评估不同策略和市场环境下的风险覆盖规则

# 07_regime_risk.py


```py
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# %% [markdown]
# # Risk Overlays and Stability Across Regimes
#
# **Docker image**: `ml4t`
#
# Ch19 catalogues the risk-overlay machinery (stop-loss, trailing stop,
# daily loss limit, drawdown breaker, time exit, vol target). This notebook
# reads each case study's overlay backtests directly out of the registry
# and asks two cross-cutting questions: which categories of overlay add
# value on average, and where do overlays improve both Sharpe *and*
# drawdown rather than buy one with the other.
#
# **Learning Objectives**:
# - Compare baseline against risk-managed Sharpe for the case studies that carry
#   Ch19 overlay backtests, however many that currently is - the count is printed
#   when the overlays load
# - Identify which rule categories help vs hurt by case study
# - Understand the mechanism by which a tight stop can cost a cross-asset
#   strategy more than it saves
#
# **Book Reference**: Chapter 20, Section 20.7 (Risk overlays)
#
# **Prerequisites**: Run [`01_aggregate_synthesis`](01_aggregate_synthesis.ipynb) first.
# Each case study's registry must contain Ch19 `risk_overlay`-stage backtests
# and the upstream Ch17 allocation baseline (or Ch16 signal-stage fallback).

# %%
"""Ch20 Risk Overlays — cross-case-study comparison from registry."""

import json

import matplotlib.pyplot as plt
import numpy as np
import polars as pl
from IPython.display import Markdown, display
from matplotlib.patches import Patch as QPatch

from case_studies.utils.analytics import (
    CASE_STUDY_IDS,
    SHORT_NAMES,
    load_chapter_backtests,
)
from case_studies.utils.strategy_analysis import rank_one
from utils.style import show_with_alt

# %% tags=["parameters"]
# 0 = all
MAX_CASE_STUDIES = 0

# %%
CS_LIST = CASE_STUDY_IDS[:MAX_CASE_STUDIES] if MAX_CASE_STUDIES else CASE_STUDY_IDS

# %% [markdown]
# ## Risk Configuration Classifier
#
# Map backtest spec risk-config names to standardized rule categories
# for cross-study comparison.


# %%
def classify_overlay(name: str) -> str:
    """Map risk config name to a rule category."""
    name_lower = name.lower()
    if "trailing" in name_lower:
        return "trailing_stop"
    if "mae_mfe" in name_lower or "calibrated" in name_lower:
        return "mae_mfe_calibrated"
    if "stop_loss" in name_lower or "sl_" in name_lower:
        return "stop_loss"
    if "take_profit" in name_lower or "tp_" in name_lower:
        return "take_profit"
    if "daily" in name_lower or "loss_limit" in name_lower or "period_loss" in name_lower:
        return "daily_limit"
    if "bar_loss" in name_lower:
        return "daily_limit"
    if "dd_breaker" in name_lower or "max_dd" in name_lower:
        return "dd_breaker"
    if "vol_target" in name_lower or "vol_stop" in name_lower:
        return "vol_target"
    if "combined" in name_lower or "chain" in name_lower or "full" in name_lower:
        return "combined"
    if "time_exit" in name_lower:
        return "time_exit"
    return "other"


# %%
def extract_risk_name(spec_json: str) -> str:
    """Extract risk config name from backtest spec."""
    spec = json.loads(spec_json)
    return spec.get("strategy", {}).get("risk", {}).get("name", "unknown")


