Hello, Carol,
could you please explain the method about using PCA to hedge interest risk?
if I can calculate three main components from current series of zero bonds (assume they are good enough to explain the yield curve,say 99.9%), how can i hedge the interest rate change?
In my opinion,
for instance, today I get the three components value by using today's zero bonds' prices; if the yield changed tomorrow, those components will change as well. but how to use it to hedge my fix income securities?
or should i change the three components to get new yield curve?
kind of confusing.
Thanks
Luis
PCA Hedging
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 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: PCA Hedging
Hi Luis
A principal component PC is a linear combination of returns on tradable assets; in other words, a PC represents the return on a bond portfolio.
Assuming you used zerocoupon rates for the PCA, then a PC is the return on a zerocoupon bond portfolio (of whatever rating you chose).
So I guess you question is how to replicate the return on a portfolio of zerocoupon bonds? If so, the answer is (a) buy a portfolio of nonzero coupons and swap them with a bank for zerocoupons, or (b) ask a bond trader to strip the coupon for you.
Carol
A principal component PC is a linear combination of returns on tradable assets; in other words, a PC represents the return on a bond portfolio.
Assuming you used zerocoupon rates for the PCA, then a PC is the return on a zerocoupon bond portfolio (of whatever rating you chose).
So I guess you question is how to replicate the return on a portfolio of zerocoupon bonds? If so, the answer is (a) buy a portfolio of nonzero coupons and swap them with a bank for zerocoupons, or (b) ask a bond trader to strip the coupon for you.
Carol
Re: PCA Hedging
Hi, Carol,
Thanks for your quick reply.
So if I use risky bonds 's data, i can still use them to do PCA, is that correct?
say, if i have a portfolio of three Risky coupon bonds and I did PCA on them then get the components.
Now my question is how to hedge those bonds from change of the interest rate.
Do the coefficient of the components can be treated as the weight of each bond?
or i need to calculate the duration of each PC to hedge?
Many thanks.
Luis
Thanks for your quick reply.
So if I use risky bonds 's data, i can still use them to do PCA, is that correct?
say, if i have a portfolio of three Risky coupon bonds and I did PCA on them then get the components.
Now my question is how to hedge those bonds from change of the interest rate.
Do the coefficient of the components can be treated as the weight of each bond?
or i need to calculate the duration of each PC to hedge?
Many thanks.
Luis

 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: PCA Hedging
Well, technologically you can apply PCA to any square symmetric pd matrix whatever
Provided they are of the same credit rating, PCA should work well if you do it on the changes in [u:qaklstfy]any[/u:qaklstfy] interest rates (i.e. log returns on any bonds)
Normally, the purpose is dimension reduction more than orthogonalization (and certainly in this exampe) so you only need to do PCA when you have lots of maturities and you want to cut the risk factor space dimension down to 3
No need for duration. EG suppose PC1 = 0.5*daily change in 3m interest rate + 0.1*dailychange in 6m interest rate and your hedged portfolio is Y  1.3*PC1 that is same as position of 0.5*1.3 in 3m IR AND position of 1.3*0.1 in 6m IR
Provided they are of the same credit rating, PCA should work well if you do it on the changes in [u:qaklstfy]any[/u:qaklstfy] interest rates (i.e. log returns on any bonds)
Normally, the purpose is dimension reduction more than orthogonalization (and certainly in this exampe) so you only need to do PCA when you have lots of maturities and you want to cut the risk factor space dimension down to 3
No need for duration. EG suppose PC1 = 0.5*daily change in 3m interest rate + 0.1*dailychange in 6m interest rate and your hedged portfolio is Y  1.3*PC1 that is same as position of 0.5*1.3 in 3m IR AND position of 1.3*0.1 in 6m IR
Re: PCA Hedging
Hello Carol,
Firstly, I'm a brazilian trader and fan of your work.
I am a bit confused with the heding strategy using PCA.
Lets say i did the PCA in the correlation matrix of a serie of normalized returns, and get as result for the first 2 PCs:
PC1= 0,5.3m 0,4.6m 0,3.9m. (shift)
PC2=0,8.3m 0,1.6m 0,4.9m
To immunize against the PC1 and PC2 a portofolio composed with "n" zerocoupons riskfree fixed bonds, with present value = PV , how should i proceed?
I should choose k securities that neutralize the impacts of the PCs in the yield curve and so on the PV of the portfolio, right?
But my question is, how should i observe the impacts of the PCs in the yield curve, so i could se the variance in the PV of the portofolio?
I mean, i did the PCA in the normalized returns of the serie of yields, and i got the PC1= 0,5.3m 0,4.6m 0,3.9m. (shift).What does that mean exactly in the original yield curve? If wanna check the impact of PC1 in the curve (not into the normalized returns), how should i proceed?
many thanks
Alex
Firstly, I'm a brazilian trader and fan of your work.
I am a bit confused with the heding strategy using PCA.
Lets say i did the PCA in the correlation matrix of a serie of normalized returns, and get as result for the first 2 PCs:
PC1= 0,5.3m 0,4.6m 0,3.9m. (shift)
PC2=0,8.3m 0,1.6m 0,4.9m
To immunize against the PC1 and PC2 a portofolio composed with "n" zerocoupons riskfree fixed bonds, with present value = PV , how should i proceed?
I should choose k securities that neutralize the impacts of the PCs in the yield curve and so on the PV of the portfolio, right?
But my question is, how should i observe the impacts of the PCs in the yield curve, so i could se the variance in the PV of the portofolio?
I mean, i did the PCA in the normalized returns of the serie of yields, and i got the PC1= 0,5.3m 0,4.6m 0,3.9m. (shift).What does that mean exactly in the original yield curve? If wanna check the impact of PC1 in the curve (not into the normalized returns), how should i proceed?
many thanks
Alex

