Explain Correlation-weighted HS example

frogs

New Member
Subscriber
Its been a few years out of school so my math is a bit rusty. So I'm not following dowd's example 4.1 for Correlation Weighted HS in the notes, page 26.
Some questions: how did you get A = [1, 0, p, sqrt(1-p^2)]
How did you get that the correlation of A bar is 0.9?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
HI @frogs It is straight-up (literally) from Dowd's example in his Chapter 4. Matrix A is defined by Choleski's decomposition; in Dowd's Chapter 8 he more clearly defines it


so if the historical correlation, rho = 0.30, then you can below (my super quick XLS below) how A*A^T = correlation matrix; i.e., my a(22) = sqrt(1- 0.30^2) = 0.954.

1665793711103.png

Then A^(-1) is is just using the desired rho = 0.90 as merely a given (arbitrary in this example, where the point is to "filter" the returns vector with a higher correlation). So he ends up with a matrix that is the product, A*A^(T); e.g., row 2, column 2 = [ 0.90 * 0 + sqrt(1 - 0.90^2) * 1 ] * [1/ (sqrt(1 - 0.30^2)] = 0.4569. This matrix multiplies by (aka, "filters") the actual historical return vector (from the 0.30 correlation) to a desired correlation of 0.90. If i get time next week, I'll share an XLS, there's just no way shortcut around the matrix math in Dowd if you really want to follow Cholesky's decomposition,. Thanks,
 
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