VaR Calculation - Covariance matrix is not positive semidefinite

Question

This is a basic question.

I have three assets, equally weighted, and all the mutual covariances are -1. Then, the covariance matrix looks like -

 1  -1  -1
-1   1  -1
-1  -1   1

Now, to calculate the VaR, I need to calculate the portfolio variance.

Am I correct in concluding that I can't calculate the portfolio variance because this matrix is not positive semidefinite? Here is some R code -

v = matrix(c(1, -1, -1, -1, 1, -1, -1, -1, 1), ncol=3)
eigen(v)
  > $eigenvalues
  > 2  2 -1

library(micEcon)
semidefiniteness(v)
  > FALSE

My next question is: Given ANY symmetric matrix by a user, how do I figure out if I can use it to calculate portfolio variance (or the covariance matrix)?

Additionally, given the three assets, I can use them to create a weighted time series for the portfolio and calculate the mean and variance of that, and use that to calculate the VaR. How is that different from calculating VaR using teh covariance method?

Answer 1

The given matrix can not represent a covariance matrix since it would imply that asset 1 is negatively correlated to asset 2 and asset 3. But asset 2 is negatively correlated to asset 3 which contradicts the first statement.

In general a covariance matrix has to be positive semi-definite and symmetric, and conversely every positive semi-definite symmetric matrix is a covariance matrix.