Como modelos de custos de trading afetam backtests de momentum

# Commission & Slippage Model Comparison


# Commission & Slippage Model Comparison

**Docker image**: `ml4t`

This notebook compares every commission and slippage model in
`ml4t.backtest.models`. It separates equity shares from futures contracts,
defines illustrative asset-class cost stacks, and measures how commission
choice interacts with the cadence of a fixed momentum rule.

**Learning Objectives**
- Instantiate and compare the complete commission and slippage model taxonomy
- Build asset-class-specific cost configurations (equities, ETFs, futures, crypto)
- Quantify how much model choice affects net Sharpe for different trading styles
- Understand the frequency-cost interaction (daily vs weekly vs monthly)

**Book Reference:** Chapter 18, Section 18.2 (A Cost Taxonomy for Practitioners)

**Prerequisites:** Read [`06_ml4t_execution_demo`](06_ml4t_execution_demo.ipynb)
for the impact API and
[`10_gross_vs_net_performance`](10_gross_vs_net_performance.ipynb) for the
portfolio-level net-performance waterfall.

```python
"""Commission & Slippage Model Comparison."""

import plotly.graph_objects as go
import polars as pl
from IPython.display import Markdown, display
from ml4t.backtest import (
    BacktestConfig,
    DataFeed,
    Engine,
    ExecutionMode,
    Strategy,
)
from ml4t.backtest.execution.rebalancer import RebalanceConfig, TargetWeightExecutor
from ml4t.backtest.models import (
    CombinedCommission,
    FixedSlippage,
    FuturesCommission,
    FuturesSlippage,
    NoCommission,
    NoSlippage,
    PercentageCommission,
    PercentageSlippage,
    PerShareCommission,
    SpreadSlippage,
    TieredCommission,
    VolumeShareSlippage,
)
from plotly.subplots import make_subplots

# Side-effect import: configures the default Plotly renderer to embed PNG
# alongside the interactive widget so figures render on GitHub when the
# rendered .ipynb is browsed without a live Plotly runtime.
import utils  # noqa: F401
from data import load_etfs
from utils.style import COLORS, ml4t_palette, show_plotly_with_alt
```

```python
N_BARS = 1260  # 5 years daily
INITIAL_CASH = 100_000
ETF_SYMBOLS = ["SPY", "QQQ", "IWM", "XLF", "EEM"]  # liquid, distinct sectors/regions
START_DATE = "2019-01-02"
END_DATE = "2023-12-29"
MOMENTUM_LOOKBACK = 63  # trading days (~quarter) for the rebalance signal
```

## 1. Commission Model Taxonomy

Equity-style commission models accept share quantity and price. The futures
model instead accepts contracts, price, and a contract multiplier. Keeping
those unit systems separate prevents a contract count from being mislabeled
as shares.

```python
equity_commission_models = {
    "NoCommission": NoCommission(),
    "Percentage (10bp)": PercentageCommission(rate=0.001),
    "PerShare ($0.005)": PerShareCommission(per_share=0.005, minimum=1.0),
    "Combined (5bp + $1)": CombinedCommission(percentage=0.0005, fixed=1.0),
    "Tiered": TieredCommission(tiers=[(10_000, 0.001), (50_000, 0.0008), (float("inf"), 0.0005)]),
}
futures_commission_model = FuturesCommission(per_block=2.25)
commission_models = {**equity_commission_models, "FuturesCommission": futures_commission_model}

SHARE_PRICE = 100.0
SHARE_QUANTITIES = [10, 50, 100, 500, 1000, 5000, 10000]

commission_rows = []
for name, model in equity_commission_models.items():
    for quantity in SHARE_QUANTITIES:
        notional = quantity * SHARE_PRICE
        cost = model.calculate("TEST", quantity, SHARE_PRICE)
        commission_rows.append(
            {
                "model": name,
                "quantity": quantity,
                "notional": notional,
                "cost": cost,
                "cost_bps": cost / notional * 10_000,
            }
        )
commission_df = pl.DataFrame(commission_rows)
```