# %% [markdown]
# ## Load Risk Overlay Results from Registry
#
# Each case study is reduced to its single highest-Sharpe overlay row, selected by
# row rather than by name. Overlay names such as `trailing_3pct` cover many risk
# parameterizations within one case study, so a name selects several rows, and
# anything downstream that treats one row as one case study then plots several
# points under one label.
#
# Ch19 backtests apply different risk overlays to the same base strategy,
# each tagged with `chapter: "ch19"`. The `risk.name` field identifies the
# overlay configuration.

# %%
ch19_raw = load_chapter_backtests(
    "ch19",
    case_studies=CS_LIST,
    metrics=["sharpe", "max_drawdown", "sortino", "total_return", "cagr"],
)

if ch19_raw.is_empty():
    msg = "No Ch19 backtest results found in any case study registry"
    raise RuntimeError(msg)

risk_df = ch19_raw.with_columns(
    overlay=pl.col("spec_json").map_elements(extract_risk_name, return_dtype=pl.Utf8),
).with_columns(
    category=pl.col("overlay").map_elements(classify_overlay, return_dtype=pl.Utf8),
)

# %%
# Baseline comes from Ch17 (allocation stage); per-CS fallback to Ch16 if absent.
_ch17_raw = load_chapter_backtests("ch17", case_studies=CS_LIST, metrics=["sharpe", "max_drawdown"])
_ch16_raw = load_chapter_backtests("ch16", case_studies=CS_LIST, metrics=["sharpe", "max_drawdown"])

_baseline_rows = []
for cs_id in CS_LIST:
    ch17_cs = (
        _ch17_raw.filter(pl.col("case_study") == cs_id)
        if not _ch17_raw.is_empty()
        else pl.DataFrame()
    )
    ch16_cs = (
        _ch16_raw.filter(pl.col("case_study") == cs_id)
        if not _ch16_raw.is_empty()
        else pl.DataFrame()
    )
    # backtest_hash decides a tie: it is unique per row, so the baseline this loop
    # picks is a function of the registry rather than of the order the frames were
    # concatenated in. Allocations whose Sharpe repeats exactly are ordinary.
    if not ch17_cs.is_empty():
        best = rank_one(ch17_cs, by="sharpe", name="backtest_hash")
        _baseline_rows.append(best.with_columns(baseline_source=pl.lit("ch17")))
    elif not ch16_cs.is_empty():
        best = rank_one(ch16_cs, by="sharpe", name="backtest_hash")
        _baseline_rows.append(best.with_columns(baseline_source=pl.lit("ch16")))

if _baseline_rows:
    _baseline_source = pl.concat(_baseline_rows)
    source_counts = _baseline_source.group_by("baseline_source").len()
    for row in source_counts.iter_rows(named=True):
        print(f"  Baseline from {row['baseline_source']}: {row['len']} case studies")
else:
    _baseline_source = pl.DataFrame()

baseline_sharpe = (
    _baseline_source.select(
        "case_study",
        baseline_sharpe=pl.col("sharpe"),
        baseline_max_dd=pl.col("max_drawdown"),
    )
    if not _baseline_source.is_empty()
    else pl.DataFrame(
        schema={"case_study": pl.Utf8, "baseline_sharpe": pl.Float64, "baseline_max_dd": pl.Float64}
    )
)

# %%
overlay_df = risk_df.join(baseline_sharpe, on="case_study", how="left").with_columns(
    sharpe_delta=pl.col("sharpe") - pl.col("baseline_sharpe"),
)

overlay_df = overlay_df.with_columns(
    is_best=(pl.col("sharpe").rank("ordinal", descending=True).over("case_study") == 1),
)

n_cs = overlay_df["case_study"].n_unique()
print(f"Loaded {len(overlay_df)} overlay results across {n_cs} case studies")
overlay_df.group_by("case_study").agg(
    n_overlays=pl.len(),
    best_sharpe=pl.col("sharpe").max(),
    baseline_sharpe=pl.col("baseline_sharpe").first(),
).sort("case_study")

