Buybacks minus heavy issuers: the anomaly that survived publication

# Frozen before any data is loaded. Every cell below reads from PREREG,
# so no value is typed twice and the plan cannot drift from the code.
#
# Provenance of every parameter.
#
# From Pontiff & Woodgate (2008), Journal of Finance 63(2), 921-945, abstract
# read on the RePEc page. No readable full text could be fetched: the Wiley
# page returns 403 and no working-paper copy was found. So only the claims
# below are theirs, and no window, threshold, or printed coefficient is
# attributed to them:
#   - "post-1970, share issuance exhibits a strong cross-sectional ability to
#     predict stock returns"
#   - before 1970 they find no statistically significant predictive ability
#     for most holding periods
#
# From Ken French's data library page for Portfolios_Formed_on_NI:
#   - NI for June of year t is the change in the log of split-adjusted shares
#     outstanding from fiscal year-end t-2 to fiscal year-end t-1
#   - portfolios formed each June on NYSE breakpoints; separate portfolios
#     for NI < 0 and NI = 0, then quintiles and deciles
#   - the page does not say whether the quintiles and deciles hold every firm
#     or only the positive-NI firms. The primary spread below is chosen so it
#     means the same thing under either reading: firms that shrank their share
#     count minus the decile that grew it most.
#
# Mine, not theirs, and named as mine:
#   - the use of French's NI measure in place of the paper's own definition,
#     which could not be read
#   - the period boundaries. 1970-01 follows the abstract's "post-1970".
#     The holdout starts in the publication month, April 2008, a date the
#     market could know, rather than at the paper's sample end, which could
#     not be read.
#   - the spread legs, cost_bps, hac_lags, and the decision rule in "success"
#
# French's sample starts in 1963-07, so the pre-1970 period here is 78 months.
# It is reported, but it is too short to test the abstract's pre-1970 claim and
# is not part of the success criterion.

PREREG = {
    "hypothesis": (
        "A value-weighted portfolio long firms that reduced their share count "
        "and short the decile of firms that increased it most earns a positive "
        "Fama-French three-factor alpha after 1970, and that alpha is still "
        "positive and reliably different from zero in the period after "
        "Pontiff and Woodgate was published."
    ),
    "universe": (
        "Ken French Portfolios_Formed_on_NI, value-weighted, built from NYSE, "
        "AMEX and NASDAQ stocks on CRSP, formed each June on net share issues "
        "with NYSE breakpoints. Built from CRSP, so no survivorship bias enters "
        "through universe selection. Includes financials and utilities, as "
        "French's file does."
    ),
    "start":     "1963-07-31",
    "end":       "2026-08-31",
    "holdout":   "2008-04-30 onwards, touched once in the last cell of Results",
    "rebalance": "annual, French forms the portfolios at the end of June",
    "params": {
        "long_leg":       "< 0 ",          # French's column name, trailing space included
        "short_leg":      "Hi 10",
        "insample_start": "1970-01-31",    # abstract: "post-1970"
        "insample_end":   "2008-03-31",    # month before JF 63(2), April 2008
        "cost_bps":       20,              # per leg, charged each July
        "hac_lags":       12,
        "factor_model":   "FF3",           # Mkt-RF, SMB, HML
    },
    "robustness": [
        # Every cut below is reported in What broke, whatever it shows.
        # None of them can change the verdict.
        "Lo 10 minus Hi 10 (the decile spread)",
        "Lo 20 minus Hi 20 (the quintile spread)",
        "equal-weighted version of the primary spread",
        "costs at 0, 20 and 50 bp per leg",
        "five-factor alpha (adds RMW and CMA) in sample and in the holdout, "
        "to see whether the investment factor already absorbs the effect",
        "1963-07 to 1969-12, reported, too short to test",
        "decade-by-decade mean of the primary spread",
    ],
    "success": (
        "Both must hold. (a) Over 1970-01 to 2008-03 the primary spread, net "
        "of costs, has a positive FF3 alpha with Newey-West t > 2. (b) Over "
        "2008-04 to 2026-08 the same alpha is positive with Newey-West t > 2. "
        "If (a) holds and (b) does not, the note reports that the effect is "
        "not reliably present after publication, and says whether the "
        "holdout point estimate is above or below the in-sample one. "
        "If (a) fails, the anomaly was never present in this series and the "
        "holdout question does not arise."
    ),
}