 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: PCA Hedging
Hi
I attach a presentation I just finished which answers your question in quite simple terms  and a spreadsheet
Also, the spreadsheet from the much longer case study in III.2.6.3 on the CD Rom does show exactly how the calculations are done. And again, in Vol IV (if you have it) there are several applications of the PC representaion  its simply [b:3enh2jkr]X = PW'[/b:3enh2jkr].....using notation from MRA.
Cheers, Carol
I attach a presentation I just finished which answers your question in quite simple terms  and a spreadsheet
Also, the spreadsheet from the much longer case study in III.2.6.3 on the CD Rom does show exactly how the calculations are done. And again, in Vol IV (if you have it) there are several applications of the PC representaion  its simply [b:3enh2jkr]X = PW'[/b:3enh2jkr].....using notation from MRA.
Cheers, Carol

 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: PCA Hedging
...and here's the spreadsheet, Cheers!
Re: PCA Hedging
Dear Dr Alexander,
I am an IR trader and I am using the classic PV01 method to calculate my Delta risk to each point of the swap curve. I have recently applied PCA to have and a view of my exposure to the 3 major curve movements. I have used your notes and the workbook “PCA_Monthly” (that I found at the web).
At worksheet “VaR” you use as an input a “PVO1 column” and the three main eigenvectors to find:
Net sensitivities
P1 P2 P3
£8,056 £15,264 £167
Question: Intuitively, what do these three numbers (which are obviously money) mean and how do I use them. Can I say that P1 is the money that I will make if the curve moves in accordance with the move defined by the first eigenvector, P2 is the money that I will make if the curve moves in accordance with move defined by the second eigenvector ext…?.
Thanks a lot in advance.
I am an IR trader and I am using the classic PV01 method to calculate my Delta risk to each point of the swap curve. I have recently applied PCA to have and a view of my exposure to the 3 major curve movements. I have used your notes and the workbook “PCA_Monthly” (that I found at the web).
At worksheet “VaR” you use as an input a “PVO1 column” and the three main eigenvectors to find:
Net sensitivities
P1 P2 P3
£8,056 £15,264 £167
Question: Intuitively, what do these three numbers (which are obviously money) mean and how do I use them. Can I say that P1 is the money that I will make if the curve moves in accordance with the move defined by the first eigenvector, P2 is the money that I will make if the curve moves in accordance with move defined by the second eigenvector ext…?.
Thanks a lot in advance.
Re: PCA Hedging
Hi,
I cannot see the files containing the slides or the spreadsheet for PCA hedging.
I cannot see the files containing the slides or the spreadsheet for PCA hedging.
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