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NEW QUESTION: 1
Which of the following statements are true in relation to Principal Component Analysis (PCA) as applied to a system of term structures?
I. The factor weights on the first principal component will show whether there is common trend in the system II. The factors to be applied to principal components are obtained from eigenvectors of the correlation matrix III. PCA is a standard method for reducing dimensionality in data when considering a large number of correlated variables IV. The smallest absolute eigenvalues and their associated eigenvectors are the most useful for explaining most of the variation
A. I and IV
B. II and IV
C. I and III
D. I, II and III
If you have not studied PCA prior to the preparing for the PRMIA exams, you will find the handbook to be a bit difficult to understand this topic. However, PRMIA have been asking questions about PCA on their exams so it is important to conceptually know what PCA is (and be able to answer the exam questions which are unlikely to ever require you to do a calculation).
Using PCA, a complex correlated system (eg, the rates in a term structure which are all correlated to each other) is effectively transformed into a 'principal component representation'. A principal component representation expresses each of the independent variables in the system as a linear representation based on
'principal components'. Principal components are derived from the eigenvectors of the correlation matrix.
Remember that if you have a positive semi-definite correlation matrix (which will be an n x n square matrix), it will have 'n' eigenvectors, each with an eigenvalue. The eigenvectors will be orthogonal (ie perpendicular) to each other. The eigenvalues for each of the eigenvectors determine which are principal components - in descending order of the eignevalues. Therefore the eigenvector with the highest eigenvalue is the first principal component, the second highest value the second principal component and so on. If there are n variables in the system, then eigenvalue divided by n gives us the proportion of the variation explained by that particular principal component.
In the case of a PCA done on the term structure of interest rates, the first principal component is the 'trend' component, and under a principal component representation its coefficients are given by the first eigenvector.
(By the way, remember the next two principal components are tilt and curvature respectively). If these coefficients are the same (they are, for an interest rate term structure), this means all rates move up or down when the value of the first principal component moves. Thust factor weights on the first component show whether there is common trend in the system. Statement I is correct.
The factors to be applied to the principal components are indeed obtained from the eigenvectors of the correlation matrix (and not the covariance matrix), and therefore statement II is correct.
PCA is used to reduce dimensionality, this is true and in fact one of the main reasons why PCA is used at all.
Therefore statement III is correct.
Statement IV is false as it is the largest eigenvalues and not the smallest that determine the eigenvectors affecting most of the variation (through the principal components they represent).
Thus Choice 'b' is the correct answer.