Factor models: CAPM, Fama-French, PCA and attribution.

From the CAPM to six factors

The CAPM (Sharpe, 1964) says that, in equilibrium, the expected excess return of any asset is its beta times the market premium. Applied to a portfolio as a time series, the regression

separates what the portfolio earns by being exposed to the market ($\beta$) from what it earns on its own ($\alpha$). Lesson 2.6.2 used it to measure a strategy against an index. The problem is that the market is not the only premium that gets paid. Fama and French (1993) added two factors built as long-short stock portfolios: size (SMB, small minus big) and value (HML, low price relative to book value minus high). Carhart (1997) added momentum (the winners of the last twelve months, excluding the last one, minus the losers) and Fama and French (2015), profitability (RMW, profitable firms minus unprofitable ones) and investment (CMA, firms that invest little minus those that invest a lot). With all six:

flowchart TB
  C[CAPM: market] --> F3[FF3: + SMB, HML]
  F3 --> C4[Carhart: + MOM]
  F3 --> F5[FF5: + RMW, CMA]
  C4 --> F6[Six factors]
  F5 --> F6

An alpha in a model is a return that that model does not explain. Each factor added turns into exposure something that used to be alpha. If factor premia can be bought cheaply (style index funds, futures), a manager who only collects factor premia adds nothing that cannot be replicated. How factors are built from stock data (quintile sorts, Fama-MacBeth, IC) is lesson 2.3.5; here they are used ready-made, to evaluate portfolios.

The data are synthetic and come with the known answer. ff4_factores_mensuales.csv has six monthly factors from 2006 to 2025 in the format of Kenneth French's library (as decimals), consistent with the universe of lesson 4.1.4. ff4_fondos_mensuales.csv has 20 fictitious funds generated with alphas and loadings published in ff4_fondos_verdad.csv, among them a fund with real alpha, one with small alpha, several with no alpha, two with negative alpha because of fees, two nonlinear ones and one that changes style. No result in this lesson says anything about real funds.

import numpy as np
import pandas as pd
import statsmodels.api as sm

fac = pd.read_csv("ff4_factores_mensuales.csv", index_col="mes")
fondos = pd.read_csv("ff4_fondos_mensuales.csv").pivot(
    index="mes", columns="fondo", values="rendimiento")
verdad = pd.read_csv("ff4_fondos_verdad.csv", index_col="fondo")
exc = fondos.sub(fac["rf"], axis=0)       # excess over the risk-free rate
F6 = ["mkt_rf", "smb", "hml", "rmw", "cma", "mom"]

print(fac.shape, exc.shape, fac.index[0], "to", fac.index[-1])
resumen = pd.DataFrame({
    "media_%": 1200 * fac[F6].mean(),
    "vol_%": 100 * np.sqrt(12) * fac[F6].std(),
    "peor_mes_%": 100 * fac[F6].min(),
})
resumen["sharpe"] = resumen["media_%"] / resumen["vol_%"]
resumen.round(2)

(240, 7) (240, 20) 2006-01 to 2025-12
        media_%  vol_%  peor_mes_%  sharpe
mkt_rf     9.27  22.22      -28.94    0.42
smb        0.18   8.01       -8.75    0.02
hml       -2.45   8.35       -7.77   -0.29
rmw        3.52   7.67       -6.15    0.46
cma        1.69   6.94       -7.41    0.24
mom        4.05  17.64      -33.13    0.23

The synthetic market has a mean premium of 9.27% a year with 22.2% volatility and a worst month of −28.9%. Value (HML) has a negative premium in the sample, profitability (RMW) the highest Sharpe (0.46) and momentum a worst month of −33.1%, the 2009 crash that the data imitate.

