# A Guide to Portfolio Optimization with Python and Modern Portfolio Theory

Portfolio optimization plays a crucial role in modern investment strategies, and one widely recognized approach is the implementation of Modern Portfolio Theory (MPT). In this article, we will present a Python script that showcases how to optimize a stock portfolio using MPT. By leveraging Yahoo Finance data and the Scipy library, we will determine the optimal asset weights that maximize the Sharpe ratio.

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## Section 1: Define Tickers and Time Range

To begin, we define a list of tickers representing the assets to include in our portfolio. For this demonstration, we will employ five exchange-traded funds (ETFs) that span various asset classes: SPY, BND, GLD, QQQ, and VTI. Furthermore, we establish the start and end dates for our analysis, utilizing a historical time range of five years.

tickers = [‘SPY’,’BND’,’GLD’,’QQQ’,’VTI’]

end_date = datetime.today()

start_date = end_date – timedelta(days = 5*365)

Next, we create an empty DataFrame to store the adjusted close prices for each asset. By leveraging the yfinance library, we can easily download the necessary data from Yahoo Finance.

for ticker in tickers:

## Section 3: Calculate Lognormal Returns

The subsequent step involves computing the lognormal returns for each asset, removing any missing values from the calculations.

console.log( ‘Code is Poetry’ );

## Section 4: Calculate Covariance Matrix

Using the annualized log returns, we proceed to compute the covariance matrix.

cov_matrix = log_returns.cov() * 252

## Section 5: Define Portfolio Performance Metrics

To evaluate portfolio performance, we define functions that calculate the portfolio’s standard deviation, expected return, and Sharpe ratio.

def standard_deviation(weights, cov_matrix):

variance = weights.T @ cov_matrix @ weights

return np.sqrt(variance)

def expected_return(weights, log_returns):

return np.sum(log_returns.mean()*weights)*252

def sharpe_ratio(weights, log_returns, cov_matrix, risk_free_rate):

return (expected_return(weights, log_returns) – risk_free_rate) / standard_deviation(weights, cov_matrix)

## Section 6: Portfolio Optimization

In this section, we set the risk-free rate, establish a function to minimize the negative Sharpe ratio, and define constraints and bounds for the optimization process.

risk_free_rate = .02

def neg_sharpe_ratio(weights, log_returns, cov_matrix, risk_free_rate):

return -sharpe_ratio(weights, log_returns, cov_matrix, risk_free_rate)

constraints = {‘type’: ‘eq’, ‘fun’: lambda weights: np.sum(weights) – 1}

bounds = [(0, 0.4) for _ in range(len(tickers))]

initial_weights = np.array([1/len(tickers)]*len(tickers))

optimized_results = minimize(neg_sharpe_ratio, initial_weights, args=(log_returns, cov_matrix, risk_free_rate), method=’SLSQP’, constraints=constraints, bounds=bounds)

## Section 7: Analyze the Optimal Portfolio

We extract the optimal weights and calculate the expected annual return, expected volatility, and Sharpe ratio for the optimized portfolio. Finally, we create a bar chart to visualize the asset weights within the portfolio.

optimal_weights = optimized_results.x

print(“Optimal Weights:”)

for ticker, weight in zip(tickers, optimal_weights):

print(f”{ticker}: {weight:.4f}”)

optimal_portfolio_return = expected_return(optimal_weights, log_returns)

optimal_portfolio_volatility = standard_deviation(optimal_weights, cov_matrix)

optimal_sharpe_ratio = sharpe_ratio(optimal_weights, log_returns, cov_matrix, risk_free_rate)

print(f”Expected Annual Return: {optimal_portfolio_return:.4f}”)

print(f”Expected Volatility: {optimal_portfolio_volatility:.4f}”)

print(f”Sharpe Ratio: {optimal_sharpe_ratio:.4f}”)

## Display the Final Portfolio in a Plot

We create a bar chart to visualize the optimal weights of the assets in the portfolio.

import matplotlib.pyplot as plt

plt.figure(figsize=(10, 6))

plt.bar(tickers, optimal_weights)

plt.xlabel(‘Assets’)

plt.ylabel(‘Optimal Weights’)

plt.title(‘Optimal Portfolio Weights’)

plt.show()

## Bottom Line

This Python script demonstrates the application of Modern Portfolio Theory in optimizing a stock portfolio. By determining the optimal weights for each asset, we aim to maximize the portfolio’s Sharpe ratio, which provides a risk-adjusted measure of return. Employing this approach enables investors to construct well-diversified portfolios and make informed decisions when allocating their investments.

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