# %% [markdown]
# ## Baseline vs Best-Managed Comparison
#
# For each case study we compare the baseline unmanaged strategy with its
# highest-Sharpe overlay configuration. A positive Sharpe delta means that
# configuration beat the baseline, which is a weaker statement than the overlay
# category being worth applying: it is the maximum over every configuration swept
# for that case study, and the next section reports what the typical one did.

# %%
best_per_cs = (
    overlay_df.filter(pl.col("is_best"))
    .with_columns(
        managed_sharpe=pl.col("sharpe"),
        managed_max_dd=pl.col("max_drawdown"),
    )
    .select(
        "display_name",
        "overlay",
        "category",
        "baseline_sharpe",
        "managed_sharpe",
        "sharpe_delta",
        "baseline_max_dd",
        "managed_max_dd",
    )
    .sort("sharpe_delta", descending=True)
)
best_per_cs

# %%
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

plot_df = best_per_cs.drop_nulls(subset=["baseline_sharpe", "managed_sharpe"]).with_columns(
    baseline_sharpe=pl.col("baseline_sharpe").fill_nan(0.0),
    managed_sharpe=pl.col("managed_sharpe").fill_nan(0.0),
    sharpe_delta=pl.col("sharpe_delta").fill_nan(0.0),
)
cs_order = plot_df["display_name"].to_list()
baseline = [float(v) for v in plot_df["baseline_sharpe"].to_list()]
managed = [float(v) for v in plot_df["managed_sharpe"].to_list()]

x = np.arange(len(cs_order))
w = 0.35
axes[0].barh(x - w / 2, baseline, w, label="Baseline")
axes[0].barh(x + w / 2, managed, w, label="Best Overlay")
axes[0].set_yticks(x)
axes[0].set_yticklabels(cs_order, fontsize=9)
axes[0].set_xlabel("Sharpe Ratio")
axes[0].set_title("Baseline vs Best Risk Overlay")
axes[0].legend(fontsize=9)
axes[0].invert_yaxis()

deltas = [float(v) for v in plot_df["sharpe_delta"].to_list()]
colors = ["#2ecc71" if d > 0 else "#e74c3c" for d in deltas]
axes[1].barh(x, deltas, color=colors)
axes[1].set_yticks(x)
axes[1].set_yticklabels(cs_order, fontsize=9)
axes[1].set_xlabel("Sharpe Delta (Managed − Baseline)")
axes[1].set_title("Risk Overlay Impact")
axes[1].axvline(0, color="gray", linewidth=0.5, linestyle="--")
axes[1].invert_yaxis()

show_with_alt(
    fig,
    "Left: paired horizontal bars per case study giving baseline Sharpe and the "
    "Sharpe of its highest-Sharpe overlay configuration. Right: the difference "
    "between them, coloured green where positive and red where negative.",
)

# %% [markdown]
#
# %% tags=["results"]
_helped = best_per_cs.filter(pl.col("sharpe_delta") > 0)
display(
    Markdown(
        f"For {_helped.height} of {best_per_cs.height} case studies the "
        "highest-Sharpe overlay configuration clears the baseline"
        + (f" ({', '.join(_helped['display_name'].to_list())})" if _helped.height else "")
        + ". Deltas run from "
        f"{best_per_cs['sharpe_delta'].min():+.3f} to "
        f"{best_per_cs['sharpe_delta'].max():+.3f}.\n\n"
        "Taking a maximum over a sweep and asking whether it clears the baseline "
        "is close to asking whether the sweep was large enough. The population "
        "statistics in the next section are the ones that say whether applying "
        "an overlay is a good idea, because they include the configurations that "
        "would have been chosen by someone without the benefit of this table."
    )
)

# %% [markdown]
# ## Rule Category Effectiveness
#
# Group overlays by rule type and compute average Sharpe delta per category
# across case studies.

# %%
category_stats = (
    overlay_df.group_by("category")