The primary source: Kenneth French's library

The real factors are published for free in Kenneth French's data library (Dartmouth), which is the primary source: Fama and French's papers use those series and almost the whole industry takes them as the reference. Each file is a ZIP with a CSV in its own format. As consulted on October 22, 2026, F-F_Research_Data_5_Factors_2x3_CSV.zip has three lines of text (“This file was created using the 202608 CRSP database”, where the risk-free rate comes from: Ibbotson until May 2024 and the ICE BofA one-month bill index since June), a blank line, the header ,Mkt-RF,SMB,HML,RMW,CMA,RF, the months in YYYYMM format with the values in percent from July 1963 to August 2026, another blank line and a second table, Annual Factors: January-December, with the years. Momentum goes in another file, F-F_Momentum_Factor_CSV.zip, with a longer text header that warns that missing data are marked with −99.99 or −999. Lines end in CRLF.

A robust reader does not count header lines, because they change from one file to another: it looks for the first line that starts with a comma and reads up to the first one that does not start with a year. The cell tests it with a text that copies that layout with the course's synthetic values:

import io

# A file with the same layout as F-F_Research_Data_5_Factors
# _2x3.csv: text header, YYYYMM months, values in %, and then
# the annual factors. The values are the course's synthetic ones.
cab = ",Mkt-RF,SMB,HML,RMW,CMA,RF"
filas = [f"{m.replace('-', '')},"
         + ",".join(f"{100 * fac.loc[m, c]:8.2f}"
                    for c in ["mkt_rf", "smb", "hml", "rmw", "cma",
                              "rf"])
         for m in fac.index[:3]]
texto = "\r\n".join(
    ["Sample file (K. French library format).",
     "", cab, *filas, "", " Annual Factors: January-December ",
     cab, "  2006,   10.00,    1.00,    2.00,    3.00,    4.00,"
     "    5.00"])

def leer_french(texto):
    """First (monthly) table of a CSV from French's library."""
    lineas = texto.splitlines()
    ini = next(i for i, l in enumerate(lineas) if l.startswith(","))
    fin = next(i for i in range(ini + 1, len(lineas))
               if not lineas[i].strip()[:1].isdigit())
    df = pd.read_csv(io.StringIO("\n".join(lineas[ini:fin])),
                     index_col=0)
    df.index = pd.PeriodIndex(df.index.astype(str), freq="M")
    df.columns = (df.columns.str.strip().str.lower()
                  .str.replace("-", "_"))
    df = df.replace([-99.99, -999], np.nan)   # missing values
    return df / 100                            # from % to decimals

leer_french(texto)

         mkt_rf     smb     hml     rmw     cma      rf
2006-01  0.0080 -0.0235  0.0296  0.0151  0.0107  0.0034
2006-02 -0.0064 -0.0207  0.0091  0.0099  0.0265  0.0033
2006-03  0.0236 -0.0360 -0.0225  0.0560  0.0189  0.0041

With a network connection, the same reader works for the real files. The code downloads the ZIP with urllib (from the standard library), reads the CSV inside and joins the five factors with momentum by month; it is not run in the course:

import io
import urllib.request
import zipfile

BASE = ("https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/"
        "ftp/")
def descargar_french(nombre):
    """Download a ZIP from the library and return the CSV as text."""
    with urllib.request.urlopen(BASE + nombre, timeout=30) as r:
        z = zipfile.ZipFile(io.BytesIO(r.read()))
    with z.open(z.namelist()[0]) as f:
        return f.read().decode("latin-1")

ff5 = leer_french(descargar_french(
    "F-F_Research_Data_5_Factors_2x3_CSV.zip"))
mom = leer_french(descargar_french("F-F_Momentum_Factor_CSV.zip"))
ff6 = ff5.join(mom, how="inner")

Twenty funds with the known answer

The regression is the same for every fund: least squares with a constant, on monthly data, and Newey-West errors with three lags (lesson 2.1.1), which correct for heteroskedasticity and some autocorrelation of the residuals. The monthly alpha is multiplied by 12 to annualize it.

