ドローダウン、ローリング指標、ベンチマークによるポートフォリオ評価

# 01_portfolio_metrics.py


```py
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# %% [markdown]
# # Portfolio Performance Analysis
#
# **Docker image**: `ml4t`
#
# `ml4t-diagnostic` computes the risk-return metrics, drawdown episodes and
# benchmark-relative statistics a portfolio report is built from, and replaces the
# unmaintained pyfolio. The strategy
# under analysis is the highest-validation-Sharpe ETF allocation backtest as recorded in
# `case_studies/etfs/run_log/registry.db` - resolved at runtime via
# `resolve_best_backtest_runs(...)` so the metrics always reflect the current
# best-Sharpe allocator on validation, not a baked-in hash. Daily portfolio
# returns are loaded from the case study's run-log, so every metric below traces
# to a registered, cost-aware engine backtest on real ETF data. The validation
# ranking chooses the artifact; the displayed return series is a strategy diagnostic,
# not a fresh unbiased model-selection estimate.
#
# **Learning Objectives**:
# - Compute summary statistics (Sharpe, Sortino, Calmar, max drawdown)
# - Analyze rolling performance metrics over configurable windows
# - Create drawdown analysis with recovery period tracking
# - Compare strategy performance against benchmarks (alpha, beta, IR)
# - Perform event analysis during market stress periods
#
# **Book Reference**: Chapter 17, Section 17.3 (Portfolio evaluation metrics)
#
# **Prerequisites**: The ETF case study (`case_studies/etfs/`) must have been run
# to produce `run_log/backtest/<hash>/daily_returns.parquet`.

# %%
"""Compute risk-return metrics, drawdowns, and stress-period analysis for ETF portfolios."""

import hashlib
import json

import numpy as np
import pandas as pd
import plotly.express as px
import plotly.graph_objects as go
import polars as pl

# ml4t-diagnostic imports
from ml4t.diagnostic.evaluation import PortfolioAnalysis
from ml4t.diagnostic.evaluation.portfolio_analysis import (
    alpha_beta,
    annual_return,
    annual_volatility,
    calmar_ratio,
    conditional_var,
    information_ratio,
    max_drawdown,
    omega_ratio,
    stability_of_timeseries,
    value_at_risk,
)
from ml4t.diagnostic.metrics import sharpe_ratio, sortino_ratio
from ml4t.diagnostic.visualization import (
    combine_figures_to_html,
    create_portfolio_dashboard,
)

from case_studies.utils.registry.queries import resolve_best_backtest_runs
from data import load_etfs
from utils.paths import get_case_study_dir, get_output_dir
from utils.reproducibility import set_global_seeds
from utils.style import COLORS, show_plotly_with_alt

# %% [markdown]
# Five settings decide what is analysed. `BACKTEST_HASH` left unset means the notebook picks the
# strategy from the case study's registry rather than naming one: it takes the allocation-stage
# backtest with the highest validation Sharpe on the 21-session forward-return target, so the
# analysis follows whatever that case study currently produces instead of a hash that goes stale
# the next time it is rebuilt. Setting it to a twelve-character hash pins one instead.
#
# `BENCHMARK_SYMBOL` is what alpha, beta and the capture ratios are measured against, and it has
# to be something the strategy could plausibly have been held instead of - SPY, a broad US equity
# fund, for a strategy allocating across US-listed ETFs. `ETF_LABEL` names the target the
# case study's models predicted, which is what makes the registry lookup unambiguous when a case
# study fits more than one.

# %% tags=["parameters"]
BACKTEST_HASH = None
BENCHMARK_SYMBOL = "SPY"
ETF_CASE_STUDY = "etfs"
ETF_LABEL = "fwd_ret_21d"
SEED = 42

# %%
set_global_seeds(SEED)

OUTPUT_DIR = get_output_dir(17, "portfolio_metrics")
OUTPUT_DIR.mkdir(parents=True, exist_ok=True)

# %% [markdown]
# ## 1. Strategy and Benchmark Returns
#
# We load the daily portfolio return series from a registered ETF case-study
# allocation backtest and use SPY as the benchmark. Rather than hard-code a
# `prediction_hash` or `backtest_hash`, the notebook queries
# `registry.db` via `resolve_best_backtest_runs()` and selects the allocation-stage
# run with the highest validation Sharpe. The strategy is therefore whatever
# the current case-study pipeline ranks first among ETF allocators on real OOS
# data; see `case_studies/etfs/run_log/backtest/<hash>/spec.json` for its full
# specification. The backtest artifact already includes its registered commission,
# slippage, and next-bar execution assumptions; this notebook does not subtract costs again.

# %%
# Record the immutable registry before selecting its registered artifact.
registry_path = get_case_study_dir(ETF_CASE_STUDY) / "run_log" / "registry.db"
registry_sha256 = hashlib.sha256(registry_path.read_bytes()).hexdigest()
print(f"ETF registry SHA-256: {registry_sha256}")

# Resolve the backtest hash from the registry unless one was injected by Papermill.
if BACKTEST_HASH is None:
    best_runs = resolve_best_backtest_runs(
        case_study=ETF_CASE_STUDY,
        label=ETF_LABEL,
        split="validation",
        stage="allocation",
        top_n=1,
    )
    if best_runs.is_empty():
        raise RuntimeError(
            f"No allocation-stage backtests found for {ETF_CASE_STUDY}/{ETF_LABEL} on "
            "validation. Run the ETF case study (Ch16/17 stages) before this notebook."
        )
    best_row = best_runs.row(0, named=True)
    BACKTEST_HASH = best_row["backtest_hash"]
    backtest_spec = json.loads(best_row["spec_json"])
    execution = backtest_spec["backtest_config"]["execution"]
    commission = backtest_spec["backtest_config"]["commission"]
    slippage = backtest_spec["backtest_config"]["slippage"]
    allocator = backtest_spec["strategy"]["allocation"]
    if execution["execution_mode"] != "next_bar":
        raise RuntimeError("Selected backtest is not a causal next-bar execution artifact")
    print(
        f"Resolved highest-validation-Sharpe allocation backtest: {BACKTEST_HASH} "
        f"(prediction={best_row['prediction_hash']}, Sharpe={best_row['sharpe']:.3f})"
    )
    print(
        f"Registered assumptions: allocator={allocator['method']}, top_k={allocator['top_k']}, "
        f"execution={execution['execution_mode']}/{execution['execution_price']}, "
        f"commission={commission['model']}, slippage={slippage['model']}"
    )
else:
    print(f"Using user-pinned BACKTEST_HASH={BACKTEST_HASH}")

# %%
# Load strategy daily returns from the registered backtest
backtest_dir = get_case_study_dir(ETF_CASE_STUDY) / "run_log" / "backtest" / BACKTEST_HASH
returns_path = backtest_dir / "daily_returns.parquet"
if not returns_path.exists():
    raise FileNotFoundError(
        f"Backtest daily returns not found at {returns_path}. Re-run the ETF case "
        "study (see case_studies/etfs/) before this notebook."
