Linear Algebra for Traders

When you work with one asset, simple formulas are enough. With dozens or thousands of assets, you need a way to handle all of them at once. Linear algebra, the mathematics of vectors and matrices, does exactly that. It powers portfolio risk calculations, factor models, regressions, principal component analysis and much of machine learning. You do not need to be a mathematician, but understanding a few core ideas makes quantitative finance far easier to follow.

Vectors and matrices#

Portfolio return and risk#

With weights w (a vector), expected returns μ (a vector) and covariance matrix Σ:

portfolio expected return = wᵀ μ
portfolio variance = wᵀ Σ w

Covariance matrices#

A covariance matrix collects the variances of all assets and their pairwise covariances. For 500 stocks, it has 500 × 500 entries. Estimating it reliably is hard, because there are so many parameters relative to the data. Techniques such as shrinkage and factor models help. See Bayesian Statistics and Factor Models.

Eigenvalues and principal component analysis#

Principal component analysis (PCA) finds the directions in which a set of assets moves together most. It uses the eigenvalues and eigenvectors of the covariance or correlation matrix.

Regression in matrix form#

Multiple regression can be written compactly:

β = (Xᵀ X)⁻¹ Xᵀ y

where X is the matrix of explanatory variables and y the vector of outcomes. This formula gives the ordinary least squares estimates for any number of variables. See Regression Analysis.

Optimisation#

Mean variance portfolio optimisation finds weights that minimise wᵀ Σ w for a target return, subject to constraints. Matrix maths makes these problems solvable for large portfolios. See Portfolio Optimization and Optimization.

Practical tools#

In Python, NumPy handles vectors and matrices efficiently:

import numpy as np
w = np.array([0.5, 0.3, 0.2])
vol = np.array([0.16, 0.06, 0.15])
corr = np.array([[1.0, -0.2, 0.1],
                 [-0.2, 1.0, 0.3],
                 [0.1, 0.3, 1.0]])
cov = np.outer(vol, vol) * corr
port_vol = np.sqrt(w @ cov @ w)
print(round(port_vol, 4))

See NumPy and Pandas for Traders.

Common pitfalls#

• Ill conditioned matrices: nearly collinear assets make inverses unstable.
• Estimation error: small errors in Σ or μ produce extreme optimised weights.
• Shape mismatches in code.

Where to learn more#

A practical route is to learn NumPy array operations, then work through covariance matrices, regression and PCA on real return data. Seeing how changing one correlation changes portfolio volatility builds intuition faster than formulas alone. See Python for Trading.

Frequently asked questions#

Why do traders need linear algebra?#

To handle many assets at once in portfolio risk, factor models, regressions, PCA and optimisation.

How is portfolio variance calculated with matrices?#

As wᵀ Σ w, where w is the vector of weights and Σ is the covariance matrix.

What is PCA used for in finance?#

To find the main common drivers of a set of assets, such as the market factor in stocks or level, slope and curvature in yield curves.

Next, learn the calculus behind sensitivities in Calculus for Traders.