PCA Hedging

[afterworkguinness]

Hi,
Can you breakdown how Tuckman arrives at the required face values of the 2 and 10 year swap rates to hedge the 5 year swap rate ? Thanks

He sets out the hedge like so:

-Face-2year*(DV01-2year/100)*Change_in_2year - Face-10year*(DV01-10year/100) * Change_in_10year - Face-5year *(DV01-5year)*Change_in_5year = 0

We are given the changes in the 2, 5 and 10 year rates from two different components, and Tuckman is constructing a hedge to hedge against both.

I can't figure out how he obtained the face values required for the hedge.

EDIT:
I think I can make my question a bit stronger so here it goes:

Tuckman says the face value required of the two year is short 120.26 and the face value required of the 10 year is short 34.06. I can't figure out how these values were arrived at. With the two variable regression hedge, we found the requisite face values like so

Face_10year = (DV01_20year/DV01_10year)*Beta_10year
Face_30year = (DV01_20year/DV01_30year)*Beta_30year

But with the PCA hedge, we are trying to construct a hedge against two components not just one. I tired the above aproach but added the expected change in the 2 year rate from both PCs to compute Face_2year and the same approach for the Face_10year and obtained different results.

 

[ami44]

Hi afterworkguinness,

the formula you cite is for calculating the P&L when you know the face values already.

To calculate the face values you need the DV01 and the exposure of the three rates to the PCs, lets call them PC1_2year, PC2_2year, PC1_5year, ....
Then you solve these two equations:
F2 * DV01_2 * PC1_2year + F5 * DV01_5 * PC1_5year + F10 * DV01_10 * PC1_10year = 0
F2 * DV01_2 * PC2_2year + F5 * DV01_5 * PC2_5year + F10 * DV01_10 * PC2_10year = 0

You have two unknown F2 and F10. You know F5 because its the amount you want to hedge.
The rationale behind the equations is, that the total exposure to each PC must be 0.

 

[afterworkguinness]

Hi @ami44 ,
Thanks for you reply, though I still can't see how to solve for both the 2 year face value and 10 year face value; the equations you listed each have two unknowns.

 

[ami44]

You have two unkown and two equations.
First solve the first equation for F2,
F2 = (F5 * DV01_5 * PC1_5year + F10 * DV01_10 * PC1_10year) / (DV01_2 * PC1_2year)
than plug that into the second equation and eliminating F2 from it. Than solve that for F10.

But that is a bit tedious, I doubt that we have to do that in the exam.
The point is, its solvable.

 

[afterworkguinness]

Maybe I'm not seeing the forest for the trees, it's been an arduous day of studying, but the above solution says to solve for the unknown face value of the 2 year I need to know the unknown face value of the 10 year ... + F10 * DV01_10 * PC1_10Year...

What am I missing ?

Thanks for your time.

 

[ami44]

You solve the first equation for F2
F2 = (F5 * DV01_5 * PC1_5year + F10 * DV01_10 * PC1_10year) / (DV01_2 * PC1_2year)
As you said, you still don't know what F2 is, since you don't know F10.
What you have is an expression for F2 that depends on F10.
In the second equation you substitute F2 with that expression
[ (F5 * DV01_5 * PC1_5year + F10 * DV01_10 * PC1_10year) / (DV01_2 * PC1_2year) ] * DV01_2 * PC2_2year + F5 * DV01_5 * PC2_5year + F10 * DV01_10 * PC2_10year = 0

This is now an equation with only one unknown F10. Now you can calculate F10 and use that value to calculate F2 with the help of the first equation.
The standard way to solve a system of n linear equations is the Gauss algorithm which is a little bit cleverer than what I just described.

But as I said, I doubt we have to do any of that in the exam.

 

[afterworkguinness]

@ami44 thanks very much.