01 — Data Exploration: US Equities

Objective

Validate the historical data pipeline and document basic empirical properties of daily returns for a small illustrative basket of liquid US equities.

Data source

Field	Value
Provider	Alpaca Markets (historical bars API)
Adjustment	Split and dividend adjusted (Adjustment.ALL)
Frequency	Daily (1d bars)
Universe	14 hand-picked liquid US equities across tech, financials, healthcare, consumer, and energy
Date range	Jan 2024 — present

Known limitations

• The selected universe is not a point-in-time universe and is not intended to support claims about an investable cross-sectional strategy. It is used for package validation and exploratory analysis.
• Survivorship bias is present — all names were selected with hindsight.

Main outputs

1. Normalized price chart
2. Return distributions and fat-tail analysis
3. Cross-sector correlation heatmap
4. Rolling volatility and volatility clustering evidence
5. Summary performance metrics table

Note: AI tools were used for cosmetic and presentation edits to this notebook.

1. Imports

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from scipy.stats import norm

from qre.analytics.metrics import full_summary, summary
from qre.data.historical import HistoricalDataStore

# Plot style
plt.rcParams.update({
    "figure.facecolor": "white",
    "axes.facecolor": "#fafafa",
    "axes.grid": True,
    "grid.alpha": 0.3,
    "axes.spines.top": False,
    "axes.spines.right": False,
    "font.size": 11,
})

%matplotlib inline

2. Loading data

14 liquid US equities spanning five sectors: technology, financials, healthcare, consumer staples, and energy.

tickers = [
    "NVDA",   # Nvidia
    "AAPL",   # Apple
    "MSFT",   # Microsoft
    "GOOGL",  # Alphabet (Google)
    "AMZN",   # Amazon
    "META",   # Meta (Facebook)
    "JPM",    # JPMorgan Chase
    "GS",     # Goldman Sachs
    "V",      # Visa
    "XOM",    # ExxonMobil
    "JNJ",    # Johnson & Johnson
    "WMT",    # Walmart
    "PG",     # Procter & Gamble
    "UNH",    # UnitedHealth
]

store = HistoricalDataStore()

# Load all tickers into wide DataFrames (no API keys needed)
panel = store.load_panel(tickers, "1d")
closes = panel["close"]

print(f"Shape: {closes.shape[0]} trading days x {closes.shape[1]} tickers")
print(f"Date range: {closes.index.min().date()} to {closes.index.max().date()}")
closes.head()

Shape: 590 trading days x 14 tickers
Date range: 2024-01-02 to 2026-05-08

	NVDA	AAPL	MSFT	GOOGL	AMZN	META	JPM	GS	V	XOM	JNJ	WMT	PG	UNH
timestamp
2024-01-02 05:00:00+00:00	48.14	183.56	364.59	137.04	149.93	343.59	163.01	369.59	254.57	94.85	149.66	51.77	139.35	514.26
2024-01-03 05:00:00+00:00	47.54	182.19	364.32	137.78	148.47	341.79	162.30	363.39	253.69	95.64	150.60	51.77	138.51	516.83
2024-01-04 05:00:00+00:00	47.97	179.87	361.71	135.27	144.57	344.42	163.38	364.50	255.30	94.81	150.28	51.27	139.26	520.06
2024-01-05 05:00:00+00:00	49.07	179.15	361.52	134.62	145.24	349.21	164.20	367.82	255.38	95.10	150.75	50.93	138.11	512.40
2024-01-08 05:00:00+00:00	52.22	183.48	368.34	137.70	149.10	355.87	163.96	370.12	258.18	93.51	151.12	51.43	139.30	511.58

3. Normalized price chart

Index all prices to a base of 100 on the first trading day to make cumulative performance visually comparable across different price levels.

normalized = closes / closes.iloc[0] * 100

fig, ax = plt.subplots(figsize=(14, 7))
for col in normalized.columns:
    ax.plot(normalized.index, normalized[col], linewidth=1.2, label=col)

ax.axhline(100, color="grey", linestyle="--", linewidth=0.8, alpha=0.5)
ax.set_title("Normalized Closing Prices (Base = 100)", fontsize=14, fontweight="bold")
ax.set_xlabel("Date")
ax.set_ylabel("Indexed to 100")
ax.legend(loc="upper left", fontsize=8, ncol=2)
plt.tight_layout()

Observations:

• Alpaca Client API uses Adjustment.ALL to return adjusted prices accounting for stock splits and dividends on ex-days. NVDA and WMT had 10-for-1 stock splits in mid-2024.
• Raw (unadjusted) prices are important for execution but not for return computation or signal generation.
• Over the selected sample, NVDA had the largest cumulative return among the basket. This should not be interpreted as an ex ante result — the universe is illustrative and was selected with full hindsight.

