MAANG Stocks: Analysis¶

This notebook downloads one year of daily adjusted close prices for MAANG stocks (Meta, Apple, Amazon, Netflix, Google). It

computes daily simple returns and log returns,
summarizes their distributional properties,
compares empirical distributions to a normal approximation using sample mean and standard deviation,
evaluates serial dependence using autocorrelation and Ljung-Box tests.

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import datetime

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import yfinance as yf
from scipy.stats import jarque_bera, kurtosis, norm, skew
from statsmodels.stats.diagnostic import acorr_ljungbox
from statsmodels.tsa.stattools import acf

# Define the MAANG tickers for analysis
tickers = ['META', 'AAPL', 'AMZN', 'NFLX', 'GOOGL']

# Download one year of daily adjusted close prices
end_date = datetime.date.today()
start_date = end_date - datetime.timedelta(days=365)
prices = yf.download(tickers, start=start_date, end=end_date, progress=False, actions=False)["Close"]
prices = prices.dropna(how='all')
import datetime

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import yfinance as yf
from scipy.stats import jarque_bera, kurtosis, norm, skew
from statsmodels.stats.diagnostic import acorr_ljungbox
from statsmodels.tsa.stattools import acf

# Define the MAANG tickers for analysis
tickers = ['META', 'AAPL', 'AMZN', 'NFLX', 'GOOGL']

# Download one year of daily adjusted close prices
end_date = datetime.date.today()
start_date = end_date - datetime.timedelta(days=365)
prices = yf.download(tickers, start=start_date, end=end_date, progress=False, actions=False)["Close"]
prices = prices.dropna(how='all')

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# Compute daily simple returns and log returns
returns = prices.pct_change().dropna()
log_returns = np.log(prices / prices.shift(1)).dropna()
# Compute daily simple returns and log returns
returns = prices.pct_change().dropna()
log_returns = np.log(prices / prices.shift(1)).dropna()

Distributional Properties¶

data.std(ddof=1): unbiased sample standard deviation with Bessel's correction (degrees of freedom = $n-1$): $$ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2}\,. $$ See https://pandas.pydata.org/docs/reference/api/pandas.Series.std.html
skew(data, bias=False): unbiased sample skewness. The adjusted estimator is: $$ G_1 = \frac{\sqrt{n(n-1)}}{n-2} \frac{m_3}{m_2^{3/2}}\,, $$ where $m_k = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^k$ is the k-th central moment and $n$ is the sample size. See https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.skew.html
kurtosis(data, fisher=True, bias=False): unbiased excess kurtosis. The adjusted estimator is: $$ G_2 = \frac{(n-1)}{(n-2)(n-3)}\left((n+1)g_2 + 6\right)\,, $$ where $g_2 = \frac{m_4}{m_2^2} - 3$ is the biased excess kurtosis estimator, and $m_k$ is the k-th central moment. See https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kurtosis.html
jarque_bera(data): Jarque-Bera test statistic for normality: $$ \mathrm{JB} = \frac{n}{6} \left(S^2 + \frac{(K-3)^2}{4} \right)\,, $$ where (S) is skewness and (K) is kurtosis. Returns (JB, p-value). See https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.jarque_bera.html

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pd.set_option("display.multi_sparse", True)


summary = []
for label, series in [('Returns', returns), ('Log Returns', log_returns)]:
    for ticker in tickers:
        data = series[ticker]
        stat = {
            'Ticker': ticker,
            'Type': label,
            'Mean': data.mean(),
            'Std Dev': data.std(ddof=1),
            'Skewness': skew(data, bias=False),
            'Kurtosis': kurtosis(data, fisher=True, bias=False),
        }
        jb_stat, jb_pvalue = jarque_bera(data)
        stat['Jarque-Bera'] = jb_stat
        stat['JB p-value'] = jb_pvalue
        summary.append(stat)

summary_df = pd.DataFrame(summary)
summary_df = summary_df.sort_values(by=['Ticker', 'Type'])
summary_df = summary_df.round(6)
summary_df
pd.set_option("display.multi_sparse", True)


summary = []
for label, series in [('Returns', returns), ('Log Returns', log_returns)]:
    for ticker in tickers:
        data = series[ticker]
        stat = {
            'Ticker': ticker,
            'Type': label,
            'Mean': data.mean(),
            'Std Dev': data.std(ddof=1),
            'Skewness': skew(data, bias=False),
            'Kurtosis': kurtosis(data, fisher=True, bias=False),
        }
        jb_stat, jb_pvalue = jarque_bera(data)
        stat['Jarque-Bera'] = jb_stat
        stat['JB p-value'] = jb_pvalue
        summary.append(stat)

summary_df = pd.DataFrame(summary)
summary_df = summary_df.sort_values(by=['Ticker', 'Type'])
summary_df = summary_df.round(6)
summary_df

