PCA and Immunization

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konti
Posts: 2
Joined: Thu Feb 28, 2013 2:30 pm

PCA and Immunization

Postby konti » Sun Mar 03, 2013 6:28 pm

Dear professor Alexander,

I am writing you concerning my research on application of PCA to immunization of interest rate risk in the pension fund’s asset-liability portfolio. In my research I am trying to construct a hedge that satisfies the following criteria. First, immunization of the asset-liability portfolio. Second, a constant hedge ratio and a stable P&L ratio on a day to day basis. Application of your examples from the book volume II to my situation, gave raise to the following questions.

First, in your example analysis you use daily yield changes as input frequency for the covariance matrix. Also, you use two years of historical data to construct your principal components. Varying the frequency of data causes a proportional change in variance of the PC’s. Now since I am interested in minimizing the adverse effects of short-term (day basis) movements in the yield curve, I conclude that daily changes is the right frequency here? However, it is not clear how many historical points I should use. For example, principal components from two years of daily yield changes would include periods of different monetary policies and economic environments but will not reflect prevailing market circumstances. In contrast, principal components from two months of daily yield changes pronounce prevailing market circumstances but are subject to cyclical policy change risk and in addition, as I expect, subject to strong deterioration of hedge performance, measured in stability of the hedge and P&L ratios. However, as I am interested in day to day stability of my hedge and P&L ratios and given an environment of intensive monitoring and if needed, frequent rebalancing, a shorter period of historical yield changes should better fit my situation?
S
econd, while the swap curve is only quoted up to fifty years, cash flow vectors of asset and liability portfolios go sometimes up to hundred years into the future. If PCA is performed on the swap curve, this gives rise to the problem of unequal sized DV01 vectors and eigenvectors. However, according to the official rules after 50 years the zero curve is kept constant. Therefore, I assume it is plausible to bring back the post 50 year bucket DV01’s back to year 50 for bonds, swaps and liabilities? In addition, recent policy changes introduced a new curve for liabilities, the Ultimate Forward Rate (UFR) curve. Up to twenty years the UFR curve is the same as the forward curve. From twenty years onwards, using non-linear Smith-Wilson (based Hermes polynomials) extrapolation method, it is being extrapolated to the Ultimate Forward Rate ( 4.2 percent in year sixty), afterwards curve is kept constant. Since in this setting it is not possible to bring the post 50 year bucket DV01’s to year 50, how should I approach the problem of Unequal sized DV01 and eigenvectors?

Third, in the aforementioned problem I want to perform the PCA on the swap curve. However, in the setting where my portfolio consists out of bonds, swaps and liabilities, will be there any quantitative difference if I opt using zero or forward curve for PC construction?

Fourth, in example II.2.8 you describe a portfolio which consists out of multiple asset classes where you consequently apply a multiple curve PCA. Is there any ground for this in the portfolio consisting of bonds, swaps and liabilities where the latter one is discounted with the Ultimate Forward Rate (UFR) curve? Thank you in advance for answering this long list of questions!

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: PCA and Immunization

Postby coalexander » Sun Mar 10, 2013 11:55 am

Hi

Apologies for the late reply -- I have been travelling, so I hope my comments are still useful.

1) I agree that in your case 2 years is too long an historical period. I would use the sample size as an input parameter, N, and look at final results for different N, with N = 50, 100, 250, 500. Shorter samples will give better PCA results and more effective immunization, but you need enough data to prevent too much re-balancing...

2) Yes, it plausible to bring back the post 50 year bucket DV01’s back to year 50 for bonds, swaps and liabilities for reasons you set out. I would stick with standard forward curve or swap curve -- the UFR curve is same up to 20 years, where almost all the action is in the variability of the curve, and the rest is artificially constructed.

3) This is an empirical question. You should get very similar results from both curves, but this depends somewhat on the currency ...I would use BOTH curve and the report on differences, if any. If you find negligible difference in results that itself is useful to other researchers.

4) I would not bother with UFR for reasons given.

Cheers, Carol

konti
Posts: 2
Joined: Thu Feb 28, 2013 2:30 pm

Re: PCA and Immunization

Postby konti » Mon Mar 11, 2013 9:28 am

Dear professor Alexander,

Thank you for your reply. However, second point of your answer still leaves me with the immidiate problem of how to deal with UFR curve in the PCA context. To clearify, its not my own choice to discount the liabilities with the UFR. Given new regulations I am bound to do so. Yes, it is true that the most variablity takes place before the 20 year point, however, given the fluctuations on the 20 year point there will still be some variability left between 20 year point and T2. Since risk managers are bound to report the hedge ratio based on the UFR curve, my question is how I can account for the varibility between the 20 year point and T2 in the context of PCA? This is also the main reason why I asked you whether multi-curve PCA is a sensible approach to this problem. Once again, thank you in advance for considering my problem.

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: PCA and Immunization

Postby coalexander » Tue Mar 12, 2013 3:35 pm

Hi again

In multiple curve PCA the perfect collinearity at shorter end between UFR and forward curve might cause problems because your correlation matrix will have many zero eigenvalues ... but you could use UFR at maturities greater than 20 years only, add that to the risk factors...worth a try! Good luck, and let me know how you get on. Carol


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