4. Return distributions

Compute simple daily returns and examine their statistical properties. A key question: how well does the normal distribution describe equity returns?

returns = closes.pct_change().dropna()

print(f"Return matrix: {returns.shape[0]} days x {returns.shape[1]} tickers")
returns.describe().round(4)

Return matrix: 589 days x 14 tickers

	NVDA	AAPL	MSFT	GOOGL	AMZN	META	JPM	GS	V	XOM	JNJ	WMT	PG	UNH
count	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000	589.0000
mean	0.0030	0.0009	0.0003	0.0020	0.0012	0.0013	0.0012	0.0017	0.0005	0.0008	0.0007	0.0017	0.0001	-0.0002
std	0.0310	0.0173	0.0151	0.0191	0.0197	0.0239	0.0153	0.0185	0.0132	0.0144	0.0109	0.0140	0.0112	0.0252
min	-0.1697	-0.0925	-0.0999	-0.0750	-0.0898	-0.1133	-0.0749	-0.0921	-0.0774	-0.0720	-0.0759	-0.0653	-0.0501	-0.2238
25%	-0.0136	-0.0069	-0.0068	-0.0084	-0.0094	-0.0107	-0.0057	-0.0073	-0.0052	-0.0074	-0.0048	-0.0057	-0.0059	-0.0091
50%	0.0034	0.0011	0.0008	0.0027	0.0011	0.0010	0.0015	0.0015	0.0012	0.0010	0.0006	0.0015	0.0006	0.0006
75%	0.0199	0.0086	0.0084	0.0116	0.0126	0.0121	0.0093	0.0115	0.0075	0.0100	0.0063	0.0087	0.0065	0.0114
max	0.1872	0.1533	0.1013	0.1022	0.1198	0.2032	0.1154	0.1310	0.0826	0.0500	0.0619	0.0954	0.0414	0.1198

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

aapl_rets = returns["AAPL"]
aapl_rets.hist(
    bins=50, alpha=0.6, label="AAPL actual",
    ax=ax, density=True, color="steelblue",
)

x = np.linspace(float(aapl_rets.min()), float(aapl_rets.max()), 200)
ax.plot(x, norm.pdf(x, float(aapl_rets.mean()), float(aapl_rets.std())),
        "r-", linewidth=2, label="Normal fit")

ax.set_title(
    "AAPL Daily Return Distribution vs. Normal",
    fontsize=14, fontweight="bold",
)
ax.set_xlabel("Daily Return")
ax.set_ylabel("Density")
ax.legend()
plt.tight_layout()

print(f"AAPL excess kurtosis: {float(aapl_rets.kurtosis()):.2f}")
print(f"AAPL skewness:       {float(aapl_rets.skew()):.2f}")

AAPL excess kurtosis: 12.56
AAPL skewness:       0.88

Leptokurtosis (fat tails + sharp peak) — one of the most well-documented properties of financial returns.

A normal distribution spreads probability relatively evenly across the range. Empirical equity returns concentrate in two places:

1. At the mean — most days produce very small moves.
2. In the tails — when large moves occur, they are larger than the normal distribution predicts.

The region in between (~1 standard deviation) is underpopulated relative to the normal — fewer "medium-sized" moves than expected.

Implications:

• VaR models that assume normality systematically underestimate tail risk. The 2008 crisis was a "25-sigma event" under Gaussian assumptions — essentially impossible, yet it occurred.
• Excess kurtosis measures this departure. The normal distribution has kurtosis = 3 (excess kurtosis = 0). Financial returns typically exhibit excess kurtosis of 5–10+.

Note: pandas .kurtosis() returns excess kurtosis (Fisher's definition, subtracting 3), so any value > 0 indicates fatter tails than the normal distribution.

5. Correlation analysis

Pairwise Pearson correlations of daily returns. Correlation structure reveals sector factors, common macro exposures, and potential diversification opportunities.

corr = returns.corr()

fig, ax = plt.subplots(figsize=(12, 9))
ax.grid(False)
im = ax.imshow(corr.to_numpy(), cmap="RdBu_r", vmin=-1, vmax=1, aspect="equal")

ax.set_xticks(range(len(tickers)))
ax.set_yticks(range(len(tickers)))
ax.set_xticklabels(tickers, rotation=45, ha="right", fontsize=9)
ax.set_yticklabels(tickers, fontsize=9)

for i in range(len(tickers)):
    for j in range(len(tickers)):
        val = float(corr.iloc[i, j])
        color = "white" if abs(val) > 0.6 else "black"
        ax.text(j, i, f"{val:.2f}", ha="center", va="center", fontsize=7, color=color)

cbar = plt.colorbar(im, fraction=0.046, pad=0.04)
cbar.set_label("Pearson Correlation", fontsize=10)
ax.set_title("Daily Return Correlation Matrix", fontsize=14, fontweight="bold")
plt.tight_layout()

