Take-home D — Cholesky builds correlated data

Section 09 covers Cholesky as a solver. This is the other half: Cholesky as a sampler.

Cov(x) = Cov(Lz) = L Cov(z) Lᵀ ≈ L I Lᵀ = L Lᵀ = Σ — independent noise in, correlated noise out. That is the mechanism behind every Monte Carlo simulation that needs correlated assets, sensors or scenarios.

On the workshop’s three-asset book, the correlated simulation’s terminal standard deviation is ≈18.8 against ≈13.6 for the independent one — 39% more spread. Its 5th percentile is ≈79.7 against ≈86.9, and its 1st percentile ≈70.7 against ≈79.9.

This is not a general law that correlation increases risk. It is specific to this book, where every pair is positively correlated. What generalizes is only that assuming independence when assets are not independent distorts the tails.