用月度 ETF 五分位分组检验跨资产动量

# 01_where_ideas_come_from.py


```py
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# %% [markdown]
# # Where a trading idea meets the data — cross-asset momentum EDA
#
# **Docker image**: `ml4t`
#
# **Purpose**: Before we freeze a setup or fit a model, we test whether one concrete
# idea — **cross-asset momentum** on the 100-ETF universe — actually leaves a
# footprint in the data. This is the exploratory step that sits between a *story*
# ("what rose recently keeps rising") and a *strategy*. We use the single most
# informative EDA tool in this domain: **sort by the signal, then look at forward
# returns per bucket.**
#
# ## The idea and its mechanism
#
# The idea: assets that went up over the past year tend to keep earning more than the
# rest of the universe over the next month, *across* asset classes (equities, sectors, countries, bonds,
# commodities). Before any statistics, name the mechanism — *who is on the other
# side, and why are they happy to trade with us?* This idea sits in the "behind"
# family: we are betting others are slow or a step behind, not that we are paid to
# bear a risk.
#
# - **SLOW**: macro information diffuses across siloed markets with a lag; capital
#   moves to the trend slowly. That gradual adjustment *is* the return.
# - **WRONG**: investors under-react to news and chase performance, extending trends
#   past fundamentals.
#
# So this is a SLOW/WRONG blend. That also tells us the failure mode in advance:
# momentum reverses sharply at turning points, and — because everyone can read the
# same price history — the edge is the kind that **decays**. Keep that hypothesis in
# mind; the data will have something to say about it.
#
# ## Learning objectives
#
# - Build a monthly cross-sectional panel from the ETF daily data.
# - Construct the classic **12-1 month momentum** signal (skip the most recent month).
# - Run the **quintile conditional-return sort** and read it for *monotonicity,
#   magnitude, and shape*.
# - Stress the result by **era** — the difference between a footprint and an artifact.
# - Leave with a measured reading that motivates the rest of the chapter: freeze the
#   setup, validate walk-forward, and respect costs and effective sample size.
#
# **Book reference**: Chapter 6 §6.1–§6.2 (from idea to evidence; mapping strategies
# and sources of edge); Chapter 7 (defining a tradable label and the conditional
# feature–label relationship).
#
# **Prerequisites**: `data` package on `PYTHONPATH`; ETF parquet present at
# `ML4T_DATA_PATH/etfs/market/`. Run `python data/etfs/market/download.py` if missing.

# %%
"""Cross-asset momentum EDA — from a story to a measured footprint."""

import matplotlib.pyplot as plt
import numpy as np
import polars as pl

from data import load_etfs
from utils.style import COLORS, add_message_title, show_with_alt

# %% tags=["parameters"]
# Production defaults — Papermill injects overrides for CI
LOOKBACK = 12  # months of past return in the signal
SKIP = 1  # skip the most recent month (the "12-1" convention)
N_QUANTILES = 5  # quintiles
MIN_NAMES = 30  # minimum eligible ETFs in a month to include it

# %% [markdown]
# ## 1. Build the monthly panel
#
# The signal and the outcome both live at a **monthly** cadence: we rank the
# universe once a month and hold for the next month. We take each ETF's last close
# of the month. `close` from `load_etfs()` is split/dividend-adjusted, so simple
# returns are total returns.

# %%
prices = load_etfs().select("timestamp", "symbol", "close")
monthly = (
    prices.sort("timestamp")
    .group_by_dynamic("timestamp", every="1mo", group_by="symbol")
    .agg(pl.col("close").last())
    .with_columns(pl.col("timestamp").dt.month_end())
    .sort(["symbol", "timestamp"])
)

n_days, n_symbols = prices["timestamp"].n_unique(), prices["symbol"].n_unique()
span = monthly["timestamp"]
print(f"Daily panel:   {n_days:>4} days x {n_symbols} ETFs")
print(f"Monthly panel: {span.n_unique():>4} months x {n_symbols} ETFs")
print(f"Span:          {span.min():%Y-%m} to {span.max():%Y-%m}")

