FRM Exam Part I · Machine Learning and Prediction

Dimension Reduction and Principal Components Analysis for FRM Part I

Updated 11 October 2026 · Fact-checked

Principal components analysis (PCA) is an unsupervised method that turns many correlated features into a few uncorrelated components, ordered by the share of total variance each explains. To solve questions, divide each eigenvalue by the sum of eigenvalues, add the shares cumulatively, and keep enough components to reach your target.

Understand Dimension Reduction and Principal Components Analysis

Dimension reduction means replacing many input variables with a smaller set that keeps most of the useful information. You do this because many features are correlated, models with too many inputs overfit, and data with fewer dimensions is cheaper to store, plot and interpret.

PCA is the standard method. It builds new variables called principal components. Each one is a weighted sum of the original features. The weights are called loadings. The first component points in the direction of greatest variance in the data. The second component captures the most remaining variance, subject to being uncorrelated (orthogonal) with the first. The third does the same, and so on.

Each component has an eigenvalue, which equals the variance of the data along that component. The eigenvalues add up to the total variance. If features are standardised first, total variance equals the number of features. The proportion of variance explained by a component is its eigenvalue divided by the sum of all eigenvalues. A scree plot graphs eigenvalues (or variance explained) against component number. You look for the elbow, where the curve flattens, and keep the components before it. You can also keep enough components to reach a cumulative target such as 90%.

PCA is unsupervised: it ignores any target variable. It is sensitive to scale, so standardise features first when they use different units. In finance, the classic use is the yield curve. Changes in rates across many maturities are highly correlated. The first three components are usually read as level (a parallel shift), slope (steepening or flattening) and curvature (the belly moving against the ends). Together they typically explain most of the variation, which lets you hedge with three factors instead of many rates.

The cost is interpretability. Components are mixtures of features, and a component with high variance is not guaranteed to be the most useful for predicting a given outcome.

Key formulas to remember

Proportion of variance explained: PVE(k) = λk ÷ (λ1 + λ2 + ... + λp); λk is the eigenvalue of component k, and p is the number of original features. The denominator is total variance.
Cumulative variance explained: Cumulative PVE(m) = (λ1 + ... + λm) ÷ (λ1 + ... + λp); Use it to choose m, the number of components kept, to reach a target such as 90%.
Principal component score: PCk = w1k·x1 + w2k·x2 + ... + wpk·xp; The weights w are the loadings. Features are usually standardised first. Loadings of a component, taken as a vector, have unit length.
Total variance with standardised features: Σλ = p; This holds when you use the correlation matrix. An eigenvalue above 1 then explains more than an average single feature (the Kaiser rule of thumb).
Orthogonality: Correlation(PCi, PCj) = 0 for i ≠ j; Components are uncorrelated by construction.

How to solve Dimension Reduction and Principal Components Analysis questions

Use this method for any PCA question on variance explained, number of components or interpretation.

1Identify what is given: eigenvalues, variance explained, a scree plot, or loadings.
2Find total variance. Sum the eigenvalues, or use p if the features are standardised and eigenvalues are not all shown.
3Compute each component's share: eigenvalue ÷ total variance.
4Add the shares in order to get cumulative variance explained.
5Apply the stopping rule in the question: a cumulative target, the elbow of the scree plot, or eigenvalue above 1.
6Interpret the loadings. Same-sign loadings across all maturities suggest level. Opposite signs at short and long ends suggest slope. Middle against ends suggests curvature.
7Check the answer: shares must sum to 100%, components must be ordered by variance, and the number kept cannot exceed p.

Quickest way: Share, cumulate, compare

When to use it: Use when the question gives eigenvalues or percentages and asks how many components to keep or how much variance is retained.

Write the eigenvalues in order and add them for the total.
Divide by the total only for the components you need.
Add running shares until you cross the target.
For a scree plot question, pick the component just before the curve flattens.
Eliminate options that call components correlated, ignore scaling, or say PCA uses the target variable.