def regresion(y, X, hac=True):
    """OLS with a constant; Newey-West errors (3 months) if hac."""
    m = sm.OLS(y, sm.add_constant(X))
    if hac:
        return m.fit(cov_type="HAC", cov_kwds={"maxlags": 3})
    return m.fit()

filas = {}
for f in exc:
    capm = regresion(exc[f], fac[["mkt_rf"]])
    ff = regresion(exc[f], fac[F6])
    filas[f] = {
        "alfa_verdad_%": 100 * verdad.loc[f, "alfa_anual"],
        "alfa_capm_%": 1200 * capm.params.iloc[0],
        "beta_capm": capm.params.iloc[1],
        "alfa_ff6_%": 1200 * ff.params.iloc[0],
        "t_ff6": ff.tvalues.iloc[0],
        "r2_ff6": ff.rsquared,
    }
tab = pd.DataFrame(filas).T
tab.round(2)

     alfa_verdad_%  alfa_capm_%  beta_capm  alfa_ff6_%  t_ff6  r2_ff6
F01           -0.1         0.04       1.00        0.05   0.55    1.00
F02           -1.0        -0.24       1.02       -0.24  -0.56    0.99
F03            0.0        -1.60       1.01       -0.72  -0.74    0.97
F04            0.0        -1.18       1.07       -1.05  -1.02    0.97
F05            0.0         3.31       0.85        0.09   0.12    0.97
F06            0.0         1.72       0.95       -1.67  -1.52    0.97
F07            0.0        -0.65       1.28        0.64   0.46    0.96
F08            3.0         3.52       0.98        3.29   2.86    0.96
F09            1.5         3.04       1.00        2.90   2.57    0.95
F10            0.0         1.45       0.73       -0.09  -0.13    0.98
F11            2.0         3.21       0.03        2.24   2.05    0.38
F12            0.0         2.88       0.67        3.13   4.15    0.96
F13            6.0        -1.19       0.54       -1.21  -1.82    0.96
F14            0.0        -3.70       1.01       -2.79  -1.98    0.94
F15            1.0         2.24       0.57        0.25   0.32    0.94
F16           -2.5        -1.99       1.00       -2.18  -3.15    0.99
F17            0.0         0.42       0.78       -0.39  -0.55    0.97
F18            0.0        -1.17       1.08        2.06   2.55    0.98
F19           -1.5        -0.09       2.00       -0.27  -0.53    1.00
F20            0.0         0.99       0.99        1.12   0.76    0.94

The header figure puts the true alpha (hollow square) next to the one measured with six factors (filled square) and its 95% interval. For most funds the interval contains the truth. Four readings of the table:

• The CAPM gets style funds wrong. F05 (quality) measures a CAPM alpha of 3.31% and 0.09% with six factors; F06 (momentum), 1.72% and −1.67%; F03 (deep value), −1.60% and −0.72%. The CAPM alpha of a style fund is, above all, the premium of its style in the sample.
• R² separates active funds from index funds. F01 (broad index) and F19 (2x leveraged) have an R² of 1.00; F11, a market-neutral long-short fund, 0.38: almost all of its risk is idiosyncratic.
• The measured alpha has a large error. With 20 years, the standard error of the annual alpha is the idiosyncratic volatility divided by √20: for a fund with 5% idiosyncratic volatility, 1.1 points. F09, with a true 1.5%, measures 2.90%; F19, with a true −1.5%, measures −0.27%.
• Three funds come out wrong for different reasons. F12 and F13 are nonlinear and F14 changes style; they are covered further down.

Loadings are recovered much better than alphas, because the factors move much more than their mean:

cargas = pd.DataFrame(
    {f: regresion(exc[f], fac[F6]).params[1:] for f in exc},
    index=F6).T
err = cargas - verdad[[f"b_{c}" for c in F6]].values
lineales = [f for f in exc if f not in ("F12", "F13", "F14")]
print("mean absolute error of the loadings (17 linear funds):")
err.loc[lineales].abs().mean().round(3)

mean absolute error of the loadings (17 linear funds):
mkt_rf    0.007
smb       0.023
hml       0.040
rmw       0.028
cma       0.042
mom       0.008
dtype: float64

In the 17 linear funds, the mean absolute error of the market loading is 0.007 and that of the others between 0.008 and 0.042. A factor regression is a good instrument for measuring exposures and a bad one for measuring alphas.