    .agg(
        n_configs=pl.len(),
        n_case_studies=pl.col("case_study").n_unique(),
        mean_sharpe_delta=pl.col("sharpe_delta").mean(),
        median_sharpe_delta=pl.col("sharpe_delta").median(),
        # A share under a name saying percent: the table printed 0.068 where
        # the chart beside it printed "7% positive".
        pct_positive=(100 * (pl.col("sharpe_delta") > 0).sum() / pl.len()),
    )
    .sort("mean_sharpe_delta", descending=True)
)
category_stats

# %%
fig, ax = plt.subplots(figsize=(10, 6.5))

cat_plot = category_stats.drop_nulls(subset=["mean_sharpe_delta"]).with_columns(
    mean_sharpe_delta=pl.col("mean_sharpe_delta").fill_nan(0.0),
    pct_positive=pl.col("pct_positive").fill_nan(0.0).fill_null(0.0),
)
category_display = {
    "daily_limit": "Daily loss limit",
    "dd_breaker": "Drawdown breaker",
    "time_exit": "Time exit",
    "stop_loss": "Stop loss",
    "trailing_stop": "Trailing stop",
}
cats = cat_plot["category"].to_list()
display_cats = [category_display.get(c, c) for c in cats]
means = [float(v) for v in cat_plot["mean_sharpe_delta"].to_list()]
pct_pos = [float(v or 0) for v in cat_plot["pct_positive"].to_list()]
colors = ["#2ecc71" if m > 0 else "#e74c3c" for m in means]

bars = ax.barh(range(len(cats)), means, color=colors, height=0.65)
ax.set_yticks(range(len(cats)))
ax.set_yticklabels(display_cats, fontsize=11)
ax.set_xlabel("Mean Sharpe Delta vs Baseline")
ax.set_title("Risk Rule Category Effectiveness (Across All Case Studies)")
ax.axvline(0, color="gray", linewidth=0.5, linestyle="--")

# Widen x-limits so "X% positive" labels don't crowd the bar tips.
x_lo, x_hi = ax.get_xlim()
ax.set_xlim(x_lo - 0.1 * (x_hi - x_lo), x_hi + 0.18 * (x_hi - x_lo))

for i, (bar, pct) in enumerate(zip(bars, pct_pos, strict=False)):
    width = bar.get_width()
    ax.text(
        width + 0.02 if width >= 0 else width - 0.02,
        i,
        f"{pct:.0f}% positive",
        va="center",
        ha="left" if width >= 0 else "right",
        fontsize=9,
    )

ax.invert_yaxis()
ax.margins(y=0.08)
show_with_alt(
    fig,
    "Horizontal bars of the mean Sharpe change against baseline for each overlay "
    "category, annotated with the share of configurations in that category that "
    "improved on the baseline.",
)

# %% [markdown]
# ## Rule Category × Case Study Heatmap
#
# For each combination of rule category and case study, the largest Sharpe change
# any configuration in that cell achieved. A cell is a maximum over however many
# configurations that combination swept, so cells backed by more configurations
# are higher for that reason alone and the heatmap is not a like-for-like
# comparison across cells.

# %%
heatmap_data = (
    overlay_df.group_by("category", "display_name")
    .agg(best_delta=pl.col("sharpe_delta").max())
    .pivot(on="display_name", index="category", values="best_delta")
    .sort("category")
)

cs_cols = [n for n in SHORT_NAMES.values() if n in heatmap_data.columns]
hm_matrix = heatmap_data.select(cs_cols).to_pandas()
hm_matrix.index = heatmap_data["category"].to_list()

fig, ax = plt.subplots(figsize=(12, 6))

vmax = max(abs(hm_matrix.min().min()), abs(hm_matrix.max().max()))
im = ax.imshow(
    hm_matrix.values,
    cmap="RdYlGn",
    aspect="auto",
    vmin=-vmax,
    vmax=vmax,
)

ax.set_xticks(range(len(cs_cols)))
ax.set_xticklabels(cs_cols, rotation=45, ha="right", fontsize=9)
ax.set_yticks(range(len(hm_matrix.index)))
ax.set_yticklabels(hm_matrix.index, fontsize=10)

for i in range(len(hm_matrix.index)):
    for j in range(len(cs_cols)):
        val = hm_matrix.iloc[i, j]
        if np.isnan(val):
            continue
        color = "white" if abs(val) > vmax * 0.6 else "black"
        ax.text(j, i, f"{val:+.2f}", ha="center", va="center", fontsize=8, color=color)

fig.colorbar(im, ax=ax, label="Best Sharpe Delta", shrink=0.8)
ax.set_title("Best Risk Overlay Effect by Rule Category × Case Study")
show_with_alt(
    fig,
    "Heatmap of the best Sharpe change achieved by each overlay category within "
    "each case study, with blank cells where that category was not swept.",
)