PREREG

import sys
import warnings

# Library import notices are not findings, and they would otherwise be
# published as output. Silenced before the imports that raise them.
warnings.filterwarnings("ignore")

import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt

sys.path.append("..")
from lib import style, data

style.use()
SEED = 0
np.random.seed(SEED)

AS_OF = "2026-10-14"          # as-of date of every cached pull in this note
P = PREREG["params"]
DECILES = ["Lo 10"] + [f"Dec {i}" for i in range(2, 10)] + ["Hi 10"]
GROUPS = ["< 0 ", "ZERO"] + DECILES

window = slice(PREREG["start"], PREREG["end"])
vw  = data.to_month_end(data.famafrench("Portfolios_Formed_on_NI", AS_OF, table=0)).loc[window]
ew  = data.to_month_end(data.famafrench("Portfolios_Formed_on_NI", AS_OF, table=1)).loc[window]
ff3 = data.to_month_end(data.famafrench("F-F_Research_Data_Factors", AS_OF)).loc[window]
ff5 = data.to_month_end(data.famafrench("F-F_Research_Data_5_Factors_2x3", AS_OF)).loc[window]

for name, d in [("NI, value-weighted", vw), ("NI, equal-weighted", ew),
                ("FF3 factors", ff3), ("FF5 factors", ff5)]:
    print(f"{name:20} {len(d):4} months  {d.index.min().date()} -> {d.index.max().date()}")

used = pd.concat([vw[GROUPS + ["Lo 20", "Hi 20"]], ew[GROUPS + ["Lo 20", "Hi 20"]]], axis=1)
print(f"\nmissing values in the portfolio columns used: {int(used.isna().sum().sum())}")
print(f"French's -99.99 missing-data code in those columns: {int((used <= -99.99).sum().sum())}")

FACTORS = {"FF3": ["Mkt-RF", "SMB", "HML"],
           "FF5": ["Mkt-RF", "SMB", "HML", "RMW", "CMA"]}

def spread(long=P["long_leg"], short=P["short_leg"], table=None, cost_bps=P["cost_bps"]):
    """Long minus short, in percent per month, with the annual round trip
    charged in July, the month after French rebuilds the portfolios: out of
    one leg and into the other, so two legs at cost_bps each."""
    table = vw if table is None else table
    gross = table[long] - table[short]
    charge = pd.Series(0.0, index=gross.index)
    charge[gross.index.month == 7] = 2 * cost_bps / 100.0
    return gross - charge

def regress(r, model=P["factor_model"], lags=P["hac_lags"]):
    """Time-series regression on the factors, Newey-West standard errors."""
    f = ff5 if model == "FF5" else ff3
    X = sm.add_constant(f.loc[r.index, FACTORS[model]])
    return sm.OLS(r, X).fit(cov_type="HAC", cov_kwds={"maxlags": lags})

def row(r, model=P["factor_model"]):
    """n, mean and its NW t, alpha and its NW t. All in percent per month."""
    m = sm.OLS(r.values, np.ones(len(r))).fit(cov_type="HAC", cov_kwds={"maxlags": P["hac_lags"]})
    fit = regress(r, model)
    return dict(n=len(r), mean=r.mean(), t_mean=float(m.tvalues[0]),
                alpha=fit.params["const"], t_alpha=fit.tvalues["const"])

def fmt(label, d, w=34):
    return (f"{label:{w}} {d['n']:>4} {d['mean']:+8.3f} {d['t_mean']:+7.2f}"
            f" {d['alpha']:+8.3f} {d['t_alpha']:+7.2f}")

HEADER = f"{'':34} {'n':>4} {'mean%':>8} {'t':>7} {'alpha%':>8} {'t':>7}"

pre70_end = (pd.Timestamp(P["insample_start"]) - pd.offsets.MonthEnd(1)).date().isoformat()
holdout_start = PREREG["holdout"][:10]
PRE70    = slice(PREREG["start"], pre70_end)
INSAMPLE = slice(P["insample_start"], P["insample_end"])
HOLDOUT  = slice(holdout_start, PREREG["end"])

primary = spread()
print(f"primary spread: '{P['long_leg'].strip()}' minus '{P['short_leg']}', value-weighted,"
      f" net of {P['cost_bps']}bp per leg each July")
print(f"in sample  {P['insample_start']} -> {P['insample_end']}")
print(f"holdout    {holdout_start} -> {PREREG['end']}")