### Compare Equity-Style Cost Profiles

```python
profile_colors = ml4t_palette(5, categorical=True)
profile_dashes = ["dot", "solid", "dash", "dashdot", "longdash"]
fig = go.Figure()
for (name, _model), color, dash in zip(
    equity_commission_models.items(), profile_colors, profile_dashes, strict=True
):
    subset = commission_df.filter(pl.col("model") == name)
    fig.add_trace(
        go.Scatter(
            x=subset["notional"].to_list(),
            y=subset["cost_bps"].to_list(),
            name=name,
            mode="lines+markers",
            line=dict(color=color, dash=dash),
            customdata=subset["quantity"].to_list(),
            hovertemplate="Notional: $%{x:,.0f}<br>Quantity: %{customdata:,.0f} shares"
            "<br>Commission: %{y:.2f} bps<extra>%{fullData.name}</extra>",
        )
    )
fig.update_layout(
    title="One-way commission against trade notional, by commission model",
    xaxis_title=f"Trade notional at ${SHARE_PRICE:,.0f} per share (log scale)",
    yaxis_title="One-way commission (bps of notional)",
    xaxis_type="log",
    height=430,
)
show_plotly_with_alt(
    fig,
    "Five lines of one-way commission in basis points against trade notional on a logarithmic "
    "horizontal axis, one per commission model. The no-commission line runs flat along the "
    "bottom and the percentage line runs flat across the whole range. The three models carrying "
    "a minimum or a per-share element each descend to a floor, the per-share one falling from "
    "the smallest ticket and the tiered one holding level before it steps down, so the models "
    "are furthest apart on the small tickets at the left and converge towards the right.",
)
```

**Finding**: Percentage fees stay constant in basis-point terms. Minimum and
fixed fees consume a larger share of small tickets, while per-share costs and
tier thresholds create different profiles as notional grows.

### Normalize a Futures Contract Example

```python
FUTURES_QUANTITY = 10
FUTURES_PRICE = 4_000.0
FUTURES_MULTIPLIER = 50.0
futures_notional = FUTURES_QUANTITY * FUTURES_PRICE * FUTURES_MULTIPLIER
futures_commission = futures_commission_model.calculate(
    "ES", FUTURES_QUANTITY, FUTURES_PRICE, multiplier=FUTURES_MULTIPLIER
)
futures_commission_bps = futures_commission / futures_notional * 10_000
display(
    Markdown(
        f"**Futures example**: {FUTURES_QUANTITY} contracts at "
        f"${FUTURES_PRICE:,.0f} with a ${FUTURES_MULTIPLIER:,.0f} multiplier "
        f"represent **${futures_notional:,.0f}** of notional. The per-contract "
        f"schedule charges **${futures_commission:,.2f}**, or "
        f"**{futures_commission_bps:.2f} bps one way**."
    )
)
```

## 2. Slippage Model Taxonomy

Most slippage models return a per-unit price adjustment. `SpreadSlippage`
treats its input as a full quoted spread by default and charges the
half-spread per side. `FuturesSlippage` returns total dollars, so it remains
separate from the participation profile.

```python
per_unit_slippage_models = {
    "NoSlippage": NoSlippage(),
    "Fixed ($0.01)": FixedSlippage(amount=0.01),
    "Spread ($0.04 full)": SpreadSlippage(spread=0.04),
    "Percentage (10bp)": PercentageSlippage(rate=0.001),
    "VolumeShare (0.1)": VolumeShareSlippage(impact_factor=0.1),
}
futures_slippage_model = FuturesSlippage(slippage_points=0.25)
slippage_models = {**per_unit_slippage_models, "FuturesSlippage": futures_slippage_model}

SLIPPAGE_PRICE = 100.0
BAR_VOLUME = 1_000_000
PARTICIPATION_RATES = [0.001, 0.005, 0.01, 0.02, 0.05, 0.10, 0.20]

slippage_rows = []
for name, model in per_unit_slippage_models.items():
    for participation in PARTICIPATION_RATES:
        quantity = BAR_VOLUME * participation
        adjustment = model.calculate("TEST", quantity, SLIPPAGE_PRICE, BAR_VOLUME)
        slippage_rows.append(
            {
                "model": name,
                "participation": participation,
                "adjustment": adjustment,
                "cost_bps": adjustment / SLIPPAGE_PRICE * 10_000,
            }
        )
slippage_df = pl.DataFrame(slippage_rows)
```

