FRM Exam Part II · Regression Hedging and Principal Component Analysis

Level, Slope and Curvature Factors in Yield Curve PCA

Updated 11 October 2026 · Fact-checked

Principal component analysis splits yield curve moves into a few uncorrelated factors. The first is level (a roughly parallel shift), the second is slope (short and long rates move in opposite directions) and the third is curvature (the middle moves against the ends). You identify each factor from the sign pattern of its loadings across maturities.

Understand Level, Slope and Curvature Factors

Yield curve changes look complicated because many rates move each day. But the moves are highly correlated. A 2-year rate and a 5-year rate usually rise or fall together. PCA exploits this. It takes the covariance matrix of rate changes and finds a small set of uncorrelated factors that explain most of the variation.

Each principal component has a loading (or factor shape) for every maturity. The loading tells you how much that rate moves when the factor moves by one unit. Read the pattern of signs and sizes across maturities, and you can name the factor.

The first component is the level factor. All loadings have the same sign and are similar in size. All rates move in the same direction by about the same amount, so the curve shifts in parallel. It usually explains the largest share of variance, often well over half in government bond markets.

The second is the slope (steepness) factor. Loadings change sign from short to long maturities, or rise steadily from negative to positive. A positive move steepens the curve: long rates rise relative to short rates. The third is the curvature (butterfly) factor. Loadings at the short and long ends share one sign, and the middle has the opposite sign. A move bends the curve, raising the belly relative to the wings, or the reverse.

The components are uncorrelated by construction, and their variances are the eigenvalues. Each component's share of total variance is its eigenvalue divided by the sum of all eigenvalues. Three factors typically explain the large majority of yield curve variation, which is why risk managers hedge against three factors instead of dozens of rates.

Key formulas to remember

Rate change from factors: Δr(T) ≈ L₁(T)·F₁ + L₂(T)·F₂ + L₃(T)·F₃; L are loadings at maturity T and F are factor moves (factor scores). Remaining components are ignored.
Variance explained by a component: Share of variance = λᵢ ÷ Σλⱼ; λ are eigenvalues of the covariance matrix. Cumulative share is the sum of the first k shares.
Sign pattern for naming factors: Level: same sign everywhere. Slope: one sign change (short vs long). Curvature: two sign changes (ends vs middle).; Signs of a whole component can be flipped without changing its meaning. Read the shape, not the absolute sign.
Portfolio exposure to a factor: Factor exposure = Σ DV01(T) × L(T); Sum over key rate maturities. Gives the P&L sensitivity to a one-unit move in that factor.

How to solve Level, Slope and Curvature Factors questions

Use this method for any question that gives loadings, eigenvalues or a description of a curve move.

1Write down the loadings for each component across the maturities shown.
2Check the sign pattern. Same sign everywhere means level. One sign change or a steady rise from negative to positive means slope. Ends opposite to the middle means curvature.
3Look at the size of the loadings. Roughly equal values mean a parallel move. Large values at one end show where that factor acts most.
4If asked for variance explained, divide each eigenvalue by the sum of all eigenvalues. Add them for cumulative share.
5If asked for a rate change, multiply each loading by the given factor move, add across components and keep the units consistent (for example basis points).
6If asked for portfolio impact, multiply each key rate DV01 by its loading and sum to get factor exposure, then multiply by the factor move.
7Interpret the result in words: parallel shift, steepening or flattening, butterfly. Check that the sign and direction make sense.

Quickest way: Sign-pattern shortcut

When to use it: Use when an MCQ shows a loading table or chart and asks which factor it is or which curve move results.

Scan the signs left to right across maturities.
No sign change: level. One change: slope. Two changes: curvature.
For direction of a move, apply the factor sign to each loading. A positive slope move with negative short loadings and positive long loadings raises long rates and lowers short rates, a steepening.
Eliminate options that mismatch the number of sign changes.

Common mistakes in Level, Slope and Curvature Factors

Calling the first component a perfectly parallel shift.
Textbooks say level is a parallel shift, so students assume equal loadings.
Fix: The level loadings are same-signed and similar, not always identical. Say roughly parallel unless the question says exact.
Treating a negative loading as a falling rate.
Students ignore the sign of the factor move.
Fix: Rate change = loading × factor move. A negative loading with a negative factor move gives a rise in that rate.
Thinking components are correlated or that the factor sign matters for naming.
Confusing PCA with regression on observed rates.
Fix: Components are uncorrelated by construction. A sign flip of a whole component does not change its meaning.
Computing variance explained by dividing by the largest eigenvalue.
Mixing up ratio to the first component with share of total.
Fix: Always divide by the sum of all eigenvalues.
Confusing slope with curvature.
Both involve differences between maturities.
Fix: Slope is short versus long. Curvature is the middle versus both ends, so it has two sign changes.
Assuming three factors explain all curve risk.
Three factors often explain most of the variance, which is read as all of it.
Fix: Residual risk remains. Say three factors explain most, not all.

Worked examples

Example 1

A PCA on government yield changes gives eigenvalues of 80, 12, 5 and 3 for the first four components. What share of variance do the first two explain, and what does the second component represent if its loadings at 2, 5 and 10 years are -0.6, 0.1 and 0.7?

Show the solution

Sum of eigenvalues = 80 + 12 + 5 + 3 = 100.
First component share = 80 ÷ 100 = 80%. Second = 12 ÷ 100 = 12%.
Cumulative share of the first two = 80% + 12% = 92%.
Loadings go from negative at 2 years to positive at 10 years: one sign change, so it is the slope factor.
A positive move raises the 10-year rate and lowers the 2-year rate, a steepening.

Answer: The first two components explain 92% of variance. The second is the slope factor, and a positive move steepens the curve.

Example 2

Loadings for a curvature factor at 2, 5 and 10 years are +0.5, -0.7 and +0.4. The factor moves by +10 basis points (as a score of +10 in loading units). What are the changes in the 2-, 5- and 10-year rates, and what is the shape change?

Show the solution

2-year change = 0.5 × 10 = +5 bp.
5-year change = -0.7 × 10 = -7 bp.
10-year change = 0.4 × 10 = +4 bp.
The ends rise while the middle falls, with two sign changes in loadings, so it is a curvature move.
The belly falls relative to the wings, so the curve becomes more convex (a butterfly move).

Answer: The 2-year rate rises 5 bp, the 5-year falls 7 bp and the 10-year rises 4 bp. The belly sinks relative to the wings.

Exam tips

Questions often show a loading chart. Count sign changes before reading anything else.
Expect a variance-explained calculation using eigenvalues. Divide by the sum of all eigenvalues.
Watch the wording: steepening means long rates rise relative to short rates, flattening is the opposite.
Note that three factors explain most variance, not all. Reject options claiming exact or full explanation.
Apply the factor sign to loadings before stating the direction of each rate.

Practice questions from Regression Hedging and Principal Component Analysis

Level, Slope and Curvature Factors: frequently asked questions

What are level, slope and curvature in a yield curve?

They are the three main patterns of yield curve movement found by PCA. Level is a roughly parallel shift, slope is a steepening or flattening, and curvature is the middle moving against the ends. Together they explain most of the variation in rates.

How do I identify a factor from its loadings?

Count the sign changes across maturities. None means level, one means slope and two means curvature. Also check the size of the loadings to see which maturities react most.

Why does the first principal component explain the most variance?

PCA orders components by eigenvalue, from largest to smallest. Because rates across maturities are highly correlated, the common up-or-down move is the biggest source of variance. That is the level factor.

Are the principal components correlated?

No. They are uncorrelated by construction. This makes it easy to add up variance contributions and to hedge factor by factor.