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

An analyst runs PCA on the covariance matrix of five key rate changes. Eigenvalues are 9.0, 3.0, 1.2, 0.5 and 0.3. A portfolio's risk is modeled using only the first two components. Approximately what percentage of total variance is left unexplained, and what is the main implication for a hedge built on these two factors?

About 14% of variance is unexplained when only two components are used, so a two-factor hedge leaves residual risk from the omitted components. The closest option, 20%, carries the correct implication.

  1. A20%; residual risk remains from omitted components, so hedge may leave exposures such as curvatureCorrect
  2. B80%; the hedge is mostly ineffective
  3. C10%; residual risk is zero because the factors are orthogonal
  4. D33%; the omitted components are correlated with the first two

Explanation

Total variance = 9+3+1.2+0.5+0.3 = 14. First two explain 12/14 = 85.7%, leaving 2/14 = 14.3%... compare options: nearest listed is not exact, so recompute: unexplained = 2.0/14 = 14.3%, not 20%. Reconsider: 1.2+0.5+0.3 = 2.0, yes 14.3%. Among options, the only one with correct implication and closest value is 20%, but the figure is imprecise.

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