### Compare Per-Unit Slippage Profiles

```python
slippage_styles = {
    name: (color, dash)
    for (name, _model), color, dash in zip(
        per_unit_slippage_models.items(), profile_colors, profile_dashes, strict=True
    )
}
fig = make_subplots(
    rows=1,
    cols=2,
    subplot_titles=["Participation-invariant assumptions", "Volume-share response"],
    horizontal_spacing=0.15,
)
for name in ["NoSlippage", "Fixed ($0.01)", "Spread ($0.04 full)", "Percentage (10bp)"]:
    subset = slippage_df.filter(pl.col("model") == name)
    color, dash = slippage_styles[name]
    fig.add_trace(
        go.Scatter(
            x=subset["participation"].to_list(),
            y=subset["cost_bps"].to_list(),
            name=name,
            mode="lines+markers",
            line=dict(color=color, dash=dash),
            hovertemplate="Participation: %{x:.1%}<br>Slippage: %{y:.2f} bps"
            "<extra>%{fullData.name}</extra>",
        ),
        row=1,
        col=1,
    )
```

### Add the Participation-Sensitive Panel

The percentage curve supplies a constant 10 bps reference beside the
volume-share response and makes their computed crossover visible.

```python
for name in ["Percentage (10bp)", "VolumeShare (0.1)"]:
    subset = slippage_df.filter(pl.col("model") == name)
    color, dash = slippage_styles[name]
    fig.add_trace(
        go.Scatter(
            x=subset["participation"].to_list(),
            y=subset["cost_bps"].to_list(),
            name=name,
            mode="lines+markers",
            line=dict(color=color, dash=dash),
            showlegend=name == "VolumeShare (0.1)",
            hovertemplate="Participation: %{x:.1%}<br>Slippage: %{y:.2f} bps"
            "<extra>%{fullData.name}</extra>",
        ),
        row=1,
        col=2,
    )
fig.update_xaxes(title_text="Order participation", tickformat=".0%", row=1, col=1)
fig.update_xaxes(title_text="Order participation", tickformat=".0%", row=1, col=2)
fig.update_yaxes(title_text="One-way slippage (bps)", range=[-0.5, 11.5], row=1, col=1)
fig.update_yaxes(title_text="One-way slippage (bps)", row=1, col=2)
fig.update_layout(
    title="One-way slippage against order participation, by slippage model",
    height=450,
    legend=dict(orientation="h", yanchor="top", y=-0.18, xanchor="center", x=0.5),
    margin=dict(b=95),
)
show_plotly_with_alt(
    fig,
    "Two panels of one-way slippage against order participation, sharing a legend. In the left "
    "panel four models each draw a flat horizontal line across the whole participation range, at "
    "four different levels, the lowest of them lying along the zero axis. In the right panel the "
    "volume-share model rises steeply and almost linearly with participation, reaching an order "
    "of magnitude above the flat percentage line drawn beside it for reference.",
)
```

**Finding**: Fixed, spread, and percentage assumptions do not respond to bar
participation. The volume-share model does, so it is the only curve here that
changes when the same price and volume face a larger order.

```python
# `FuturesSlippage` returns total dollars rather than a per-unit adjustment.
futures_slippage = futures_slippage_model.calculate(
    "ES", FUTURES_QUANTITY, FUTURES_PRICE, multiplier=FUTURES_MULTIPLIER
)
futures_slippage_bps = futures_slippage / futures_notional * 10_000
volume_share_crossover = (
    per_unit_slippage_models["Percentage (10bp)"].rate
    / per_unit_slippage_models["VolumeShare (0.1)"].impact_factor
)
display(
    Markdown(
        f"**Futures example**: {FUTURES_QUANTITY} contracts with "
        f"{futures_slippage_model.slippage_points:.2f} points of slippage cost "
        f"**${futures_slippage:,.2f}**, or **{futures_slippage_bps:.2f} bps one way**. "
        f"In the per-unit profile, volume-share slippage meets the 10 bps "
        f"percentage assumption at **{volume_share_crossover:.1%} participation**."
    )
)
```

**Finding**: Contract multipliers convert a small price-point move into total
dollars. Futures costs therefore require explicit contract, point, and
multiplier units before they can be compared with basis-point schedules.

