Statistical analysis CAC40(^FCHI)

What is CAC40?

CAC40(^FCHI) is the French stock market. It measures of the 40 most significant stocks.

Statistical analysis of the CAC 40, or the CAC 40 Index, involves examining historical data and applying various statistical techniques to gain insights into the performance, volatility, and characteristics of this benchmark stock market index representing the 40 largest companies listed on Euronext Paris. Here's how you can conduct a statistical analysis of the CAC 40:

Data Retrieval:

Begin by obtaining historical CAC 40 index data. You can typically access this data from financial data providers, stock exchanges, or financial websites that offer historical market data.

Data Exploration:

Import the data into statistical analysis software like Python (using Pandas and NumPy) or statistical packages like R. Start by exploring the data's structure and contents. This step involves checking for missing values, and outliers, and understanding the data's time frame.

Descriptive Statistics:

Calculate and examine basic statistical measures to describe the CAC 40's historical performance, including:

Mean (average) daily returns

Standard deviation (volatility) of returns

Skewness (asymmetry) and kurtosis (tail heaviness) of return distributions

Maximum and minimum values

These statistics provide insights into the index's historical behavior.

Time Series Analysis:

Conduct time series analysis to explore patterns, trends, and seasonality in CAC 40 returns. Time series analysis may involve techniques like autoregressive integrated moving average (ARIMA) modeling or seasonal decomposition of time series (STL decomposition).

Volatility Analysis:

Calculate and analyze volatility measures such as historical volatility and implied volatility (if options data is available). Volatility analysis helps assess risk and can be used for options pricing and risk management.

Correlation Analysis:

Investigate the relationships between CAC 40 returns and other variables. For instance, you can calculate correlations with economic indicators, interest rates, or exchange rates to understand the index's sensitivity to external factors.

Hypothesis Testing:

Formulate and test hypotheses about the CAC 40's performance and behavior. You can use statistical tests to determine if certain events or factors have had a statistically significant impact on the index.

Regression Analysis:

Perform regression analysis to model and predict CAC 40 returns based on independent variables. For example, you might explore how changes in interest rates or GDP growth affect the index.

Monte Carlo Simulations (Optional):

Use Monte Carlo simulations to model potential future scenarios for the CAC 40 based on historical data and assumed statistical distributions. This can be valuable for risk assessment and portfolio optimization.

Visualization:

Create visualizations, such as line charts, histograms, and scatter plots, to present your findings effectively. Visualizations can help in conveying statistical insights to a broader audience.

Interpretation and Reporting:

Interpret the results of your statistical analysis and prepare reports or presentations summarizing your findings. Clearly communicate any insights, trends, or implications for investment strategies.

Continuous Monitoring:

Regularly update your analysis to adapt to changing market conditions and incorporate new data as it becomes available.

Statistical analysis of the CAC 40 is valuable for investors, traders, and financial analysts looking to make informed decisions, manage risk, and understand the behavior of one of Europe's most prominent stock market indices.

How to get historical data cac40 in Python?

Some code has been written below to collect data from Yahoo Finance.

df = yf.download('^FCHI',

start='1985-01-01',

end='2021-07-28',

progress=False)

df = df.loc[:, ['Adj Close']]

df.rename(columns={'Adj Close':'adj_close'}, inplace=True)

df['simple_rtn'] = df.adj_close.pct_change()

df['log_rtn'] = np.log(df.adj_close/df.adj_close.shift(1))

df[['simple_rtn','log_rtn']].tail(20)

How to see Cac40 log and simple data in Python?

Get simple and log the return

Define Realise volatility

def realized_volatility(x):

 return np.sqrt(np.sum(x**2))

df_rv = df.groupby(pd.Grouper(freq='M')).apply(realized_volatility)

df_rv.rename(columns={'log_rtn': 'rv'}, inplace=True)

df_rv.rv = df_rv.rv * np.sqrt(12)

fig, ax = plt.subplots(2, 1, sharex=True)

ax[0].plot(df)

ax[1].plot(df_rv)

