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Hi, does anyone have an intuitive explanation for the principal components analysis? Any help will be greatly appreciated!
PCA is a reasonably complex concept. Fortunately, it is not tested too extensively on the PRMIA exam though there have been questions about orthogonality etc which are related (https://riskprep.com/all-tutorials/36-exam-2/123-eigenvectors-eigenvalues-and-orthogonality).
That said, PCA is an important tool, not just used in finance but in other fields as well, such as data mining primarily for reducing the number of independent variables to reduce complexity (‘dimension reduction’).
I know I am not helping much but let me put an article or something up in the coming weeks that provides a conceptual view of what PCA is and where it can be used, and of course one can figure the math out if ever actually required.
And in the meantime if someone else has anything to share on PCA, please do post it here, will be greatly appreciated.
-Mukul Pareek
This may or may not help your intuition on this matter. Say you’ve got a yield curve with points on the term structure for the overnight rate, 30 day rate, 60 day rate, 90 day rate, 180 day rate, 1 year, 2 year, 5 year, 10 year, 25 year. That’s 10 points. Rates on a particular date for these 10 points could be (5.0,5.1,5.2,5.3, 5.4, 5.6,5.8,6.1,6.5,6.9) say. The next day they might be a bit different. Anyway, over many days we collect rates for these term structure points. The rates on each day have 10 components, ranging from “overnight” through to “25 years”. Conceptually at least, you could decide to plot the rates for each day as a SINGLE POINT in a 10-dimensional space. Let’s say the coordinate system for this space is X1, X2, X3, ….X10. Then for the rates above [(5.0,5.1,5.2,5.3, 5.4, 5.6,5.8,6.1,6.5,6.9)], this is a SINGLE POINT in 10-space with coordinates X1=5.0, X2=5.1, …..X10=6.9). So we can proceed to plot all our rates, for each day, in this 10-space. If we have rates for 200 days, we will end up with 200 points.
In practice, i.e., empirically, what happens is that these points don’t really “fill out” this 10 dimensional space. For the most part they are confined a 3-dimensional subspace (at least for interest rates, not necessarily for other multivariable data). What PCA does is to find a new coordinate system for this subspace – PC1, PC2, PC3. It’s not any old coordinate system, it has features such as PC1 explaining most of the variability in the rates data, and then selecting PC2, ranked second in explaining variability and also orthogonal to PC1, etc.
PC1 is generally associated with a change in the overall level of rates, PC2 with the “tilt” in rates (difference between short and long rates) and PC3 with “bow” (how straight or how curved is the yield curve). This is explained on p.151 of the Handbook for Exam III.
Mathematically/numerically it is much simpler to have to deal with just 3 uncorrelated risk factors than the original 10 correlated risk factors.
Personally, I’m still trying to get a better understanding myself of this area. The above is just my vague recollection of doing this in statistics classes at university, so don’t rely on it too much.Hello,
here is some tutorial I found useful regarding this topic:
http://quantturk.com/wp-content/uploads/2011/09/PCA-tutorial1.pdf
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