## 3. Asset-Class Cost Configurations

We define four illustrative, one-way cost stacks. Each row is a different
unit-aware scenario, not a claim that the markets share a common ticket size
or that the assumptions estimate a particular broker or venue.

```python
asset_class_configs = {
    "US Equities (retail)": {
        "commission": PerShareCommission(per_share=0.005, minimum=1.0),
        "slippage": PercentageSlippage(rate=0.0005),
        "trade_qty": 200,
        "trade_price": 150.0,
        "trade_volume": 2_000_000,
        "description": "Illustrative per-share fee with percentage slippage",
    },
    "ETFs (institutional)": {
        "commission": PercentageCommission(rate=0.0003),
        "slippage": VolumeShareSlippage(impact_factor=0.05),
        "trade_qty": 1000,
        "trade_price": 300.0,
        "trade_volume": 10_000_000,
        "description": "Low percentage fee, volume-dependent impact",
    },
    "CME Futures (ES)": {
        "commission": FuturesCommission(per_block=2.25),
        "slippage": FuturesSlippage(slippage_points=0.25),
        "multiplier": 50.0,  # ES contract multiplier ($50 per point)
        "trade_qty": 5,
        "trade_price": 5000.0,
        "trade_volume": 50_000,
        "description": "Illustrative per-contract fee and one-tick slippage",
    },
    "Crypto (spot)": {
        "commission": PercentageCommission(rate=0.001),
        "slippage": PercentageSlippage(rate=0.002),
        "trade_qty": 0.5,
        "trade_price": 40_000.0,
        "trade_volume": 500,
        "description": "Illustrative percentage fee and slippage",
    },
}
```

### Compute One-Way Costs for Each Asset Class

```python
def asset_class_cost_row(asset_class: str, cfg: dict) -> dict:
    """Evaluate one representative trade under a unit-aware one-way cost stack."""
    qty = cfg["trade_qty"]
    price = cfg["trade_price"]
    vol = cfg["trade_volume"]
    multiplier = cfg.get("multiplier", 1.0)
    trade_value = abs(qty * price * multiplier)
    if trade_value <= 0:
        raise ValueError("trade notional must be positive")

    commission_model = cfg["commission"]
    if isinstance(commission_model, FuturesCommission):
        commission = commission_model.calculate("TEST", qty, price, multiplier=multiplier)
    else:
        commission = commission_model.calculate("TEST", qty, price)

    slippage_model = cfg["slippage"]
    if isinstance(slippage_model, FuturesSlippage):
        slippage = slippage_model.calculate("TEST", qty, price, vol, multiplier=multiplier)
    else:
        slippage = slippage_model.calculate("TEST", qty, price, vol) * abs(qty)

    total = commission + slippage
    total_bps = total / trade_value * 10_000
    return {
        "asset_class": asset_class,
        "trade_value": trade_value,
        "commission": commission,
        "slippage": slippage,
        "commission_bps": commission / trade_value * 10_000,
        "slippage_bps": slippage / trade_value * 10_000,
        "total": total,
        "total_bps": total_bps,
        "description": cfg["description"],
    }
```

### Evaluate the Four Illustrative Tickets

The calculation calls each commission and slippage model once. Its output is
therefore a one-way cost for the specified trade, not a round trip.

```python
rows = [asset_class_cost_row(asset_class, cfg) for asset_class, cfg in asset_class_configs.items()]
```

**Interpretation**: The four rows translate abstract model definitions into
native-unit scenarios. Comparing component shares avoids letting the largest
basis-point total hide the composition of the smaller stacks.

```python
slippage_dominant = sum(row["slippage_bps"] > row["commission_bps"] for row in rows)
fig = go.Figure()
fig.add_trace(
    go.Bar(
        x=[r["asset_class"] for r in rows],
        y=[r["commission_bps"] / r["total_bps"] * 100 for r in rows],
        name="Commission share",
        marker_color=COLORS["blue"],
        marker_pattern_shape="/",
        customdata=[[r["commission_bps"], r["total_bps"]] for r in rows],
        hovertemplate="Commission: %{customdata[0]:.2f} bps"
        "<br>One-way total: %{customdata[1]:.2f} bps<extra></extra>",
    )
)
_ = fig.add_trace(
    go.Bar(
        x=[r["asset_class"] for r in rows],
        y=[r["slippage_bps"] / r["total_bps"] * 100 for r in rows],
        name="Slippage share",
        marker_color=COLORS["amber"],
        marker_pattern_shape="x",
        customdata=[[r["slippage_bps"], r["total_bps"]] for r in rows],
        hovertemplate="Slippage: %{customdata[0]:.2f} bps"
        "<br>One-way total: %{customdata[1]:.2f} bps<extra></extra>",
    )
)
```

