Covariance Matrix

[Liming]

Dear David,

I have a question about the covariance matrix concept presented in page http://www.bionicturtle.com/learn/article/covariance_matrix_65_minutes_frm_quant_video/ . To me, it seems that it has stopped abruptly in the "V" matrix , leaving unsolved the final calculation of the total covariance V (V=DCD) for the three-asset portfolio. So my questions are as follows:
1) should V = summation across the whole table of V? that is: 0.04 + 0.016*2 +0.015*2 + 0.01 + 0.0045*2 + 0.0225 = 0.1435 ?

2) the portfolio covariance should considers permutation, and that's why we can see 0.016 twice and likewise for the other two pairs?

Thank you!

Cheers!
Liming
29/09/2009

 

[David Harper CFA FRM]

Hi Liming,

I attached the XLS in case you want to look at,
here is GOOG version: http://spreadsheets.google.com/ccc?key=0AkH--fhd7VQNdGFvQ1dTZWxIby1TbVVTNDl5QW9pR2c&hl=en

i don't think it would possible to compute a single portfolio covariance (either $ or %^2): we would need the weights (or positions) of the three assets.
(i.e., you can imagine if our portfolio changes the mix of assets, we can get an infinite variety of portfolio variances)
Rather, the V = D*C*D is meant to be matrix notation as in:

3x3 covariance matrix = 3x3 diagonal matrix * 3x3 correlation matrix * 3x3 diagonal matrix

re"2) the portfolio covariance should considers permutation, and that’s why we can see 0.016 twice and likewise for the other two pairs?"
Yes, that is exactly correct: in the covariance matrix, each "pairwise" cell has an identical twin because, e.g,
covariance (asset 1, asset 3) = covariance (asset 3, asset 1)
and the diagonals contains variances because covariance (asset 1, asset 1) = variance (asset1)!

you may find helpful the learning XLS @ http://www.bionicturtle.com/premium/spreadsheet/4.a.2_delta_normal_var/
this reproduced Jorion's example of 3-asset portfolio
except this does include exposures so it uses the covariance matrix to produce a final "single" portfolio volatilty

David