Observations:

• Tech cluster (~0.4 pairwise correlation): NVDA, MSFT, GOOGL, AMZN, META — consistent with a common technology/growth factor.
• Financials cluster: GS/JPM correlation of 0.78 (highest in the basket) — both are investment banks with shared exposure to interest rates, deal flow, and credit markets. ~0.4 correlation with V (financial services).
• Defensive cluster: PG, WMT, JNJ show slight positive correlation — consumer staples and healthcare tend to co-move during risk-off episodes.
• Diversifier: UNH shows minimal correlation with the rest of the basket.
• Energy: XOM shows only slight correlation with financials (JPM, GS, V); largely independent of the tech cluster.
• Market factor: Most pairwise correlations are positive, reflecting the common market factor. Long-only portfolios cannot diversify away this systematic risk.

Highest correlations (all >= 0.5): GS/JPM, META/AMZN, AMZN/MSFT, AMZN/GOOGL, MSFT/NVDA.

6. Rolling volatility

21-day rolling annualized volatility for a subset of names. Volatility is not constant — it clusters in regimes, which has direct implications for position sizing and risk management.

rolling_vol = returns.rolling(window=21).std() * np.sqrt(252)

highlight = ["AAPL", "NVDA", "JPM", "UNH"]
colors = ["#1f77b4", "#ff7f0e", "#2ca02c", "#d62728"]

fig, ax = plt.subplots(figsize=(14, 7))
for ticker, color in zip(highlight, colors):
    ax.plot(rolling_vol[ticker], label=ticker, linewidth=1.2, color=color)

ax.set_title("21-Day Rolling Annualized Volatility", fontsize=14, fontweight="bold")
ax.set_ylabel("Annualized Volatility")
ax.set_xlabel("Date")
ax.legend(fontsize=10)
plt.tight_layout()

Volatility clustering: large moves tend to follow large moves, and small moves follow small moves. First documented by Mandelbrot (1963).

Mechanism: When a shock occurs (earnings surprise, central bank decision, geopolitical event), it creates uncertainty. Traders reprice risk, hedging activity increases, stop-losses trigger — this feeds on itself until the market digests the new information and settles.

Rolling-window artefact: The 21-day window acts as a memory buffer. A single large daily return enters the window and stays in the calculation for 21 days; when it exits, volatility can drop abruptly. This explains the sharp edges and plateau-like structure in the plots. The multi-window comparison below helps distinguish this artefact from genuine clustering.

Implications for quantitative trading:

• Position sizing should adapt to the current volatility regime — smaller positions during high-vol periods.
• Volatility-targeting strategies need real-time volatility estimates, which motivates GARCH-family models (explored in later phases).
• The standard $\sqrt{252}$ annualization assumes constant volatility — clearly an approximation.

windows = [5, 21, 63]
fig, axes_arr = plt.subplots(3, 1, figsize=(14, 10), sharex=True)
axes = np.asarray(axes_arr).flatten()

for ax, window in zip(axes, windows):
    vol = returns["NVDA"].rolling(window=window).std() * np.sqrt(252)
    ax.plot(vol, color="steelblue", linewidth=1.0)
    ax.set_ylabel("Ann. Vol")
    ax.set_title(
        f"NVDA {window}-Day Rolling Volatility",
        fontsize=11, fontweight="bold",
    )

axes[-1].set_xlabel("Date")
fig.suptitle(
    "Effect of Window Length on Rolling Volatility",
    fontsize=14, fontweight="bold", y=1.01,
)
plt.tight_layout()

Separating artefact from signal: The multi-window comparison (5, 21, 63 days) above helps disentangle the rolling-window artefact from genuine volatility persistence. If clustering were purely an artefact of the window, shorter windows would show no persistence at all. Instead, even the 5-day window shows extended high-volatility episodes.

