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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

  1. Equal Weight vs Minimum Variance
  2. Create a 5-asset portfolio with equal weights
  3. Calculate the minimum variance portfolio

  4. Compare volatilities

  5. Risk Contribution Analysis

  6. Calculate marginal risk for each asset
  7. Compute risk contribution percentages

  8. Identify which assets contribute most to risk

  9. Principal Component Analysis

  10. Perform PCA on a correlation matrix
  11. Determine how many factors explain 90% of variance

  12. Interpret the factor loadings

  13. Efficient Frontier

  14. Generate 1000 random portfolios
  15. Calculate return and volatility for each
  16. 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!