Attribution: where the return comes from

With the estimated loadings, the mean excess return of a fund breaks down exactly into alpha plus loading times the mean of each factor (the regression with a constant passes through the means):

F10 is the textbook case: a CAPM alpha of 1.45%.

def atribucion(f, factores):
    """Annual mean return: alpha + loading × mean of each factor."""
    m = regresion(exc[f], fac[factores], hac=False)
    partes = 1200 * m.params[1:] * fac[factores].mean()
    partes["alfa"] = 1200 * m.params.iloc[0]
    partes["total"] = 1200 * exc[f].mean()
    return partes

pd.DataFrame({"capm": atribucion("F10", ["mkt_rf"]),
              "ff6": atribucion("F10", F6)}).round(2)

        capm   ff6
alfa    1.45 -0.09
cma      NaN  0.44
hml      NaN -1.24
mkt_rf  6.79  7.51
mom      NaN  0.34
rmw      NaN  1.21
smb      NaN  0.07
total   8.25  8.25

The mean excess return of 8.25% a year is the same in both columns; the split changes. With the CAPM, 6.79 points are market and 1.45 alpha. With six factors, the market rises to 7.51 (F10's market beta with the other factors is higher than its CAPM beta), profitability adds 1.21, investment 0.44, momentum 0.34, value subtracts 1.24 and the alpha is left at −0.09. The fund had exposures to premia that the CAPM does not measure; it had no alpha.

Many funds, some of them lucky

Testing the alpha of 20 funds means running 20 tests. At a 5% level, even if none had alpha, one would be expected to come out significant by chance. Lesson 2.1.8 explains the corrections; here they are applied with the truth in view:

from statsmodels.stats.multitest import multipletests

p = pd.Series({f: regresion(exc[f], fac[F6]).pvalues.iloc[0]
               for f in exc})
holm = multipletests(p, alpha=0.05, method="holm")
bh = multipletests(p, alpha=0.05, method="fdr_bh")
sig = pd.DataFrame({"alfa_verdad_%": tab["alfa_verdad_%"],
                    "t": tab["t_ff6"], "p": p,
                    "holm": holm[0], "fdr_bh": bh[0]})
sig[sig["p"] < 0.05].round(4)

     alfa_verdad_%       t       p   holm  fdr_bh
F08            3.0  2.8593  0.0042  False    True
F09            1.5  2.5705  0.0102  False    True
F11            2.0  2.0466  0.0407  False   False
F12            0.0  4.1459  0.0000   True    True
F14            0.0 -1.9828  0.0474  False   False
F16           -2.5 -3.1471  0.0016   True    True
F18            0.0  2.5541  0.0106  False    True

Seven funds have p < 0.05. With the truth known:

Holm controls the probability of making any false positive among the 20 and keeps F12 and F16: it removes the false positive F18, but also the real alphas of F08, F09 and F11. Benjamini-Hochberg controls the proportion of false discoveries and accepts five: three with real or real negative alpha, the false positive F18 and the spurious F12. No correction fixes F12: its t of 4.15 is real, but it measures a badly posed model, not skill.

F18 is a “contrarian” fund with a loading of −0.4 on momentum and zero alpha. In this sample its residual had a positive mean of the order of two standard errors. With 20 funds, it is to be expected that this happens to one.

flowchart TB
  A["Alpha with t > 2"] --> M{Many funds?}
  M -- yes --> C[Holm or BH]
  M -- no --> N{Nonlinear?}
  C --> N
  N -- yes --> T[Add TM or HM]
  N -- no --> E{Stable loading?}
  T --> E
  E -- no --> V[Rolling windows]
  E -- yes --> R[Credible alpha]

Market timing and option selling

Two funds have returns that are not linear in the market:

• F12 does market timing. It raises its exposure before rallies and cuts it before declines. The result is a convex relationship with the market; Treynor and Mazuy (1966) modeled it with a quadratic term: $r - r_f = \alpha + \beta\,MKT + \gamma\,MKT^2 + \dots$. Truth: $\gamma = 0.5$, $\alpha = 0$.
• F13 sells puts. It collects a fixed premium and, when the market falls, loses more than its normal beta. Henriksson and Merton (1981) modeled it with a $\min(MKT, 0)$ term: a different beta in declines. Truth: $\alpha = 6\%$ a year (the premium), normal beta 0.4 and an extra beta of 0.25 in declines.