# %% [markdown]
# ## 2. The signal and the outcome
#
# **Signal — 12-1 momentum**: the cumulative return from 12 months ago to *one month
# ago*. Skipping the most recent month is deliberate: it sidesteps the short-term
# reversal that contaminates a raw 12-month return (that reversal is a *different*
# mechanism — liquidity provision — with the opposite sign).
#
# **Outcome — forward 1-month return**: what we actually earn holding the next month.
# Note the two clocks are already distinct: a 12-month **lookback** feeding a 1-month
# **holding period**. Keeping them separate is a core discipline of the strategy
# research framework (§6.3–§6.5).

# %%
monthly = monthly.with_columns(
    momentum=(pl.col("close").shift(SKIP) / pl.col("close").shift(LOOKBACK) - 1).over("symbol"),
    fwd_ret=(pl.col("close").shift(-1) / pl.col("close") - 1).over("symbol"),
)

# %% [markdown]
# ## 3. The simplest test: sort into quintiles, look at forward returns
#
# Each month we rank every eligible ETF by its momentum signal and split them into
# five equal buckets (quintile 1 = worst past year, quintile 5 = best), form an
# **equal-weight portfolio** of each bucket, and record its return over the *next*
# month. If the mechanism has a footprint, the buckets should line up: higher past
# momentum → higher forward return.
#
# We report each bucket two ways:
#
# - **Gross** — its raw forward return.
# - **Net of the market** — the bucket minus the equal-weight universe that month
#   (its *relative* performance). This strips out beta — the return you get just for
#   being invested — and leaves the momentum *tilt*. Note this is net of the
#   **benchmark**, not net of **trading costs**: we have not subtracted a basis point
#   of cost anywhere in this notebook.
#
# We are reading three things:
#
# - **Monotonicity** — does the relationship order correctly across *all* buckets,
#   not just top-vs-bottom?
# - **Magnitude** — is the top-minus-bottom spread economically material?
# - **Shape** — is the effect smooth (premium-like) or concentrated in the tails
#   (event-like)?

# %% [markdown]
# The eligible cross-section keeps the months where both the signal and the outcome are
# present and at least `MIN_NAMES` ETFs qualify. Sorting by `(timestamp, momentum,
# symbol)` breaks a tie in the signal by symbol, so the rank is stable from run to run.
# The quintile is then cut at the rank quantiles $1 + k(n-1)/5$, which is what pandas'
# `qcut` does on a rank: buckets stay balanced and no tie merges two of them.

# %%
elig = (
    monthly.drop_nulls(["momentum", "fwd_ret"])
    .filter(pl.len().over("timestamp") >= MIN_NAMES)
    .sort(["timestamp", "momentum", "symbol"])
)
rank = pl.col("momentum").rank("ordinal").over("timestamp")
n_names = pl.len().over("timestamp")
quintile = pl.lit(1)
for k in (1, 2, 3, 4):
    quintile = quintile + (rank > 1 + 0.2 * k * (n_names - 1)).cast(pl.Int32)
elig = elig.with_columns(quintile=quintile)
qs = list(range(1, N_QUANTILES + 1))
print(f"Cross-sectional observations: {elig.height:,} over {elig['timestamp'].n_unique()} months")

# %% [markdown]
# Each bucket becomes an equal-weight portfolio in each month, alongside the equal-weight
# universe as a benchmark, and both are averaged across months. The horizon reported is
# the one the design uses: the sort ranks once a month and holds for one month, so a
# monthly return is the honest unit. Raising a single-month cross-sectional sort to the
# twelfth power would assume the edge recurs in every month of the year, and the by-era
# split below is the reason not to assume it.

# %%
benchmark = elig.group_by("timestamp").agg(bench=pl.col("fwd_ret").mean())
buckets = elig.group_by(["timestamp", "quintile"]).agg(ret=pl.col("fwd_ret").mean())
wide = (
    buckets.pivot(values="ret", index="timestamp", on="quintile")
    .join(benchmark, on="timestamp")
    .sort("timestamp")
)

bench_pct = benchmark["bench"].mean() * 100  # %/month
gross = [wide[str(q)].mean() * 100 for q in qs]  # %/month, quintile 1..5
net_bps = [(wide[str(q)] - wide["bench"]).mean() * 1e4 for q in qs]  # bps/month, net of benchmark

print(f"\nBenchmark (equal-weight universe): {bench_pct:.2f}% / month")
print("Forward 1-month return by momentum quintile:")
print(
    pl.DataFrame(
        {
            "quintile": qs,
            "gross_%/mo": [round(g, 3) for g in gross],
            "net_bps/mo": [round(b, 1) for b in net_bps],
        }
    )
)