[<matplotlib.lines.Line2D at 0x7f703738a710>,
 <matplotlib.lines.Line2D at 0x7f7037333a50>,
 <matplotlib.lines.Line2D at 0x7f7037333c10>]

realized_volatility

fig, ax = plt.subplots(3, 1, figsize=(24, 20), sharex=True)

df.adj_close.plot(ax=ax[0])

ax[0].set(title = 'CAC40 time series',

ylabel = 'Stock price ($)')

df.simple_rtn.plot(ax=ax[1])

ax[1].set(ylabel = 'Simple returns (%)')

df.log_rtn.plot(ax=ax[2])

ax[2].set(xlabel = 'Date',

ylabel = 'Log returns (%)')

[Text(0, 0.5, 'Log returns (%)'), Text(0.5, 0, 'Date')]

Plot price cac40, simple and log price

import cufflinks as cf
from plotly.offline import iplot, init_notebook_mode
init_notebook_mode()
df_rolling = df[['simple_rtn']].rolling(window=21) \
.agg(['mean', 'std'])
df_rolling.columns = df_rolling.columns.droplevel()
df_outliers = df.join(df_rolling)
def indentify_outliers(row, n_sigmas=3):
   x = row['simple_rtn']
   mu = row['mean']
   sigma = row['std']
   if (x > mu + 3 * sigma) | (x < mu - 3 * sigma):
    return 1
   else:
    return 0
df_outliers['outlier'] = df_outliers.apply(indentify_outliers,
axis=1)
outliers = df_outliers.loc[df_outliers['outlier'] == 1,
['simple_rtn']]
fig, ax = plt.subplots()
ax.plot(df_outliers.index, df_outliers.simple_rtn,
color='blue', label='Normal')
ax.scatter(outliers.index, outliers.simple_rtn,
color='red', label='Anomaly')
ax.set_title("CAC40 returns")
ax.legend(loc='lower right')
<matplotlib.legend.Legend at 0x7f70323a8910>
CAC40 return
r_range = np.linspace(min(df.log_rtn), max(df.log_rtn), num=1000)
mu = df.log_rtn.mean()
sigma = df.log_rtn.std()
norm_pdf = scs.norm.pdf(r_range, loc=mu, scale=sigma)
fig, ax = plt.subplots(1, 2, figsize=(16, 8))
# histogram
sns.distplot(df.log_rtn, kde=False, norm_hist=True, ax=ax[0])
ax[0].set_title('Distribution of CAC40 returns', fontsize=16)
ax[0].plot(r_range, norm_pdf, 'g', lw=2,
label=f'N({mu:.2f}, {sigma**2:.4f})')
ax[0].legend(loc='upper left');
/usr/local/lib/python3.7/dist-packages/seaborn/distributions.py:2557: FutureWarning:

`distplot` is a deprecated function and will be removed in a future version. Please adapt your code to use either `displot` (a figure-level function with similar flexibility) or `histplot` (an axes-level function for histograms).

Distribution of CAC40 returns
df.log_rtn.plot(title='Daily CAC40 returns')
<matplotlib.axes._subplots.AxesSubplot at 0x7f7032068b50>
'Daily CAC40 returns
df = yf.download(['^FCHI', '^VIX'],
start='1985-01-01',
end='2021-07-28',
progress=False)
adjacent price data cac40
df = df[['Adj Close']]
df.columns = df.columns.droplevel(0)
df = df.rename(columns={'^FCHI': 'Cac40', '^VIX': 'vix'})
df['log_rtn'] = np.log(df.Cac40 / df.Cac40.shift(1))
df['vol_rtn'] = np.log(df.vix / df.vix.shift(1))
df.dropna(how='any', axis=0, inplace=True)
corr_coeff = df.log_rtn.corr(df.vol_rtn)
ax = sns.regplot(x='log_rtn', y='vol_rtn', data=df,
line_kws={'color': 'red'})
ax.set(title=f'FCHI vs. VIX ($\\rho$ = {corr_coeff:.2f})',
ylabel='VIX log returns',
xlabel='FTSE100 log returns')
[Text(0, 0.5, 'VIX log returns'),
 Text(0.5, 0, 'FTSE100 log returns'),
 Text(0.5, 1.0, 'FCHI vs. VIX ($\\rho$ = -0.42)')]
CAC40 AND VIX correlation chart
We can see that both the negative slope of the regression line and a strong negative correlation between the two series confirm the existence of the leverage effect in the return series
 more updates to follow CAC40