Table of contents

The problem is a solve

Many portfolio calculations are written with an inverse. For example, the unconstrained minimum-variance direction is proportional to Σ−11\Sigma^{-1}\mathbf 1. That notation describes the mathematics, but computing an explicit inverse is usually the wrong numerical instruction.

What we actually need is the solution to

Σx=1.\Sigma x=\mathbf 1.

If Σ\Sigma is symmetric positive definite, factor it once as Σ=LL⊤\Sigma=LL^\top, then solve two triangular systems.

Python
import numpy as np

def minimum_variance_weights(cov: np.ndarray) -> np.ndarray:
    ones = np.ones(cov.shape[0])
    chol = np.linalg.cholesky(cov)
    y = np.linalg.solve(chol, ones)
    direction = np.linalg.solve(chol.T, y)
    return direction / direction.sum()

Why this is better

Cholesky uses the structure we already believe the covariance matrix has. It does less work than a generic inverse and avoids materializing a matrix we never needed. The code also says what the calculation means: solve for a direction, then normalize it.

The diagnostic benefit

A failed factorization is useful information. It tells us the estimated covariance matrix is not positive definite at the precision we are using. Silently applying a generic inverse can conceal that modeling problem behind unstable weights.