# Criterion (a): in sample, 1970 to the month before publication.
r = primary.loc[INSAMPLE]
d = row(r)
print("net of costs, % per month, Newey-West t\n")
print(HEADER)
print(fmt("in sample, primary spread", d))

fit = regress(r)
print("\nFF3 loadings of the spread in sample")
for k in FACTORS["FF3"]:
    print(f"  {k:7} {fit.params[k]:+6.2f}  (t {fit.tvalues[k]:+6.2f})")

passed_a = d["alpha"] > 0 and d["t_alpha"] > 2
print(f"\ncriterion (a): alpha > 0 and t > 2  ->  {'PASS' if passed_a else 'FAIL'}")

def group_alphas(sl, table=None):
    table = vw if table is None else table
    out = {}
    for g in GROUPS:
        ex = (table[g] - ff3["RF"]).loc[sl]
        fit = regress(ex)
        out[g] = (fit.params["const"], fit.bse["const"])
    return pd.DataFrame(out, index=["alpha", "se"]).T

ga = group_alphas(INSAMPLE)
labels = ["< 0", "0"] + [str(i) for i in range(1, 11)]
x = np.arange(len(GROUPS))
legs = {P["long_leg"], P["short_leg"]}
colors = [style.ACCENT if g in legs else style.SERIES[2] for g in GROUPS]

fig, ax = plt.subplots(figsize=(8, 4.4))
ax.bar(x, ga["alpha"], color=colors, width=0.7)
ax.axvline(1.5, color=style.MUTED, lw=0.5, alpha=0.5)
ax.errorbar(x, ga["alpha"], yerr=2 * ga["se"], fmt="none", ecolor=style.MUTED, lw=0.8, capsize=2)
ax.axhline(0, color=style.MUTED, lw=0.7)
ax.set_xticks(x, labels)
ax.set_xlabel("Net share issues: repurchased (< 0), unchanged (0), then deciles 1 to 10")
ax.set_ylabel("FF3 alpha, % per month")
ax.set_title(f"In sample, {P['insample_start'][:7]} to {P['insample_end'][:7]}: "
             "alpha by issuance group, value-weighted, ±2 s.e.")
plt.show()

# Criterion (b): the holdout. This cell is the only place it is evaluated.
d_in  = row(primary.loc[INSAMPLE])
d_out = row(primary.loc[HOLDOUT])
print("net of costs, % per month, Newey-West t\n")
print(HEADER)
print(fmt("in sample", d_in))
print(fmt("holdout", d_out))

passed_b = d_out["alpha"] > 0 and d_out["t_alpha"] > 2
print(f"\ncriterion (a)  {'PASS' if passed_a else 'FAIL'}")
print(f"criterion (b)  {'PASS' if passed_b else 'FAIL'}")
print(f"holdout alpha is {d_out['alpha'] / d_in['alpha']:.0%} of the in-sample alpha")

VARIANTS = [
    (f"primary, {P['cost_bps']}bp",       lambda: spread(),                                "FF3"),
    ("primary, 0bp",                       lambda: spread(cost_bps=0),                      "FF3"),
    ("primary, 50bp",                      lambda: spread(cost_bps=50),                     "FF3"),
    ("Lo 10 minus Hi 10",                  lambda: spread("Lo 10", "Hi 10"),                "FF3"),
    ("Lo 20 minus Hi 20",                  lambda: spread("Lo 20", "Hi 20"),                "FF3"),
    ("primary, equal-weighted",            lambda: spread(table=ew),                        "FF3"),
    ("primary, five-factor alpha",         lambda: spread(),                                "FF5"),
]

print(f"{'':30} {'in sample':>18} {'holdout':>18}")
print(f"{'variant':30} {'alpha%':>9} {'t':>7}  {'alpha%':>9} {'t':>7}")
for label, f, model in VARIANTS:
    s = f()
    a, b = row(s.loc[INSAMPLE], model), row(s.loc[HOLDOUT], model)
    print(f"{label:30} {a['alpha']:+9.3f} {a['t_alpha']:+7.2f}  {b['alpha']:+9.3f} {b['t_alpha']:+7.2f}")

