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# What is posiitve semi-definite matrix? (R16.P1.T2.HULL_CH11)

##### Active Member
Hi,
In reference to R16.P1.T2.HULL_CH11:Topic: VARIANCE_COVARIANCE_MATRIX_+VE_SEMI_DEFINITE

Can anyone explain what is meant by the "Variance-Covariance Matrix" In order to be "Internally Consistent" has to satisfy the condition of w * C * wT =>+Ve Semi Definite" ...? What is "w" here...?

I understand the meaning of the "Variance-Covariance Matrix" ..but am lost of the meaning a matrix being "Internally Consistent" and what it means to satisfy the condition of being +Ve Semi Definite"..if some one could illustrate a matrix with numbers, I would be very grateful...

#### Nicole Seaman

Staff member
Subscriber
Hi,
In reference to R16.P1.T2.HULL_CH11:Topic: VARIANCE_COVARIANCE_MATRIX_+VE_SEMI_DEFINITE

Can anyone explain what is meant by the "Variance-Covariance Matrix" In order to be "Internally Consistent" has to satisfy the condition of w * C * wT =>+Ve Semi Definite" ...? What is "w" here...?

I understand the meaning of the "Variance-Covariance Matrix" ..but am lost of the meaning a matrix being "Internally Consistent" and what it means to satisfy the condition of being +Ve Semi Definite"..if some one could illustrate a matrix with numbers, I would be very grateful...

Someone else may be able to help more, but there are discussions here related to your questions that may be helpful: https://www.bionicturtle.com/forum/...ma-and-garch-1-1-models-hull.8339/#post-33793. I just searched the keywords "matrix internally consistent" and it brought up our practice questions that are related to these concepts. I find that many times, the discussions under the practice question threads are great for elaborating on the notes Thanks,

Nicole

#### berrymucho

##### Member
Hi,
In reference to R16.P1.T2.HULL_CH11:Topic: VARIANCE_COVARIANCE_MATRIX_+VE_SEMI_DEFINITE

Can anyone explain what is meant by the "Variance-Covariance Matrix" In order to be "Internally Consistent" has to satisfy the condition of w * C * wT =>+Ve Semi Definite" ...? What is "w" here...?

I understand the meaning of the "Variance-Covariance Matrix" ..but am lost of the meaning a matrix being "Internally Consistent" and what it means to satisfy the condition of being +Ve Semi Definite"..if some one could illustrate a matrix with numbers, I would be very grateful...
Basically, this is a requirement to guarantee that the variance of a portfolio remains non-negative (i.e. ≥0), since the variance of a portfolio is calculated as Var(P) = w' × Cov × w. In this case the w's are the asset allocation weights (adding up to 1) but in all generality they can be scaled to any number, the inequality still holds. Not all (symmetric) matrix Cov would satisfy w' × Cov × w ≥ 0, for all w's. A Cov matrix that is "internally consistent" does. This is something to keep a eye on when you calculate covariances from a sample, the sampling errors might give you a matrix that is not internally consistent. All subsequent calculations involving this matrix would be wrong or fail. However there are procedures to adjust matrices to force positive-semi-definiteness. Hope this helps, this can be a (very) technical topic.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari @berrymucho explained it better than i can, sincerely To illustrate, I just retrieved an example of portfolio variance (based on matrix multiplication), XLS is here https://www.dropbox.com/s/5dj4pz4bjyrnhef/1004-semi-definite.xlsx?dl=0
This implements berry's w' × Cov × w = portfolio variance of 0.6579% in the upper panel (notice how the same weights are transposed). Then to violate the positive semi-definite condition, I lazily changed the variance of Asset #1 to a negative (@berrymucho please confirm, I think any such violation is an example, even a negative diagonal value?) so that the resulting portfolio variance is negative. So that's a violation, and practically speaking, a negative variance is not possible, as evidenced by the error for a volatility. I hope that's helpful!) ##### Active Member
Thanks so much @Nicole Seaman for pointing me to the thread on the topic...I had searched by "positive semi definite" and had scanned all the existing threads....but was not able to find the underlying explanation as to what was meant my the Covariance matrix being "Internally Consistent " ....  Also why the condition w' × Cov × w had to be satisfied in order for the Cov Matrix to be "Internally Consistent"...  will keep looking though....to see if I could get a better understanding of this....

oh..some of the above statements no longer applies since now I see the Gurus explaining this topic...am about to dig in...  #### berrymucho

##### Member
@gargi.adhikari @berrymucho explained it better than i can, sincerely To illustrate, I just retrieved an example of portfolio variance (based on matrix multiplication), XLS is here https://www.dropbox.com/s/5dj4pz4bjyrnhef/1004-semi-definite.xlsx?dl=0
This implements berry's w' × Cov × w = portfolio variance of 0.6579% in the upper panel (notice how the same weights are transposed). Then to violate the positive semi-definite condition, I lazily changed the variance of Asset #1 to a negative (@berrymucho please confirm, I think any such violation is an example, even a negative diagonal value?) so that the resulting portfolio variance is negative. So that's a violation, and practically speaking, a negative variance is not possible, as evidenced by the error for a volatility. I hope that's helpful!) Hi @David Harper CFA FRM , @gargi.adhikari,

Yes, I was thinking similarly about the description, in words, of what non-semi-positive-defineteness would mean in an example. Let's take a 2 assets portfolio with a covariance matrix given by [{0.3 -0.5},{-0.5 0.2}]. It's symmetric, and the variances of the 2 assets, taken individually, are positive, however the variance for a 50%-50% portfolio will be negative... In this case the covariance between the 2 assets (off-diagonal term) is larger than the variance of the individual assets: the 2 assets when interacting with each other produce a greater dispersion than their own taken separately if you will... so the portfolio would be "so diversified" that its variance becomes negative... equivalently, imagine a correlation>1... this of course doesn't make sense. This is what happens in practice, the off-diagonal terms may exceed the variances on the diagonal. The sample variances on the diagonal won't be negative since they are a sum of squares, but they may be underestimated, and/or the off-diagonal terms over-estimated. This is really a numerical issue, generally the inconsistencies are small, but small or big the square root of a negative number is going to fail.
If you work with data and coding, this issue will come to bite you pretty quickly so you take care of testing your matrices. Otherwise, on paper, this is not going to derail your algebra until you have concrete numbers to work with.

#### berrymucho

##### Member
Thanks so much @Nicole Seaman for pointing me to the thread on the topic...I had searched by "positive semi definite" and had scanned all the existing threads....but was not able to find the underlying explanation as to what was meant my the Covariance matrix being "Internally Consistent " ....  Also why the condition w' × Cov × w had to be satisfied in order for the Cov Matrix to be "Internally Consistent"...  will keep looking though....to see if I could get a better understanding of this....

oh..some of the above statements no longer applies since now I see the Gurus explaining this topic...am about to dig in...  It looks like you may actually be asking for the linkage between "internally consistent" and the mathematical expression "w'×Cov×w≥0", more than the meaning of positive-definiteness itself perhaps. To my knowledge "internally consistent" means "w'×Cov×w≥0" by definition as it relates to the properties of covariance matrices (please somebody correct me). Internal consistency has a broader meaning in statistics though, e.g. people who rank high on the "I like cake" question should not rank high on the "I hate cake" question at the same time. Back to the matrices, you can see it in terms of the off-diagonal terms should be consistent with respect to the diagonal terms, as I discussed in my other post.

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