Common mistakes in Dimension Reduction and Principal Components Analysis

Dividing an eigenvalue by the number of features instead of the sum of eigenvalues.
With standardised data the sum equals p, so students apply it even when data are not standardised.
Fix: Always divide by the sum of all eigenvalues unless the question says the features are standardised.
Saying components are correlated with each other.
Students confuse components with the original correlated features.
Fix: Remember PCA produces uncorrelated, orthogonal components by construction.
Forgetting to standardise features measured in different units.
Students focus on the maths and ignore scale.
Fix: A feature with large units dominates the variance. Standardise first, or use the correlation matrix.
Choosing components by the largest loading rather than the eigenvalue order.
Loadings and eigenvalues are mixed up.
Fix: Eigenvalues rank the components and give variance explained. Loadings show what each component is made of.
Treating PCA as a supervised method that predicts a target.
PCA is often used before a regression, so it feels supervised.
Fix: PCA only looks at the features. High-variance components may not predict the target best.
Reading the scree elbow as the point where the curve starts to fall.
The word elbow is ambiguous.
Fix: Keep the components before the point where the curve flattens into a gentle tail.

Worked examples

Example 1

A PCA on 6 standardised features gives eigenvalues 3.0, 1.5, 0.6, 0.5, 0.3 and 0.1. How many components are needed to explain at least 85% of total variance?

Show the solution

Total variance = 3.0 + 1.5 + 0.6 + 0.5 + 0.3 + 0.1 = 6.0, which equals p as expected.
PC1 share = 3.0 ÷ 6.0 = 50%. Cumulative = 50%.
PC2 share = 1.5 ÷ 6.0 = 25%. Cumulative = 75%.
PC3 share = 0.6 ÷ 6.0 = 10%. Cumulative = 85%.
85% meets the 'at least 85%' target, so three components are enough.

Answer: Three components explain exactly 85% of total variance.

Example 2

In a PCA of changes in nine Treasury yields, the eigenvalues of the first three components are 7.2, 1.1 and 0.4, and total variance is 9.0. The first component has positive loadings of similar size on all maturities. (a) What share of variance do the first two components explain? (b) How is PC1 interpreted?

Show the solution

PC1 share = 7.2 ÷ 9.0 = 80%.
PC2 share = 1.1 ÷ 9.0 = 12.22%.
Cumulative for two components = 80% + 12.22% = 92.22%, or (7.2 + 1.1) ÷ 9.0 = 8.3 ÷ 9.0 = 92.22%.
Same-sign loadings of similar size mean all yields move together in the same direction, which is a parallel shift.

Answer: (a) About 92.2% of variance. (b) PC1 is the level factor, a parallel shift of the yield curve.

Exam tips

Expect arithmetic: eigenvalue ÷ sum, then cumulative totals. Do it in the order given and stop once you reach the target.
For yield curve questions, link the loading pattern to level, slope or curvature, and remember level usually explains the most variance.
Watch the wording on scree plots: the elbow marks where extra components add little, so you keep those before it.
Remember PCA is unsupervised, needs scaled inputs, and gives uncorrelated components. These facts are common distractor material.
State the trade-off in conceptual questions: fewer dimensions reduce overfitting and noise, but components are harder to interpret.

Practice questions from Machine Learning and Prediction

Dimension Reduction and Principal Components Analysis in other exams

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

Dimension Reduction and Principal Components Analysis: frequently asked questions

What is the purpose of PCA in machine learning?

PCA reduces many correlated features to a few uncorrelated components that keep most of the variance. This lowers noise, helps limit overfitting and makes data easier to visualise and process. The price is that components are harder to explain than the original features.

How do I interpret a PCA scree plot?

A scree plot shows each component's eigenvalue or variance explained in order. Find the elbow where the curve flattens. Components before the elbow carry most of the information, so you keep them and drop the tail.

What are the PCA factors in the yield curve?

The first three components of yield changes are usually read as level, slope and curvature. Level is a parallel shift, slope is steepening or flattening, and curvature is the middle moving against the ends. Together they typically explain most of the variation in rates.

Do I need to standardise data before PCA?

Yes, when features have different units or scales. PCA picks directions of maximum variance, so a large-scale feature would dominate. Standardising, or using the correlation matrix, gives each feature equal weight.