# %% [markdown]
# ## Drawdown Protection
#
# Compare max drawdown reduction across case studies. An overlay can reduce
# drawdown at the cost of Sharpe or improve both, and the table below separates
# the two rather than reporting drawdown alone.

# %%
dd_improvement = (
    overlay_df.filter(pl.col("is_best"))
    .with_columns(
        dd_reduction=(
            (pl.col("baseline_max_dd").abs() - pl.col("max_drawdown").abs())
            / pl.col("baseline_max_dd").abs()
            * 100
        )
    )
    .with_columns(managed_max_dd=pl.col("max_drawdown"))
    .select(
        "display_name",
        "overlay",
        "baseline_max_dd",
        "managed_max_dd",
        "dd_reduction",
        "sharpe_delta",
    )
    .drop_nulls(subset=["dd_reduction", "sharpe_delta"])
    .sort("dd_reduction", descending=True)
)
dd_improvement

# %%
dd_plot = dd_improvement.filter(
    pl.col("dd_reduction").is_not_null() & pl.col("sharpe_delta").is_not_null()
).with_columns(
    dd_reduction=pl.col("dd_reduction").fill_nan(0.0),
    sharpe_delta=pl.col("sharpe_delta").fill_nan(0.0),
)

if not dd_plot.is_empty():
    fig, ax = plt.subplots(figsize=(10, 5))
    names = dd_plot["display_name"].to_list()
    dd_red = [float(v) for v in dd_plot["dd_reduction"].to_list()]
    s_delta = [float(v) for v in dd_plot["sharpe_delta"].to_list()]

    colors = ["#2ecc71" if s > 0 else "#e74c3c" for s in s_delta]
    ax.barh(range(len(names)), dd_red, color=colors)
    ax.set_yticks(range(len(names)))
    ax.set_yticklabels(names, fontsize=10)
    ax.set_xlabel("Max Drawdown Reduction (%)")
    ax.set_title("Drawdown Reduction vs Sharpe Impact")
    ax.axvline(0, color="gray", linewidth=0.5, linestyle="--")
    ax.invert_yaxis()

    for i, (dd, sd) in enumerate(zip(dd_red, s_delta, strict=False)):
        label = f"Sharpe {sd:+.2f}"
        ax.text(dd + 0.5, i, label, va="center", fontsize=8)

    show_with_alt(
        fig,
        "Horizontal bars of the percentage reduction in maximum drawdown achieved "
        "by the best overlay in each case study, annotated with the Sharpe change "
        "that came with it.",
    )
else:
    print("No drawdown improvement data available")

