Properties and effects of a covariance matrix

[fullofquestions]

I've looked around and have not found a full answer. The question is as follows:

"A covariance matrix must be positive semi-definite so that:"

- eigenvalues are not negative (this is true)
- portfolio variances are not -
- portfolio variances are not 0
- Cholesky factorization is possible

What are the main implications of a positive semi-definite covariance matrix?

 

[David Harper CFA FRM]

Hi FoQ,

FYI, I have an example of Cholesky decomposition on the second sheet of the structured monte carlo @
http://www.bionicturtle.com/premium/spreadsheet/4.a.5_structured_monte_carlo/
(it's purpose is to transform the covariance matrix into a (lower) matrix A' that helps produced correlated normals)

Semi-definite matrix is a necessary condition for the Cholesky decompose; i.e., the matrix needs to be semi-definite for us to do this
e.g., if you go to my second sheet and replace a diagonal in the covariance matrix with a negative (note that violates semi-definite) then you will get errors...

...but that would imply a negative variance (i.e., variances are on the diagonal of a covariance matrix!)...that cannot happen b/c (x^2) can't be negative...so i would not think of an "implication" but rather semi-definite is a *necessary condition* of the covariance matrix required for us to do the Cholesky decompose...

David