Out[3]:

	Ticker	Type	Mean	Std Dev	Skewness	Kurtosis	Jarque-Bera	JB p-value
6	AAPL	Log Returns	0.001540	0.014244	0.193638	1.606197	26.467302	0.000002
1	AAPL	Returns	0.001643	0.014288	0.268632	1.600978	27.724448	0.000001
7	AMZN	Log Returns	0.000545	0.018975	-0.055314	3.473778	118.682638	0.000000
2	AMZN	Returns	0.000725	0.018987	0.099364	3.583976	126.658372	0.000000
9	GOOGL	Log Returns	0.002902	0.018267	1.063048	3.909024	196.698681	0.000000
4	GOOGL	Returns	0.003074	0.018507	1.195354	4.509515	259.031316	0.000000
5	META	Log Returns	-0.000677	0.022468	-0.090636	7.438957	548.145389	0.000000
0	META	Returns	-0.000426	0.022462	0.220936	7.388606	542.394973	0.000000
8	NFLX	Log Returns	-0.001624	0.020942	-0.081695	9.554998	905.382883	0.000000
3	NFLX	Returns	-0.001405	0.020923	0.286153	10.509491	1098.840831	0.000000

The table above summarizes the first- and second-moment statistics, skewness, excess kurtosis, and Jarque-Bera normality test results for both simple and log returns. The Jarque-Bera test is especially useful for detecting departures from the normal distribution due to skewness and kurtosis.

Empirical Distribution with Normal Overlay¶

Plot the empirical distribution of log returns for each ticker and overlay a normal density using the sample mean and standard deviation. This helps visualize departures from normality.

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fig, axes = plt.subplots(len(tickers)+1, 1, figsize=(4, 4*len(tickers)))
axes = axes.flatten()

for idx, ticker in enumerate(tickers):
    data = log_returns[ticker].dropna()
    mu = data.mean()
    sigma = data.std(ddof=1)
    x = np.linspace(data.min(), data.max(), 200)
    ax = axes[idx]
    ax.hist(data, bins=30, density=True, alpha=0.6, color=f'C{idx}', edgecolor='black')
    ax.plot(x, norm.pdf(x, loc=mu, scale=sigma), color='black', linewidth=2, label='Normal approx')
    ax.axvline(mu, color='gray', linestyle='--', linewidth=1)
    ax.set_title(f'{ticker} Empirical Distribution')
    ax.set_xlabel('Log Return')
    ax.set_ylabel('Density')
    ax.legend()

axes[-1].set_visible(False)
plt.tight_layout()
plt.show()
fig, axes = plt.subplots(len(tickers)+1, 1, figsize=(4, 4*len(tickers)))
axes = axes.flatten()

for idx, ticker in enumerate(tickers):
    data = log_returns[ticker].dropna()
    mu = data.mean()
    sigma = data.std(ddof=1)
    x = np.linspace(data.min(), data.max(), 200)
    ax = axes[idx]
    ax.hist(data, bins=30, density=True, alpha=0.6, color=f'C{idx}', edgecolor='black')
    ax.plot(x, norm.pdf(x, loc=mu, scale=sigma), color='black', linewidth=2, label='Normal approx')
    ax.axvline(mu, color='gray', linestyle='--', linewidth=1)
    ax.set_title(f'{ticker} Empirical Distribution')
    ax.set_xlabel('Log Return')
    ax.set_ylabel('Density')
    ax.legend()

axes[-1].set_visible(False)
plt.tight_layout()
plt.show()

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Autocorrelation Analysis¶

Compute the autocorrelation function (ACF) of log returns and then assess serial dependence with the Ljung-Box test.

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nlags = 5
acf_results = {}
for ticker in tickers:
    acf_results[ticker] = acf(log_returns[ticker], nlags=nlags, fft=False)

acf_df = pd.DataFrame(acf_results, index=[f'Lag {i}' for i in range(nlags + 1)])
acf_df
nlags = 5
acf_results = {}
for ticker in tickers:
    acf_results[ticker] = acf(log_returns[ticker], nlags=nlags, fft=False)

acf_df = pd.DataFrame(acf_results, index=[f'Lag {i}' for i in range(nlags + 1)])
acf_df

Out[5]:

