Why Portfolio Variance Requires a Positive Semidefinite Covariance Matrix

Summary

The document raises a portfolio optimization problem in which a variance calculation produces a negative result. It identifies the covariance matrix as a possible source: if the matrix is not positive semidefinite, some portfolio weight vector can yield a negative quadratic-form variance. The questioner asks how to construct a valid covariance matrix in a spreadsheet and what positive semidefiniteness means, noting that individual covariances can be negative.

No answer or worked construction is included, so the document does not provide spreadsheet steps or diagnose the source of the matrix inconsistency. Its useful point is the distinction between allowing negative pairwise covariance and requiring the full covariance matrix to produce nonnegative variance for every portfolio. The question also notes a claim that complete observations cannot produce such an invalid estimated covariance matrix, but supplies no resolution; the issue remains open in this text.

Key ideas

• A portfolio variance is a quadratic form in portfolio weights and the covariance matrix.
• A covariance matrix must be positive semidefinite for all portfolio variances to be nonnegative.
• Individual covariances may be negative without making the full covariance matrix invalid.
• The document asks whether spreadsheet-built covariances can violate this condition but gives no answer.
• The supplied text does not explain how incomplete observations or spreadsheet calculations might cause the problem.

Tags

# negative portfolio variance? Creating a positive semi definite matrix in excel


# negative portfolio variance? Creating a positive semi definite matrix in excel












I am attempting a portfolio optimization model and ended up generating negative portfolio variance using 2WaWbσaσbcorrel(a,b) or 2WaWb*Cov(a,b)

From reading the linked article where other users had an issue, I’m seeing that it is because the covariance matrix is not semi definite positive:

Negative variance?

The solutions offered are for code, but I I need to use excel. Is there a way to generate a true covariance matrix within excel?

I’m also trying to wrap my head around what exactly semi-definite positive means and why what I’ve done won’t work. I understand that the portfolio variance cannot be negative. Within the linked post, another user states, “there exists no data set (with complete observations) from which you could have estimated such a covariance matrix”, but I don’t see why as covariance can be negative.

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