01       0.0120         -0.0200 
10 1962-04-01      -0.00880         0.00426
# ℹ 740 more rows
test <- factors_ff3_monthly |>
  inner_join(factors_replicated, join_by(date)) |>
  mutate(
    across(c(smb_replicated, hml_replicated), ~round(., 4))
  )

Small-minus-Big

model_smb <- lm(smb ~ smb_replicated, data = test)
summary(model_smb)

Call:
lm(formula = smb ~ smb_replicated, data = test)

Residuals:
       Min         1Q     Median         3Q        Max 
-0.0205101 -0.0014497  0.0000327  0.0014360  0.0147423 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)    -0.0000994  0.0001283  -0.775    0.439    
smb_replicated  0.9880963  0.0042202 234.137   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.003508 on 748 degrees of freedom
Multiple R-squared:  0.9865,    Adjusted R-squared:  0.9865 
F-statistic: 5.482e+04 on 1 and 748 DF,  p-value: < 2.2e-16

High-minus-Low

model_hml <- lm(hml ~ hml_replicated, data = test)
summary(model_hml)

Call:
lm(formula = hml ~ hml_replicated, data = test)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.023527 -0.002818 -0.000200  0.002187  0.033922 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)    0.0003464  0.0002155   1.607    0.108    
hml_replicated 0.9620722  0.0071073 135.364   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.005876 on 748 degrees of freedom
Multiple R-squared:  0.9608,    Adjusted R-squared:  0.9607 
F-statistic: 1.832e+04 on 1 and 748 DF,  p-value: < 2.2e-16

Fama-French Five-Factor Model

other_sorting_variables <- compustat |>
  mutate(sorting_date = ymd(str_c(year(datadate) + 1, "0701"))) |>
  select(gvkey, sorting_date, be, op, inv) |>
  inner_join(market_equity, 
             join_by(gvkey, sorting_date)) |>
  mutate(bm = be / me) |>
  select(permno, sorting_date, me, be, bm, op, inv)

sorting_variables <- size |>
  inner_join(
    other_sorting_variables, 
    join_by(permno, sorting_date)
    ) |>
  drop_na() |>
  distinct(permno, sorting_date, .keep_all = TRUE)
portfolios <- sorting_variables |>
  group_by(sorting_date) |>
  mutate(
    portfolio_size = assign_portfolio(
      data = pick(everything()),
      sorting_variable = size,
      percentiles = c(0, 0.5, 1)
    )) |> 
  group_by(sorting_date, portfolio_size) |> 
  mutate(
    across(c(bm, op, inv), ~assign_portfolio(
      data = pick(everything()), 
      sorting_variable = ., 
      percentiles = c(0, 0.3, 0.7, 1)),
      .names = "portfolio_{.col}"
    )
  ) |>
  ungroup() |> 
  select(permno, sorting_date, 
         portfolio_size, portfolio_bm,
         portfolio_op, portfolio_inv)

portfolios <- crsp_monthly |>
  mutate(sorting_date = case_when(
    month(date) <= 6 ~ ymd(str_c(year(date) - 1, "0701")),
    month(date) >= 7 ~ ymd(str_c(year(date), "0701"))
  )) |>
  inner_join(portfolios, join_by(permno, sorting_date))

HML - Value

portfolios_value <- portfolios |>
  group_by(portfolio_size, portfolio_bm, date) |>
  summarize(
    ret = weighted.mean(ret_excess, mktcap_lag), 
    .groups = "drop"
  )

factors_value <- portfolios_value |>
  group_by(date) |>
  summarize(
    hml_replicated = mean(ret[portfolio_bm == 3]) -
      mean(ret[portfolio_bm == 1])
  )

RMW - Profitabilität

portfolios_profitability <- portfolios |>
  group_by(portfolio_size, portfolio_op, date) |>
  summarize(
    ret = weighted.mean(ret_excess, mktcap_lag), 
    .groups = "drop"
  ) 

factors_profitability <- portfolios_profitability |>
  group_by(date) |>
  summarize(
    rmw_replicated = mean(ret[portfolio_op == 3]) -
      mean(ret[portfolio_op == 1])
  )

CMA - Investment

portfolios_investment <- portfolios |>
  group_by(portfolio_size, portfolio_inv, date) |>
  summarize(
    ret = weighted.mean(ret_excess, mktcap_lag), 
    .groups = "drop"
  )

factors_investment <- portfolios_investment |>
  group_by(date) |>
  summarize(
    cma_replicated = mean(ret[portfolio_inv == 1]) -
      mean(ret[portfolio_inv == 3])
  )

