FRM Exam Part II · Regression Hedging and Principal Component Analysis

Principal Component Analysis Fundamentals for FRM Part II

Updated 11 October 2026 · Fact-checked

PCA rewrites correlated risk factors, such as yield changes, as uncorrelated factors called principal components. Each component is an eigenvector of the covariance matrix. Its eigenvalue is the variance it explains. Variance explained = eigenvalue ÷ sum of all eigenvalues. Rank components by eigenvalue and keep the top few.

Understand Principal Component Analysis Fundamentals

Interest rates at different maturities move together. If you track ten key rates, you have ten risk factors, but they are highly correlated. PCA reduces this to a few uncorrelated factors that capture most of the movement.

The input is the covariance matrix (or correlation matrix) of the changes in the risk factors, for example daily changes in 2-year, 5-year, 10-year and 30-year yields. PCA finds the directions in which the data varies most. Each direction is an eigenvector. It is a set of weights (loadings) on the original variables. The variance of the data along that direction is the matching eigenvalue.

Eigenvectors of a symmetric covariance matrix are orthogonal, so the components are uncorrelated. The eigenvalues add up to the total variance, which is the trace of the covariance matrix. So the share of variance explained by component k is its eigenvalue divided by the sum of all eigenvalues. You order components from largest to smallest eigenvalue.

For government yield curves, the first three components are usually read as level (parallel shift), slope (steepening or flattening) and curvature (butterfly). These typically explain most of the variance, so a risk manager can hedge or measure risk with three factors instead of ten or more. This is why PCA is used for interest rate risk.

Two cautions. PCA is purely statistical, so the factors come from past data and may change. Also, if you use a covariance matrix, variables with larger variance dominate. A correlation matrix puts every variable on the same scale. The component scores (the time series of each factor) are uncorrelated with each other, and the variance of a component's score equals its eigenvalue.

Key formulas to remember

Eigenvalue equation: Σv = λv; Σ is the covariance matrix, v is an eigenvector (loadings), λ is its eigenvalue, the variance explained by that component.
Variance explained by component k: λk ÷ (λ1 + λ2 + ... + λn); Multiply by 100 for a percentage. The denominator equals the total variance, the sum of the diagonal of Σ.
Cumulative variance explained: (λ1 + ... + λk) ÷ Σλ; Used to decide how many components to keep.
Total variance: Trace(Σ) = Σ of the variances = Σ of the eigenvalues; Use it to check that your eigenvalues are consistent.
Component score: PCk = v1k·x1 + v2k·x2 + ... + vnk·xn; The xi are the changes in the original factors. Scores of different components are uncorrelated.
Orthogonality and normalisation: vi · vj = 0 for i ≠ j; Σ(vik)² = 1; Loadings are scaled so each eigenvector has unit length.
Volatility of a component: σk = √λk; Standard deviation of the component score.

How to solve Principal Component Analysis Fundamentals questions

Use this method for any PCA question on variance, loadings or hedging.

1Identify the input: a covariance matrix or a correlation matrix, and the risk factors it describes.
2List the eigenvalues. If they are not given, check that they sum to the trace of the matrix.
3Sort the eigenvalues from largest to smallest. This ranks the components.
4Compute variance explained for each component as eigenvalue ÷ sum of eigenvalues. Add them up for cumulative variance.
5Read the eigenvector loadings. Same-sign loadings across maturities mean level. Sign change from negative to positive means slope. Positive, negative, positive means curvature.
6To find a portfolio's exposure to a component, multiply the position changes by the loadings and sum. Volatility of that component is √λ.
7Check the answer: shares must sum to 100%, and eigenvectors must be orthogonal with unit length.

Quickest way: Eigenvalue shortcut

When to use it: When the question gives eigenvalues or variances and asks for percentage explained or number of components needed.

Add all eigenvalues once to get the total.
Divide the one you need by that total.
For cumulative questions, add the largest eigenvalues until you pass the stated percentage.
If a loading vector is asked for, check unit length before choosing an option: the squares must sum to 1.
Eliminate options that give shares above 100% or components not sorted by size.