	META	AAPL	AMZN	NFLX	GOOGL
Lag 0	1.000000	1.000000	1.000000	1.000000	1.000000
Lag 1	0.016932	0.095279	0.089386	0.059752	0.065097
Lag 2	-0.019335	-0.008767	-0.010252	0.073500	-0.016948
Lag 3	-0.075196	0.052104	0.027675	0.024893	0.064377
Lag 4	-0.005060	-0.030008	-0.019361	-0.006440	0.046376
Lag 5	0.144932	-0.017792	0.091321	-0.021232	0.047239

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fig, ax = plt.subplots(figsize=(12, 6))
for ticker in tickers:
    ax.plot(range(nlags + 1), acf_results[ticker], marker='o', label=ticker, linewidth=2)

ax.axhline(0, color='black', linestyle='--', linewidth=0.8)
ax.set_xlabel('Lag')
ax.set_ylabel('Autocorrelation')
ax.set_title('ACF of Log Returns for MAANG Stocks')
ax.set_xticks(range(nlags + 1))
ax.legend()
ax.grid(alpha=0.3)
plt.tight_layout()
plt.show()
fig, ax = plt.subplots(figsize=(12, 6))
for ticker in tickers:
    ax.plot(range(nlags + 1), acf_results[ticker], marker='o', label=ticker, linewidth=2)

ax.axhline(0, color='black', linestyle='--', linewidth=0.8)
ax.set_xlabel('Lag')
ax.set_ylabel('Autocorrelation')
ax.set_title('ACF of Log Returns for MAANG Stocks')
ax.set_xticks(range(nlags + 1))
ax.legend()
ax.grid(alpha=0.3)
plt.tight_layout()
plt.show()

Ljung-Box Test for Serial Correlation¶

The Ljung-Box test checks whether there is statistically significant autocorrelation across multiple lags simultaneously. A p-value below 0.05 suggests evidence against the null hypothesis of no autocorrelation.

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results = []
for ticker in tickers:
    lb_test = acorr_ljungbox(log_returns[ticker], lags=nlags, return_df=True)
    lb_test['Ticker'] = ticker
    results.append(lb_test)

lb_df = pd.concat(results, ignore_index=False)
lb_df = lb_df.reset_index().rename(columns={'index': 'Lag'})
lb_df['Lag'] = lb_df['Lag'] + 1
lb_df = lb_df[['Ticker', 'Lag', 'lb_stat', 'lb_pvalue']].rename(columns={
    'lb_stat': 'LB Statistic',
    'lb_pvalue': 'p-value'
})
lb_df
results = []
for ticker in tickers:
    lb_test = acorr_ljungbox(log_returns[ticker], lags=nlags, return_df=True)
    lb_test['Ticker'] = ticker
    results.append(lb_test)

lb_df = pd.concat(results, ignore_index=False)
lb_df = lb_df.reset_index().rename(columns={'index': 'Lag'})
lb_df['Lag'] = lb_df['Lag'] + 1
lb_df = lb_df[['Ticker', 'Lag', 'lb_stat', 'lb_pvalue']].rename(columns={
    'lb_stat': 'LB Statistic',
    'lb_pvalue': 'p-value'
})
lb_df

Out[7]:

	Ticker	Lag	LB Statistic	p-value
0	META	2	0.072253	0.788084
1	META	3	0.166844	0.919963
2	META	4	1.603413	0.658616
3	META	5	1.609945	0.807004
4	META	6	6.990275	0.221365
5	AAPL	2	2.287778	0.130396
6	AAPL	3	2.307226	0.315495
7	AAPL	4	2.996958	0.392095
8	AAPL	5	3.226664	0.520635
9	AAPL	6	3.307751	0.652655
10	AMZN	2	2.013546	0.155901
11	AMZN	3	2.040141	0.360570
12	AMZN	4	2.234723	0.525142
13	AMZN	5	2.330350	0.675249
14	AMZN	6	4.466465	0.484383
15	NFLX	2	0.899754	0.342848
16	NFLX	3	2.266710	0.321951
17	NFLX	4	2.424137	0.489157
18	NFLX	5	2.434716	0.656363
19	NFLX	6	2.550181	0.768920
20	GOOGL	2	1.067935	0.301412
21	GOOGL	3	1.140616	0.565351
22	GOOGL	4	2.193547	0.533221
23	GOOGL	5	2.742206	0.601850
24	GOOGL	6	3.313803	0.651729

Ljung-Box Interpretation¶

If the Ljung-Box test fails to reject the null hypothesis, the log returns are consistent with a white noise process. A white noise series is defined as a sequence of random variables with constant mean, constant variance, and zero autocorrelation at all non-zero lags.

In this notebook, the Ljung-Box results do not provide strong evidence of predictive lagged structure in the MAANG log returns, supporting the idea that the series behaves like white noise over this sample period.