hard · Quantitative Finance prob-stats
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)?
- Zero correlation across every pair of assets, since positive semi-definiteness is assumed to force all off-diagonal covariance entries to vanish exactly to zero.
- A positive semi-definite matrix is assumed to carry complex-valued eigenvalues, which would make the simulated return vector x non-real in general.
- 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 quadratic form x^top Σ x is assumed to turn strictly negative for some choice of vector x, so no valid square-root factorization can exist.
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