# The top-minus-bottom spread is identical gross or net — the benchmark cancels.
long_short = (wide["5"] - wide["1"]).drop_nulls()  # tradable Q5-Q1, dollar-neutral
spread_bps = net_bps[-1] - net_bps[0]
ls_t = long_short.mean() / long_short.std() * np.sqrt(long_short.len())
print(f"\nTop-minus-bottom spread: {spread_bps:.0f} bps / month (same gross or net)")
print(
    f"Tradable long-short (Q5-Q1): {long_short.mean() * 1e4:.0f} bps / month, "
    f"t = {ls_t:.2f} (n = {long_short.len()}) — before costs"
)

# %%
fig, (axg, axn) = plt.subplots(1, 2, figsize=(11, 4.6))
labels = [str(q) for q in qs]

bg = axg.bar(labels, gross, color=COLORS["slate"], width=0.68)
bg[-1].set_color(COLORS["amber"])
axg.axhline(bench_pct, ls="--", lw=1, color="grey")
add_message_title(
    axg,
    "Forward 1-month return by momentum quintile, gross",
    subtitle="100-ETF universe, monthly rebalance; dashed line is the equal-weight universe",
)
axg.set_xlabel("Momentum quintile")
axg.set_ylabel("Forward 1-month return, %")
for x, v in zip(labels, gross):
    axg.text(x, v + 0.02, f"{v:.2f}", ha="center", va="bottom", fontsize=10)
axg.margins(y=0.16)

bn = axn.bar(labels, net_bps, color=COLORS["slate"], width=0.68)
bn[-1].set_color(COLORS["amber"])
axn.axhline(0, ls="--", lw=1, color="grey")
add_message_title(
    axn,
    "The same buckets net of the equal-weight universe",
    subtitle="Bucket return minus the universe return in the same month, in basis points",
)
axn.set_xlabel("Momentum quintile  (1 = worst  →  5 = best)")
axn.set_ylabel("Net of market, bps / month")
for x, v in zip(labels, net_bps):
    axn.text(
        x,
        v + (0.8 if v >= 0 else -0.8),
        f"{v:.0f}",
        ha="center",
        va="bottom" if v >= 0 else "top",
        fontsize=10,
    )
axn.margins(y=0.20)
show_with_alt(
    fig,
    "Two bar panels over the five momentum quintiles. The left panel plots each "
    "bucket's gross forward 1-month return in percent against a dashed line at the "
    "equal-weight universe return; the right panel plots the same buckets net of that "
    "universe, in basis points, against a dashed zero line. The top quintile is "
    "highlighted in amber in both panels and every bar is labelled with its value.",
)

# %% [markdown]
# **Read it.** Gross, all five buckets sit close together and close to the dashed
# benchmark line: most of what they earn is beta, the return for being invested at all
# while the universe drifted up. The right panel takes the universe out and leaves the
# *tilt*: the lower buckets net negative, the upper ones net positive, ordering roughly
# with past momentum. The printed table above carries the levels; what the figure is for
# is the ordering and the size of the gaps against the benchmark. The top-minus-bottom
# spread is identical gross or net, because the benchmark cancels, and the long-short
# t-statistic is printed beside it - read it before reading the spread. On the full
# sample the footprint is there and it is small, and one average over two decades can
# hide a lot, so the next section cuts the sample by era.

# %% [markdown]
# ## 4. Is it lumpy but present, or an artifact? Split by era
#
# A real mechanism is usually *lumpy but present*: still there when the sample is cut
# by time, even if it is not equally strong everywhere. An effect that lives entirely in
# one period is telling you what it actually is. We split the sample in half and rerun
# the sort. Recall the mechanism's own warning: information-based edges **decay** as
# everyone learns to read the same price history.

# %%
wide_era = wide.with_columns(
    era=pl.when(pl.col("timestamp") < pl.date(2016, 1, 1))
    .then(pl.lit("2006–2015"))
    .otherwise(pl.lit("2016–2025"))
)
rows = []
for era in ["2006–2015", "2016–2025"]:
    e = wide_era.filter(pl.col("era") == era)
    q1 = (e["1"] - e["bench"]).mean() * 1e4
    q5 = (e["5"] - e["bench"]).mean() * 1e4
    ls_e = (e["5"] - e["1"]).drop_nulls()
    rows.append(
        {
            "era": era,
            "Q1_net_bps": round(q1),
            "Q5_net_bps": round(q5),
            "spread_bps": round(q5 - q1),
            "ls_t": round(ls_e.mean() / ls_e.std() * np.sqrt(ls_e.len()), 2),
            "months": e.height,
        }
    )
era_tbl = pl.DataFrame(rows)
print(era_tbl)

# %%
fig, ax = plt.subplots(figsize=(9, 5))
eras = era_tbl["era"].to_list()
q1_bps, q5_bps, spreads = (
    era_tbl["Q1_net_bps"].to_list(),
    era_tbl["Q5_net_bps"].to_list(),
    era_tbl["spread_bps"].to_list(),
)
x = np.arange(len(eras))
w = 0.38
ax.bar(x - w / 2, q1_bps, w, label="Worst quintile (Q1)", color=COLORS["slate"])
ax.bar(x + w / 2, q5_bps, w, label="Best quintile (Q5)", color=COLORS["amber"])
