Lernen Portfolio Optimization with Constraints | Risk Analysis and Portfolio Optimization

Portfolio optimization is a fundamental task in financial analysis, aiming to allocate investments across different assets to achieve the best possible balance between expected return and risk. The concept of the efficient frontier is central to this process. It represents the set of optimal portfolios that offer the highest expected return for a specific level of risk. Constraints play a crucial role in portfolio optimization. Common constraints include prohibiting short selling (no negative asset weights) and requiring full investment (the sum of all asset weights equals one). These constraints ensure that the resulting portfolio is practical and aligns with real-world investment policies.


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import numpy as np
import pandas as pd
from scipy.optimize import minimize

# Simulated expected returns and covariance matrix for three assets
expected_returns = np.array([0.08, 0.12, 0.10])
cov_matrix = np.array([
    [0.005, -0.010, 0.004],
    [-0.010, 0.040, -0.002],
    [0.004, -0.002, 0.023]
])

assets = ['Asset A', 'Asset B', 'Asset C']

def portfolio_stats(weights, returns, cov):
    port_return = np.dot(weights, returns)
    port_volatility = np.sqrt(np.dot(weights.T, np.dot(cov, weights)))
    return port_return, port_volatility

def objective(weights, returns, cov):
    # Minimize negative Sharpe ratio (assume risk-free rate = 0 for simplicity)
    port_return, port_volatility = portfolio_stats(weights, returns, cov)
    return -port_return / port_volatility

constraints = (
    {'type': 'eq', 'fun': lambda x: np.sum(x) - 1},  # Full investment
)
bounds = tuple((0, 1) for _ in assets)  # No short selling

initial_guess = np.array([1/3, 1/3, 1/3])

result = minimize(
    objective,
    initial_guess,
    args=(expected_returns, cov_matrix),
    method='SLSQP',
    bounds=bounds,
    constraints=constraints
)

optimal_weights = result.x
optimal_port_return, optimal_port_volatility = portfolio_stats(optimal_weights, expected_returns, cov_matrix)

print("Optimal Weights:")
for asset, weight in zip(assets, optimal_weights):
    print(f"{asset}: {weight:.2%}")
print(f"Expected Portfolio Return: {optimal_port_return:.2%}")
print(f"Portfolio Volatility: {optimal_port_volatility:.2%}")

In this optimization process, you set up a mean-variance optimization for three assets by defining their expected returns and covariance matrix. The objective is to maximize the Sharpe ratio, which balances return against risk. The constraints ensure that the sum of the asset weights equals one (full investment) and that each weight is between zero and one (no short selling). The optimization algorithm searches for the combination of asset weights that yields the highest expected return for the lowest risk, given these constraints. The resulting optimal weights tell you how much of your portfolio to allocate to each asset. Interpreting these weights helps you decide on a practical investment strategy that aligns with your risk preferences and investment policy.


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import matplotlib.pyplot as plt

# Generate portfolios along the efficient frontier
num_portfolios = 100
results = np.zeros((2, num_portfolios))
weight_records = []

for i in range(num_portfolios):
    # Random weights, normalized to sum to 1, no short selling
    weights = np.random.rand(len(assets))
    weights /= np.sum(weights)
    if np.all(weights >= 0):  # Enforce no short selling
        port_return, port_volatility = portfolio_stats(weights, expected_returns, cov_matrix)
        results[0, i] = port_volatility
        results[1, i] = port_return
        weight_records.append(weights)

plt.figure(figsize=(8, 5))
plt.scatter(results[0, :], results[1, :], c=results[1, :] / results[0, :], marker='o')
plt.scatter(optimal_port_volatility, optimal_port_return, marker='*', color='r', s=200, label='Optimal Portfolio')
plt.xlabel('Portfolio Volatility (Risk)')
plt.ylabel('Expected Return')
plt.title('Efficient Frontier (No Short Selling)')
plt.colorbar(label='Sharpe Ratio')
plt.legend()
plt.show()

War alles klar?

Danke für Ihr Feedback!

Abschnitt 3. Kapitel 4

Fragen Sie AI

Fragen Sie alles oder probieren Sie eine der vorgeschlagenen Fragen, um unser Gespräch zu beginnen

Suggested prompts:

Can you explain what the efficient frontier plot tells me about my investment options?

How would the results change if short selling were allowed?

What does the Sharpe ratio represent in this context?

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import numpy as np
import pandas as pd
from scipy.optimize import minimize

# Simulated expected returns and covariance matrix for three assets
expected_returns = np.array([0.08, 0.12, 0.10])
cov_matrix = np.array([
    [0.005, -0.010, 0.004],
    [-0.010, 0.040, -0.002],
    [0.004, -0.002, 0.023]
])

assets = ['Asset A', 'Asset B', 'Asset C']

def portfolio_stats(weights, returns, cov):
    port_return = np.dot(weights, returns)
    port_volatility = np.sqrt(np.dot(weights.T, np.dot(cov, weights)))
    return port_return, port_volatility

def objective(weights, returns, cov):
    # Minimize negative Sharpe ratio (assume risk-free rate = 0 for simplicity)
    port_return, port_volatility = portfolio_stats(weights, returns, cov)
    return -port_return / port_volatility

constraints = (
    {'type': 'eq', 'fun': lambda x: np.sum(x) - 1},  # Full investment
)
bounds = tuple((0, 1) for _ in assets)  # No short selling

initial_guess = np.array([1/3, 1/3, 1/3])

result = minimize(
    objective,
    initial_guess,
    args=(expected_returns, cov_matrix),
    method='SLSQP',
    bounds=bounds,
    constraints=constraints
)

optimal_weights = result.x
optimal_port_return, optimal_port_volatility = portfolio_stats(optimal_weights, expected_returns, cov_matrix)

print("Optimal Weights:")
for asset, weight in zip(assets, optimal_weights):
    print(f"{asset}: {weight:.2%}")
print(f"Expected Portfolio Return: {optimal_port_return:.2%}")
print(f"Portfolio Volatility: {optimal_port_volatility:.2%}")


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import matplotlib.pyplot as plt

# Generate portfolios along the efficient frontier
num_portfolios = 100
results = np.zeros((2, num_portfolios))
weight_records = []

for i in range(num_portfolios):
    # Random weights, normalized to sum to 1, no short selling
    weights = np.random.rand(len(assets))
    weights /= np.sum(weights)
    if np.all(weights >= 0):  # Enforce no short selling
        port_return, port_volatility = portfolio_stats(weights, expected_returns, cov_matrix)
        results[0, i] = port_volatility
        results[1, i] = port_return
        weight_records.append(weights)

plt.figure(figsize=(8, 5))
plt.scatter(results[0, :], results[1, :], c=results[1, :] / results[0, :], marker='o')
plt.scatter(optimal_port_volatility, optimal_port_return, marker='*', color='r', s=200, label='Optimal Portfolio')
plt.xlabel('Portfolio Volatility (Risk)')
plt.ylabel('Expected Return')
plt.title('Efficient Frontier (No Short Selling)')
plt.colorbar(label='Sharpe Ratio')
plt.legend()
plt.show()

War alles klar?

Danke für Ihr Feedback!

Abschnitt 3. Kapitel 4

Portfolio Optimization with Constraints

1. What is the efficient frontier in portfolio theory?

2. Why are constraints important in portfolio optimization?

3. Which Python libraries are commonly used for numerical optimization in finance?

Portfolio Optimization with Constraints

1. What is the efficient frontier in portfolio theory?

2. Why are constraints important in portfolio optimization?

3. Which Python libraries are commonly used for numerical optimization in finance?