AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - Linear Algebra"
python "linear_algebra_tutorial.py"

Quantitative Methods – Linear Algebra¶

Overview¶

Linear algebra is the mathematical foundation for portfolio optimization, risk modeling, factor analysis, and quantitative finance. This utility teaches essential concepts through practical financial applications.

Key Concepts¶

Vectors¶

Portfolio weights: Allocation across assets
Returns: Expected returns as vectors
Dot product: Portfolio expected return calculation
Vector norms: Risk and distance metrics

Matrices¶

Returns matrix: Assets × time periods
Covariance matrix: Asset risk relationships
Correlation matrix: Normalized dependencies
Matrix operations: Addition, multiplication, transpose

Covariance & Correlation¶

Covariance: Measure of joint variability
Correlation: Normalized covariance (-1 to 1)
Portfolio variance: σ²_p = w^T Σ w
Diversification benefit: Reduce risk through low correlation

Eigenvalues & Eigenvectors¶

Principal Component Analysis (PCA): Dimensionality reduction
Risk factors: Identify major sources of portfolio risk
Variance explained: How much variation each factor captures
Factor models: Multi-factor risk attribution

Matrix Operations¶

Inverse: Solve for optimal portfolios
Determinant: Check invertibility
Transpose: Switch rows and columns
Solving systems: Optimize with constraints

Key Examples¶

Portfolio Expected Return¶

weights = np.array([0.3, 0.3, 0.4])
expected_returns = np.array([0.10, 0.12, 0.08])

# Dot product
portfolio_return = weights @ expected_returns # 9.8%

Portfolio Variance¶

# σ²_portfolio = w^T Σ w
portfolio_variance = weights @ cov_matrix @ weights
portfolio_vol = np.sqrt(portfolio_variance)

Minimum Variance Portfolio¶

# w_mvp = (Σ^-1 1) / (1^T Σ^-1 1)
cov_inv = np.linalg.inv(cov_matrix)
ones = np.ones(n_assets)
weights_mvp = (cov_inv @ ones) / (ones @ cov_inv @ ones)

Principal Component Analysis¶

# Find major risk factors
eigenvalues, eigenvectors = np.linalg.eig(correlation_matrix)

# First eigenvalue = proportion of variance from market factor
market_factor_weight = eigenvalues[0] / sum(eigenvalues)

Files¶

linear_algebra_tutorial.py: Comprehensive tutorial with examples
README.md: This file

How to Run¶

pip install numpy
python linear_algebra_tutorial.py

Financial Applications¶

1. Portfolio Optimization¶

Use matrix operations to find optimal portfolio weights: - Minimum variance portfolio: Lowest risk - Maximum Sharpe portfolio: Best risk-adjusted return - Efficient frontier: Trade-off between risk and return

2. Risk Attribution¶

Decompose portfolio risk by asset/factor: - Marginal risk: Change in risk from small weight change - Risk contribution: Each asset's contribution to total variance - Factor exposure: Sensitivity to market factors

3. Factor Models¶

Multi-factor risk models (Fama-French, etc.): - Market factor: Overall market beta - Size factor: Small-cap vs large-cap - Value factor: Value vs growth stocks

4. Dimensionality Reduction¶

Simplify complex portfolios: - PCA: Reduce 100+ stocks to 5-10 factors - Factor investing: Exposures to systematic factors - Risk budgeting: Allocate risk across factors

Mathematical Foundations¶

Portfolio Variance Formula¶

σ²_portfolio = Σᵢ Σⱼ wᵢ wⱼ σᵢⱼ

In matrix notation:
σ²_p = w^T Σ w

Where:
- w = vector of portfolio weights
- Σ = covariance matrix
- σᵢⱼ = covariance between assets i and j

Eigendecomposition¶

Σ = Q Λ Q^T

Where:
- Σ = covariance matrix
- Q = matrix of eigenvectors
- Λ = diagonal matrix of eigenvalues

Minimum Variance Portfolio¶

min w^T Σ w
subject to: w^T 1 = 1

Solution: w_mvp = (Σ^-1 1) / (1^T Σ^-1 1)

Practice Problems¶

Equal Weight vs Minimum Variance
Create a 5-asset portfolio with equal weights
Calculate the minimum variance portfolio
Compare volatilities
Risk Contribution Analysis
Calculate marginal risk for each asset
Compute risk contribution percentages
Identify which assets contribute most to risk
Principal Component Analysis
Perform PCA on a correlation matrix
Determine how many factors explain 90% of variance
Interpret the factor loadings
Efficient Frontier
Generate 1000 random portfolios
Calculate return and volatility for each
Plot efficient frontier

References¶

Markowitz Portfolio Theory: "Portfolio Selection" (1952)
Linear Algebra: Gilbert Strang's textbook
NumPy Documentation: https://numpy.org/doc/
Quantitative Portfolio Management: Attilio Meucci

Key Takeaways¶

Covariance matrix is central to portfolio risk Matrix multiplication computes portfolio metrics efficiently Eigenvalues reveal dominant risk factors Matrix inverse solves for optimal weights PCA reduces dimensionality while preserving information

Master linear algebra to build sophisticated portfolio optimization and risk management systems!

Continue in Quantitative Methods¶

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