# %% [markdown]
# ## Quadrant Analysis: Where Overlays Earn Their Keep
#
# The quadrant analysis is the cleanest cross-cutting summary: each case
# study is placed by its (Sharpe delta, drawdown-reduction) coordinate.
# The win-win quadrant is the only one that justifies overlay deployment
# without a tradeoff conversation.

# %%
regime_rows = []

for cs_id in overlay_df["case_study"].unique().sort().to_list():
    cs_overlay = overlay_df.filter(pl.col("case_study") == cs_id)
    best_overlay_name = cs_overlay.filter(pl.col("is_best")).select("overlay").head(1)
    if best_overlay_name.is_empty():
        continue

    display_name = cs_overlay["display_name"].first()
    baseline_sr = cs_overlay["baseline_sharpe"].first()
    best_sr = cs_overlay.filter(pl.col("is_best")).select("sharpe").head(1).item()
    best_dd = cs_overlay.filter(pl.col("is_best")).select("max_drawdown").head(1).item()
    baseline_dd = cs_overlay["baseline_max_dd"].first()

    if baseline_sr is None or best_sr is None:
        continue

    sharpe_delta = best_sr - baseline_sr if baseline_sr is not None else 0
    dd_delta = (
        (abs(baseline_dd) - abs(best_dd)) if baseline_dd is not None and best_dd is not None else 0
    )

    regime_rows.append(
        {
            "case_study": cs_id,
            "display_name": display_name,
            "baseline_sharpe": float(baseline_sr) if baseline_sr is not None else 0,
            "managed_sharpe": float(best_sr) if best_sr is not None else 0,
            "sharpe_delta": float(sharpe_delta),
            "baseline_dd": float(abs(baseline_dd)) if baseline_dd is not None else 0,
            "managed_dd": float(abs(best_dd)) if best_dd is not None else 0,
            "dd_improvement": float(dd_delta),
            "dd_reduction_pct": float(dd_delta / abs(baseline_dd) * 100)
            if baseline_dd and abs(baseline_dd) > 0.001
            else 0,
        }
    )

regime_df = pl.DataFrame(regime_rows)

# %%
if regime_df.height >= 3:
    fig, axes = plt.subplots(1, 2, figsize=(14, 6))

    ax = axes[0]
    s_deltas = regime_df["sharpe_delta"].to_list()
    dd_improv = regime_df["dd_reduction_pct"].to_list()
    names = regime_df["display_name"].to_list()

    for sd, ddi, name in zip(s_deltas, dd_improv, names, strict=False):
        if sd > 0 and ddi > 0:
            color = "#2ecc71"
        elif sd <= 0 and ddi > 0:
            color = "#f39c12"
        elif sd > 0 and ddi <= 0:
            color = "#3498db"
        else:
            color = "#e74c3c"

        ax.scatter(sd, ddi, c=color, s=80, edgecolors="white", zorder=5)
        ax.annotate(
            name,
            (sd, ddi),
            textcoords="offset points",
            xytext=(6, 4),
            fontsize=8,
            color="#475569",
        )

    ax.axhline(0, color="gray", linewidth=0.8, linestyle="--")
    ax.axvline(0, color="gray", linewidth=0.8, linestyle="--")
    ax.set_xlabel("Sharpe Delta (managed − baseline)")
    ax.set_ylabel("Drawdown Reduction (%)")
    ax.set_title("(a) Overlay Effectiveness Quadrants")

    quad_legend = [
        QPatch(facecolor="#2ecc71", label="Win-win (↑SR, ↓DD)"),
        QPatch(facecolor="#f39c12", label="DD reduced, SR cost"),
        QPatch(facecolor="#e74c3c", label="Over-constrained"),
    ]
    ax.legend(handles=quad_legend, fontsize=8, frameon=False, loc="lower left")

    ax = axes[1]
    base_dd = regime_df["baseline_dd"].to_list()

    colors_b = ["#2ecc71" if d > 0 else "#e74c3c" for d in s_deltas]
    ax.scatter(base_dd, s_deltas, c=colors_b, s=80, alpha=0.8, edgecolors="white", zorder=5)
    ax.axhline(0, color="gray", linewidth=0.8, linestyle="--")

    for name, x, y in zip(names, base_dd, s_deltas, strict=False):
        ax.annotate(
            name,
            (x, y),
            textcoords="offset points",
            xytext=(6, 4),
            fontsize=8,
            color="#475569",
        )

    ax.set_xlabel("Baseline Max Drawdown (absolute)")
    ax.set_ylabel("Sharpe Delta from Best Overlay")
    ax.set_title("(b) Do Overlays Help More in High-Risk Strategies?")

    fig.suptitle("Cross-Dataset Overlay Effectiveness", fontsize=11, y=1.02)
    show_with_alt(
        fig,
        "Panel a: each case study placed by Sharpe change against drawdown "
        "reduction, with quadrants labelled for the four combinations. Panel b: "
        "Sharpe change against the depth of the baseline drawdown.",
    )
else:
    print("Insufficient data for regime-conditional analysis.")