    )

strategy_df = (
    pl.read_parquet(returns_path)
    .rename({"daily_return": "strategy"})
    .with_columns(pl.col("timestamp").cast(pl.Date))
    .sort("timestamp")
)
# Trim the leading pre-trade window where daily_return == 0 (before the first rebalance)
first_active = strategy_df.filter(pl.col("strategy") != 0)["timestamp"].min()
if first_active is None:
    raise RuntimeError(f"Registered backtest {BACKTEST_HASH} contains no active strategy returns")
strategy_df = strategy_df.filter(pl.col("timestamp") >= first_active)

strategy_returns = strategy_df.to_pandas().set_index("timestamp")["strategy"].astype(float)
strategy_returns.index = pd.to_datetime(strategy_returns.index)

# %%
# Load SPY benchmark over the same window
spy_lo, spy_hi = strategy_returns.index.min(), strategy_returns.index.max()
spy_data = (
    load_etfs()
    .filter(
        (pl.col("symbol") == BENCHMARK_SYMBOL)
        & (pl.col("timestamp") >= pl.lit(str(spy_lo.date())).str.to_datetime())
        & (pl.col("timestamp") <= pl.lit(str(spy_hi.date())).str.to_datetime())
    )
    .sort("timestamp")
)
spy_prices = spy_data.select(["timestamp", "close"]).to_pandas().set_index("timestamp")["close"]
spy_returns = spy_prices.pct_change().dropna()

# Align both series on the intersection of dates
common = strategy_returns.index.intersection(spy_returns.index)
strategy_returns = strategy_returns.loc[common].rename("Strategy")
spy_returns = spy_returns.loc[common]

print(
    f"Strategy: {len(strategy_returns):,} days  ({strategy_returns.index[0].date()} → {strategy_returns.index[-1].date()})"
)
print(f"Benchmark: {BENCHMARK_SYMBOL}, {len(spy_returns):,} days")

# %% [markdown]
# ## 2. Setting up the analysis
#
# Every metric below is computed from the same three inputs - the strategy's daily returns, the
# benchmark's, and the dates they fall on - so they are bound once here rather than passed
# separately to each function. Two of the arguments are choices rather than data.
#
# The **risk-free rate** is what gets subtracted before a ratio is taken. It is set to zero here,
# which makes every Sharpe and Sortino below an excess-return-over-cash figure only to the extent
# that cash paid nothing; over a window covering 2022-23 it did not, so these ratios are
# marginally flattering. Applying the same rate to the benchmark does **not** leave the comparison
# unchanged. Sharpe is $(\mu - r_f)/\sigma$, so raising the rate by $\Delta$ costs each series
# $\Delta/\sigma$ - and that is *larger* for the series with the smaller volatility. The steadier
# of the two loses more of its ratio, which is what can reorder them: a low-volatility series
# ranked first at $r_f = 0$ can fall behind a more volatile one once a realistic cash rate is
# subtracted from both. Sortino is not invariant either, and it is not a plain division:
# `periodic_sortino_ratio` measures downside deviation on returns *after* subtracting the rate, so
# raising the rate shrinks the numerator and grows the denominator at the same time. It still only
# ever moves the ratio down - that holds for a strategy earning less than cash too, where the
# numerator is negative and both effects might have been expected to fight - but by an amount that
# is not $\Delta$ over anything, so read the two ratios' sensitivities as different in kind rather
# than assuming Sharpe's arithmetic carries over. What a common rate does leave alone is the
# information ratio, which is computed on
# active returns - the difference between the two series, from which any rate applied to both
# cancels before the ratio is taken.
#
# **Periods per year** is what an annualization multiplies by. Daily equity returns are quoted
# against 252 trading sessions. Switching to 365 does not move every annualized number by the same
# amount, because two different scalings are at work: volatility and the ratios built on it scale
# with the square root of the count, so they rise by the square root of the ratio of the two
# counts, roughly a fifth, while a compounded
# annual return raises one plus the periodic return to the count itself and therefore moves with
# the underlying growth rate rather than by a fixed factor.

# %%
# Create analysis object
RISK_FREE_RATE = 0.0  # annual; set it to a cash rate to measure excess return over cash
PERIODS_PER_YEAR = 252  # trading sessions in a year, the grid these returns are quoted on

analysis = PortfolioAnalysis(
    returns=strategy_returns.values,
    benchmark=spy_returns.values,
    dates=strategy_returns.index,
    risk_free=RISK_FREE_RATE,
    periods_per_year=PERIODS_PER_YEAR,
)

print(f"Daily returns analysed: {len(strategy_returns):,} sessions")
print(f"Measured against: {BENCHMARK_SYMBOL}")
print(f"Annual risk-free rate subtracted before every ratio: {RISK_FREE_RATE:.2%}")
print(f"Periods a year used to annualize: {PERIODS_PER_YEAR}")

# %% [markdown]
# ## 3. Summary Statistics
#
# The headline table is the smallest set that does not mislead: a return, the volatility it was
# earned at, two ratios of one against the other, the worst loss along the way, and three
# benchmark-relative figures. Reporting the return alone says nothing about what was risked to
# get it; reporting the Sharpe ratio alone hides how far the path fell before recovering.

# %%
# Compute all metrics.
metrics = analysis.compute_summary_stats()

headline_metrics = pd.DataFrame(
    {
        "Total Return": [f"{metrics.total_return:.2%}"],
        "Annual Return": [f"{metrics.annual_return:.2%}"],
        "Annual Volatility": [f"{metrics.annual_volatility:.2%}"],
        "Sharpe": [f"{metrics.sharpe_ratio:.3f}"],
        "Sortino": [f"{metrics.sortino_ratio:.3f}"],
        "Max Drawdown": [f"{metrics.max_drawdown:.2%}"],
        "Alpha": [f"{metrics.alpha:.2%}" if metrics.alpha is not None else "N/A"],
        "Beta": [f"{metrics.beta:.3f}" if metrics.beta is not None else "N/A"],
        "Information Ratio": [
            f"{metrics.information_ratio:.3f}" if metrics.information_ratio is not None else "N/A"
        ],
    },
    index=[BACKTEST_HASH],
)
headline_metrics

