Using Change-Based PCA to Assess Interest Rate Richness and Cheapness
Summary
The discussion addresses how to use principal component analysis for relative-value analysis when interest rate levels are nonstationary. One proposed method applies PCA to daily rate changes, cumulatively sums the resulting factors, and regresses rate levels on those cumulative factors. Researchers can then examine whether the regression residuals are stationary and potentially use large positive or negative residuals to identify bonds whose yields appear rich or cheap relative to the model. A factor error correction model is offered as a related approach for forecasting rates and informing bond positioning.
The replies also describe a common alternative: apply PCA directly to rate levels for rich-cheap analysis, while using change-based PCA for risk management. These are suggestions rather than results from a data analysis; the document supplies no empirical test or investment performance evidence. Residual stationarity testing, panel unit root methods, model specification, and possible adjustments for a zero lower bound all require care, and the discussion does not establish that any one approach will work across markets or samples.
Key ideas
- PCA on daily rate changes can produce factors whose cumulative sums serve as explanatory variables for rate levels.
- Regressing rate levels on cumulative factors creates residuals that can be tested for stationarity.
- Large residuals may be used as relative-value signals, with positive residuals interpreted as yields above model-implied levels.
- A factor error correction model can extend the approach by forecasting rates for bond positioning.
- Level-based PCA is also commonly suggested for rich-cheap analysis, while change-based PCA can support risk management.
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# Reproducing levels when PCA has been done on changes # Reproducing levels when PCA has been done on changes I want to use PCA for rich/cheap analysis of interest rates. For this I did the PCA on the time series of daily difference in interest rates, which is stationary. I cant do pca on levels, as they are not stationary and I have not been able to find any suitable transformation to convert it to stationary either. I can then reproduce the daily changes using selected eigen vectors. But since I want to make a conclusion about the levels of the original rates(rich/cheap), how should I reproduce the levels from these PCA-based changes? And if doing this even makes sense statistically. I can post my data if that helps. ## Answer by John (score 4) https://quant.stackexchange.com/a/18311 The literature on cointegration in large datasets or panels is really the only place where I've seen this sort of issue discussed. Breitung and Pesaran, among other places, talks about it. I would recommend applying the PCA to the rate changes (perhaps with some kind of zero lower bound adjustment). Then, take the cumulative sum of each of the factors. This is effectively like the cumulative factor you would be using in a cointegration test or factor error correction model. Next, regress the rates (in levels) against the cumulative factors. At this point, you can test the residuals of this regression for stationarity (I would expect the residuals to be mean-reverting, there are a lot of nuances to panel unit root testing that I'm ignoring). You have a number of options when creating an investment strategy based on this information. The most obvious is to just buy (sell) the bonds where the residuals are most positive (negative). Highly positive residuals indicates that the yields are more than what the model would suggest and should fall. Obviously, there are a lot of variations you can make on this. Another alternative is to estimate a factor error correction model. A factor error correction model bears a lot of similarity to the approach described in the second paragraph, but just extends it a little. Once you estimate the model, you can forecast rates, and use that to determine your bond positioning. ## Answer by Helin (score 4) https://quant.stackexchange.com/a/18313 I just want to mention that it's highly prevalent to apply PCA to rate levels in rich/cheap analyses. Personally I prefer that... There's an old MS publication that discusses this very topic and the recommendation is to use level PCA for rich/cheap, and to use change PCA for risk management. There's a really good Salomon paper (Principles of Principal Component) that discusses this in depth as well.
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