Foundation course in quantitative foundations
Linear Algebra for Finance
Linear algebra for finance, taught on the objects quants use every day: return vectors, covariance matrices, factor loadings and regressions, with each idea computed in Python as well as on paper.
Linear algebra for finance is the language of portfolios, risk and regression. A portfolio is a vector of weights, its risk is a quadratic form in a covariance matrix, and nearly every factor model and estimator in this catalogue is a matrix calculation. This course teaches that language on the data it is used for.
Why do quants need linear algebra?
Because the work is done in matrices. Mean-variance optimisation inverts a covariance matrix. Principal component analysis finds the few movements that explain most of a yield curve. Monte Carlo simulation of correlated assets uses a Cholesky factor. Without the algebra these methods are black boxes, and their failures, such as an unstable inverse, go unnoticed.
How is the course taught?
Audio lessons cover the ideas, explainer films show the geometry, and the workbooks carry worked examples by hand and in Python, with cheat sheets for every decomposition. The graded problem set is marked by the faculty with written feedback.
Who is this course for?
New quants and engineers moving into finance who need the algebra behind portfolio and risk models. It is part of the Foundation Certificate in Computational Finance, which opens both the Professional Diploma in Algorithmic Trading and the Fellowship in Computational Finance.
What you will be able to do
- Work with vectors and matrices by hand and in NumPy, including products, inverses and solving linear systems.
- Build and check a covariance matrix, and explain when it is not positive definite and what to do about it.
- Use eigenvalues, eigenvectors and principal component analysis to describe yield curves and equity returns.
- Derive and compute least squares estimates in matrix form.
Syllabus, module by module
Vectors and portfolios
Weights and returns as vectors, dot products, norms and the return of a portfolio.
Matrices and linear systems
Products, transposes, inverses, rank and solving for hedge ratios.
Covariance and correlation matrices
Building them from data, positive definiteness, and repairing a broken matrix.
Eigenvalues and eigenvectors
What they mean for risk, and the spectral decomposition of a covariance matrix.
Principal component analysis
Level, slope and curvature of the yield curve; market and sector modes in equities.
Least squares and decompositions
Regression in matrix form, QR and Cholesky, and their use in simulation.
How the certificate is earned
Quizzes at the end of each unit and a graded problem set in which you build a covariance matrix, run a principal component analysis of yield changes and solve a minimum-variance portfolio. On a pass you receive the Course certificate in Linear Algebra for Finance, issued by the Institute with a verification code. Read how certification works.
Before you start
School algebra. The course teaches the Python it uses.
This course counts toward
- Foundation Certificate in Computational FinanceThe entry quantitative finance certificate and the first semester of both four-semester credentials. Nine courses that settle the mathematics, statistics, data and market knowledge every later course assumes, closed by a capstone.
- Professional Diploma in Algorithmic TradingThe top credential on the practitioner path. Four semesters, 41 courses and a 160-hour capstone covering everything between the mathematics and a book you could defend to a risk committee.
- Fellowship in Computational FinanceThe top credential on the researcher path. Four semesters, 38 courses, and a dissertation supervised in writing and examined by two faculty members.
Questions applicants ask
How much linear algebra does a quant need?
Enough to read and build covariance matrices, regressions and factor models without guessing. That means products, inverses, eigenvalues and the main decompositions, which is exactly what this course covers.
Is this a pure mathematics course?
No. Every idea is introduced on a financial object, such as a portfolio or a yield curve, and computed in Python. Proofs appear only where they explain why a method works or fails.