# %% [markdown]
# ### What each metric is sensitive to
#
# The nine above answer different questions, and each is blind to something the next one sees.
# What separates them is the denominator: what each one divides return by decides which kind of
# bad outcome it can register at all.
#
# | Metric | What it divides return by | What it cannot see |
# |--------|---------------------------|--------------------|
# | **Sharpe ratio** | total volatility | which side the volatility came from - an upside surprise penalises it as much as a loss |
# | **Sortino ratio** | volatility of losses only | how long a loss lasted, or how deep the path went |
# | **Calmar ratio** | the largest peak-to-trough loss | everything except that one episode |
# | **Omega ratio** | the probability-weighted mass of losses | nothing about ordering: the same returns shuffled give the same number |
# | **Tail ratio** | the size of the worst losses against the largest gains | the middle of the distribution, which is most of it |
# | **Alpha** | nothing - it is a return, net of what the benchmark's moves explain | whether the deviations that produced it were large or small |
# | **Information ratio** | the volatility of the deviations from the benchmark | the direction of the market the deviations were taken in |
# | **Up capture** | the benchmark's return in its rising periods | anything about falling ones |
# | **Down capture** | the benchmark's return in its falling periods | anything about rising ones |
#
# There is no threshold that makes one of these good. A Sharpe of 1 is unremarkable for a
# strategy trading at daily frequency and hard to reach for one holding positions for a quarter,
# and the same number computed over three years and over thirty means different things because
# the estimate's own standard error shrinks with the square root of the sample. What the table is
# for is reading them together: a strategy that scores well on all of them is doing something
# different from one that scores well on Sharpe alone.

# %% [markdown]
# ## 4. Rolling Metrics
#
# A single Sharpe ratio over the whole backtest is an average over every market the strategy
# traded through. Recomputing it inside a moving window shows whether that average describes a
# steady process or two different regimes with a crossover in the middle - which is the question
# an investor with a finite horizon is actually asking.
#
# Three windows are used, and the choice is a trade-off between resolution and noise. Twenty-one
# sessions is about a calendar month: it responds within weeks and its Sharpe estimate swings
# wildly, because a ratio estimated from twenty-one observations has a standard error close to
# the ratio itself. Sixty-three sessions is a quarter. Two hundred and fifty-two is a year, which
# smooths through most single episodes and therefore reacts to a regime change several months
# after it happens. Reading the three together is what separates a change in the strategy from a
# short run of luck.

# %%
# Compute rolling metrics
ROLLING_WINDOWS = [21, 63, 252]  # about a month, a quarter and a year of sessions

rolling = analysis.compute_rolling_metrics(
    windows=ROLLING_WINDOWS,
    metrics=["sharpe", "volatility", "returns"],
)

print(f"Sharpe, volatility and return recomputed over windows of {rolling.windows} sessions.")

# %%
# Plot rolling Sharpe ratio
fig = go.Figure()

for window in ROLLING_WINDOWS:
    if window in rolling.sharpe:
        sharpe_series = rolling.sharpe[window]
        fig.add_trace(
            go.Scatter(
                x=strategy_returns.index,
                y=sharpe_series.to_numpy(),
                name=f"{window}d Rolling",
                opacity=0.8,
            )
        )

fig.add_hline(y=0, line_dash="dash", line_color=COLORS["neutral"], opacity=0.3)
fig.add_hline(
    y=1, line_dash="dot", line_color=COLORS["amber"], opacity=0.3, annotation_text="Sharpe = 1"
)
fig.add_hline(
    y=2, line_dash="dot", line_color=COLORS["positive"], opacity=0.3, annotation_text="Sharpe = 2"
)

fig.update_layout(
    title="Rolling Sharpe ratio over 21-, 63- and 252-session windows",
    xaxis_title="Date",
    yaxis_title="Sharpe Ratio",
    height=450,
    legend=dict(yanchor="top", y=0.99, xanchor="right", x=0.99),
)
show_plotly_with_alt(
    fig,
    "Three lines of rolling Sharpe ratio against date, over 21, 63 and 252 sessions. The 21-session line swings across the whole vertical range while the 252-session line stays within a narrow band.",
)

# %%
# Plot rolling volatility (annualized)
fig = go.Figure()

for window in ROLLING_WINDOWS:
    if window in rolling.volatility:
        vol_series = rolling.volatility[window]
        fig.add_trace(
            go.Scatter(
                x=strategy_returns.index,
                y=vol_series.to_numpy() * 100,
                name=f"{window}d Rolling",
                opacity=0.8,
            )
        )

fig.update_layout(
    title="Annualized rolling volatility over 21-, 63- and 252-session windows",
    xaxis_title="Date",
    yaxis_title="Volatility (%)",
    height=400,
)
show_plotly_with_alt(
    fig,
    "Three lines of annualized rolling volatility against date, over 21, 63 and 252 sessions, with the shortest window spiking far above the other two during market stress.",
)

# %% [markdown]
# ## 5. Drawdown Analysis
#
# A **drawdown** is the loss from a peak in cumulative value to the lowest point before that peak
# is regained, measured as a fraction of the peak. It is the loss an investor who bought at the
# worst moment actually lived through, which is why it drives the decision to abandon a strategy
# in a way that volatility does not: volatility is symmetric and a drawdown is not.
#
# Three numbers describe one: how deep it went, how long it took to reach the bottom, and how
# long it took to climb back to the old peak. The last of the three is the one usually left out,
# and it is the one that decides whether a strategy is holdable - the same loss recovered in two
# months and recovered in four years is the same number and not the same experience. Both
# durations below are counted in trading sessions, not calendar days, because that is the grid
# the return series is quoted on.

# %%
# Compute drawdown analysis
drawdown = analysis.compute_drawdown_analysis(top_n=5)

deepest = pd.DataFrame(
    [
        {
            "Depth": f"{dd.depth * 100:.2f}%",
            "Peak": dd.peak_date.date(),
            "Valley": dd.valley_date.date(),
            "Recovered": dd.recovery_date.date() if dd.recovery_date else "not yet",
            "Peak to valley (sessions)": dd.duration_days,
            "Valley to peak (sessions)": dd.recovery_days
            if dd.recovery_days is not None
            else "still under",
        }
        for dd in drawdown.top_drawdowns
    ],
    index=pd.RangeIndex(1, len(drawdown.top_drawdowns) + 1, name="Rank"),
)
deepest


# %% [markdown]
# `compute_drawdown_analysis` above returns the episodes; the chart below needs the value on every
# date, which is the same quantity evaluated continuously. `DrawdownResult` already carries it as
# `underwater_curve`, and it is written out here as well because it is three lines - compound the
# returns, carry the running maximum, take the shortfall from it - and seeing the definition once
# is what makes every drawdown figure in this chapter readable. The benchmark needs the same
# curve, and no analysis object was built for it, so the function earns its place twice over.
#
# Computing it both ways is also the cheapest check available that the definition above is the
# one the library uses, so the two are asserted equal rather than assumed to agree.


# %%
def compute_drawdown_series(returns: pd.Series) -> pd.Series:
    """Compute drawdown time series."""
    cumulative = (1 + returns).cumprod()
    running_max = cumulative.expanding().max()
    drawdown = (cumulative - running_max) / running_max
    return drawdown


dd_series = compute_drawdown_series(strategy_returns)
dd_benchmark = compute_drawdown_series(spy_returns)

np.testing.assert_allclose(
    dd_series.to_numpy(), drawdown.underwater_curve.to_numpy(), rtol=0, atol=1e-12
)
print("Hand-derived underwater curve matches DrawdownResult.underwater_curve.")

