JUNRAN WANG’s Untitled project

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.

!pip install xlrd !pip install yfinance !pip install pyPortfolioOpt import pandas as pd import xlrd import numpy as np import yfinance as yf from pypfopt import risk_models from pypfopt import expected_returns from pypfopt.efficient_frontier import EfficientFrontier

Read csv document, convert the first column to a proper datetime format, and set the converted datetime as the index of the dataframe.

file_path = '/work/F-F_Research_Data_5_Factors_2x3_daily.xls' factors_data = pd.read_excel(file_path) factors_data['Unnamed: 0'] = pd.to_datetime(factors_data['Unnamed: 0'], format='%Y%m%d') factors_data.set_index('Unnamed: 0', inplace=True) factors_data.index.name = 'Date' print(factors_data)

Create the maximum Sharpe portfolio's daily return dataframe.

assets = ['AAPL', 'BAC', 'VZ', 'KO', 'XOM', 'TLT', 'IEI', 'GLD'] data = yf.download(assets, start='2018-08-30', end='2023-08-30', progress=False)['Adj Close'].dropna() mu = expected_returns.mean_historical_return(data) Sigma = risk_models.sample_cov(data) ef = EfficientFrontier(mu, Sigma) ef.max_sharpe() cleaned_weights_maxsharpe = np.array(list(ef.clean_weights().values())) data_maximum_sharpe = data.dot(cleaned_weights_maxsharpe) daily_returns = pd.DataFrame(data_maximum_sharpe.pct_change().dropna(), columns=['Daily Returns']) print(daily_returns)

Merge the maximum Sharpe portfolio's daily return dataframe with Fama French's five return factors.

merged_data = pd.merge(daily_returns, factors_data, how="inner", on="Date") print(merged_data)

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

import matplotlib.pyplot as plt import seaborn as sns

Calculate the correlation Matrix for Fama-French factors.

merged_data_without_RF = merged_data.drop(columns=['RF']) corr_matrix = merged_data_without_RF.corr() sns.set(font_scale=0.5) plt.figure(figsize=(5,2.5)) sns.heatmap(corr_matrix, annot=True, linewidths=.5) plt.title('Correlation Matrix 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?

!pip install statsmodels import statsmodels.api as sm

To facilitate the comparison of whether the intercept is significantly different from the risk-free rate, I set the dependent variable in the regression to be the daily returns minus the risk-free rate, which is the excess returns. This way, I only need to test if the intercept is significantly different from zero.

Y = merged_data['Daily Returns']-merged_data['RF']

The regression coefficient for the 'Mkt-RF' factor is 0.0040, indicating that the portfolio is relatively sensitive to this factor. The p-value of intercept is 0.000, indicating that the intercept is significantly different from the risk-free rate.

X_market_rf = merged_data['Mkt-RF'] X_market_rf = sm.add_constant(X_market_rf) model_market_rf = sm.OLS(Y, X_market_rf).fit() print(model_market_rf.summary())

The regression coefficient for the 'SMB' factor is -3.18e-06, indicating that the portfolio is not sensitive to this factor. However, the p-value of 'SMB' factor is 0.994, indicating that the coefficient is not significant. The p-value of intercept is 0.000, indicating that the intercept is significantly different from the risk-free rate.

X_smb = merged_data['SMB'] X_smb = sm.add_constant(X_smb) model_smb = sm.OLS(Y, X_smb).fit() print(model_smb.summary())

The regression coefficient for the 'HML' factor is -0.0022, indicating that the portfolio is relatively sensitive to this factor. The p-value of intercept is 0.000, indicating that the intercept is significantly different from the risk-free rate.

X_hml = merged_data['HML'] X_hml = sm.add_constant(X_hml) model_hml = sm.OLS(Y, X_hml).fit() print(model_hml.summary())

The regression coefficient for the 'RMW' factor is -0.0019, indicating that the portfolio is relatively sensitive to this factor. The p-value of intercept is 0.000, indicating that the intercept is significantly different from the risk-free rate.

X_rmw = merged_data['RMW'] X_rmw = sm.add_constant(X_rmw) model_rmw = sm.OLS(Y, X_rmw).fit() print(model_rmw.summary())

The regression coefficient for the 'CMA' factor is -0.0029, indicating that the portfolio is relatively sensitive to this factor. The p-value of intercept is 0.000, indicating that the intercept is significantly different from the risk-free rate.

X_cma = merged_data['CMA'] X_cma = sm.add_constant(X_cma) model_cma = sm.OLS(Y, X_cma).fit() print(model_cma.summary())

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?

From the regression results, it can be observed that ‘SMB’ and ‘RMW’ are not statistically significant and thus should be excluded. The portfolio shows the highest sensitivity to 'CMA' and 'Mkt-RF'. Additionally, the p-value of the intercept is 0.000, indicating that the intercept is significantly different from the risk-free rate.

X_all_factors = merged_data[['Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA']] X_all_factors = sm.add_constant(X_all_factors) model_all_factors = sm.OLS(Y, X_all_factors).fit() print(model_all_factors.summary())