Data Lab 4

1. Revisit the all-weather portfolio you crafted. Create the maximum Sharpe portfolio's daily return dataframe and then merge it with Fama French's five return factors.

2. Examine visually the correlation between portfolio and factor returns.

3. Regress the portfolio return on each factor and assess the portfolio's sensitivity to each factor. For the curious, optional challenge, how do you test whether the intercept (i.e., alpha) is significantly different from the risk-free rate for a single-factor regression?

4. Regress the portfolio return on all factors and assess the portfolio's sensitivity to factors. For the curious, optional challenge, how do you test whether the intercept (i.e., alpha) is significantly different from the risk-free rate for a multi-factor regression?

5. Optional Bonus. Construct a multi-factor pricing model for assets based on Arbitrage Pricing Theory. The Arbitrage Pricing Theory (APT) is a theory of asset pricing that holds that an asset’s returns can be forecasted with the linear relationship between an asset’s expected returns and the macroeconomic (e.g., GDP, changes in inflation, yield curve changes, changes in interest rates, market sentiments, exchange rates) or firm-specific statistical factors that affect the asset’s risk. Hint: You can draw these variables straight into your Jupyter notebook via Refinitiv API.

Setting up the workspace

!pip install pandas !pip install yfinance !pip install statsmodels==0.14.0 import yfinance as yf import matplotlib.pyplot as plt import pandas as pd import numpy as np import statsmodels.api as sm import seaborn as sns

Question 1

FamaFrench = pd.read_csv('/work/F-F_Research_Data_5_Factors_2x3_daily.txt', parse_dates=['Unnamed: 0']) FamaFrench = FamaFrench.rename(columns={'Unnamed: 0': 'Date'})

FamaFrench.set_index('Date', inplace=True) print(FamaFrench)

# Define the stock symbols of all the stocks in our portfolio stock_symbols = ["PM", "WMT", "BRK-A", "AMZN", "GOOG"] # Define the period as 5 years start_date = "2018-09-01" end_date = "2023-09-01" # Get the stock data from yahoo finance stock_data = yf.download(stock_symbols, start=start_date, end=end_date) # Calculate daily adjusted returns daily_returns = stock_data['Adj Close'].pct_change() daily_returns = daily_returns[1:] print(daily_returns) print(type(daily_returns))

# Take the weights for the different portfolios from Data Lab 2 max_sharpe_weights = [0.0, 0.17837, 0.30187, 0.04334, 0.47643]

# Calculate the portfolio returns by taking the dot product of weights and returns for each day portfolio_returns = pd.DataFrame() portfolio_returns["Daily Return"] = daily_returns.dot(max_sharpe_weights)

print(portfolio_returns) print(type(portfolio_returns))

# Merge the dataframes based on the 'Date' index merged_data = pd.merge(portfolio_returns, FamaFrench, left_index=True, right_index=True, how='inner') print(merged_data)

Question 2

First, we visualize the correlation through a heatmap.

# Calculate the correlation matrix correlation_matrix = merged_data.corr() # Create a heatmap to visualize the correlation plt.figure(figsize=(10, 8)) sns.heatmap(correlation_matrix, annot=True, cmap='coolwarm', linewidths=0.5) plt.title("Correlation Heatmap between Portfolio and Factor Returns") plt.show()

Secondly, we visualize the correlation between the portolio and the individual factors as a rolling time-series.

# List of factor columns factor_columns = ['Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA', 'RF'] # Calculate rolling correlations for each factor rolling_correlations = {} window_size = 30 # You can adjust the window size as needed for factor in factor_columns: rolling_correlations[factor] = merged_data['Daily Return'].rolling(window=window_size).corr(merged_data[factor]) # Create a plot for each factor's rolling correlation plt.figure(figsize=(12, 6)) for factor in factor_columns: plt.plot(merged_data.index, rolling_correlations[factor], label=factor) plt.title('Rolling Correlations Between Portfolio and Factors') plt.xlabel('Date') plt.ylabel('Correlation') plt.legend() plt.grid(True) plt.show()

Question 3

X = sm.add_constant(merged_data['Mkt-RF']) # Add a constant term for the intercept y = merged_data['Daily Return'] # Perform the regression model = sm.OLS(y, X).fit() # Print the regression summary print(model.summary()) factors = ['Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA'] for factor in factors: X = sm.add_constant(merged_data[factor]) y = merged_data['Daily Return'] model = sm.OLS(y, X).fit() print(f"Factor: {factor}") print(model.summary()) print("\n")

Question 4

X = sm.add_constant(merged_data[['Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA', 'RF']]) # Add a constant term for the intercept y = merged_data['Daily Return'] # Perform the multiple linear regression model = sm.OLS(y, X).fit() # Print the regression summary print(model.summary())