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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?

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Be careful to distinguish the variance of a portfolio and its VaR that stand for Value At Risk. I guess your are trying to compute the Variance $\sigma^2$, and this question is hence barely on-topic, since it is basic quantitative finance. –  SRKX Aug 19 '13 at 7:23
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1 Answer 1

up vote 5 down vote accepted

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.

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Is it true that if we want to simulate three time series from a covariance matrix, the matrix must be positive semidefinite? –  Lokesh Luha Aug 18 '13 at 14:30
    
Also, what about symmetric matrices which are not positive semi-definite? For such matrices, should I create a weighted time series for the Portfolio and look at the variance of that to calculate VaR? Does that make sense? –  Lokesh Luha Aug 18 '13 at 14:31
    
A covariance matrix has to be positive semi-definite (and symmetric). If it is not then it does not qualify as a covariance matrix. Yes you can calculate the VaR from the portfolio time series or you can construct the covariance matrix from the asset time series (it will be positive semi-definite if done correctly) and calculate the portfolio VaR from that. –  RRG Aug 18 '13 at 14:38
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