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This is my first question and I would very much appreciate any help.

For a project I am trying to compare different hedging techniques for hedging a long portfolio of bonds.

I have a history of yields for US govt bonds, from which I have computed the first three PCA factors in Python and Excel, please see below.

First row is eigenvalue/sum of eigenvalues, then eigenvalue, then eigenvectors.

Explained variance, eigenvalues and eigenvectors

My questions are:

1) To hedge a portfolio of bonds against one factor, I can multiply the dollar durations of each bond in the (long-only) portfolio by its component of the eigenvector, then hedge by shorting the sum of these, divided by the eigenvalue component of my hedging instrument? eg. if I wanted to hedge the 1m with the 3m against the first factor I should short 0.33*dollarduration(1m)/0.35*dollarduration(3m) of the 3m?

2) Then the same for the next factor and associated eigenvector with another hedging instrument as they are independent?

3) I know these eigenvectors are not the loadings, these are the eigenvectors multiplied by the sqrt of the eigenvalue(?) - but since we are multiplying them all by the same number it doesn't matter which we use for calculating the hedge ratios?

4) Is it correct that the interpretation of the loadings (for levels PCA), as opposed to the eigenvectors, is the change in yield associated with a 1 SD change in the factor?

5) Levels or changes to yields? I have found levels to work quite a lot better, but I know people advocate using changes for PCA.

6) What is the minimum history over which one should compute the factors? I tried 1y, 2y, 3y and they don't seem to be that stable.

7) The hedge ratios implied by linear regression (bond to hedge regressed against hedging instrument) seemed to be almost identical to those implied using my methodology from above and the first factor - is this expected?

Thank you very much!

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  • $\begingroup$ I know this is from more than a year ago, but would anyone be able to help with this? Particularly 1, 2, 3, and 5? $\endgroup$ Sep 21 at 0:42
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There's an old Salomon paper called "Principles of Principal Components: A Fresh Look at Risk, Hedging and Relative Value" which might answer your questions. You can find it online by Googling the title. HTH.

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