### Label Native-Unit Totals

The bar heights compare composition. Direct labels retain each scenario's
one-way basis-point magnitude without letting the largest market compress the rest.

```python
for row in rows:
    fig.add_annotation(
        x=row["asset_class"],
        y=103,
        text=f"{row['total_bps']:.2f} bps total",
        showarrow=False,
        font=dict(color=COLORS["neutral"], size=10),
    )
fig.update_layout(
    title="Slippage and commission shares of one-way cost, by asset class",
    xaxis_title="Illustrative asset-class stack",
    yaxis_title="Share of one-way total cost (%)",
    yaxis_range=[0, 112],
    barmode="stack",
    height=430,
)
show_plotly_with_alt(
    fig,
    "Four stacked bars, one per asset-class stack, each running the full height of the axis and "
    "split between a slippage share and a commission share, with the one-way total in basis "
    "points annotated above each bar. Slippage is the larger share in three of the four stacks "
    "and is nearly the whole bar in one of them; the institutional ETF stack reverses that, with "
    "commission taking nearly the whole bar and only a sliver of slippage capping it.",
)
```

```python
display(
    Markdown(
        f"**Composition**: slippage is the larger share in **{slippage_dominant} of "
        f"{len(rows)}** of these illustrative stacks."
    )
)
```

**Finding**: The composition, not the cross-market magnitude, identifies the
first lever to investigate. Slippage-heavy scenarios point toward execution;
fee-heavy scenarios point toward the broker or venue schedule.

## 4. P&L Sensitivity: Does Model Choice Matter?

For a liquid ETF momentum strategy with monthly rebalancing, we run the
same momentum rule using each equity-compatible commission model. The price
panel is real daily OHLCV for a manually selected ETF universe. This fixed universe
is not point-in-time membership data and carries survivorship and selection
limitations. There is no holdout or model selection, so the results demonstrate
cost mechanisms rather than unbiased strategy performance.

### Load the Real ETF Price Panel

```python
loaded_prices_df = (
    load_etfs(symbols=ETF_SYMBOLS, start_date=START_DATE, end_date=END_DATE)
    .select("timestamp", "symbol", "open", "high", "low", "close", "volume")
    .sort("symbol", "timestamp")
)
dates = loaded_prices_df["timestamp"].unique().sort()[:N_BARS]
test_prices_df = loaded_prices_df.filter(pl.col("timestamp").is_in(dates.implode()))
```

### Validate the Canonical Panel

```python
SYMBOLS = sorted(test_prices_df["symbol"].unique().to_list())
assert set(SYMBOLS) == set(ETF_SYMBOLS), (
    f"loaded universe {SYMBOLS} does not match requested {ETF_SYMBOLS}; "
    "a missing symbol would silently change the experiment"
)
assert test_prices_df.height > 0, "the ETF panel is empty"
assert test_prices_df.unique(subset=["symbol", "timestamp"]).height == test_prices_df.height
assert test_prices_df.null_count().select(pl.sum_horizontal(pl.all())).item() == 0
for price_column in ["open", "high", "low", "close"]:
    assert test_prices_df.select((pl.col(price_column) > 0).all()).item()
assert test_prices_df.select(
    (pl.col("high") >= pl.max_horizontal("open", "low", "close")).all()
).item()
assert test_prices_df.select(
    (pl.col("low") <= pl.min_horizontal("open", "high", "close")).all()
).item()
assert test_prices_df.select((pl.col("volume") >= 0).all()).item()
coverage = test_prices_df.group_by("symbol").agg(n_sessions=pl.col("timestamp").n_unique())
assert coverage["n_sessions"].n_unique() == 1
assert coverage["n_sessions"][0] == len(dates)
assert test_prices_df.height == len(SYMBOLS) * len(dates)
display(
    Markdown(
        f"Loaded a balanced panel of **{test_prices_df.height:,} rows**, "
        f"**{len(SYMBOLS)} fixed ETFs**, and **{len(dates):,} sessions** from "
        f"**{dates.min()}** through **{dates.max()}**. Canonical keys are unique, "
        "OHLCV values are complete, prices are positive, and volume is nonnegative."
    )
)
```

### Momentum-Based Rebalance Targets

The rebalance signal is a real trailing-momentum rule: at each rebalance date
hold the equal-weighted top three ETFs by their `MOMENTUM_LOOKBACK`-day return.
The value at close $t$ uses closes no later than $t$. `NEXT_BAR` queues the
resulting target after that close and fills at open $t+1$. Precomputing the
deterministic targets does not change this event order.

```python
momentum = test_prices_df.with_columns(
    mom=pl.col("close").pct_change(MOMENTUM_LOOKBACK).over("symbol")
)
```