A more rigorous test is the autocorrelation of squared returns — if $r_t^2$ is positively autocorrelated, volatility is genuinely persistent regardless of any smoothing window. The plot below provides visual evidence; formal testing via GARCH models is deferred to later phases.

nvda_sq = returns["NVDA"] ** 2

fig, ax = plt.subplots(figsize=(14, 5))
ax.bar(
    nvda_sq.index, nvda_sq.to_numpy(),
    color="steelblue", width=1.0, alpha=0.7,
)
ax.set_title(
    "NVDA Squared Daily Returns (Proxy for Daily Variance)",
    fontsize=14, fontweight="bold",
)
ax.set_ylabel("Return$^2$")
ax.set_xlabel("Date")
plt.tight_layout()

7. Performance summary

Full metrics including benchmark-relative analytics (vs SPY) and distributional characteristics, computed via qre.analytics.metrics.full_summary. All ratios use 252 trading days per year. Sharpe, Sortino, and alpha assume zero risk-free rate.

# Load SPY as benchmark
spy_closes = store.load("SPY", "1d")["close"]
spy_ret = spy_closes.pct_change().dropna()

metrics_table = pd.DataFrame({
    ticker: full_summary(returns[ticker], spy_ret)
    for ticker in tickers
}).T.round(4)

# Also add SPY itself (standalone metrics only)
spy_standalone = summary(spy_ret)
metrics_table.loc["SPY"] = pd.Series(
    full_summary(spy_ret, spy_ret)
).round(4)

highlight_cols = [
    "sharpe_ratio", "annualized_return", "alpha", "beta",
]
metrics_table.style.format("{:.4f}").background_gradient(
    cmap="RdYlGn", subset=highlight_cols,
)

	annualized_return	annualized_volatility	sharpe_ratio	sortino_ratio	max_drawdown	calmar_ratio	win_rate	profit_factor	beta	alpha	tracking_error	information_ratio	up_capture	down_capture	skewness	excess_kurtosis	value_at_risk_95	cvar_95	tail_ratio	expected_tail_ratio	best_day	worst_day
NVDA	0.8978	0.4918	1.5490	1.5910	-0.3688	2.4343	0.5467	1.3108	2.0643	0.3182	0.4018	1.3611	2.3879	2.0490	0.1923	5.0864	-0.0439	-0.0668	1.0391	1.0600	0.1872	-0.1697
AAPL	0.2216	0.2749	0.8644	0.8912	-0.3336	0.6641	0.5399	1.1771	1.1483	-0.0091	0.2055	0.1106	1.0605	1.0472	0.8758	12.5624	-0.0259	-0.0381	0.9907	1.0826	0.1533	-0.0925
MSFT	0.0571	0.2404	0.3513	0.3383	-0.3391	0.1684	0.5263	1.0661	0.9653	-0.1230	0.1840	-0.7089	0.9632	1.1301	-0.1466	7.9667	-0.0243	-0.0355	0.8811	0.8984	0.1013	-0.0999
GOOGL	0.5827	0.3032	1.6660	1.7422	-0.2981	1.9551	0.5586	1.3397	1.1249	0.2634	0.2446	1.1869	1.3372	1.0403	0.4798	4.7704	-0.0257	-0.0402	1.1103	1.1573	0.1022	-0.0750
AMZN	0.2916	0.3134	0.9729	0.9937	-0.3088	0.9445	0.5348	1.1851	1.3845	0.0074	0.2297	0.3920	1.4529	1.4629	0.2406	5.0119	-0.0278	-0.0427	1.0513	1.0649	0.1198	-0.0898
META	0.2780	0.3790	0.8337	0.9028	-0.3415	0.8141	0.5246	1.1728	1.4549	0.0034	0.3075	0.3288	1.4265	1.4135	1.1959	13.6335	-0.0300	-0.0503	1.1044	1.1329	0.2032	-0.1133
JPM	0.3021	0.2426	1.2096	1.1517	-0.2442	1.2368	0.5620	1.2489	0.9216	0.0954	0.1928	0.4075	0.9706	0.8549	0.0718	8.7500	-0.0226	-0.0374	0.9030	0.8590	0.1154	-0.0749
GS	0.4885	0.2930	1.5042	1.5248	-0.3090	1.5812	0.5586	1.3141	1.3153	0.1581	0.2096	1.0774	1.4572	1.2833	0.4118	8.4618	-0.0241	-0.0410	1.0965	1.0289	0.1310	-0.0921
V	0.1010	0.2091	0.5647	0.5370	-0.2038	0.4958	0.5501	1.1104	0.6889	-0.0299	0.1844	-0.5248	0.5984	0.6126	0.0527	8.2801	-0.0181	-0.0317	0.9813	0.9131	0.0826	-0.0774
XOM	0.1976	0.2279	0.9057	0.8598	-0.1892	1.0446	0.5348	1.1626	0.2981	0.1424	0.2497	-0.0338	0.1565	-0.0790	-0.5417	1.8833	-0.0234	-0.0337	1.0030	0.8557	0.0500	-0.0720
JNJ	0.1822	0.1734	1.0524	1.0647	-0.1445	1.2614	0.5348	1.2030	0.0244	0.1772	0.2335	-0.1389	0.0835	-0.1407	-0.1500	6.3041	-0.0157	-0.0225	1.1264	1.1178	0.0619	-0.0759
WMT	0.4849	0.2223	1.8898	2.0448	-0.2192	2.2119	0.5518	1.4086	0.4562	0.3221	0.2273	0.9029	0.4817	0.0501	0.7341	6.6254	-0.0183	-0.0283	1.1583	1.2797	0.0954	-0.0653
PG	0.0214	0.1772	0.2082	0.1958	-0.2115	0.1012	0.5280	1.0361	0.1183	0.0115	0.2259	-0.7879	0.0911	0.0675	-0.4159	2.1279	-0.0178	-0.0265	0.9758	0.8954	0.0414	-0.0501
UNH	-0.1214	0.4008	-0.1129	-0.0981	-0.6139	-0.1978	0.5178	0.9766	0.2857	-0.1066	0.4143	-0.6279	0.1530	0.2594	-2.4420	21.9436	-0.0297	-0.0660	1.0022	0.7573	0.1198	-0.2238
SPY	0.2296	0.1599	1.3729	1.3417	-0.1876	1.2244	0.5750	1.3009	1.0000	0.0000	0.0000	0.0000	1.0000	1.0000	1.0293	22.4020	-0.0153	-0.0227	0.8351	0.9597	0.1050	-0.0585

