FRM Exam Part I · Modeling Non-Parallel Term Structure Shifts and Hedging

Principal Components Analysis of the Yield Curve

Updated 11 October 2026 · Fact-checked

PCA breaks yield curve changes into uncorrelated factors ranked by variance explained. The first three are usually level (parallel shift), slope (steepening or flattening) and curvature (change in the belly). Together they typically explain most of the variance, with level the largest. Each factor has an eigenvector of loadings and an eigenvalue equal to its variance.

Understand Principal Components Analysis of Term Structure

A yield curve has many points: 1-year, 2-year, 5-year, 10-year, 30-year and more. These rates move together, but not perfectly. Tracking every rate separately is messy. Principal components analysis (PCA) finds a small set of new factors that capture most of the movement.

PCA works on the covariance (or correlation) matrix of rate changes. It produces eigenvectors and eigenvalues. Each eigenvector is a set of loadings, one per maturity, that defines a factor. Each eigenvalue is the variance of that factor. Factors are uncorrelated with each other and ordered from largest eigenvalue to smallest.

For government yield curves, the pattern is stable across markets such as US Treasuries and EUR curves. Factor 1, level, has loadings of the same sign across maturities, so all rates move in the same direction. This is a roughly parallel shift and explains the most variance. Factor 2, slope, has loadings that change sign from short to long maturities. Short rates fall while long rates rise, or the reverse. Factor 3, curvature, has loadings with one sign at the short and long ends and the opposite sign in the middle. It describes a bend or twist in the belly.

The share of variance explained by a factor is its eigenvalue divided by the sum of all eigenvalues. Because the sum of eigenvalues equals the total variance of the rate changes, the shares add to 100%. In practice the first factor explains the large majority, the first two explain nearly all, and the first three explain almost everything. The exact figures depend on the data, so use the numbers given in the question.

For risk management, PCA reduces the dimension of the problem. You can measure and hedge exposure to three factors instead of dozens of rates. This also fits the theme of this chapter: a single duration hedge covers only the level shift, while slope and curvature risks need extra hedging instruments.

Key formulas to remember

Variance explained by a factor: Share of factor i = λᵢ ÷ Σλⱼ; λᵢ is the eigenvalue of factor i. The denominator is the total variance, the sum of all eigenvalues.
Cumulative variance explained: Cumulative share (first k factors) = (λ₁ + … + λₖ) ÷ Σλⱼ; Add the shares of the top k factors. They are uncorrelated, so variances add.
Total variance: Σλⱼ = Σ variance of each rate change (when PCA uses the covariance matrix); With a correlation matrix, the total equals the number of variables.
Factor standard deviation: σᵢ = √λᵢ; The eigenvalue is a variance. Take the square root for a volatility in the same units as the rate changes.
Rate change from factors: Δrate(T) = Σ (loading of factor i at T × factor score i); Loadings come from eigenvectors. The factor scores are uncorrelated.
Eigenvector normalization: Σ(loadings of one factor)² = 1; Use this to check or fill in a missing loading.

How to solve Principal Components Analysis of Term Structure questions

Use this method for any PCA-of-the-yield-curve question. Most ask you to identify a factor, compute variance explained, or apply loadings.

1Identify what is given: eigenvalues, eigenvectors (loadings), or a described pattern of rate moves.
2If asked to name a factor, read the sign pattern of the loadings. Same sign everywhere is level. A sign change from short to long is slope. Signs that change twice (ends versus middle) is curvature.
3If asked about variance explained, add all eigenvalues to get the total variance.
4Divide the relevant eigenvalue (or the sum of the top k) by the total.
5If asked about the effect of a shock, multiply each loading by the factor shock and read off the change at each maturity.
6If asked about volatility, take the square root of the eigenvalue.
7Check the answer: shares must lie between 0 and 100%, and the first factor must have the largest eigenvalue.

Quickest way: Sign-pattern and share shortcut

When to use it: Use it when the question shows a table of loadings or eigenvalues and you have little time.

Scan the loadings for signs. All the same: level. One sign change: slope. Two sign changes: curvature.
Sum the eigenvalues once. Write the total down.
Divide only the eigenvalue you need. Do not compute every share.
Eliminate options that put curvature as the largest factor or that give shares above 100%.
For a shock question, multiply loading by shock for the one maturity asked and ignore the rest.