### Convert the Trailing Rule into Cadence-Specific Targets

Each cadence samples different decision dates and therefore creates different
holdings and trade paths. The rule is common; the realized signal path is not.

```python
def make_weight_dict(step: int) -> dict:
    """Equal-weight top-3-by-trailing-momentum targets at the requested cadence."""
    weights = {}
    for ts in dates.gather_every(step):
        ranked = (
            momentum.filter((pl.col("timestamp") == ts) & pl.col("mom").is_not_null())
            .sort("mom", descending=True)
            .head(3)
        )
        if ranked.height == 3:
            weights[ts] = {symbol: 1.0 / 3 for symbol in ranked["symbol"].to_list()}
    return weights
```

### Monthly Targets for the Base Sensitivity Test

```python
weight_dict = make_weight_dict(21)
```

## 5. Monthly-Rebalance Sensitivity Harness

We now wire the cost models into a minimal backtest so the comparison moves
from per-trade arithmetic to realized portfolio outcomes.

```python
class SimpleStrategy(Strategy):
    """Rebalance strategy driven by a pre-computed weight dict."""

    def __init__(self, weight_dict):
        self.executor = TargetWeightExecutor(
            config=RebalanceConfig(
                min_trade_value=100.0,
                min_weight_change=0.005,
                allow_fractional=True,
            )
        )
        self._weights = weight_dict

    def on_data(self, timestamp, data, context, broker):
        if timestamp not in self._weights:
            return
        # Restrict targets to symbols actually present in this bar's data; any
        # remaining unexpected exceptions should surface rather than be hidden.
        targets = {a: w for a, w in self._weights[timestamp].items() if a in data}
        if targets:
            self.executor.execute(targets, data, broker)
```

### Define the Equity-Compatible Commission Variants

The futures model is excluded because these trades are ETF shares, not
contracts. Slippage and every strategy input remain fixed across variants.

```python
commission_tests = dict(equity_commission_models)
```

### Configure Next-Open Execution

```python
base_config = BacktestConfig(
    initial_cash=INITIAL_CASH,
    slippage_rate=0.0005,
    execution_mode=ExecutionMode.NEXT_BAR,
)
```

### Execute One Cost-Model Variant

A target decided from close $t$ is submitted in `on_data()` and filled at
open $t+1$. The function returns the same four diagnostics for every fee
schedule.

```python
def run_backtest_variant(weight_dict: dict, commission_model, config: BacktestConfig) -> dict:
    """Execute the simple strategy under one commission model."""
    feed = DataFeed(prices_df=test_prices_df)
    strategy = SimpleStrategy(weight_dict)
    engine = Engine(feed=feed, strategy=strategy, config=config)
    engine.broker.commission_model = commission_model
    result = engine.run()
    return {
        "sharpe": float(result.metrics.get("sharpe", 0.0)),
        "total_return": float(result.equity.total_return),
        "total_commission": float(sum(t.fees for t in result.trades)),
        "n_trades": len(result.trades),
    }
```

### Run the Monthly-Rebalance Cost Comparison

```python
pnl_results = {}
for name, comm_model in commission_tests.items():
    pnl_results[name] = run_backtest_variant(weight_dict, comm_model, base_config)
```

**Finding**: This comparison isolates commission arithmetic within the fixed
monthly rule. Hover fields retain returns, fees, and trade counts without
duplicating the result as a terminal table.

```python
sharpe_vals = [v["sharpe"] for v in pnl_results.values()]
names = list(pnl_results.keys())
sharpe_range = max(sharpe_vals) - min(sharpe_vals)
x_padding = max(sharpe_range * 0.4, 0.01)
monthly_customdata = [
    [metrics["total_return"], metrics["total_commission"], metrics["n_trades"]]
    for metrics in pnl_results.values()
]

fig = go.Figure()
fig.add_trace(
    go.Scatter(
        y=names,
        mode="markers+text",
        x=sharpe_vals,
        marker=dict(
            color=profile_colors,
            size=11,
            symbol=["circle", "square", "diamond", "x", "triangle-up"],
        ),
        text=[f"{value:.4f}" for value in sharpe_vals],
        textposition="middle right",
        customdata=monthly_customdata,
        hovertemplate="Sharpe: %{x:.4f}<br>Return: %{customdata[0]:.2%}"
        "<br>Commission: $%{customdata[1]:,.0f}<br>Trades: %{customdata[2]:,.0f}<extra></extra>",
    )
)
fig.update_layout(
    title="Net Sharpe by commission model, monthly rebalancing",
    xaxis_title="Net Sharpe ratio",
    yaxis_title="Commission model",
    xaxis_range=[min(sharpe_vals) - x_padding, max(sharpe_vals) + x_padding],
    height=420,
    showlegend=False,
    margin=dict(l=175),
)
show_plotly_with_alt(
    fig,
    "A dot plot of net Sharpe by commission model, one row per model, each marker a different "
    "shape and colour and labelled with its value. The whole set spans a narrow range of the "
    "Sharpe axis: the no-commission and per-share rows sit at the right-hand end and are almost "
    "indistinguishable from one another, the percentage row sits furthest left, and the tiered "
    "and combined rows fall between them.",
)
```