# %%
# Plot underwater curve
fig = go.Figure()

fig.add_trace(
    go.Scatter(
        x=dd_series.index,
        y=dd_series * 100,
        name="Strategy",
        fill="tozeroy",
        line=dict(color=COLORS["negative"], width=1),
    )
)

fig.add_trace(
    go.Scatter(
        x=dd_benchmark.index,
        y=dd_benchmark * 100,
        name="Benchmark (SPY)",
        line=dict(color=COLORS["neutral"], width=1, dash="dash"),
    )
)

fig.update_layout(
    title="Underwater curves for the strategy and the SPY benchmark",
    xaxis_title="Date",
    yaxis_title="Drawdown (%)",
    height=400,
)
show_plotly_with_alt(
    fig,
    "Two underwater curves against date, the strategy filled to zero and the benchmark dashed, each showing the percentage below its own running peak.",
)

# %%
# Drawdown distribution
fig = px.histogram(
    dd_series * 100,
    nbins=50,
    title="Distribution of the strategy's daily drawdown",
    labels={"value": "Drawdown (%)", "count": "Frequency"},
)
fig.add_vline(
    x=metrics.max_drawdown * 100,
    line_dash="dash",
    line_color=COLORS["negative"],
    annotation_text=f"Max DD: {metrics.max_drawdown * 100:.1f}%",
)
fig.update_layout(height=350, showlegend=False)
show_plotly_with_alt(
    fig,
    "Histogram of the strategy's daily drawdown, concentrated near zero with a thin tail reaching the maximum drawdown marked by a vertical line.",
)

# %% [markdown]
# ## 6. Monthly and Annual Returns
#
# Daily returns compounded to month and year ends answer a question the aggregate figures do
# not: how much of the total came from how few periods. A strategy whose annual return is
# carried by two months is a different proposition from one earning the same amount steadily,
# and the two are indistinguishable in a Sharpe ratio computed over the whole sample.

# %%
# Compute monthly returns
monthly = analysis.compute_monthly_returns()
monthly_df = monthly.to_pandas()  # Convert Polars to pandas

print("Monthly Return Statistics:")
print(f"  Mean:   {monthly_df['monthly_return'].mean() * 100:.2f}%")
print(f"  Std:    {monthly_df['monthly_return'].std() * 100:.2f}%")
print(f"  Best:   {monthly_df['monthly_return'].max() * 100:.2f}%")
print(f"  Worst:  {monthly_df['monthly_return'].min() * 100:.2f}%")

# %% [markdown]
# The heatmap below is the same monthly series as a year-by-month grid. Read down a column to see
# whether a calendar month is systematically good or bad, and across a row to see how much of a
# year's return came from how few of its months.

# %%
monthly_df["year"] = monthly_df["year"].astype(int)
monthly_df["month"] = monthly_df["month"].astype(int)

heatmap_data = (
    monthly_df.pivot(index="year", columns="month", values="monthly_return") * 100
)  # Convert to percentage

# Month names
month_names = ["Jan", "Feb", "Mar", "Apr", "May", "Jun", "Jul", "Aug", "Sep", "Oct", "Nov", "Dec"]

fig = go.Figure(
    data=go.Heatmap(
        z=heatmap_data.values,
        x=month_names,
        y=heatmap_data.index,
        colorscale=[
            [0.0, COLORS["negative"]],
            [0.5, COLORS["silver"]],
            [1.0, COLORS["positive"]],
        ],
        zmid=0,
        text=np.round(heatmap_data.values, 1),
        texttemplate="%{text:.1f}%",
        textfont={"size": 10},
        colorbar=dict(title="Return %"),
    )
)

fig.update_layout(
    title="Monthly return by year and calendar month",
    xaxis_title="Month",
    yaxis_title="Year",
    height=400,
)
show_plotly_with_alt(
    fig,
    "Heatmap of monthly returns, years down the vertical axis and calendar months across, red for losses and green for gains, with each cell labelled.",
)

# %%
# Annual returns comparison
annual = analysis.compute_annual_returns()
annual_df = annual.to_pandas()  # Convert Polars to pandas

# Also compute benchmark annual
spy_annual = spy_returns.groupby(spy_returns.index.year).apply(lambda x: (1 + x).prod() - 1)

# Create comparison
fig = go.Figure()

fig.add_trace(
    go.Bar(
        x=annual_df["year"],
        y=annual_df["annual_return"] * 100,
        name="Strategy",
        marker_color=COLORS["blue"],
    )
)

fig.add_trace(
    go.Bar(
        x=spy_annual.index,
        y=spy_annual.values * 100,
        name="Benchmark (SPY)",
        marker_color=COLORS["silver_muted"],
    )
)

fig.update_layout(
    title="Annual return, strategy against the SPY benchmark",
    xaxis_title="Year",
    yaxis_title="Return (%)",
    barmode="group",
    height=400,
)
show_plotly_with_alt(
    fig,
    "Grouped bars of annual return for the strategy and for SPY, one pair per calendar year.",
)

