Applying cholesky decomposition to variance covariance matrix?

Question

Hello,

I have got 2 asset's times series of 2400 data each and I have to:

1st: get my variance covariance matrix (which will be 2x2 matrix size wright?)
2nd: I need to apply cholesky decomposition to get my new "D" matrix and then calculate: D'D=E
-if my variance covariance 2x2 matrix is (1.5471; 0.03356)
(0.0356; 0.03082),
how do I apply the cholesky decomposition?and why do I use
the cholesky decomposition? what it the sense of using it?


thank you

Answer 1

1. That sounds right, if I'm understanding the question correctly.



2. If you apply the Cholesky algorithm (see first link) to the matrix A = [ [ a b ] [ b c ] ], you get:



L = [ [ √a 0 ] [ b/√a √(c - b²/a) ] ]



You can verify that L L* = A.



In your case, and using your notation:



L = (1.2438; 0) (2.6981; 0.1735)



3. The reason why computing the Cholesky decomposition is useful is that L is, in a sense, the "square root" of A. If you think of the covariance matrix as the variance of a multivariate distribution, then the Cholesky decomposition is the square root of the variance, that is, it's the standard deviation.



Anywhere that you would use the standard deviation in univariate statistics, you would use L here.

Answer 2

You need to know the weights of each of the stocks in the market portfolio. Then to get the betas you: a) Calculate the variance of the market portfolio using the weights and the variance covariance matrix b) Divide each covariance by the variance of the market portfolio for the beta Note that a is a fairly lengthy process for a test.

Answer 3

Cholesky Decomposition Example 2x2