Linear algebra essentials in scipy.linalg

Matrices as the machinery of portfolios

A portfolio problem is a matrix problem. The covariance matrix $\Sigma$ holds the variances and co-movements of the assets, and almost every task asks you to do something with it. scipy.linalg.solve(A, b) solves $A\mathbf x=\mathbf b$ for the unknown $\mathbf x$. You should not compute the inverse inv(A) and multiply: the direct solve does less work and loses less accuracy. The Cholesky factorization, linalg.cholesky(S, lower=True), splits a covariance matrix as $\Sigma=LL^{\top}$, where $L$ is lower triangular, which means zeros above the diagonal. If $\mathbf z$ is a vector of independent standard normal numbers, then $L\mathbf z$ has covariance $\Sigma$, and this is how you simulate correlated returns.

The eigenvalues of $\Sigma$ tell you how many independent sources of risk there really are. linalg.eigvalsh(S) returns them for a symmetric matrix, smallest first. For two assets with correlation $\rho$ the correlation matrix has eigenvalues $1+\rho$ and $1-\rho$, so highly correlated assets have one big eigenvalue and one tiny one. The ratio of the largest to the smallest eigenvalue is the condition number. A large condition number means the matrix is nearly singular, that is, nearly impossible to invert, and a small change in the data can swing the answer of a solve a lot. Optimisers that take the inverse of $\Sigma$ are sensitive to exactly this.

• Solve, do not invert: solve(A, b) is faster and more accurate than inv(A) @ b.
• Cholesky makes correlation: multiply independent normals by $L$ to get returns with covariance $\Sigma$.
• Watch the condition number: highly correlated assets make $\Sigma$ nearly singular and solutions unstable.

Lab · what's graded

• The Cholesky factor20 pts
• Solving a system10 pts
• A condition number10 pts
• A factor's share of variance10 pts
• What each routine is for15 pts
• Simulating correlated returns10 pts
• Why not invert25 pts