# %% [markdown]
#
# %% [markdown]
# Panel (a) places each case study by what its best overlay did to Sharpe and to
# maximum drawdown. Four outcomes are possible and the quadrants name all four,
# whether or not this registry has a case study in each: both improve, drawdown
# falls at the cost of Sharpe, Sharpe rises at the cost of drawdown, or both
# worsen. The second is the ordinary insurance trade, paid for in return.
#
# Panel (b) asks whether an overlay helps more when the baseline drawdown is
# deeper. That is the plausible mechanism - a rule that cuts losing positions has
# more to cut - but with this many case studies the panel shows the coordinates
# and settles nothing.

# %% [markdown]
# ## Key Takeaways

# %% tags=["results"]
_cat = category_stats.sort("median_sharpe_delta", descending=True)
_neg_med = _cat.filter(pl.col("median_sharpe_delta") < 0)
_total_configs = int(_cat["n_configs"].sum())
_top_rate = _cat.sort("pct_positive", descending=True).row(0, named=True)
_best = best_per_cs.sort("sharpe_delta", descending=True).row(0, named=True)
_dd_change = (
    100 * (_best["managed_max_dd"] - _best["baseline_max_dd"]) / abs(_best["baseline_max_dd"])
)
display(
    Markdown(
        f"**Across {_total_configs} overlay configurations in "
        f"{_cat.height} categories**, the median Sharpe change is negative in "
        f"{_neg_med.height} of them"
        + (f" ({', '.join(_neg_med['category'].to_list())})" if _neg_med.height else "")
        + ". The typical overlay costs Sharpe: it truncates trades that would "
        "have recovered and adds turnover.\n\n"
        f"**Highest positive rate**: {_top_rate['category']}, improving on "
        f"baseline in {_top_rate['pct_positive']:.1f} percent of its "
        f"{_top_rate['n_configs']} configurations. The categories run from "
        f"{_cat['pct_positive'].min():.1f} to {_cat['pct_positive'].max():.1f} "
        "percent, so on none of them is improvement the common case.\n\n"
        f"**Largest single improvement**: {_best['display_name']} with "
        f"{_best['overlay']}, Sharpe {_best['baseline_sharpe']:.2f} to "
        f"{_best['managed_sharpe']:.2f} ({_best['sharpe_delta']:+.2f}), maximum "
        f"drawdown {_best['baseline_max_dd']:.1%} to "
        f"{_best['managed_max_dd']:.1%}, a {abs(_dd_change):.0f} percent "
        "reduction. That is one configuration selected as the top of a sweep on "
        "validation data, which is where a claim like it belongs on the evidence "
        "and not in a deployment decision."
    )
)

# %% [markdown]
# What holds regardless of which case studies are loaded:
#
# - **An overlay is not free.** It cuts the left tail and the right tail
#   together, and it trades more. A category whose median configuration loses
#   Sharpe is the normal finding, not a broken sweep.
# - **The highest-Sharpe configuration of a sweep is not the expected outcome of applying
#   the rule.** Both are reported above, and it is the population statistic that
#   should inform whether to use an overlay, because that is the distribution a
#   future choice is drawn from.
# - **Drawdown reduction and Sharpe are separate outcomes.** An overlay that cuts
#   drawdown while costing Sharpe is buying insurance, and whether that is worth
#   it depends on what the drawdown would have cost, which is a question about
#   the mandate rather than about the backtest.
# - **The default is no overlay.** The evidence needed to depart from it is a
#   configuration that improves both metrics and continues to do so out of
#   sample, and nothing here has been tested out of sample.
#
# ## Known Limitations
#
# - Only case studies with Ch19 overlay backtests appear; a case study that ran no
#   overlay sweep is absent rather than shown as zero. The loaded count is printed
#   above.
# - Every Sharpe here is a validation-fold number, and the overlay was chosen by
#   looking at it. The improvement of a best-of-sweep configuration is inflated by
#   the size of the sweep, and no deflation is applied.
# - The baseline is the allocation-stage strategy, with a signal-stage fallback
#   where no allocation baseline exists. A delta measured against a fallback
#   baseline is not comparable with one measured against an allocation baseline.
# - Maximum drawdown is a single realized path statistic with no interval. Two
#   configurations differing by a few percentage points of drawdown are not
#   distinguishable on this evidence.
#
# **Next**: [`08_recommendations`](08_recommendations.ipynb) for per-case-study
# recommendations and a practitioner decision matrix.

```