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Identify which factors dominate term structure movements, and describe principal components analysis.

FRM® Part I · Valuation and Risk Models · Modeling Non-Parallel Term Structure Shifts and Hedging · FRM1.VR.13.a

The idea

Yield curve changes look complicated, but a few factors drive nearly all of them. Principal components analysis (PCA) finds them statistically. It takes the covariance matrix of historical changes in rates at many maturities and decomposes it into eigenvectors. Each eigenvector is a factor, its loadings show how much each maturity moves per unit of the factor, and its eigenvalue is the factor's variance. The factors are uncorrelated by construction and are ranked by variance explained. The share of variance a factor explains is Share = its eigenvalue ÷ the sum of all the eigenvalues. For government curves, three factors dominate: level (all rates move the same way, usually the largest), slope (short and long rates move in opposite directions) and curvature (the middle moves against the wings). Exam point: the factors are data driven, not chosen by the analyst, and a high explained share describes the historical curve, not every portfolio.

Formulas

Share of yield curve variance explained by a principal component

Share = λᵢ ÷ Σλ

λᵢ
the eigenvalue (variance) of factor i
Σλ
the sum of all the eigenvalues, the total variance

Eigenvalues are variances, so shares add across factors. Using their square roots, or dropping small eigenvalues from the total, misstates the share.

A practice question

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

Empirical studies of government yield curves show that a small number of factors explains most of the variation in the term structure. Which set of factors is generally found to dominate?

  1. Duration, convexity and credit spreadWhy not: Lists price sensitivities of a bond and a spread, not the factors that move the curve.
  2. Short rate, inflation and term premiumWhy not: Names economic drivers of rates; the factors found by PCA are statistical shapes of curve movement.
  3. Level, slope and curvature - correct
  4. Level, volatility and liquidityWhy not: Level is correct, but volatility and liquidity are not curve shape factors; the second and third are slope and curvature.

The three dominant factors found in PCA of government yield curves are level (parallel move), slope (steepening or flattening) and curvature (the middle moving against the wings). The other lists mix in bond risk measures or economic variables.

A harder one

PCA is run on daily changes in six government yields. The eigenvalues of the covariance matrix, in squared basis points, are 31.5, 6.3, 1.4, 0.5, 0.2 and 0.1. What share of the total variance of the yield changes is explained by the first two factors?

  1. 78.8%Why not: Counts only the first factor (31.5 / 40), not the first two.
  2. 98.0%Why not: Adds the third eigenvalue as well (39.2 / 40), giving the share of three factors.
  3. 75.4%Why not: Adds standard deviations (square roots of eigenvalues) instead of variances; the explained share is based on variances.
  4. 94.5% - correct

Total variance = sum of eigenvalues = 40.0. First two factors: (31.5 + 6.3) / 40.0 = 94.5%. The share explained is an eigenvalue ratio, so it is in variance terms.

The hardest tier

A risk manager finds that the first three principal components explain 97% of the variance of historical daily yield curve changes. Which conclusion is most appropriate when assessing the risk of a specific portfolio?

  1. The 97% describes historical curve moves in aggregate; a portfolio concentrated in one maturity or in weaker factors can have more risk unexplained. - correct
  2. Because the factors are uncorrelated, the unexplained 3% diversifies away in every portfolio and can safely be ignored.Why not: Uncorrelated factors do not make the residual vanish; a portfolio can load heavily on the residual.
  3. The factor loadings are fixed over time, so the PCA needs no re-estimation as market conditions and rate regimes change.Why not: Loadings and variances are estimated from a sample and shift with the rate regime; they should be monitored.
  4. Hedging the portfolio's exposure to the level factor alone removes about 97% of its interest rate risk, since level dominates.Why not: Assumes the three-factor share belongs to the level factor; level typically explains the most but not all of the 97%.

The explained-variance share is a property of the historical rate changes across the whole curve. A given portfolio can have exposures that line up with the weak factors or with one maturity, so its unexplained share can be larger than 3%.

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