X = fac[F6].copy()
X["mkt2"] = fac["mkt_rf"] ** 2                  # Treynor-Mazuy
X["mkt_neg"] = np.minimum(fac["mkt_rf"], 0)     # Henriksson-Merton

tm = regresion(exc["F12"], X[F6 + ["mkt2"]])
hm = regresion(exc["F13"], X[F6 + ["mkt_neg"]])
pd.DataFrame({
    "F12_lineal": [tab.loc["F12", "alfa_ff6_%"], np.nan],
    "F12_TM": [1200 * tm.params.iloc[0], tm.params.iloc[-1]],
    "F13_lineal": [tab.loc["F13", "alfa_ff6_%"], np.nan],
    "F13_HM": [1200 * hm.params.iloc[0], hm.params.iloc[-1]],
}, index=["alfa_%", "coef_no_lineal"]).round(2)

                F12_lineal  F12_TM  F13_lineal  F13_HM
alfa_%                3.13    0.68       -1.21    5.83
coef_no_lineal         NaN    0.44         NaN    0.24

The linear model gets both wrong, and in opposite directions. To F12 it attributes 3.13% of alpha: the mean of $MKT^2$ is positive and, without the quadratic term, it ends up in the constant. With Treynor-Mazuy the alpha drops to 0.68% (not significant) and $\gamma$ comes out at 0.44, close to 0.5. To F13 it attributes −1.21%, and with Henriksson-Merton the alpha comes out at 5.83% and the extra beta at 0.24: the truth.

So what is F13's 6%? What it collects for insuring others against declines. What it costs:

b_mkt = hm.params.iloc[1]
extra = hm.params.iloc[-1]
print(f"normal beta {b_mkt:.2f}, extra beta in declines {extra:.2f}")
coste_put = 1200 * extra * X["mkt_neg"].mean()
alfa_hm = 1200 * hm.params.iloc[0]
print(f"alpha (premium collected):    {alfa_hm:6.2f} %")
print(f"mean of extra·min(mkt, 0):    {coste_put:6.2f} %")
peores = fac["mkt_rf"].nsmallest(3).index
(100 * pd.DataFrame({"mkt_rf": fac.loc[peores, "mkt_rf"],
                     "F13": exc.loc[peores, "F13"]})).round(2)

normal beta 0.40, extra beta in declines 0.24
alpha (premium collected):      5.83 %
mean of extra·min(mkt, 0):     -5.68 %
         mkt_rf    F13
mes
2018-10  -28.94 -18.54
2011-09  -21.28 -12.79
2025-03  -20.74 -13.54

The extra beta of 0.24 in declines has an average cost of 5.68% a year, almost the whole alpha. In the three worst months of the synthetic market (−28.9%, −21.3% and −20.7%), F13 loses 18.5%, 12.8% and 13.5%. The figure shows the shape: averages by market bucket over a kinked line with slope 0.40 in rallies and 0.64 in declines. An “alpha” collected in exchange for tail risk is not skill; it is a risk premium under another name (lesson 3.6.1). The linear regression, on top of that, hides it.

Style drift

F14 was a value fund (a loading of 0.8 on HML) until December 2015 and since January 2016 has been a growth fund (−0.3). A single regression over the whole sample averages the two:

movil = {}
for i in range(36, len(exc) + 1):
    v = slice(i - 36, i)
    m = regresion(exc["F14"].iloc[v], fac[F6].iloc[v], hac=False)
    movil[exc.index[i - 1]] = m.params["hml"]
movil = pd.Series(movil)
mitades = {t: regresion(exc["F14"].loc[a:b], fac[F6].loc[a:b],
                        hac=False).params["hml"]
           for t, (a, b) in {"2006-2015": ("2006-01", "2015-12"),
                             "2016-2025": ("2016-01", "2025-12")
                             }.items()}
print("whole sample:", round(cargas.loc["F14", "hml"], 2),
      "· by halves:", {k: round(float(v), 2)
                       for k, v in mitades.items()})
movil.iloc[::24].round(2)

whole sample: 0.22 · by halves: {'2006-2015': 0.85, '2016-2025': -0.35}
2008-12    0.94
2010-12    0.84
2012-12    0.86
2014-12    0.79
2016-12    0.51
2018-12   -0.39
2020-12   -0.35
2022-12   -0.36
2024-12   -0.35
dtype: float64