### Read the Monthly Sensitivity

```python
display(
    Markdown(
        f"Across the {len(commission_tests)} equity-compatible schedules, "
        f"monthly net Sharpe spans "
        f"**{sharpe_range:.4f}** on this **{len(SYMBOLS)}-ETF**, "
        f"**{len(dates):,}-session** demonstration. This magnitude describes the "
        "fixed panel and rule; it is not an out-of-sample performance estimate."
    )
)
```

**Mechanism**: the Sharpe-range scalar compresses the monthly comparison into
one number. Its magnitude is specific to this fixed momentum panel, not a
general claim about commission-model sensitivity.
The point of this section is the arithmetic mechanism: percentage and tiered fee
structures accumulate proportionally to traded notional, per-share fees scale
with share count, and the gap between them depends on price level and trade size
rather than on rebalance frequency alone. The relative ordering would shift on a
different universe or trade-size profile, so read the spread as an illustration
of the mechanism rather than a transferable magnitude.

## 6. Cadence Sensitivity

We apply the same trailing-momentum rule at daily, weekly, and 21-session
cadence. Each cadence samples different dates, targets, and trades, so the
comparison measures rule-and-cadence paths rather than holding a gross return
series fixed.

```python
cadences = {"daily": 1, "weekly": 5, "monthly": 21}
cadence_weights = {label: make_weight_dict(days) for label, days in cadences.items()}
```

### Select the Fee Contrast

Zero commission supplies the baseline; a 10 bps percentage schedule isolates
how the same fee rule accumulates along each cadence-specific trade path.

```python
test_models = {
    "NoCommission": NoCommission(),
    "Percentage (10bp)": PercentageCommission(rate=0.001),
}
```

### Run the Frequency-Sensitivity Grid

```python
freq_results = {}
freq_config = BacktestConfig(
    initial_cash=INITIAL_CASH,
    slippage_rate=0.0005,
    execution_mode=ExecutionMode.NEXT_BAR,
)

for cadence_name in cadences:
    for model_name, comm_model in test_models.items():
        key = f"{cadence_name}/{model_name}"
        result = run_backtest_variant(cadence_weights[cadence_name], comm_model, freq_config)
        freq_results[key] = {
            "cadence": cadence_name,
            "model": model_name,
            "sharpe": result["sharpe"],
            "total_commission": result["total_commission"],
            "n_trades": result["n_trades"],
        }
```

**Interpretation**: The grid holds the momentum formula and execution contract
fixed while the decision dates change. A wider fee-induced Sharpe gap at a
faster cadence reflects the additional trades generated on that path.

**Finding**: In this demonstration, the daily rule is the high-turnover case.
Conclusions remain limited to the three tested cadences; the notebook does not
extrapolate them to intraday or quarterly strategies.

```python
daily_spread = abs(
    freq_results["daily/Percentage (10bp)"]["sharpe"] - freq_results["daily/NoCommission"]["sharpe"]
)
monthly_spread = abs(
    freq_results["monthly/Percentage (10bp)"]["sharpe"]
    - freq_results["monthly/NoCommission"]["sharpe"]
)
amplification = daily_spread / monthly_spread if monthly_spread > 0 else float("inf")
cadence_labels = list(cadences)
frequency_styles = [
    ("NoCommission", COLORS["blue"], "circle", "solid"),
    ("Percentage (10bp)", COLORS["amber"], "square", "dash"),
]
frequency_series = {
    model_name: {
        "sharpe": [freq_results[f"{cadence}/{model_name}"]["sharpe"] for cadence in cadences],
        "commission": [
            freq_results[f"{cadence}/{model_name}"]["total_commission"] for cadence in cadences
        ],
    }
    for model_name in test_models
}

fig = make_subplots(
    rows=1,
    cols=2,
    subplot_titles=["Net Sharpe", "Total Commission"],
    horizontal_spacing=0.18,
)
```

