Individual Data Lab 2

- Portfolio Construction (portfolio optimization)

Note 1: Review what you have learned in DataCamp assignments (e.g., Introduction to Portfolio Analysis in Python).¶ Note 2: To deep dive into what each function does, use the help function, e.g., run "help(npf.npv)."

1. Craft an all-weather portfolio. Pick your own portfolio of stocks following an investment theme that has stood and will stand the test of time. Extract the data from Yahoo Finance/Refinitiv/any other platforms and explore the statistical distribution of each stock's past 5-year returns.

# Install yahoo finance API package !pip install yfinance

# Import import yfinance as yf import pandas as pd import numpy as np import datetime as dt import matplotlib.pyplot as plt # Define a list of stock tickers for your portfolio # 10 companies: Apple, Microsoft, Google, Amazon, NVIDIA, Tesla, Meta, UnitedHealth, Walmart, JPMorgan Chase stock_tickers = ["AAPL", "MSFT", "GOOGL", "AMZN", "NVDA", "TSLA", "META", "UNH", "WMT", "JPM"] # start and end dates for past 5-year start_date = dt.datetime(2018,9,8) end_date = dt.datetime(2023,9,8) # Download historical stock price data from Yahoo Finance stocks_price = yf.download(stock_tickers, start=start_date, end=end_date) # Extract adjusted closing prices adj_prices = stocks_price["Adj Close"] # Calculate the daily returns for each stock [.dropna()removes missing values)] daily_returns = adj_prices.pct_change().dropna() # Explore: # print(stocks_price) # stocks_price.info() # stocks_price.describe() # stocks_price.head() # Explore the statistical distribution of each stock's past 5-year returns. for ticker in stock_tickers: stock_returns = daily_returns[ticker] print(f"Stock Name: {ticker}") print(f"Mean: {stock_returns.mean()}") print(f"Std. dev / Volatility: {stock_returns.std()}") print(f"Skewness: {stock_returns.skew()}") print(f"Kurtosis: {stock_returns.kurtosis()}") print()

daily_returns.head()

2. Build an equally-weighted all-weather portfolio of your stock picks. Then, deep dive into the return characteristics of the portfolio (i.e., annualized portfolio return and volatility, portfolio's Sharpe ratio, portfolio's daily return profile - mean, volatility, skewness, kurtosis). Finally, draw the cumulative return of your all-weather portfolio for the past five years.

# ---------- Function : portChar. -------------- # portChar function outputs the return characteristics of the portfolio # 1. annualized portfolio return and volatility # 2. portfolio's Sharpe ratio # 3. portfolio's daily return profile - mean, volatility, skewness, kurtosis # 4. portfolio's cumulative return def portChar(daily_returns, weights): # 1. annualized portfolio return and volatility, # Calculate average return for each stock meanDailyReturns = daily_returns.mean() # Calculate average portfolio return portReturn = np.sum(meanDailyReturns * weights) # annualized portfolio return --- 252 trading days in one calendar year annualized_portReturn = portReturn * 252 print("Portfolio's Annualized Return: ", str(np.round(annualized_portReturn, 4) * 100) + '%') # Construct a covariance matrix for the daily returns data cov_matrix = (daily_returns.cov()) * 252 # Calculate the variance with the formula port_variance = np.dot(weights.T, np.dot(cov_matrix, weights)) # Calculate the standard deviation of portfolio port_stddev = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights))) # annualized portfolio volatility --- 252 trading days in one calendar year annual_portStdDev = port_stddev * np.sqrt(252) print("Portfolio's Annualized Volatility/Std. Dev: ", np.round(annual_portStdDev, 3)) # 2. portfolio's Sharpe ratio # 10 Year Treasury Rate (I:10YTCMR) # Risk-Free-Rate 4.26% for Sep 08 2023 risk_free = 0.042 port_SharpeRatio = (annualized_portReturn - risk_free) / annual_portStdDev print("Portfolio's Sharpe Ratio: ", str(np.round(port_SharpeRatio, 4) * 100), '%') # 3. portfolio's daily return profile - mean, volatility, skewness, kurtosis # portfolio's daily returns port_daily_Return = daily_returns.dot(weights) print("Portfolio daily return - Mean : ", str(np.round(port_daily_Return.mean(), 4) * 100) + '%') print("Portfolio daily return - Standard Deviation : ", str(np.round(port_daily_Return.std(), 4) * 100) + '%') print("Portfolio daily return - skewness : ", np.round(port_daily_Return.skew(), 4)) print("Portfolio daily return - Kurtosis : ", np.round(port_daily_Return.kurtosis(), 4)) # 4. portfolio's cumulative return # # Create portfolio returns column # daily_returns['Portfolio']= daily_returns.dot(weights) # # Compound the [daily] portfolio percentage returns over time # port_daily_Cum_Return = (1 + daily_returns).cumprod() port_daily_Cum_Return = (1 + port_daily_Return).cumprod() # RETURN ----------- return daily_returns.index, port_daily_Cum_Return # --------------------- Excute the portChar function ------------------ # Weights - an equally-weighted all-weather portfolio weights = np.repeat(1 / len(stock_tickers), len(stock_tickers)) index, port_Cum_Return = portChar(daily_returns, weights) # draw the cumulative return of your all-weather portfolio for the past five years. fig, ax = plt.subplots() ax.plot(index, port_Cum_Return, color='blue', label="portfolio") plt.title('Cumulative Return of the All-Weather Portfolio for the past 5 years') plt.xlabel('Date') plt.ylabel('Cumulative Return') # ax.xaxis.set_major_locator(plt.dates.YearLocator()) plt.grid(True) plt.legend() plt.show()