d = row(primary.loc[PRE70])
print(f"\nbefore 1970, primary spread, {PREREG['start'][:7]} to {pre70_end[:7]}:"
      f" {d['n']} months, alpha {d['alpha']:+.3f}, t {d['t_alpha']:+.2f}")

fit5 = {name: regress(primary.loc[sl], "FF5") for name, sl in [("in sample", INSAMPLE), ("holdout", HOLDOUT)]}
print("FF5 loadings of the primary spread\n")
print(f"{'':8} {'in sample':>16} {'holdout':>16}")
for k in ["const"] + FACTORS["FF5"]:
    a, b = fit5["in sample"], fit5["holdout"]
    name = "alpha" if k == "const" else k
    print(f"{name:8} {a.params[k]:+7.2f} (t {a.tvalues[k]:+5.1f}) {b.params[k]:+7.2f} (t {b.tvalues[k]:+5.1f})")

decade = (primary.index.year // 10) * 10
print("primary spread by decade, net, % per month\n")
print(HEADER)
for dec_start, r in primary.groupby(decade):
    lab = f"{r.index.min():%Y-%m} to {r.index.max():%Y-%m}"
    print(fmt(lab, row(r), w=34))

cum = primary.cumsum()
fig, ax = plt.subplots(figsize=(8, 4.4))
ax.axvline(pd.Timestamp(holdout_start), color=style.MUTED, lw=0.9, ls="--")
ax.plot(cum.index, cum.values, color=style.ACCENT, lw=1.6)
ax.axhline(0, color=style.MUTED, lw=0.7)
ax.text(pd.Timestamp(holdout_start), ax.get_ylim()[1], "  holdout starts, April 2008", va="top", color=style.MUTED)
ax.set_ylabel("Cumulative net return, sum of monthly %")
ax.set_title("Repurchasers minus heaviest issuers, value-weighted, net of costs")
style.check_points(ax)
plt.show()

gi = group_alphas(INSAMPLE)
gh = group_alphas(HOLDOUT)
print("FF3 alpha by group, value-weighted, % per month (alpha / s.e. in brackets)\n")
print(f"{'group':6} {'in sample':>18} {'holdout':>18}")
for g, lab in zip(GROUPS, labels):
    a, b = gi.loc[g], gh.loc[g]
    print(f"{lab:6} {a['alpha']:+8.3f} ({a['alpha']/a['se']:+5.2f})  {b['alpha']:+8.3f} ({b['alpha']/b['se']:+5.2f})")
fig, ax = plt.subplots(figsize=(8, 4.4))
# Lines join the ten deciles only. Repurchased and unchanged are separate
# groups, not points on the same scale, so they are drawn as markers alone.
dec = x[2:]
for g, color, lw, lab in [(gi, style.SERIES[1], 1.2, f"In sample, {P['insample_start'][:7]} to {P['insample_end'][:7]}"),
                          (gh, style.ACCENT, 1.8, f"Holdout, {holdout_start[:7]} to {PREREG['end'][:7]}")]:
    ax.plot(dec, g["alpha"].iloc[2:], color=color, lw=lw, marker="o", ms=4, label=lab)
    ax.plot(x[:2], g["alpha"].iloc[:2], color=color, lw=0, marker="o", ms=5)
ax.axvline(1.5, color=style.MUTED, lw=0.5, alpha=0.5)
ax.axhline(0, color=style.MUTED, lw=0.7)
ax.set_xticks(x, labels)
ax.set_xlabel("Net share issues: repurchased (< 0), unchanged (0), then deciles 1 to 10")
ax.set_ylabel("FF3 alpha, % per month")
ax.set_title("FF3 alpha by issuance group, value-weighted")
ax.legend(loc="lower left")
plt.show()

# What makes this note reproducible after an upstream source updates.
import matplotlib, statsmodels
print(f"python       {sys.version.split()[0]}")
print(f"pandas       {pd.__version__}")
print(f"numpy        {np.__version__}")
print(f"statsmodels  {statsmodels.__version__}")
print(f"matplotlib   {matplotlib.__version__}")
print(f"seed         {SEED}")
print(f"data as-of   famafrench {AS_OF}  (Portfolios_Formed_on_NI t0 and t1,"
      " F-F_Research_Data_Factors, F-F_Research_Data_5_Factors_2x3)")