!pip install PyPortfolioOpt !pip install pandas numpy matplotlib yfinance import numpy as np import pandas as pd import yfinance as yf from pypfopt.efficient_frontier import EfficientFrontier from pypfopt import risk_models from pypfopt import expected_returns import matplotlib.pyplot as plt

# Define the list of stock tickers tickers = ['AAPL', 'AMZN', 'JNJ', 'JPM', 'XOM'] # Download historical data from Yahoo Finance data = yf.download(tickers, start='2018-09-03', end='2023-09-03')['Adj Close'] # Calculate expected returns and sample covariance mu = expected_returns.mean_historical_return(data) S = risk_models.sample_cov(data) # Create a new EfficientFrontier object for the maximum Sharpe ratio portfolio ef_max_sharpe = EfficientFrontier(mu, S) # Calculate the expected returns, volatility, and Sharpe ratio for the maximum Sharpe ratio portfolio max_sharpe_weights = ef_max_sharpe.max_sharpe() max_sharpe_performance = ef_max_sharpe.portfolio_performance(risk_free_rate=0.0537) # On August 31, 2023, the yield on short-term U.S. Treasury bonds is 5.37%. # Calculate daily returns based on portfolio weights portfolio_returns_max_sharpe = (data.pct_change() * max_sharpe_weights).sum(axis=1) # Calculate daily return statistics mean_daily_return_max_sharpe = portfolio_returns_max_sharpe.mean() volatility_max_sharpe = portfolio_returns_max_sharpe.std() skewness_max_sharpe = portfolio_returns_max_sharpe.skew() kurtosis_max_sharpe = portfolio_returns_max_sharpe.kurtosis() cumulative_returns_max_sharpe = (portfolio_returns_max_sharpe + 1).cumprod()

# 1. Create the maximum Sharpe portfolio's daily return dataframe and then merge it with Fama French's five return factors # Create a DataFrame to store daily returns daily_returns_df = pd.DataFrame(portfolio_returns_max_sharpe, columns=['Daily_Return']) print(daily_returns_df.head())

!pip install xlrd # Read Fama-French factors data from Excel fama_french_data = pd.read_excel('/work/F-F_Research_Data_5_Factors_2x3_daily.xls') # Check the date column is in datetime format fama_french_data['Date'] = pd.to_datetime(fama_french_data['Date']) print(fama_french_data.head())

# Merge the portfolio returns with Fama-French data using the 'Date' column merged_data = pd.merge(daily_returns_df, fama_french_data, on='Date', how='inner') print(merged_data.head())

import seaborn as sns #2. Examine visually the correlation between portfolio and factor returns. # Calculate the correlation matrix between 'Daily_Return' and factor returns correlation_matrix = merged_data[['Daily_Return', 'Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA']].corr() # Create a heatmap to visualize the correlation matrix plt.figure(figsize=(10, 6)) sns.heatmap(correlation_matrix, annot=True, cmap='coolwarm', fmt=".2f", linewidths=0.5) plt.title("Correlation Heatmap: Daily_Return vs. Factor Returns") plt.show()

!pip install statsmodels

#3. Regress the portfolio return on each factor and assess the portfolio's sensitivity to each factor. import statsmodels.api as sm # Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['Mkt-RF'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform a simple linear regression model = sm.OLS(y, X) result = model.fit() # Print the regression summary print(result.summary())

import statsmodels.api as sm # Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['SMB'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform a simple linear regression model = sm.OLS(y, X) result = model.fit() # Print the regression summary print(result.summary())

import statsmodels.api as sm # Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['HML'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform a simple linear regression model = sm.OLS(y, X) result = model.fit() # Print the regression summary print(result.summary())

import statsmodels.api as sm # Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['RMW'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform a simple linear regression model = sm.OLS(y, X) result = model.fit() # Print the regression summary print(result.summary())

import statsmodels.api as sm # Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['CMA'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform a simple linear regression model = sm.OLS(y, X) result = model.fit() # Print the regression summary print(result.summary())

#3.2 how do you test whether the intercept (i.e., alpha) is significantly different from the risk-free rate for a multi-factor regression import statsmodels.api as sm from scipy import stats # Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['Mkt-RF'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform the linear regression model = sm.OLS(y, X) result = model.fit() # Get the coefficients and standard errors alpha = result.params['const'] alpha_se = result.bse['const'] RF = merged_data['RF'] # Calculate the t-statistic t_statistic = (alpha - RF) / alpha_se # Calculate the p-value element-wise p_values = 2 * (1 - stats.t.cdf(abs(t_statistic), df=result.df_resid)) # Check if any p-value is less than the significance level alpha_level = 0.05 # Significance level if any(p_values < alpha_level): print("Reject the null hypothesis. Alpha is significantly different from the risk-free rate.") else: print("Fail to reject the null hypothesis. Alpha is not significantly different from the risk-free rate.")