for xi, (q1, q5, sp) in enumerate(zip(q1_bps, q5_bps, spreads)):
    ax.text(
        xi - w / 2, q1, f"{q1:.0f}", ha="center", va="bottom" if q1 >= 0 else "top", fontsize=10
    )
    ax.text(
        xi + w / 2, q5, f"{q5:.0f}", ha="center", va="bottom" if q5 >= 0 else "top", fontsize=10
    )
    ax.text(xi, max(q1, q5) + 3, f"spread {sp:.0f} bps", ha="center", va="bottom", fontsize=11)
ax.axhline(0, lw=0.8, color="grey")
ax.set_xticks(x)
ax.set_xticklabels(eras)
ax.set_ylabel("Forward 1-month return, net of market, bps / month")
add_message_title(
    ax,
    "Quintile 1 and quintile 5 net of the universe, by decade",
    subtitle="Monthly average in basis points; the labelled gap is quintile 5 minus quintile 1",
)
ax.legend(frameon=False, loc="lower right")
ax.margins(y=0.2)
show_with_alt(
    fig,
    "A grouped bar chart with one pair of bars per decade: the worst momentum quintile "
    "in slate and the best in amber, each plotted as its monthly average net of the "
    "equal-weight universe in basis points, against a zero line. Every bar is labelled "
    "with its value and the quintile 5 minus quintile 1 gap is annotated above each pair.",
)

# %% [markdown]
# **Read it.** Net of the market, the gap between the top and bottom quintile is wide
# in the first decade and much narrower in the second; both spreads are in the printed
# table. An information-based edge fading as it becomes widely known is one reading of
# that, and it is the reading the mechanism named at the top of the notebook predicts.
# Two things keep it at suggestive rather than shown: the tradable long-short is not
# statistically significant in either era, which the `ls_t` column reports, and a
# two-bucket split is itself a coarse cut. The exercises move the boundary and ask how
# stable the fade is.

# %% [markdown]
# ## 5. One feasibility check: how fast does the signal turn over?
#
# Costs are decided by turnover, and turnover is governed by how much the ranking
# changes month to month. We measure the month-over-month rank autocorrelation of
# the signal: a high autocorrelation means this month's ranking is largely last month's
# ranking, so we trade little.

# %% [markdown]
# The Spearman rank correlation is the Pearson correlation of the ranks. Each pair of
# adjacent months is joined on the symbols present in both, kept only where at least
# `MIN_NAMES` symbols survive the join, and the per-pair correlations are averaged.

# %%
mom = monthly.select("timestamp", "symbol", "momentum").drop_nulls("momentum")
months = mom["timestamp"].unique().sort().to_list()
month_index = {t: i for i, t in enumerate(months)}
mom = mom.with_columns(mi=pl.col("timestamp").replace_strict(month_index, return_dtype=pl.Int32))
this_month = mom.select("symbol", "momentum", "mi")
prev_month = mom.select(
    "symbol", pl.col("momentum").alias("mom_prev"), (pl.col("mi") + 1).alias("mi")
)
pairs = (
    this_month.join(prev_month, on=["symbol", "mi"])
    .filter(pl.len().over("mi") >= MIN_NAMES)
    .with_columns(
        r_curr=pl.col("momentum").rank("average").over("mi"),
        r_prev=pl.col("mom_prev").rank("average").over("mi"),
    )
)
autocorr = pairs.group_by("mi").agg(rho=pl.corr("r_curr", "r_prev"))["rho"].mean()
print(f"Mean month-over-month rank autocorrelation: {autocorr:.3f}")
print("(High → the ranking is persistent → turnover and costs are modest.)")

# %% [markdown]
# ## 6. What the EDA established, and what comes next
#
# What it established, and what it did **not**:
#
# - Net of the market, the mechanism leaves a **top-concentrated** tilt: enough to be
#   worth engineering, not enough to trade on its own. The tradable long-short is
#   **not statistically significant** at the t-statistic printed above, and all of this
#   is **before costs**.
# - It is **regime-dependent**: the spread is much narrower in the second decade than
#   in the first. Evaluation must respect *when* the edge lived, which is why we
#   validate **walk-forward** rather than on a shuffled or full-sample average.
# - The signal is **persistent**, at the rank autocorrelation printed above, so
#   turnover is low and costs are unlikely to erase the tilt outright. We have still
#   not subtracted a single basis point.
#
# Nothing here is a backtest. It is visual confirmation that there is *something* to
# model, plus a set of cautions. The rest of the chapter turns this into a disciplined
# evaluation: **freeze the setup** (§6.3), **validate it walk-forward without leakage**
# (§6.5), and keep the search **countable and auditable** (§6.7). Chapter 7 then
# defines a **tradable label** and accounts for **overlap and effective sample size**
# before trusting any significance.

```