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Cholesky Decomposition
Definition
Cholesky decomposition factors a symmetric positive-definite matrix — in portfolio simulation, the correlation matrix — into a lower triangular matrix and its transpose. When this lower triangular matrix is multiplied by a vector of independent random normal draws, the resulting returns exhibit exactly the correlations specified in the original matrix. This is essential for realistic portfolio simulation because asset classes do not move independently: public and private equity tend to decline together during market stress, and failures to model these correlations lead to systematic underestimation of portfolio risk.
In the Context of Endowment Management
For endowment portfolios with significant allocations to correlated risk assets (public equity, private equity, real assets), Cholesky decomposition is the methodological backbone that ensures simulation results reflect the real-world tendency of risky assets to decline simultaneously during market stress. Without it, tail risk estimates would be materially understated.