With the whole sample, the value loading comes out at 0.22: a “slightly value” fund that never existed. By halves it comes out at 0.85 and −0.35, close to the truth. The 36-month rolling regression sees the change, but late: it crosses zero in September 2018, almost three years later. In addition, the misspecified average leaves an alpha of −2.79% with t −1.98, which is not in the fund: it is the consequence of imposing a fixed loading on a fund that does not have one. Rolling windows, piecewise regressions or time-varying loading models (the Kalman filter of lesson 2.1.6) are the tools for this.

Risk attribution of a portfolio

Attribution also works for risk. With a factor model $\Sigma = B\,\Sigma_f\,B' + D$, the variance of a portfolio $w$ is $x'\Sigma_f x + w'Dw$ with $x = B'w$, the portfolio's exposure to each factor. By Euler, as in lesson 4.1.4, each factor contributes $x_k(\Sigma_f x)_k / \sigma_p^2$ and the rest is idiosyncratic.

The universe of lesson 4.1.4 was generated with a published ten-factor model: true loadings in universo4_verdad.csv, factor volatilities and correlations by regime in universo4_factores_verdad.csv and the realized daily factor returns in universo4_factores_diarios.parquet. The true attribution of the calm regime and the one estimated with daily regressions on the calm sessions can be computed and compared:

precios = pd.read_parquet("universo4_precios.parquet")
precios = precios.set_index("fecha")
diarios = pd.read_parquet("universo4_factores_diarios.parquet")
diarios = diarios.set_index("fecha")
r = precios.pct_change()
r.iloc[0] = precios.iloc[0] / 100 - 1
cols = [c for c in precios if c not in ("MONET", "RF_T13")]
ex_d = r[cols].sub(diarios["rf"], axis=0)
nombres = [c[2:] for c in diarios if c.startswith("f_")]
f_d = diarios[[f"f_{k}" for k in nombres]].set_axis(nombres, axis=1)

# truth: B, factor covariance and D of the calm regime
uv = pd.read_csv("universo4_verdad.csv", index_col="activo").loc[cols]
B = uv[[f"b_{k}" for k in nombres]].values
fv = pd.read_csv("universo4_factores_verdad.csv")
fv = fv[fv["regimen"] == "calma"].set_index("factor").loc[nombres]
C = fv[[f"corr_{k}" for k in nombres]].values
Sf = np.outer(fv["vol_anual"], fv["vol_anual"]) * C
D = np.diag(uv["vol_idio"] ** 2)
cvr = pd.read_csv("universo4_cov_verdad.csv")
cvr = cvr[cvr["regimen"] == "calma"].set_index("activo")
dif = B @ Sf @ B.T + D - cvr.loc[cols, cols].values
print(f"max |BΣfB' + D − Σ_calm|: {np.abs(dif).max():.1e}")

# estimate: daily regressions on the calm sessions
calma = diarios["regimen"] == "calma"
Xf = sm.add_constant(f_d[calma])
coef = np.linalg.lstsq(Xf.values, ex_d[calma].values, rcond=None)[0]
B_e = coef[1:].T
res = ex_d[calma].values - Xf.values @ coef
Sf_e = 252 * np.cov(f_d[calma].values.T)
D_e = np.diag(252 * res.var(axis=0, ddof=Xf.shape[1]))

def riesgo_factores(w, B, Sf, D):
    x = B.T @ w                          # exposure to each factor
    var = x @ Sf @ x + w @ D @ w
    rc = x * (Sf @ x) / var              # contribution of each factor
    return pd.Series(np.r_[rc, w @ D @ w / var],
                     index=nombres + ["idiosyncratic"])