# %% [markdown]
# ## 7. Benchmark-Relative Analysis
#
# Regressing the strategy's daily returns on the benchmark's splits them in two. **Beta** is the
# slope: how much the strategy moved, on average, for each unit the market moved. **Alpha** is
# the intercept, annualized: the part of the return the market's moves do not explain.
#
# Beta is what decides whether alpha is interesting. A strategy with a beta of one and no alpha
# has reproduced the index; one with a beta of one half and no alpha has reproduced half of it
# and kept half the capital idle. The two cutoffs below - a fifth below one and a fifth above -
# are conventional labels for "materially less exposed than the market" and "materially more",
# and there is nothing special about them beyond marking a fifth of the market's own movement
# in each direction.

# %%
alpha, beta = alpha_beta(
    strategy_returns.values, spy_returns.values, periods_per_year=PERIODS_PER_YEAR
)

print(f"Alpha: {alpha * 100:.2f}% (annualized)")
print(f"Beta:  {beta:.3f}")

# Interpretation
if beta < 0.8:
    beta_interp = "Defensive (low market exposure)"
elif beta > 1.2:
    beta_interp = "Aggressive (high market exposure)"
else:
    beta_interp = "Neutral (market-like exposure)"
print(f"       {beta_interp}")

# %% [markdown]
# **Tracking error** is the volatility of the difference between the two return series, and the
# **information ratio** divides the average of that difference by it. Together they ask whether
# the deviations from the benchmark were worth taking: a strategy can exceed its benchmark by a
# wide margin through deviations so volatile that the excess is indistinguishable from luck.
#
# Read the information ratio the way a t-statistic is read, because over $T$ years that is what
# it is: multiplied by $\sqrt{T}$ it gives roughly the t-statistic of the average active return.
# An information ratio of one half sustained over four years is therefore about one standard
# error from zero, which is why the number needs a horizon attached before it means anything.

# %%
active_returns = strategy_returns.values - spy_returns.values
tracking_error = active_returns.std(ddof=1) * np.sqrt(PERIODS_PER_YEAR)
ir = information_ratio(
    strategy_returns.values, spy_returns.values, periods_per_year=PERIODS_PER_YEAR
)

years = len(strategy_returns) / PERIODS_PER_YEAR
print(f"Tracking Error:    {tracking_error * 100:.2f}% (annualized)")
print(f"Information Ratio: {ir:.3f}  over {years:.1f} years")
print(f"Implied t-statistic on the average active return: {ir * np.sqrt(years):.2f}")

# %% [markdown]
# **Capture ratios** split the comparison by the direction the benchmark moved. Up capture is
# what the strategy returned across the periods the benchmark rose, as a fraction of what the
# benchmark returned over those same periods; down capture is the same across the periods it
# fell. A strategy capturing four fifths of the upside and half the downside is doing something
# a single alpha number cannot express.
#
# They are conventionally quoted on monthly periods rather than daily ones, so both series are
# compounded to month ends before the ratio is taken. The frequency is part of the definition:
# the same strategy scores differently at daily and monthly frequency, because a month in which
# the benchmark ends up while falling for three weeks counts as an up period at one and as a mix
# at the other.
#
# The ratio is taken between the two *average* returns over those periods. `ml4t.diagnostic`
# exposes an `up_down_capture` that instead divides the two compounded wealth factors, and for
# down periods that inverts the reading: a strategy losing half as much as the benchmark in every
# month the benchmark falls has captured half the downside, and the compounded form reports more
# than the whole of it, because both products are below one and the larger numerator makes the
# ratio exceed one. A down capture above one is supposed to mean the strategy fell harder than
# the market. Take the ratio of the means.

# %%
strat_monthly = strategy_returns.resample("ME").apply(lambda x: (1 + x).prod() - 1)
bench_monthly = spy_returns.resample("ME").apply(lambda x: (1 + x).prod() - 1)
up_months, down_months = bench_monthly > 0, bench_monthly < 0
up_capture = strat_monthly[up_months].mean() / bench_monthly[up_months].mean()
down_capture = strat_monthly[down_months].mean() / bench_monthly[down_months].mean()

print("Capture ratios, on monthly periods:")
print(f"  Up Capture:   {up_capture * 100:.1f}%")
print(f"  Down Capture: {down_capture * 100:.1f}%")

# The gap between the two is the asymmetry: positive means the strategy kept more of the
# benchmark's up months than of its down months.
capture_spread = up_capture - down_capture
print(f"  Spread (up - down): {capture_spread * 100:.1f}pp")

# %%
# Rolling Beta
rolling_beta = pd.Series(index=strategy_returns.index, dtype=float)

window = PERIODS_PER_YEAR  # one year of sessions
for i in range(window, len(strategy_returns)):
    strat_window = strategy_returns.iloc[i - window : i].values
    bench_window = spy_returns.iloc[i - window : i].values
    _, beta_i = alpha_beta(strat_window, bench_window)
    rolling_beta.iloc[i] = beta_i

fig = go.Figure()

fig.add_trace(
    go.Scatter(
        x=rolling_beta.index,
        y=rolling_beta,
        name="Rolling 1Y Beta",
        line=dict(color=COLORS["copper"], width=2),
    )
)

fig.add_hline(
    y=1, line_dash="dash", line_color=COLORS["neutral"], annotation_text="Beta = 1 (Market)"
)
fig.add_hline(y=0, line_dash="dot", line_color=COLORS["neutral"], opacity=0.3)

fig.update_layout(
    title="Rolling one-year beta against the SPY benchmark",
    xaxis_title="Date",
    yaxis_title="Beta",
    height=400,
)
show_plotly_with_alt(
    fig,
    "Rolling one-year beta against SPY plotted against date, with reference lines at one and zero.",
)

# %% [markdown]
# ## 8. Risk Metrics (VaR, CVaR)
#
# **Value at Risk** at a confidence level is the loss that a stated fraction of periods exceeded.
# Read at one day in twenty, it is the daily loss that one trading day in twenty was worse than;
# read at one day in a hundred, it is the loss the worst day in a hundred exceeded. Both levels
# below are empirical quantiles of the return series itself: the returns are sorted and the
# quantile is taken, with no distribution assumed.

# %%
# Value at Risk
var_95 = value_at_risk(strategy_returns.values, confidence=0.95)
var_99 = value_at_risk(strategy_returns.values, confidence=0.99)

print("Value at Risk, from the sample's own return distribution:")
print(f"  95% VaR: {var_95 * 100:.2f}% (daily)")
print(f"  99% VaR: {var_99 * 100:.2f}% (daily)")
breaches = int((strategy_returns.values < var_95).sum())
print(f"  Days worse than the VaR: {breaches:,} of {len(strategy_returns):,}")

