hard · Quantitative Finance

Ignoring degenerate ties, what is the rank of hatΣ and the consequence for the minimum-variance weights?

1. 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
2. hatΣ has rank n=3 generically since there are 3 observations, so it is invertible but ill-conditioned, inflating the weights
3. hatΣ has full rank 3 because covariance matrices are positive definite by construction, so hatΣ^-1 exists but the weights are unstable
4. hatΣ has rank at most n=3 but the demeaning is irrelevant to rank; it is invertible whenever no two assets are perfectly correlated

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