Portfolio optimization aims to maximize returns and minimize risks by constructing an optimal asset allocation. Python’s powerful libraries like NumPy and CVXPY enable solving this optimization problem, which is subject to constraints like target return and weight restrictions, using techniques like quadratic programming.
Big investors, such as Warren Buffet and Peter Lynch, follow the portfolio approach. Even mutual funds ( after you take out taxes ) work on the same principle of portfolio theory. This article will study the Modern Portfolio theory and its optimization.
Portfolio optimization in Python involves using libraries like NumPy and CVXPY to maximize returns and minimize risks by adjusting asset weights based on the covariance matrix and expected returns, ensuring the sum of weights equals one and all weights are non-negative.
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Understanding Portfolios
A portfolio is essentially a collection of assets with different or the same weights. It is usually constructed using the Capital Asset Pricing Model(CAPM) and utility theory, and as mentioned before, it is done to maximize returns and minimize risks. Let us now look at the formula for how a portfolio is constructed.
In the above formula, Wi and Ri represent each asset’s weights and historical returns. The summation of all the assets is our portfolio return. Let us now look at how a portfolio is optimized.
Optimizing Portfolios with Python
We use Operations Research concepts to maximize our portfolios. Our objective function is the expected return on the portfolio. Sometimes, instead of maximizing returns, we have target returns. After that, we calculate and minimize the portfolio variance and define constraints.
The sum of weights of all assets should be equal to 1, and they should be non-negative. It becomes an optimization problem in Operations, and then a Simple Greedy or Brute Force approach may be applied. Let us now look at the code and understand it step by step.
import numpy as np
import cvxpy as cp
We have imported two Python libraries, which will be used in further calculations. The Cvxpy library, our portfolio, is generally used for optimization problems.
def portfolio_optimization(expected_returns, cov_matrix, target_return):
num_assets = len(expected_returns)
# Define the variables
weights = cp.Variable(num_assets)
# Define the objective function (minimize portfolio variance)
portfolio_variance = cp.quad_form(weights, cov_matrix)
objective = cp.Minimize(portfolio_variance)
# Define the constraints
constraints = [
cp.sum(weights) == 1, # Sum of weights equals 1
weights >= 0, # Non-negativity constraint on weights
weights.T @ expected_returns == target_return # Target return constraint
]
In the above block, we have declared variables, expected returns, a covariance matrix, and our targeted return. We have also created an objective function, the portfolio return. Additionally, we have added constraints, like the sum of all assets should be 100% and the weights should be non-negative.
# Formulate and solve the optimization problem
problem = cp.Problem(objective, constraints)
problem.solve()
# Retrieve the optimal portfolio weights
optimal_weights = weights.value
return optimal_weights
In the above block, we have used the called modules to optimize our objective function value. We have used the field of operations research in finance.
# Example usage
if __name__ == "__main__":
# Example data
expected_returns = np.array([0.08, 0.12, 0.1]) # Expected returns of assets
cov_matrix = np.array([[0.1, 0.03, 0.05], [0.03, 0.12, 0.07], [0.05, 0.07, 0.15]]) # Covariance matrix
target_return = 0.1 # Target return
# Perform portfolio optimization
optimal_weights = portfolio_optimization(expected_returns, cov_matrix, target_return)
print("Optimal weights:", optimal_weights)
Finally, we have randomly created asset returns, which, by performing matrix operations, gives us a covariance matrix. Finally, we print the results of our optimized value.
Let us now look at the output of the code above.
Therefore, the assets with returns of 8%, 12%, and 10% should be constructed with weights of 45%, 45%, and 10% to maximize returns and minimize variance. Let us look at the code used to get data from Yahoo Finance and determine the weights for the optimized portfolio.
Putting It All Together
import pandas as pd
import numpy as np
import cvxpy as cp
def portfolio_optimization_from_excel(file_path, target_return):
# Read returns data from Excel file into DataFrame
returns_data = pd.read_excel(file_path, index_col=0)
# Calculate expected returns
expected_returns = returns_data.mean()
# Calculate covariance matrix
cov_matrix = returns_data.cov()
# Number of assets
num_assets = len(expected_returns)
# Define the variables
weights = cp.Variable(num_assets)
# Define the objective function (minimize portfolio variance)
portfolio_variance = cp.quad_form(weights, cov_matrix.values)
objective = cp.Minimize(portfolio_variance)
# Define the constraints
constraints = [
cp.sum(weights) == 1, # Sum of weights equals 1
weights >= 0, # Non-negativity constraint on weights
weights.T @ expected_returns.values == target_return # Target return constraint
]
# Formulate and solve the optimization problem
problem = cp.Problem(objective, constraints)
problem.solve()
# Retrieve the optimal portfolio weights
optimal_weights = weights.value
return optimal_weights
# Example usage
if __name__ == "__main__":
# Path to the Excel file containing returns data
file_path = "returns_data.xlsx"
# Target return
target_return = 0.1 # Adjust as needed
# Perform portfolio optimization
optimal_weights = portfolio_optimization_from_excel(file_path, target_return)
print("Optimal weights:", optimal_weights)
We can download data from Yahoo Finance and calculate the mean, variance, and covariance matrix of the assets. The process remains the same after that.
Conclusion
Portfolio optimization is crucial to investment management, enabling investors to achieve their desired returns while minimizing risk exposure. Python’s versatility and robust optimization libraries make it an ideal tool for implementing advanced portfolio optimization techniques, leveraging real-world data from sources like Yahoo Finance. How could you extend this approach to incorporate additional constraints or objectives specific to your investment goals?
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