# %% [markdown]
# VaR says where the tail begins and nothing about what is inside it: a strategy whose worst day
# loses a few percent and one whose worst day loses ten times that can share the same VaR.
# **Conditional VaR**, also called expected shortfall, is the average of the losses that did
# exceed the threshold, so it is the number that separates them. It is the reason a risk report
# quotes both.
#
# Both are read off this sample's own history. Neither is a forecast, and the one-day-in-a-hundred
# quantile of a few thousand observations rests on a few dozen of them.

# %%
cvar_95 = conditional_var(strategy_returns.values, confidence=0.95)
cvar_99 = conditional_var(strategy_returns.values, confidence=0.99)

print("Conditional VaR, the average loss on the days that breached VaR:")
print(f"  95% CVaR: {cvar_95 * 100:.2f}% (daily)")
print(f"  99% CVaR: {cvar_99 * 100:.2f}% (daily)")

# %%
# Visualize return distribution with VaR
fig = go.Figure()

# Histogram of returns
fig.add_trace(go.Histogram(x=strategy_returns * 100, nbinsx=100, name="Daily Returns", opacity=0.7))

# VaR lines - stagger annotation y-positions to prevent overlap at the top
fig.add_vline(
    x=var_95 * 100,
    line_dash="dash",
    line_color=COLORS["amber"],
    annotation_text="95% VaR",
    annotation_position="top right",
)
fig.add_vline(
    x=var_99 * 100,
    line_dash="dash",
    line_color=COLORS["negative"],
    annotation_text="99% VaR",
    annotation_position="bottom right",
)
fig.add_vline(
    x=cvar_95 * 100,
    line_dash="dot",
    line_color=COLORS["copper"],
    annotation_text="95% CVaR",
    annotation_position="top left",
)

fig.update_layout(
    title="Daily return distribution with value at risk and conditional VaR",
    xaxis_title="Daily Return (%)",
    yaxis_title="Frequency",
    height=400,
    showlegend=False,
)
show_plotly_with_alt(
    fig,
    "Histogram of daily strategy returns with vertical lines marking the value at risk at two "
    "confidence levels and the conditional value at risk further into the left tail.",
)

# %% [markdown]
# ## 9. Event Analysis
#
# Aggregate statistics average over every market the strategy traded through, so they say
# nothing about the episodes an investor remembers. Cumulating the strategy and the benchmark
# over five named windows - two crashes, two recoveries and one credit event - shows whether the
# defensive profile the beta and capture ratios suggest actually held when it was tested.
#
# The five windows are chosen after the fact, which is what makes this a description rather than
# a test. Each is bounded by the dates the episode is conventionally dated to; the strategy's
# behaviour outside them is what every other section measures.

# %%
# Define market stress periods
STRESS_PERIODS = {
    "COVID Crash (2020)": ("2020-02-19", "2020-03-23"),
    "COVID Recovery": ("2020-03-23", "2020-08-31"),
    "2022 Bear Market": ("2022-01-03", "2022-10-12"),
    "2023 Banking Crisis": ("2023-03-01", "2023-03-31"),
    "2023 Rally": ("2023-10-27", "2023-12-29"),
}


def analyze_period(returns: pd.Series, benchmark: pd.Series, start: str, end: str):
    """Analyze returns over a specific period."""
    period_ret = returns.loc[start:end]
    period_bench = benchmark.loc[start:end]

    if len(period_ret) == 0:
        return None

    cum_ret = (1 + period_ret).prod() - 1
    cum_bench = (1 + period_bench).prod() - 1
    excess = cum_ret - cum_bench

    return {
        "strategy_return": cum_ret,
        "benchmark_return": cum_bench,
        "excess_return": excess,
        "days": len(period_ret),
    }


# %%
period_results = []
for name, (start, end) in STRESS_PERIODS.items():
    result = analyze_period(strategy_returns, spy_returns, start, end)
    if result:
        result["period"] = name
        period_results.append(result)
        print(f"\n{name}:")
        print(f"  Strategy:  {result['strategy_return'] * 100:+.2f}%")
        print(f"  Benchmark: {result['benchmark_return'] * 100:+.2f}%")
        print(f"  Excess:    {result['excess_return'] * 100:+.2f}%")

# %%
# Visualize stress period performance
if period_results:
    period_df = pd.DataFrame(period_results)

    fig = go.Figure()

    fig.add_trace(
        go.Bar(
            x=period_df["period"],
            y=period_df["strategy_return"] * 100,
            name="Strategy",
            marker_color=COLORS["blue"],
        )
    )

    fig.add_trace(
        go.Bar(
            x=period_df["period"],
            y=period_df["benchmark_return"] * 100,
            name="Benchmark",
            marker_color=COLORS["silver_muted"],
        )
    )

    fig.update_layout(
        title="Cumulative return over five stress and recovery windows",
        yaxis_title="Return (%)",
        barmode="group",
        height=450,
    )
    show_plotly_with_alt(
        fig,
        "Grouped bars of cumulative return for the strategy and the benchmark over five named stress and recovery windows.",
    )

# %% [markdown]
# ## 10. Stability Analysis
#
# **Stability** here is the R-squared of a straight line fitted to the cumulative value curve
# against time. A value near one means the curve looks like a line, which is what a strategy
# compounding at a steady rate produces; a lower value means the same total return arrived in
# bursts. It is a description of the path, not of the return: a strategy that doubles in one
# month and flatlines for four years scores badly and still doubled.
#
# A high R-squared does not mean the path was comfortable. A straight line fitted to eight years
# of compounding absorbs a drawdown lasting several months without much loss of fit, so read
# this number against the drawdown table in section 5 rather than instead of it.

# %%
# Stability of returns (R² of cumulative returns vs time)
stability = stability_of_timeseries(strategy_returns.values)
print(f"Stability (R²): {stability:.3f}")

# %%
# Cumulative returns with trend line
cumulative = (1 + strategy_returns).cumprod()

# Fit trend line
x = np.arange(len(cumulative))
coeffs = np.polyfit(x, cumulative, 1)
trend = np.polyval(coeffs, x)

fig = go.Figure()

fig.add_trace(
    go.Scatter(
        x=cumulative.index,
        y=cumulative,
        name="Cumulative Return",
        line=dict(color=COLORS["blue"], width=2),
    )
)

fig.add_trace(
    go.Scatter(
        x=cumulative.index,
        y=trend,
        name=f"Trend (R² = {stability:.3f})",
        line=dict(color=COLORS["negative"], dash="dash"),
    )
)

fig.update_layout(
    title="Cumulative return with a fitted linear trend",
    xaxis_title="Date",
    yaxis_title="Cumulative Return",
    height=400,
)
show_plotly_with_alt(
    fig,
    "Cumulative return against date with a fitted straight trend line overlaid, the two diverging where the path deviates from steady compounding.",
)