SMB - Size (als letztes wegen dependent sort)

factors_size <- bind_rows(
  portfolios_value,
  portfolios_profitability,
  portfolios_investment
) |> 
  group_by(date) |>
  summarize(
    smb_replicated = mean(ret[portfolio_size == 1]) -
      mean(ret[portfolio_size == 2])
  )
factors_replicated <- factors_size |>
  full_join(factors_value, join_by(date)) |>
  full_join(factors_profitability, join_by(date)) |>
  full_join(factors_investment, join_by(date))
# A tibble: 738 × 5
   date       smb_replicated hml_replicated rmw_replicated cma_replicated
   <date>              <dbl>          <dbl>          <dbl>          <dbl>
 1 1962-07-01       -0.00370       -0.0243        0.0190         -0.0291 
 2 1962-08-01        0.00396       -0.00638       0.00798         0.00649
 3 1962-09-01       -0.0129         0.00444       0.00161         0.00203
 4 1962-10-01       -0.0263        -0.00250       0.0129          0.00765
 5 1962-11-01        0.0213         0.00944      -0.0100         -0.00209
 6 1962-12-01       -0.0217        -0.00821       0.00383        -0.00767
 7 1963-01-01        0.0292         0.0135       -0.000551        0.0176 
 8 1963-02-01        0.00434        0.0212       -0.0165          0.0124 
 9 1963-03-01       -0.00960        0.00966       0.00637        -0.00681
10 1963-04-01       -0.00891        0.00400       0.0105          0.00845
# ℹ 728 more rows
test <- factors_ff5_monthly |>
  inner_join(factors_replicated, join_by(date)) |>
  mutate(
    across(c(smb_replicated, hml_replicated, 
             rmw_replicated, cma_replicated), ~round(., 4))
  )

Small-minus-Big

model_smb <- lm(smb ~ smb_replicated, data = test)
summary(model_smb)

Call:
lm(formula = smb ~ smb_replicated, data = test)

Residuals:
       Min         1Q     Median         3Q        Max 
-0.0190136 -0.0018549  0.0002052  0.0019887  0.0137217 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)    -0.0001828  0.0001328  -1.377    0.169    
smb_replicated  0.9639056  0.0042402 227.324   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.003567 on 724 degrees of freedom
Multiple R-squared:  0.9862,    Adjusted R-squared:  0.9862 
F-statistic: 5.168e+04 on 1 and 724 DF,  p-value: < 2.2e-16

High-minus-Low

model_hml <- lm(hml ~ hml_replicated, data = test)
summary(model_hml)

Call:
lm(formula = hml ~ hml_replicated, data = test)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.044359 -0.004058 -0.000349  0.004025  0.036610 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)    0.0005772  0.0002944    1.96   0.0503 .  
hml_replicated 0.9890888  0.0100505   98.41   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.007907 on 724 degrees of freedom
Multiple R-squared:  0.9304,    Adjusted R-squared:  0.9303 
F-statistic:  9685 on 1 and 724 DF,  p-value: < 2.2e-16

Robust-minus-Weak

model_rmw <- lm(rmw ~ rmw_replicated, data = test)
summary(model_rmw)

Call:
lm(formula = rmw ~ rmw_replicated, data = test)

Residuals:
       Min         1Q     Median         3Q        Max 
-0.0199638 -0.0031132  0.0000533  0.0032781  0.0187142 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)    2.265e-05  2.014e-04   0.112     0.91    
rmw_replicated 9.502e-01  8.791e-03 108.083   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.005381 on 724 degrees of freedom
Multiple R-squared:  0.9416,    Adjusted R-squared:  0.9416 
F-statistic: 1.168e+04 on 1 and 724 DF,  p-value: < 2.2e-16

Conservative-minus-Aggressive

model_cma <- lm(cma ~ cma_replicated, data = test)
summary(model_cma)

Call:
lm(formula = cma ~ cma_replicated, data = test)

Residuals:
       Min         1Q     Median         3Q        Max 
-0.0151966 -0.0026756 -0.0001165  0.0024566  0.0214458 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)    0.0006444  0.0001676   3.845 0.000131 ***
cma_replicated 0.9627165  0.0079268 121.451  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.004492 on 724 degrees of freedom
Multiple R-squared:  0.9532,    Adjusted R-squared:  0.9531 
F-statistic: 1.475e+04 on 1 and 724 DF,  p-value: < 2.2e-16

Ausschluss von jungen Unternehmen

  • Ersten zwei Jahre des Unternehmens in der Compustat ausschließen
  • Durch IPO oder ähnliches häufig instabile Bilanzkennzahlen (überproportional volatil)
  • Oft fehlen Profitabilität (RMW) und Investment (CMA) in den Daten
    • Führt zu verzerrten Faktorberechnungen
  • Ergebnis
    • Auschluss führt zu robusterer Faktorkonstruktion
    • Qualität steigt (insbesondere bei den zusätzlichen FF5 Faktoren)

Download über tidyfinance Paket

library(tidyfinance)

factors_ff5_monthly <- download_data("factors_ff_5_2x3_monthly") |>
  select(date, smb, hml, rmw, cma)