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Describe principal component analysis (PCA) and how it helps in putting together a portfolio of hedges.

FRM® Part II · Market Risk Measurement and Management · Regression Hedging and Principal Component Analysis · FRM2.MR.11.h

The idea

Principal component analysis (PCA) takes the covariance matrix of changes in a set of rates and rewrites it as uncorrelated factors, the principal components, ranked by the variance each explains. The component variances add up to the total variance. For yield curves the first three components are usually read as a level shift, a slope change and a curvature change. Together they explain the great majority of variance, often well above 90 percent. For hedging, measure the portfolio's exposure to each of the first few components and trade instruments so those exposures net to zero, using as many hedging instruments as components hedged. Because the components are uncorrelated, the residual risk is the sum of the variances of the unhedged components. The exam point is that PCA reduces many rates to a few factors and hedges those factors.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement best describes principal component analysis (PCA) as applied to a term structure of interest rates?

  1. It regresses each rate change on all the others and calls the fitted values the components.Why not: Describes a multiple regression; PCA components are uncorrelated linear combinations, not fitted values.
  2. It requires the analyst to specify the level, slope and curvature factors in advance and estimates only their volatilities.Why not: The factors come from the data, not the analyst; the interpretation as level, slope and curvature comes after estimation.
  3. It splits each rate into a deterministic trend and a random error.Why not: Describes trend-plus-noise modelling, not a decomposition of covariance.
  4. It decomposes the covariance of rate changes into uncorrelated components ranked by variance explained, the first three typically interpreted as level, slope and curvature. - correct

PCA finds orthogonal directions of maximal variance in the covariance matrix of rate changes. For yield curves the first three are level, slope and curvature.

A harder one

A PCA of six rate series gives component variances of 7.2, 1.9, 0.5, 0.2, 0.1 and 0.1. A risk team wants to hedge enough components to cover at least 95% of total variance. How many components must it include?

  1. Two.Why not: Two components cover only 91%, below the 95% threshold.
  2. Three. - correct
  3. Four.Why not: Overshoots: three components already cover 96%.
  4. Six.Why not: Includes every component; the aim is only to reach the threshold, which three components do.

Total = 10. Cumulative shares: 72%, 91%, 96%. The third component takes the total past 95%.

The hardest tier

A portfolio has P&L sensitivities per one-standard-deviation move of USD -240,000 to the level component and USD +90,000 to the slope component. A 5-year swap has sensitivities of +38,000 (level) and -12,000 (slope) per USD 10 million face; a 30-year swap has +85,000 (level) and +32,000 (slope) per USD 10 million face. To neutralize both exposures, the position in the 5-year swap is closest to:

  1. buying USD 68.6 million face. - correct
  2. buying USD 63.2 million face.Why not: Hedges the level exposure only, 240,000/38,000 per 10 million, ignoring the slope exposure and the 30-year swap.
  3. selling USD 68.6 million face.Why not: Right size, wrong sign; the portfolio's level exposure is negative so hedge exposure must be positive.
  4. selling USD 2.4 million face.Why not: This is the position in the 30-year swap rather than the 5-year swap, so it answers for the wrong instrument.

Let x5, x30 be face in units of USD 10 million. Level: 38 x5 + 85 x30 = 240; slope: -12 x5 + 32 x30 = -90 (in thousands). Solving gives x5 = 6.856 (buy 68.6 million) and x30 = -0.2415 (sell 2.4 million).

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