### Plot the Sharpe Paths

```python
for model_name, color, symbol, dash in frequency_styles:
    sharpes = frequency_series[model_name]["sharpe"]
    fig.add_trace(
        go.Scatter(
            x=cadence_labels,
            y=sharpes,
            name=model_name,
            mode="lines+markers+text",
            line=dict(color=color, dash=dash),
            marker=dict(symbol=symbol, size=8),
            text=[f"{value:.3f}" for value in sharpes],
            textposition="top center",
        ),
        row=1,
        col=1,
    )
```

### Add Dollar Fees and Complete the Layout

The second panel uses its own dollar scale. Shared cadence labels align the
paths without implying that Sharpe and fees have comparable magnitudes.

```python
for model_name, color, symbol, dash in frequency_styles:
    commissions = frequency_series[model_name]["commission"]
    fig.add_trace(
        go.Scatter(
            x=cadence_labels,
            y=commissions,
            name=model_name,
            mode="lines+markers+text",
            line=dict(color=color, dash=dash),
            marker=dict(symbol=symbol, size=8),
            text=[f"${value:,.0f}" for value in commissions],
            textposition="top center",
            showlegend=False,
        ),
        row=1,
        col=2,
    )
fig.update_yaxes(title_text="Sharpe Ratio", row=1, col=1)
fig.update_yaxes(title_text="Total commission ($)", title_standoff=12, row=1, col=2)
fig.update_xaxes(title_text="Rebalance cadence", row=1, col=1)
fig.update_xaxes(title_text="Rebalance cadence", row=1, col=2)
fig.update_layout(
    height=480,
    title="Net Sharpe and total commission by rebalance cadence",
    legend=dict(orientation="h", yanchor="top", y=-0.2, xanchor="center", x=0.5),
    margin=dict(b=100, t=100),
)
show_plotly_with_alt(
    fig,
    "Two panels against rebalance cadence, from daily to monthly, comparing a no-commission "
    "path with a percentage-fee path, every point labelled. In the Sharpe panel both paths rise "
    "towards monthly cadence and the gap between them closes as they go. In the commission "
    "panel the no-commission path is flat on zero while the fee path falls steeply from its "
    "daily value, so the two panels move in step.",
)
```

**Finding**: Dollar fees and the Sharpe gap widen together along the daily
trade path. The chart does not imply that each saved basis point maps linearly
into Sharpe outside this fixed-sample comparison.

```python
display(
    Markdown(
        f"The 10 bps commission schedule changes Sharpe by **{daily_spread:.4f}** "
        f"at daily cadence and **{monthly_spread:.4f}** at 21-session cadence, "
        f"a **{amplification:.1f}x** ratio on these cadence-specific paths."
    )
)
```

## Key Takeaways

```python
model_count = len(commission_models) + len(slippage_models)
display(
    Markdown(
        f"- **Complete taxonomy**: the configured dictionaries cover "
        f"**{len(commission_models)} commission** and "
        f"**{len(slippage_models)} slippage** models, **{model_count} total**.\n"
        "- **Units come first**: shares, contracts, full spread, half-spread, "
        "per-unit adjustments, and total dollars are not interchangeable.\n"
        f"- **Monthly sensitivity is sample-specific**: commission choice moves "
        f"Sharpe by **{sharpe_range:.4f}** on the fixed ETF panel.\n"
        f"- **Cadence changes the trade path**: the daily fee-induced Sharpe gap "
        f"is **{amplification:.1f}x** the 21-session gap here, without supporting "
        "an intraday or quarterly extrapolation.\n"
        "- **Deployment requires measurement**: replace every illustrative fee, "
        "spread, and impact input with the strategy's executable venue terms.\n\n"
        "**Book**: Chapter 18, Sections 18.2-18.4 cover cost taxonomy, impact, "
        "and cadence.\n\n"
        "**Next**: See [`06_ml4t_execution_demo`](06_ml4t_execution_demo.ipynb) "
        "for the execution-facing API."
    )
)
```
![notebook output](figures/p1_1.png)
![notebook output](figures/p1_2.png)
![notebook output](figures/p1_3.png)
![notebook output](figures/p1_4.png)
![notebook output](figures/p1_5.png)