hard · Quantitative Finance
A quantitative researcher is using the Cholesky decomposition Σ = LL^top to simulate a vector of correlated returns x = Lz.
Why does the standard Cholesky algorithm fail if the estimated covariance matrix Σ is positive semi-definite (PSD) but not strictly positive definite (PD)?
- The lack of strict positive definiteness implies that the asset correlations are all exactly equal to zero, making the decomposition redundant.
- A positive semi-definite matrix has complex eigenvalues, which makes the resulting simulation vector x non-real.
- The recursive step for the lower-triangular entries L_ij requires division by the diagonal element L_jj, which is zero for singular matrices.
- The property x^top Σ x ≥ 0 is violated, preventing the matrix from having any valid square root decomposition.
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