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FRM Part II · FRM Exam Part II · Regression Hedging and Principal Component Analysis

A risk analyst applies principal component analysis (PCA) to the covariance matrix of daily changes in six par swap rates. Which statement best describes the first principal component extracted?

The first principal component is the unit-length weighted combination of the rate changes that captures the largest share of total variance. Later components are uncorrelated with it. It often looks like a parallel shift, but that is an empirical result, not a definition.

  1. AIt is the linear combination of rate changes with the smallest variance, orthogonal to all others
  2. BIt is the linear combination of rate changes, with unit-length weights, that explains the largest share of total varianceCorrect
  3. CIt is the average of the six rate changes, by construction
  4. DIt is the component that is most highly correlated with the second principal component

Explanation

PCA chooses weights of unit length that maximize the variance of the resulting combination; this is the first component. Components are uncorrelated by construction, so the last option is wrong. The first component often resembles a parallel shift but is not defined as a simple average.

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