3. Repeat (2) but with the maximum Sharpe portfolio and minimum volatility portfolio.

!pip install PyPortfolioOpt

3.1 With maximum Sharpe portfolio

from pypfopt.efficient_frontier import EfficientFrontier from pypfopt import risk_models from pypfopt import expected_returns # ------------- IMPORTANT. ---------------- # need daily price data as the input. - SO can not use [daily_returns], use [adj_prices] # available data： # [daily_returns] - the daily returns for each stock # [adj_prices] - extract adjusted closing prices for stocks # ---------------- ---------------- ---------------- ---------------- ---------------- # ------------- ef ---------------- # Calculate expected annualized returns and sample covariance / risk mu = expected_returns.mean_historical_return(adj_prices) Sigma = risk_models.sample_cov(adj_prices) # Calculate the Efficient Frontier with mu and S ef = EfficientFrontier(mu, Sigma) # ------------ the Maximum Sharpe ratio portfolio ------------------- raw_weights_MaxSharpe = ef.max_sharpe() cleaned_weights_MaxSharpe = ef.clean_weights() portfolio_returns_MaxSharpe = ef.portfolio_performance(verbose=True) # ---------- Function : convert_ef_weight_2arrayweight -------------- # Output of ef weights is a dictionary, extract the weight in Array type. def convert_ef_weight_2arrayweight (ef_weight, stock_tickers): list_weight = [] for ticker in stock_tickers: weight = ef_weight[ticker] list_weight.append(weight) array_weight = np.array(list_weight) return array_weight # ----------- Plot ------------- # ---get array weight for MaxSharpe array_weight_MaxSharpe = convert_ef_weight_2arrayweight(cleaned_weights_MaxSharpe, stock_tickers) print(array_weight_MaxSharpe) MaxSharpe_index, MaxSharpe_port_Cum_Return = portChar(daily_returns, array_weight_MaxSharpe) # draw the cumulative return of your all-weather portfolio for the past five years. fig, ax = plt.subplots() ax.plot(MaxSharpe_index, MaxSharpe_port_Cum_Return, color='red', label="MaxSharpe portfolio") plt.title('Cumulative Return of the [MaxSharpe Portfolio] for the past 5 years') plt.xlabel('Date') plt.ylabel('Cumulative Return') # ax.xaxis.set_major_locator(plt.dates.YearLocator()) plt.grid(True) plt.legend() plt.show()

3.2 With minimum volatility portfolio.

# ------------- ef ---------------- # Calculate expected annualized returns and sample covariance / risk mu = expected_returns.mean_historical_return(adj_prices) Sigma = risk_models.sample_cov(adj_prices) # Calculate the Efficient Frontier with mu and S ef = EfficientFrontier(mu, Sigma) # ------------ the Minimum Volatility portfolio ------------------- raw_weights_MinVol = ef.min_volatility() cleaned_weights_MinVol = ef.clean_weights() portfolio_returns_MinVol = ef.portfolio_performance(verbose=True) # ----------- Plot ------------- # ---get array weight for MinVol array_weight_MinVol = convert_ef_weight_2arrayweight(cleaned_weights_MinVol, stock_tickers) print(array_weight_MinVol) MinVol_index, MinVol_port_Cum_Return = portChar(daily_returns, array_weight_MinVol) # draw the cumulative return of your all-weather portfolio for the past five years. fig, ax = plt.subplots() ax.plot(MinVol_index, MinVol_port_Cum_Return, color='green', label="MinVol portfolio") plt.title('Cumulative Return of the [MinVol Portfolio] for the past 5 years') plt.xlabel('Date') plt.ylabel('Cumulative Return') # ax.xaxis.set_major_locator(plt.dates.YearLocator()) plt.grid(True) plt.legend() plt.show()