Common mistakes in Principal Component Analysis Fundamentals

Dividing an eigenvalue by the largest eigenvalue, or by the number of variables, instead of by the sum of eigenvalues.
Students recall 'proportion' but forget the base is the total variance.
Fix: Always use eigenvalue ÷ sum of all eigenvalues. The sum equals the trace of the covariance matrix.
Saying the eigenvector gives the variance explained.
Eigenvector and eigenvalue are confused.
Fix: The eigenvector holds the weights (direction). The eigenvalue is the variance along that direction.
Assuming the principal components are the same as the original factors or are correlated with each other.
Students forget the orthogonality property.
Fix: Components are linear combinations of the original factors and are mutually uncorrelated.
Labelling the first component as slope or curvature.
Memorised order is mixed up.
Fix: For typical yield curves, the first is level, the second slope, the third curvature. Confirm with the loading signs.
Ignoring whether the matrix is covariance or correlation.
Both are symmetric matrices and look alike.
Fix: With a correlation matrix, the eigenvalues sum to the number of variables. With a covariance matrix, they sum to total variance.
Treating PCA factors as stable or economically guaranteed.
The level, slope and curvature labels sound structural.
Fix: They are estimated from past data, can change over time, and the labels are interpretation, not part of the maths.

Worked examples

Example 1

A PCA on the covariance matrix of daily changes in four USD Treasury yields gives eigenvalues 8.0, 1.5, 0.4 and 0.1 (units: bp squared). What percentage of total variance do the first two components explain together, and what is the standard deviation of the first component?

Show the solution

Total variance = 8.0 + 1.5 + 0.4 + 0.1 = 10.0.
First two components: 8.0 + 1.5 = 9.5.
Cumulative share = 9.5 ÷ 10.0 = 95%.
Standard deviation of component 1 = √8.0 = 2.83 bp.

Answer: The first two components explain 95% of total variance. The first component has a standard deviation of about 2.83 bp per day.

Example 2

A correlation matrix of three EUR swap rates has eigenvalues 2.40, 0.45 and 0.15. The first eigenvector is (0.58, 0.58, 0.58) approximately. Interpret the first component and find how much variance it explains. How many components are needed to explain at least 90%?

Show the solution

For a correlation matrix, the sum of eigenvalues equals the number of variables: 2.40 + 0.45 + 0.15 = 3.00, which checks.
Variance explained by component 1 = 2.40 ÷ 3.00 = 80%.
Loadings are equal and positive, so all rates move in the same direction: a level (parallel shift) factor.
Component 1 plus 2: (2.40 + 0.45) ÷ 3.00 = 2.85 ÷ 3.00 = 95%, which exceeds 90%.
One component gives only 80%, so two are needed.

Answer: Component 1 is a level factor explaining 80% of variance. Two components are needed to reach at least 90% (they explain 95%).

Exam tips

Expect numeric questions that give eigenvalues and ask for percentage explained. Compute the total first.
Know the sign patterns: all the same sign is level, one sign change is slope, two sign changes is curvature.
Remember orthogonality: components are uncorrelated, so their variances add up without covariance terms.
Read the stem for covariance versus correlation. It changes what the eigenvalues add up to.
Be ready for conceptual options on limitations: PCA is data-driven, backward-looking and its factors can shift over time.

Practice questions from Regression Hedging and Principal Component Analysis

Principal Component Analysis Fundamentals in other exams

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

Principal Component Analysis Fundamentals: frequently asked questions

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

The eigenvector is the set of weights that defines a principal component. The eigenvalue is the variance of the data along that component. Larger eigenvalues mean more important components.

How do I calculate the variance explained by a principal component?

Divide that component's eigenvalue by the sum of all eigenvalues. The sum equals the total variance, the trace of the covariance matrix. Add the shares for the top components to get cumulative variance.

Why use PCA for interest rate risk?

Yields at different maturities are highly correlated, so many factors can be reduced to a few uncorrelated ones. Level, slope and curvature usually explain most of the movement. This makes hedging and risk measurement simpler and more stable than tracking each rate.

Should I use the covariance or correlation matrix?

A covariance matrix keeps the original scale, so high-variance variables dominate the first component. A correlation matrix standardises all variables first. Use the one the question specifies, and check the eigenvalue total accordingly.