Cholesky decomposition is the mathematical method that transforms independent random numbers into correlated asset returns. It ensures that simulated stock, bond, and cash returns move together (or apart) in realistic patterns — capturing the diversification effects that make multi-asset portfolios work.
Cholesky decomposition is a matrix factorization technique used in Monte Carlo simulation to generate correlated random variables from independent ones. In retirement planning, it transforms three independent random draws into correlated returns for stocks, bonds, and cash that reflect real-world co-movement between asset classes.
How It Works
The process has three steps:
- Define a correlation matrix: a table of correlation coefficients between each pair of asset classes
- Decompose the matrix: Cholesky factorization produces a lower-triangular matrix L such that L × Lᵀ = the correlation matrix
- Transform independent draws: multiply a vector of independent random returns by L to produce correlated returns
Retirement Lab implements a 3×3 Cholesky decomposition for stocks, bonds, and cash. The decomposition is computed once per simulation run, then applied to each of the thousands of iterations.
Why It Matters for Retirement Planning
Asset allocation works because different asset classes don't move in lockstep. Bonds often rise when stocks fall, providing a cushion during downturns. Without Cholesky decomposition, a simulation would treat each asset class as if it moves independently — producing unrealistic scenarios that either overstate or understate the benefit of diversification.
By enforcing the correct correlation structure, the simulation captures:
- Crisis behavior: stocks falling while bonds rally (negative correlation)
- Normal conditions: moderate positive correlation between asset classes
- The true risk profile: of a multi-asset portfolio, not just the sum of individual risks
This is a pro-tier feature in Retirement Lab — free-tier simulations use independent returns, while pro-tier simulations model correlated returns via Cholesky decomposition.
Frequently Asked Questions
- Why can't you just generate returns for each asset class independently?
- Independent random draws ignore the relationships between asset classes. In reality, stocks and bonds often move in opposite directions during crises. Without Cholesky decomposition to enforce these correlations, a simulation would miss the diversification benefit of a multi-asset portfolio — and produce unrealistic scenarios where all assets crash simultaneously at rates inconsistent with historical data.
- What is the correlation matrix used in Retirement Lab?
- Retirement Lab uses a 3x3 correlation matrix for stocks, bonds, and cash. The default values reflect historical relationships — for example, stocks and bonds have historically had low or negative correlation, meaning bonds tend to hold value or rise when stocks fall. You can customize the matrix in the pro-tier advanced settings.