w = pd.Series(0.0, index=cols)
w[["RV_EEUU", "RF_T710"]] = [0.6, 0.4]
w_n = np.full(len(cols), 1 / len(cols))
pd.DataFrame({
    "60/40_verdad": riesgo_factores(w.values, B, Sf, D),
    "60/40_estim": riesgo_factores(w.values, B_e, Sf_e, D_e),
    "1/N_verdad": riesgo_factores(w_n, B, Sf, D),
    "1/N_estim": riesgo_factores(w_n, B_e, Sf_e, D_e),
}).round(3)

max |BΣfB' + D − Σ_calm|: 5.0e-09
               60/40_verdad  60/40_estim  1/N_verdad  1/N_estim
mercado               0.976        0.974       0.786      0.777
tamano               -0.008       -0.006       0.007      0.006
valor                 0.002        0.003       0.004      0.002
tipos                -0.000        0.002      -0.011     -0.010
credito               0.000       -0.000       0.021      0.021
materias              0.000        0.000       0.107      0.114
dolar                -0.000       -0.001       0.052      0.055
emergentes            0.000       -0.001       0.019      0.021
europa                0.000        0.000       0.001      0.000
japon                 0.000       -0.000       0.003      0.002
idiosyncratic         0.030        0.029       0.011      0.011

First, the check: $B\,\Sigma_f\,B' + D$ with the published files reproduces the true calm covariance with an error of $5 \cdot 10^{-9}$, that is, the rounding of the files. Then, the attribution: in the 60/40, the market factor explains 97.6% of the risk (truth) and 97.4% (estimate); rates, nothing: the portfolio is exposed to them, but their covariance with the portfolio is almost zero, because the factor's own variance is offset by its negative correlation with the market. In the 1/N of the 36 assets, the market explains 78.6%, commodities 10.7% and the dollar 5.2%; the estimates are within one point of the truth for every factor. With a well-specified model and 20 years of daily data, risk attribution is reliable. Return attribution, as seen with the alphas, not so much.

PCA: factors without names

The Fama-French factors are observed: portfolios built with an economic rule. Principal components are statistical factors: the directions of maximum variance of the returns themselves, orthogonal to each other, with no name and no fixed sign. They are computed with the eigenvalue decomposition of the correlation matrix (lesson 0.7.3); their use for estimating covariances with little data is in lesson 4.1.2. Here the question is a different one: are the components of a universe its factors?

Z = (ex_d - ex_d.mean()) / ex_d.std()
autoval, autovec = np.linalg.eigh(np.corrcoef(Z.values.T))
autoval, autovec = autoval[::-1], autovec[:, ::-1]
pcs = Z.values @ autovec[:, :10]
print("eigenvalues:", autoval[:6].round(2))
print("variance explained by 1, 3 and 10:",
      (autoval.cumsum()[[0, 2, 9]] / autoval.sum()).round(3))

def r2(y, X):
    X = np.column_stack([np.ones(len(X)), X])
    e = y - X @ np.linalg.lstsq(X, y, rcond=None)[0]
    return 1 - e.var() / y.var()

pd.DataFrame({k: [r2(f_d[f].values, pcs[:, :k]) for f in nombres]
              for k in (1, 3, 5, 10)}, index=nombres).round(2)

eigenvalues: [20.15  6.11  2.68  1.06  0.95  0.74]
variance explained by 1, 3 and 10: [0.56  0.804 0.931]
              1     3     5     10
mercado     0.97  0.98  0.99  0.99
tamano      0.05  0.05  0.06  0.09
valor       0.00  0.06  0.08  0.78
tipos       0.05  0.96  0.98  0.98
credito     0.28  0.33  0.93  0.95
materias    0.21  0.85  0.89  0.93
dolar       0.16  0.35  0.65  0.91
emergentes  0.19  0.23  0.24  0.41
europa      0.00  0.00  0.01  0.03
japon       0.00  0.00  0.00  0.95

The first component explains 56% of the variance of the 36 assets; the first three, 80%, and the first ten, 93%. Having ten true factors does not mean that the first ten components are those ten factors. The table gives the R² of the regression of each true factor on the first $k$ components:

