Lära Introduction to Portfolio Optimization | Risk Analysis and Portfolio Management

Understanding how to optimize a portfolio is at the heart of modern investment management. The process aims to help you maximize returns while controlling risk, a fundamental challenge in finance. The efficient frontier is a key concept here: it represents the set of portfolios that offer the highest expected return for a given level of risk, or the lowest risk for a given level of return. By plotting portfolios on a risk-return graph, you can visualize the tradeoff between risk (typically measured by standard deviation or volatility) and expected return. Portfolios that lie below the efficient frontier are considered suboptimal because you could achieve a higher return for the same risk or a lower risk for the same return. The goal of portfolio optimization is to construct a mix of assets that lies on this frontier, balancing your appetite for risk with your desire for returns.


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import numpy as np
import matplotlib.pyplot as plt

# Simulate returns for 3 assets
np.random.seed(42)
mean_returns = np.array([0.10, 0.12, 0.15])
cov_matrix = np.array([
    [0.005, 0.002, 0.001],
    [0.002, 0.006, 0.003],
    [0.001, 0.003, 0.010]
])

num_portfolios = 5000
results = np.zeros((3, num_portfolios))

for i in range(num_portfolios):
    # Generate random weights and normalize
    weights = np.random.random(3)
    weights /= np.sum(weights)
    # Calculate portfolio return and risk
    portfolio_return = np.dot(weights, mean_returns)
    portfolio_std = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
    results[0,i] = portfolio_std
    results[1,i] = portfolio_return
    results[2,i] = (portfolio_return - 0.03) / portfolio_std  # Sharpe ratio (risk-free rate = 3%)

# Plot risk-return scatter
plt.scatter(results[0,:], results[1,:], c=results[2,:], cmap='viridis')
plt.xlabel('Risk (Standard Deviation)')
plt.ylabel('Expected Return')
plt.title('Simulated Portfolio Allocations')
plt.colorbar(label='Sharpe Ratio')
plt.show()

When constructing portfolios, you must consider certain constraints. One of the most important is that the sum of all portfolio weights must equal 1. This means you are fully investing your available capital, with no money left unallocated. Another common constraint is no short selling, which means each asset's weight must be greater than or equal to zero—you cannot invest a negative amount in any asset. These constraints ensure the portfolio is realistic and aligns with most investors' requirements. Portfolio optimization techniques in Python take these rules into account to generate valid and practical investment solutions.


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import numpy as np

# Generate random portfolio weights for 3 assets
weights = np.random.random(3)
print("Unnormalized weights:", weights)

# Normalize so weights sum to 1
weights /= np.sum(weights)
print("Normalized weights:", weights)
print("Sum of weights:", np.sum(weights))

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Avsnitt 2. Kapitel 6

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              123456789101112131415161718192021222324252627282930313233
            
import numpy as np
import matplotlib.pyplot as plt

# Simulate returns for 3 assets
np.random.seed(42)
mean_returns = np.array([0.10, 0.12, 0.15])
cov_matrix = np.array([
    [0.005, 0.002, 0.001],
    [0.002, 0.006, 0.003],
    [0.001, 0.003, 0.010]
])

num_portfolios = 5000
results = np.zeros((3, num_portfolios))

for i in range(num_portfolios):
    # Generate random weights and normalize
    weights = np.random.random(3)
    weights /= np.sum(weights)
    # Calculate portfolio return and risk
    portfolio_return = np.dot(weights, mean_returns)
    portfolio_std = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
    results[0,i] = portfolio_std
    results[1,i] = portfolio_return
    results[2,i] = (portfolio_return - 0.03) / portfolio_std  # Sharpe ratio (risk-free rate = 3%)

# Plot risk-return scatter
plt.scatter(results[0,:], results[1,:], c=results[2,:], cmap='viridis')
plt.xlabel('Risk (Standard Deviation)')
plt.ylabel('Expected Return')
plt.title('Simulated Portfolio Allocations')
plt.colorbar(label='Sharpe Ratio')
plt.show()


              12345678910
            
import numpy as np

# Generate random portfolio weights for 3 assets
weights = np.random.random(3)
print("Unnormalized weights:", weights)

# Normalize so weights sum to 1
weights /= np.sum(weights)
print("Normalized weights:", weights)
print("Sum of weights:", np.sum(weights))

Var allt tydligt?

Tack för dina kommentarer!

Avsnitt 2. Kapitel 6

Introduction to Portfolio Optimization

1. What is the efficient frontier?

2. Why must portfolio weights sum to 1?

3. What is the main objective of portfolio optimization?

Introduction to Portfolio Optimization

1. What is the efficient frontier?

2. Why must portfolio weights sum to 1?

3. What is the main objective of portfolio optimization?