DataLabs

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.

# Required Libraries import pandas as pd import pandas_datareader as pdr import datetime import yfinance as yf import numpy as np import matplotlib.pyplot as plt import statsmodels.api as sm # Define the list of stock tickers tickers = ['AAPL','V','XOM','KO','AEP'] # Fetch historical data for the past 5 years end_date = datetime.datetime.now() start_date = end_date - datetime.timedelta(days=5*365) data = yf.download(tickers, start = start_date, end = end_date)['Adj Close'] # Calculate daily returns for each stock stock_returns = data.pct_change().dropna() # Calculate annualized mean return and standard deviation of each stock mean_daily_returns = stock_returns.mean() mean_annual_returns = mean_daily_returns * 252 std_deviation = stock_returns.std() * np.sqrt(252) # Portfolio Weights for different portfolios(Calculated in previous datalab) max_sharpe_weights = np.array([0.92166, 0.0, 0.03532, 0.0 ,0.04302]) # Calculate Daily Portfolio Returns daily_max_sharpe_returns = stock_returns.dot(max_sharpe_weights) # Create DataFrame df_portfolios = pd.DataFrame({'Max_Sharpe': daily_max_sharpe_returns}) # Calculate Cumulative Returns cumulative_returns = (1 + df_portfolios).cumprod() # Put them side-by-side in a DataFrame ff_data = pd.read_csv('F-F_Research_Data_5_Factors_2x3_daily.CSV',skiprows=3) ff_data.set_index('Unnamed: 0', inplace=True) ff_data.index = pd.to_datetime(ff_data.index.astype(str), format='%Y%m%d') merged_data = df_portfolios.merge(ff_data, left_index=True, right_index=True) print(merged_data)

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

factors = ['Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA'] for factor in factors: plt.figure(figsize=(10,6)) plt.scatter(merged_data['Max_Sharpe'], merged_data[factor]) plt.title(f'Correlation between Max_Sharpe and {factor}') plt.xlabel('Max_Sharpe Returns') plt.ylabel(f'{factor} Returns') plt.grid(True) plt.show() import seaborn as sns # Compute the correlation matrix k = merged_df.drop(['Ticker','Year'],axis = 1, inplace=True) corr = k.corr() # Draw the heatmap plt.figure(figsize=(10,8)) sns.heatmap(corr, annot=True, cmap="coolwarm", center=0) plt.title('Correlation Heatmap') plt.show()

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?

factors = ['Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA'] sensitivities = {} for factor in factors: # Set up the independent variable with a constant term X = sm.add_constant(merged_data[factor]) # Run the regression model = sm.OLS(merged_data['Max_Sharpe'], X) results = model.fit() # Store the coefficient (sensitivity) of the factor sensitivities[factor] = results.params[factor] print(f"\nRegression results for factor {factor}:") print(results.summary()) print("\nSensitivities of Portfolio to Each Factor:") for factor, sensitivity in sensitivities.items(): print(f"{factor}: {sensitivity:.4f}")

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?

import statsmodels.api as sm X = sm.add_constant(merged_data[['Mkt-RF','SMB', 'HML','RMW','CMA']]) model = sm.OLS(merged_data['Max_Sharpe'], X).fit() b0, b1, b2, b3, b4, b5 = model.params print('Intercept (Alpha): %f' % b0) print('Sensitivities of active returns to factors:\nMkt-Rf: %f\nSMB: %f\nHML: %f\nRMW: %f\nCMA: %f' % (b1, b2, b3, b4, b5)) X_multi = merged_data[['Mkt-RF','SMB', 'HML','RMW','CMA']] X_multi = sm.add_constant(X_multi) # Adds a constant term to the predictors model_multi = sm.OLS(merged_data['Max_Sharpe'], X_multi) results_multi = model_multi.fit() print(results_multi.summary())

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.

The APT is a substitute for the Capital Asset Pricing Model (CAPM) in that both assert a linear relation between assets’ expected returns and their covariance with other random variables. (In the CAPM, the covariance is with the market portfolio’s return.) The covariance is interpreted as a measure of risk that investors cannot avoid by diversification. The slope coefficient in the linear relation between the expected returns and the covariance is interpreted as a risk premium ~ "Arbitrage Pricing Theory (Guberman and Wang 2005).