#try more variables # Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['SMB'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform the linear regression model = sm.OLS(y, X) result = model.fit() # Get the coefficients and standard errors alpha = result.params['const'] alpha_se = result.bse['const'] RF = merged_data['RF'] # Calculate the t-statistic t_statistic = (alpha - RF) / alpha_se # Calculate the p-value element-wise p_values = 2 * (1 - stats.t.cdf(abs(t_statistic), df=result.df_resid)) # Check if any p-value is less than the significance level alpha_level = 0.05 # Significance level if any(p_values < alpha_level): print("Reject the null hypothesis. Alpha is significantly different from the risk-free rate.") else: print("Fail to reject the null hypothesis. Alpha is not significantly different from the risk-free rate.")

# Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['HML'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform the linear regression model = sm.OLS(y, X) result = model.fit() # Get the coefficients and standard errors alpha = result.params['const'] alpha_se = result.bse['const'] RF = merged_data['RF'] # Calculate the t-statistic t_statistic = (alpha - RF) / alpha_se # Calculate the p-value element-wise p_values = 2 * (1 - stats.t.cdf(abs(t_statistic), df=result.df_resid)) # Check if any p-value is less than the significance level alpha_level = 0.05 # Significance level if any(p_values < alpha_level): print("Reject the null hypothesis. Alpha is significantly different from the risk-free rate.") else: print("Fail to reject the null hypothesis. Alpha is not significantly different from the risk-free rate.")

# Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['RMW'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform the linear regression model = sm.OLS(y, X) result = model.fit() # Get the coefficients and standard errors alpha = result.params['const'] alpha_se = result.bse['const'] RF = merged_data['RF'] # Calculate the t-statistic t_statistic = (alpha - RF) / alpha_se # Calculate the p-value element-wise p_values = 2 * (1 - stats.t.cdf(abs(t_statistic), df=result.df_resid)) # Check if any p-value is less than the significance level alpha_level = 0.05 # Significance level if any(p_values < alpha_level): print("Reject the null hypothesis. Alpha is significantly different from the risk-free rate.") else: print("Fail to reject the null hypothesis. Alpha is not significantly different from the risk-free rate.")

# Define the dependent variable (y) and the independent variable (X) y = merged_data['Daily_Return'] X = merged_data['CMA'] # Add a constant (intercept) to the independent variable X = sm.add_constant(X) # Perform the linear regression model = sm.OLS(y, X) result = model.fit() # Get the coefficients and standard errors alpha = result.params['const'] alpha_se = result.bse['const'] RF = merged_data['RF'] # Calculate the t-statistic t_statistic = (alpha - RF) / alpha_se # Calculate the p-value element-wise p_values = 2 * (1 - stats.t.cdf(abs(t_statistic), df=result.df_resid)) # Check if any p-value is less than the significance level alpha_level = 0.05 # Significance level if any(p_values < alpha_level): print("Reject the null hypothesis. Alpha is significantly different from the risk-free rate.") else: print("Fail to reject the null hypothesis. Alpha is not significantly different from the risk-free rate.")

#4. Regress the portfolio return on all factors and assess the portfolio's sensitivity to factors. # Define the dependent variable (y) and the independent variables (X) including all factors y = merged_data['Daily_Return'] X = merged_data[['Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA']] # Include all the factors # Add a constant (intercept) to the independent variables X = sm.add_constant(X) # Perform the multiple linear regression model = sm.OLS(y, X) result = model.fit() # Print the regression summary print(result.summary())

#4.2 test whether the intercept (i.e., alpha) is significantly different from the risk-free rate for a multi-factor regression # Define the dependent variable (y) and the independent variables (X) including all factors y = merged_data['Daily_Return'] X = merged_data[['Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA']] # Include all factors # Add a constant (intercept) to the independent variables X = sm.add_constant(X) # Perform the multiple linear regression model = sm.OLS(y, X) result = model.fit() # Get the coefficients and standard errors alpha = result.params['const'] alpha_se = result.bse['const'] RF = merged_data['RF'] # Calculate the t-statistic t_statistic = (alpha - RF) / alpha_se # Calculate the p-value element-wise p_values = 2 * (1 - abs(stats.t.cdf(t_statistic, df=result.df_resid))) # Check if any p-value is less than the significance level alpha_level = 0.05 # Significance level if any(p_values < alpha_level): print("Reject the null hypothesis. Alpha is significantly different from the risk-free rate.") else: print("Fail to reject the null hypothesis. Alpha is not significantly different from the risk-free rate.")