medium · Quantitative Finance prob-stats
A 3× 3 sample covariance matrix is estimated as hatΣ from n=3 daily return observations of 3 assets (returns demeaned). A portfolio optimizer needs hatΣ^-1.
Ignoring degenerate ties, what is the rank of hatΣ and the consequence for the minimum-variance weights?
- hatΣ has rank at most n-1=2, so it is singular and non-invertible; the optimizer finds a zero-variance (in-sample) portfolio in the null space, which is pure overfitting
- hatΣ has rank n=3 generically since there are 3 observations, so it is invertible but severely ill-conditioned, inflating the resulting minimum-variance weights
- hatΣ has full rank 3 because sample covariance matrices are always positive definite by construction, so hatΣ^-1 exists exactly, though weights stay unstable and noisy
- hatΣ has rank at most n=3, but demeaning the returns before estimation is irrelevant to rank; it remains invertible whenever no two of the three assets are perfectly correlated
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