How does Cholesky decomposition generate correlated random variables for Monte Carlo simulation?

Cholesky decomposition is the workhorse method for converting independent random variables into correlated ones. It works by decomposing a correlation (or covariance) matrix into a lower triangular matrix.

The Key Idea:

If C is a correlation matrix, Cholesky finds a lower triangular matrix L such that:

C = L x L^T

Then if Z is a vector of independent standard normals, the product L x Z gives correlated normals with the desired correlation structure.

Two-Asset Example:

Suppose Ashford Risk Analytics wants to simulate correlated returns for two assets with correlation rho = 0.6.

Correlation matrix:

C = | 1.0 0.6 |

Cholesky decomposition:

L = | 1.0 0 |

(Where 0.8 = sqrt(1 - 0.6^2) = sqrt(0.64))

Generating Correlated Returns:

1. Draw two independent standard normals: Z1 = 0.75, Z2 = -1.20
2. Apply Cholesky:

• Correlated variable 1: X1 = 1.0 x Z1 + 0 x Z2 = 0.75
• Correlated variable 2: X2 = 0.6 x Z1 + 0.8 x Z2 = 0.6(0.75) + 0.8(-1.20) = 0.45 - 0.96 = -0.51

1. Convert to returns: R_i = mu_i + sigma_i x X_i

The beautiful thing: X1 and X2 will have correlation exactly equal to 0.6 across many simulations.

Three or More Assets:

For n assets, the Cholesky matrix is n x n lower triangular. Each row builds on previous variables:

• Asset 1: uses only Z1
• Asset 2: uses Z1 and Z2 (to be correlated with Asset 1)
• Asset 3: uses Z1, Z2, and Z3 (to be correlated with both)

Important Requirements:

• The correlation matrix must be positive semi-definite (all eigenvalues >= 0)
• If it's not (e.g., from mixing different estimation windows), Cholesky fails and you need to fix the matrix first
• For very large portfolios (1000+ assets), computational efficiency matters — Cholesky is O(n^3)

FRM Exam Tips:

• Know the 2x2 case cold: L[2,1] = rho, L[2,2] = sqrt(1-rho^2)
• Understand that Cholesky preserves the Normal distribution — if inputs are Normal, outputs are multivariate Normal
• For non-normal returns, Cholesky + copula methods are needed

Practice simulation problems in our FRM Part I question bank.