# %% [markdown]
# ## 11. Full Performance Report
#
# The sections above computed each family of statistics next to the reasoning that motivates it.
# Collecting them into one table grouped by what they measure - return, risk, the ratio of one
# to the other, and the comparison against the benchmark - is the form a report takes when it
# goes to someone who did not follow the derivation.

# %%
# Build the summary table
summary_data = {"Category": [], "Metric": [], "Value": []}


def add_metric(category: str, name: str, value: str) -> None:
    summary_data["Category"].append(category)
    summary_data["Metric"].append(name)
    summary_data["Value"].append(value)


# %%
returns_metrics = [
    ("Total Return", f"{metrics.total_return * 100:.2f}%"),
    ("Annual Return (CAGR)", f"{metrics.annual_return * 100:.2f}%"),
    ("Best Month", f"{monthly_df['monthly_return'].max() * 100:.2f}%"),
    ("Worst Month", f"{monthly_df['monthly_return'].min() * 100:.2f}%"),
]
risk_metrics = [
    ("Annual Volatility", f"{metrics.annual_volatility * 100:.2f}%"),
    ("Max Drawdown", f"{metrics.max_drawdown * 100:.2f}%"),
    ("95% VaR (Daily)", f"{var_95 * 100:.2f}%"),
    ("95% CVaR (Daily)", f"{cvar_95 * 100:.2f}%"),
]
risk_adjusted_metrics = [
    ("Sharpe Ratio", f"{metrics.sharpe_ratio:.3f}"),
    ("Sortino Ratio", f"{metrics.sortino_ratio:.3f}"),
    ("Calmar Ratio", f"{metrics.calmar_ratio:.3f}"),
    ("Omega Ratio", f"{metrics.omega_ratio:.3f}"),
]
benchmark_metrics = []
if metrics.alpha is not None:
    benchmark_metrics = [
        ("Alpha", f"{metrics.alpha * 100:.2f}%"),
        ("Beta", f"{metrics.beta:.3f}"),
        ("Tracking Error", f"{tracking_error * 100:.2f}%"),
        ("Information Ratio", f"{metrics.information_ratio:.3f}"),
        ("Up Capture (monthly)", f"{up_capture * 100:.1f}%"),
        ("Down Capture (monthly)", f"{down_capture * 100:.1f}%"),
    ]
other_metrics = [
    ("Stability (R²)", f"{stability:.3f}"),
    ("Win Rate", f"{metrics.win_rate * 100:.1f}%"),
    ("Profit Factor", f"{metrics.profit_factor:.2f}"),
]

# %%
for name, value in returns_metrics:
    add_metric("Returns", name, value)
for name, value in risk_metrics:
    add_metric("Risk", name, value)
for name, value in risk_adjusted_metrics:
    add_metric("Risk-Adjusted", name, value)
for name, value in benchmark_metrics:
    add_metric("Benchmark", name, value)
for name, value in other_metrics:
    add_metric("Other", name, value)

# %%
summary_df = pd.DataFrame(summary_data)
summary_df

# %% [markdown]
# The four groups answer four questions and none of them answers another's. Read alpha together
# with beta, tracking error and the information ratio: a positive alpha earned through deviations
# whose own volatility swamps it has not established that the active positions were worth taking.
# The two capture ratios then say where the residual profile came from - holding on in rising
# markets, or losing less in falling ones.

# %% [markdown]
# ## 12. The same metrics, one at a time
#
# The class above computes everything from one binding. Each metric is also available as a plain
# function over an array of returns, which is what to reach for when checking a single number
# somewhere else without setting up an analysis object. The values below are the same ones the
# table carries, computed the other way.

# %%
# Example: Computing metrics without PortfolioAnalysis class
returns_arr = strategy_returns.values
benchmark_arr = spy_returns.values

print("Standalone Metric Functions:")
print(f"  sharpe_ratio():       {sharpe_ratio(returns_arr):.3f}")
print(f"  sortino_ratio():      {sortino_ratio(returns_arr):.3f}")
print(f"  calmar_ratio():       {calmar_ratio(returns_arr):.3f}")
print(f"  omega_ratio():        {omega_ratio(returns_arr):.3f}")
print(f"  max_drawdown():       {max_drawdown(returns_arr) * 100:.2f}%")
print(f"  annual_return():      {annual_return(returns_arr) * 100:.2f}%")
print(f"  annual_volatility():  {annual_volatility(returns_arr) * 100:.2f}%")

# %% [markdown]
# ## 13. Portfolio Dashboard (Pyfolio Replacement)
#
# The library provides `create_portfolio_dashboard()` which generates a complete
# tear sheet in a single call. This is the production replacement for pyfolio's
# `create_full_tear_sheet()`.

# %%
tear_sheet = create_portfolio_dashboard(
    analysis,
    theme="default",
    include_benchmark=True,
    height_per_row=350,
)

print("Tear sheet generated.")
print(f"  Figures included: {list(tear_sheet.figures.keys())}")

# %% [markdown]
# ### Reading what the dashboard computed
#
# The dashboard carries its own metrics object. The figures below come from it, so printing its
# numbers first is what says the two are describing the same series - they are the values already
# computed in section 3, reached by a different route.

# %%
ts = tear_sheet.metrics
print(
    f"Sharpe: {ts.sharpe_ratio:.3f}  |  Sortino: {ts.sortino_ratio:.3f}  |  Calmar: {ts.calmar_ratio:.3f}"
)
print(f"Annual Return: {ts.annual_return * 100:.2f}%  |  Max DD: {ts.max_drawdown * 100:.2f}%")
print(f"Alpha: {ts.alpha * 100:.2f}%  |  Beta: {ts.beta:.3f}  |  IR: {ts.information_ratio:.3f}")

# %% [markdown]
# `tear_sheet.show()` renders every figure in sequence. Pulling three out by name instead is what
# to do when a report needs a selection, and it is also where each figure's title and its alt
# text for a screen reader are set, neither of which the dashboard can know from the returns
# alone.

# %%
dashboard_titles = {
    "Cumulative Returns": "Cumulative return, strategy against the SPY benchmark",
    "Drawdown": "Strategy underwater curve",
    "Monthly Returns Heatmap": "Monthly return by year and calendar month",
}
dashboard_alt = {
    "Cumulative Returns": (
        "Two cumulative return paths against date, the strategy and the SPY benchmark, "
        "both compounding upward and separating over the sample."
    ),
    "Drawdown": (
        "The strategy's underwater curve against date, plotted as the percentage below the "
        "running peak, with its deepest fall during the 2020 crash."
    ),
    "Monthly Returns Heatmap": (
        "Heatmap of monthly return, years down the vertical axis and calendar months "
        "across, coloured red for losses and green for gains."