Common mistakes in Principal Components Analysis of Term Structure

Dividing an eigenvalue by the largest eigenvalue instead of the sum of all eigenvalues.
Students think of a share relative to the top factor.
Fix: The denominator is always the total variance, the sum of all eigenvalues.
Treating the eigenvalue as a standard deviation.
The word value hides that it is a variance.
Fix: Take √λ to get the factor volatility.
Calling a factor with mixed-sign loadings the level factor.
Students memorize the order (level, slope, curvature) but not the sign patterns.
Fix: Level has one sign throughout. Check signs, not just the rank of the eigenvalue.
Assuming the factors are correlated, so variances cannot be added.
Rates are highly correlated, so students carry that over to the factors.
Fix: Principal components are uncorrelated by construction. Their variances add.
Believing a duration hedge removes all yield curve risk if level explains most variance.
A large first-factor share makes the rest seem negligible.
Fix: A duration hedge covers only parallel shifts. Slope and curvature risk remain and can matter for large or long-short positions.
Stating a fixed percentage such as a set figure for level as a universal rule.
Textbook examples show specific numbers.
Fix: The shares depend on the data and period. Use the numbers in the question, and remember only the ordering and the typical dominance of level.

Worked examples

Example 1

A PCA of changes in five Treasury yields gives eigenvalues 8.0, 1.5, 0.3, 0.15 and 0.05. What proportion of total variance do the first two factors explain together?

Show the solution

Total variance = 8.0 + 1.5 + 0.3 + 0.15 + 0.05 = 10.0.
First two factors = 8.0 + 1.5 = 9.5.
Share = 9.5 ÷ 10.0 = 0.95.

Answer: 95% of total variance is explained by the first two factors.

Example 2

A factor has loadings of 0.50 (2-year), 0.50 (5-year), -0.50 (10-year) and -0.50 (30-year), with the signs of the loading going from positive to negative once. Its eigenvalue is 4. Identify the factor, and find the effect on each rate of a one-standard-deviation shock to this factor.

Show the solution

Check normalization: 0.50² + 0.50² + (-0.50)² + (-0.50)² = 0.25 + 0.25 + 0.25 + 0.25 = 1. The loadings are valid for an eigenvector.
Signs change once from short to long maturities, so this is the slope factor.
Factor standard deviation = √4 = 2.
Change in each rate = loading × shock. 2-year: 0.50 × 2 = +1.0. 5-year: 0.50 × 2 = +1.0. 10-year: -0.50 × 2 = -1.0. 30-year: -0.50 × 2 = -1.0 (in the units of the rate changes).
Interpretation: short and medium rates rise while longer rates fall, so the curve flattens.

Answer: It is the slope factor. A one-standard-deviation positive shock changes the 30-year rate by -1.0 in the units of the data, while the 2-year and 5-year rates each rise by 1.0, so the curve flattens.

Exam tips

Memorize the sign patterns: none, one change, two changes for level, slope, curvature.
Always compute the share using the sum of all eigenvalues. Check whether the question gives a total.
If the question gives eigenvalues in descending order, the first is level unless the loadings say otherwise.
Link PCA to hedging: a duration hedge covers level only. Expect options that say slope and curvature remain.
Watch whether the question asks for variance or standard deviation. Take the square root only for the latter.

Practice questions from Modeling Non-Parallel Term Structure Shifts and Hedging

Principal Components Analysis of Term Structure in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Principal Components Analysis of Term Structure: frequently asked questions

What are the three main factors in PCA of the yield curve?

They are level, slope and curvature. Level moves all rates the same way, slope moves short and long rates in opposite directions, and curvature moves the middle against the ends. Level explains the most variance.

How much variance do the factors explain?

The exact shares depend on the data. Typically the first factor explains the large majority and the first three explain almost all. In an exam, compute the share from the eigenvalues given.

What is the difference between an eigenvector and an eigenvalue in PCA?

An eigenvector holds the loadings that define a factor across maturities. An eigenvalue is the variance of that factor. Dividing it by the sum of eigenvalues gives the share of variance explained.

Why is PCA useful for hedging?

It reduces many correlated rates to a few uncorrelated factors. You can then measure and hedge exposure to level, slope and curvature with a small number of instruments instead of every rate.