• One component is already the market (R² 0.97). With three, rates (0.96) and commodities (0.85) appear.
• With five, credit (0.93); with ten, the dollar (0.91), Japan (0.95) and value (0.78).
• Size, emerging markets and Europe do not appear even with ten components (0.09, 0.41 and 0.03). The European factor, with 6% volatility, only loads on two assets (European equities hedged and unhedged); the Japanese one, with 9%, also on two, and it does appear. PCA ranks by explained variance: a narrow, low-volatility factor falls below combinations of idiosyncratic noise from other assets.

cargas_pc = pd.DataFrame(autovec[:, :3], index=cols,
                         columns=["pc1", "pc2", "pc3"])
for c in cargas_pc:
    s = cargas_pc[c].sort_values()
    print(c, "extremes:", ", ".join(
        f"{k} {v:+.2f}" for k, v in pd.concat([s.head(2),
                                               s.tail(2)]).items()))

pc1 extremes: RF_T20 -0.05, RF_T710 -0.05, SC_IND +0.21, RV_EEUU +0.22
pc2 extremes: RF_T710 -0.38, RF_T37 -0.38, ES_VAL +0.05, SC_FIN +0.06
pc3 extremes: MP_AMP -0.48, MP_ENE -0.47, RF_CIG +0.10, RF_CHY +0.14

The loadings of the first three components can be interpreted because the universe is simple: the first loads positively on all equity markets (market), the second loads almost only on Treasury bonds, all with the same sign (rates; the negative sign is arbitrary: a component and its opposite are the same) and the third on commodities. In a real universe the interpretation is harder and changes from one window to another; from the fourth component onward there is almost never a clean economic reading.

What each type of factor is for:

Common mistakes

• Stopping at the CAPM. A CAPM alpha in a style fund is the style premium.
• Factors from another region or currency. US factors for a European fund put the exchange rate and the difference between markets into the alpha.
• Reading the t of one fund among many. Among 20 funds, one without alpha comes out with t > 2 (F18).
• Linear models for strategies with options or timing. The linear regression manufactures alpha (F12) or hides it (F13). For volatility-selling strategies, the alpha is an insurance premium.
• Fixed loadings for funds that change. F14 comes out with a style it never had and a spurious negative alpha.
• Interpreting every principal component. The first is usually the market; from there on, be careful.
• Log returns. The factor regression of a portfolio uses simple excess returns, like French's factors.

Exercises

1. Repeat the six-factor regression of F08 with only the first ten years and with only the last ten. How many times does its alpha (true: 3%) come out significant? What does that say about evaluating a manager with ten years of data?

for a, b in [("2006-01", "2015-12"), ("2016-01", "2025-12")]:
    m = regresion(exc["F08"].loc[a:b], fac[F6].loc[a:b])
    print(f"{a[:4]}-{b[:4]}: alpha {1200 * m.params.iloc[0]:5.2f} %"
          f"  t {m.tvalues.iloc[0]:5.2f}")

2006-2015: alpha  3.69 %  t  2.42
2016-2025: alpha  2.47 %  t  1.43

2. Estimate F13 with the CAPM and with the Henriksson-Merton model with the market only (without the other five factors) and compare the alpha. Does the conclusion change compared with the six-factor model?

capm13 = regresion(exc["F13"], fac[["mkt_rf"]])
hm13 = regresion(exc["F13"], X[["mkt_rf", "mkt_neg"]])
print(f"CAPM: alpha {1200 * capm13.params.iloc[0]:.2f} %, "
      f"beta {capm13.params.iloc[1]:.2f}")
print(f"HM:   alpha {1200 * hm13.params.iloc[0]:.2f} %, "
      f"beta {hm13.params.iloc[1]:.2f}, "
      f"extra {hm13.params.iloc[2]:.2f}")

CAPM: alpha -1.19 %, beta 0.54
HM:   alpha 5.79 %, beta 0.40, extra 0.24

3. No code: a manager of a put-selling strategy on an index presents a six-factor alpha of −0.5% and argues that “the strategy has no alpha, but it does not cost anything either, because its beta is only 0.5”. Use F13 to explain what the linear regression is hiding and what you would ask for to evaluate it.