    ),
}
for name, title in dashboard_titles.items():
    fig = tear_sheet.figures[name]
    fig.update_layout(
        title=title,
        paper_bgcolor=COLORS["bg_light"],
        plot_bgcolor=COLORS["bg_light"],
    )
    show_plotly_with_alt(fig, dashboard_alt[name])

# %% [markdown]
# ### Saving the dashboard as a file
#
# `save_html` writes every figure and its metrics into one file that opens in a browser with no
# Python installed. `include_plotlyjs="cdn"` keeps the file small by loading the plotting library
# from the network when it opens, which is the right trade unless the file has to work offline.

# %%
# Export to HTML (self-contained, shareable file)
html_path = OUTPUT_DIR / "portfolio_dashboard.html"
tear_sheet.save_html(html_path, include_plotlyjs="cdn")
print(f"Dashboard saved to: {html_path.name}")

# %% [markdown]
# ## 14. Assembling a custom report
#
# The dashboard above is fixed: it decides which figures appear and in what order. A report going
# to someone who is not reading the notebook usually needs a different selection and a sentence
# beside each figure saying what to look at. `combine_figures_to_html` takes the figures already
# built here plus that narrative and writes one self-contained file.

# %%
# Collect figures we created earlier for a custom report
custom_figures = []
custom_sections = []

# %% [markdown]
# The first custom section focuses on cumulative growth against the benchmark.

# %%
# Add cumulative returns figure
fig_cum = go.Figure()
fig_cum.add_trace(
    go.Scatter(
        x=strategy_returns.index,
        y=(1 + strategy_returns).cumprod(),
        name="Strategy",
        line=dict(color=COLORS["blue"], width=2),
    )
)
fig_cum.add_trace(
    go.Scatter(
        x=spy_returns.index,
        y=(1 + spy_returns).cumprod(),
        name="Benchmark (SPY)",
        line=dict(color=COLORS["neutral"], width=1, dash="dash"),
    )
)
fig_cum.update_layout(
    title="Cumulative return, strategy against the SPY benchmark",
    xaxis_title="Date",
    yaxis_title="Cumulative Return",
    height=400,
)
custom_figures.append(fig_cum)
custom_sections.append(
    {
        "title": "Strategy Performance",
        "text": f"The strategy achieved a total return of {metrics.total_return * 100:.1f}% "
        f"vs benchmark return of {((1 + spy_returns).prod() - 1) * 100:.1f}%. "
        f"Sharpe ratio: {metrics.sharpe_ratio:.2f}.",
        "figure_index": 0,
    }
)

# %% [markdown]
# The second section summarizes drawdown severity and recovery behavior.

# %%
# Add drawdown figure
fig_dd = go.Figure()
fig_dd.add_trace(
    go.Scatter(
        x=dd_series.index,
        y=dd_series * 100,
        fill="tozeroy",
        line=dict(color=COLORS["negative"], width=1),
        name="Drawdown",
    )
)
fig_dd.update_layout(
    title="Strategy underwater curve",
    xaxis_title="Date",
    yaxis_title="Drawdown (%)",
    height=350,
)
custom_figures.append(fig_dd)
custom_sections.append(
    {
        "title": "Risk Analysis",
        "text": f"Maximum drawdown reached {metrics.max_drawdown * 100:.1f}%. "
        f"The strategy has a Calmar ratio of {metrics.calmar_ratio:.2f}.",
        "figure_index": 1,
    }
)

# %% [markdown]
# Export the assembled figures and narrative as a standalone HTML report.

# %%
# Generate the custom report
report_path = OUTPUT_DIR / "custom_strategy_report.html"
combine_figures_to_html(
    figures=custom_figures,
    title="Strategy Performance Report",
    sections=custom_sections,
    output_file=report_path,
    theme="default",
    include_toc=True,
)
print(f"Custom report saved to: {report_path.name}")

# %% [markdown] tags=["results"]
# ### What this run produced
#
# The strategy analysed is whichever allocation backtest the ETF case study currently ranks first
# on validation Sharpe; its hash, allocator, cost model and execution convention are printed in
# section 1, and every number in this notebook is computed from that one artifact. The summary table in
# section 11 is the full set. Three things in it are worth reading together rather than
# separately: the annualized return against the annual volatility it was earned at, the maximum
# drawdown against the Calmar ratio that divides the return by it, and the alpha against the beta
# and tracking error that say how much market exposure and how much active deviation produced it.
#
# The rolling panels in section 4 are what the aggregate figures hide. Where the 252-session
# Sharpe and the 21-session Sharpe disagree for an extended stretch, the aggregate is an average
# over two different regimes rather than a description of one process.

# %% [markdown]
# ## Key Takeaways
#
# 1. **One ratio cannot describe a return path.** Sharpe divides return by total volatility,
#    so it registers neither how far the path fell nor how long it stayed down, and it treats
#    an upside surprise as a cost. Report it alongside the maximum drawdown and the Sortino
#    ratio, which are blind to different things.
# 2. **Rolling metrics expose regime dependence.** Aggregate Sharpe can be positive
#    while rolling windows show extended negative periods - a critical warning for
#    investors with finite horizons.
# 3. **Benchmark comparison separates alpha from beta.** Alpha, beta, tracking error,
#    and information ratio must be read together. Up/down capture reveals asymmetric
#    exposure that aggregate alpha misses.
# 4. **Stress-period analysis tests robustness.** The maximum drawdown and event panels
#    turn an aggregate performance score into the timing and recovery questions that
#    matter to investors.
#
# ### Known limitations
#
# - **One strategy, one benchmark, one sample.** Every number here describes a single backtest
#   over a single history. None of them carries a confidence interval, and the Sharpe ratio in
#   particular is estimated with a standard error close to 1/sqrt(years) - so two strategies
#   differing by a few tenths over a few years have not been separated by this evidence.
# - **The strategy was chosen by validation Sharpe.** Ranking a set of candidates and then
#   reporting the statistics of the one that ranked first overstates them, because the ranking itself used the same
#   kind of noise the statistic measures. This notebook is a diagnostic of a chosen artifact, not
#   an unbiased estimate of what that artifact would earn next.
# - **Stress periods are named by hand.** The five windows in section 9 were chosen because they
#   are known episodes, which means the selection is retrospective. They show how the strategy
#   behaved in them; they do not establish how it behaves in stress generally.
# - **VaR and CVaR are empirical quantiles.** Both are read off the sample's own distribution, so
#   the one-day-in-a-hundred figures rest on a few dozen observations and neither extrapolates
#   past the worst day in the history.
# - **The risk-free rate is zero.** Ratios that subtract it are therefore excess-over-nothing
#   rather than excess-over-cash, which flatters them over any period when cash paid a return.
#
# **Next**: These metrics are used throughout Ch17-19 to evaluate allocation methods,
# transaction cost impact, and risk controls.
#
# **Book**: Section 17.3 discusses the full evaluation framework.

```