Observations:

Top performers: NVDA (90% annualized return), GOOGL (58%), WMT (49%), GS (49%). WMT stands out as the most efficient risk-adjusted performer — highest Sharpe (1.89) with relatively low volatility (22%).

Underperformer: UNH — the only negative return in the basket, worst drawdown (-61%), and profit factor below 1.0.

Patterns:

• Win rates cluster around 52–56%. Even the best-performing stocks only rise slightly more than half of all trading days. Profitability is driven by the magnitude of wins vs. losses (profit factor), not frequency.
• Risk-return tradeoff. Lower-volatility names (JNJ, PG, V) exhibit smaller drawdowns but also lower returns — the classic cross-sectional relationship.
• Drawdowns are substantial across the board. Every name experienced a peak-to-trough decline of at least 14%, and most exceeded 25%. This underscores the importance of position sizing and risk management even in a strong market environment.
• Beta decomposition. Most tech names have beta > 1.5, meaning their returns are amplified market moves. Alpha separates genuine stock-specific outperformance from beta exposure — NVDA and GOOGL show strong positive alpha even after adjusting for their high beta.

8. Skewness vs. Kurtosis: individual stocks vs. the index

How do distributional properties change when you compare the small basket of stocks with the larger market, represented by the S&P 500?

skew_kurt = metrics_table[["skewness", "excess_kurtosis"]]

fig, ax = plt.subplots(figsize=(10, 7))

# Individual stocks
stocks = skew_kurt.drop("SPY", errors="ignore")
ax.scatter(
    stocks["skewness"], stocks["excess_kurtosis"],
    s=80, color="steelblue", alpha=0.8, zorder=3,
)
for ticker in stocks.index:
    ax.annotate(
        ticker,
        (stocks.loc[ticker, "skewness"],
         stocks.loc[ticker, "excess_kurtosis"]),
        textcoords="offset points", xytext=(6, 4),
        fontsize=9, color="steelblue",
    )

# SPY highlighted
spy_row = skew_kurt.loc["SPY"]
ax.scatter(
    spy_row["skewness"], spy_row["excess_kurtosis"],
    s=150, color="#d62728", marker="*", zorder=4,
    label="SPY (index)",
)
ax.annotate(
    "SPY", (spy_row["skewness"], spy_row["excess_kurtosis"]),
    textcoords="offset points", xytext=(8, 4),
    fontsize=11, fontweight="bold", color="#d62728",
)

# Reference lines
ax.axhline(0, color="grey", linestyle="--", linewidth=0.8, alpha=0.5)
ax.axvline(0, color="grey", linestyle="--", linewidth=0.8, alpha=0.5)

ax.set_xlabel("Skewness", fontsize=12)
ax.set_ylabel("Excess Kurtosis", fontsize=12)
ax.set_title(
    "Skewness vs. Kurtosis: Individual Stocks vs. SPY",
    fontsize=14, fontweight="bold",
)
ax.legend(fontsize=10)
plt.tight_layout()