# Plot comparison # Compare the Cumulative Return between the [MaxSharpe Portfolio] & [MinVol Portfolio] for the past 5 years # draw the cumulative return of your all-weather portfolio for the past five years. fig, ax = plt.subplots() ax.plot(MaxSharpe_index, MaxSharpe_port_Cum_Return, color='red', label="MaxSharpe portfolio") ax.plot(MinVol_index, MinVol_port_Cum_Return, color='green', label="MinVol portfolio") plt.title('Cumulative Return of the [MaxSharpe Portfolio] & [MinVol Portfolio] for the past 5 years') plt.xlabel('Date') plt.ylabel('Cumulative Return') # ax.xaxis.set_major_locator(plt.dates.YearLocator()) plt.grid(True) plt.legend() plt.show()

4. [Optional Bonus] With PyPortfolioOpt, how can you further improve your all-weather portfolio construction?

Hint: The Black-Litterman model (https://pyportfolioopt.readthedocs.io/en/latest/BlackLitterman.html) to express subjective views and L2 regularisation to alleviate extreme portfolio allocation (i.e., weights).

# ------------- IMPORTANT. ---------------- # need daily price data as the input. - SO can not use [daily_returns], use [adj_prices] # available data： # [daily_returns] - the daily returns for each stock # [adj_prices] - extract adjusted closing prices for stocks # [risk_free]. Risk Free Rate. 4.26% for Sep 08 2023 # ---------------- ---------------- ---------------- ---------------- ---------------- # Import Black-Litterman model from pypfopt import BlackLittermanModel from pypfopt import CLA from pypfopt import objective_functions from pypfopt import BlackLittermanModel, CLA, objective_functions from pypfopt import expected_returns, risk_models from pypfopt.efficient_frontier import EfficientFrontier # Calculate expected returns and covariance matrix mu = expected_returns.mean_historical_return(adj_prices) Sigma = risk_models.sample_cov(adj_prices) # print(type(mu)) # print(type(Sigma)) # Create the Black-Litterman model viewdict = {"AAPL": 0.20, "MSFT": 0, "NVDA": -0.2, "TSLA": 0.131321} bl = BlackLittermanModel(Sigma, pi="equal", absolute_views=viewdict, omega="default") # bl = BlackLittermanModel(Sigma, pi="equal", absolute_views=viewdict, omega="default") # Set the risk aversion coefficient delta = 0.5 # bl.market_implied_risk_aversion(delta) # Set the views P = [[1, 0, 0, 0, 0], [0, 0, 0, 0, 1], [0, 1, -1, 0, 0], [0, 0, 0, 0, 0.5]] Q = [[0.20], [-0.30], [0], [-0.2]] # bl.bl_weights(P, Q) # Get the adjusted expected returns and covariance matrix adjusted_mu = bl.bl_returns() adjusted_cov_matrix = bl.bl_cov() # the Efficient Frontier ef = EfficientFrontier(adjusted_mu, adjusted_cov_matrix) # Apply L2 regularization to alleviate extreme portfolio allocation ef.add_objective(objective_functions.L2_reg, gamma=0.1) # Set the risk aversion coefficient -eg: if you believe stock with ticker "AAPL" will outperform the market by 5% # delta = 0.5 # BL.market_implied_risk_aversion(delta) # Set your subjective views # BL.bl_weights(delta=0.5, P=[[0, 0, 0, 0, 0]], Q=[[0.05]]) # ------------ the Black-Litterman modeled portfolio ------------------- # Get the optimized all-weather portfolio raw_weights_BlackLitterman = ef.max_sharpe() # Clean the weights for better interpretation cleaned_weights_BlackLitterman = ef.clean_weights() portfolio_returns_BlackLitterman = ef.portfolio_performance(verbose=True) # ----------- Plot ------------- # ---get array weight for MinVol array_weight_BlackLitterman = convert_ef_weight_2arrayweight(cleaned_weights_BlackLitterman, stock_tickers) BlackLitterman_index, BlackLitterman_port_Cum_Return = portChar(daily_returns, array_weight_BlackLitterman) # draw the cumulative return of your all-weather portfolio for the past five years. fig, ax = plt.subplots() ax.plot(BlackLitterman_index, BlackLitterman_port_Cum_Return, color='black', label="BlackLitterman portfolio") plt.title('Cumulative Return of the [Black-Litterman Modeled Portfolio] for the past 5 years') plt.xlabel('Date') plt.ylabel('Cumulative Return') # ax.xaxis.set_major_locator(plt.dates.YearLocator()) plt.grid(True) plt.legend() plt.show()