CFA Level II Exam · The Term Structure and Interest Rate Dynamics

Yield Curve Factor Models and Key Rate Duration

Updated 7 October 2026 · Fact-checked

Yield curve factor models explain most curve movements with three factors: level (parallel shift), steepness (twist) and curvature (butterfly). Key rate duration measures a bond's price sensitivity to a change in one maturity point. To solve questions, sum each exposure times its rate change, then apply the sign.

Understand Yield Curve Factor Models and Duration

A yield curve can change shape in many ways, but historically most of the variation is explained by three factors. Using them lets you describe risk with a few numbers instead of tracking every maturity.

Level is a parallel shift: all yields rise or fall by about the same amount. It explains the largest share of curve changes. Steepness is a twist: short rates and long rates move in opposite directions, or by different amounts, so the curve steepens or flattens. Curvature is a butterfly: the middle of the curve moves differently from the short and long ends, so the curve becomes more or less humped.

Effective duration assumes a parallel shift. It gives one number for the whole curve. That is fine for level risk, but it hides where on the curve the exposure sits. Two portfolios can have the same effective duration and behave very differently when the curve twists.

Key rate duration fixes this. You shift one point on the par or spot curve (for example the 5-year rate), keep the others unchanged, and measure the percentage price change per 1% shift. Each bond has a set of key rate durations, one per chosen maturity. Their sum approximately equals the effective duration. A bond with all its cash flow at one maturity, such as a zero-coupon bond, has almost all its key rate duration at that maturity.

To estimate return, multiply each key rate duration by the change at that point and add the results, with a negative sign. This is the factor-model way to measure yield curve risk. In a barbell versus bullet comparison, the barbell has more exposure at the ends and less in the middle, so it gains or loses relative to the bullet when curvature changes.

Key formulas to remember

Effective duration: EffDur = (PV₋ − PV₊) ÷ (2 × PV₀ × Δcurve); PV₋ and PV₊ are prices after the curve shifts down and up in parallel. Δcurve is in decimal form, e.g. 0.01.
Key rate duration: KRD(k) = (PV₋ₖ − PV₊ₖ) ÷ (2 × PV₀ × Δy); Only the rate at maturity point k is shifted. All other points stay fixed. Δy is in decimal form, e.g. 0.01.
Sum of key rate durations: Σ KRD(k) ≈ EffDur; The sum approximates effective duration because a parallel shift of the curve equals shifting every key rate by the same amount.
Approximate price change from curve change: %ΔPV ≈ −Σ [KRD(k) × Δy(k)]; Use the actual change at each key rate. Add convexity only if the question gives it.
Three-factor description: Δ yield curve ≈ level + steepness + curvature; Level = parallel shift. Steepness = twist. Curvature = butterfly.

How to solve Yield Curve Factor Models and Duration questions

Use this order for any item-set question on curve factors or key rate duration.

1Identify what moved: read the vignette for the change in each maturity point and classify it as level, steepness or curvature.
2Find the exposure: locate the key rate durations or effective duration in the exhibit, and match each to its maturity point.
3Check the units: convert rate changes in basis points to decimals or percent consistently (1 bp = 0.01%).
4Multiply each key rate duration by its rate change, then add the products.
5Apply the negative sign: rates up means price down, rates down means price up.
6Compare with effective duration if asked: for a parallel shift, price change ≈ −EffDur × shift, which approximately equals −(sum of KRDs) × shift, so the single effective duration number can be used.
7State the conclusion in words: which portfolio gains, and why (more exposure to the point that fell most).

Quickest way: Sign and weight shortcut

When to use it: Use when the question asks which portfolio or bond outperforms after a curve move, with no need for exact numbers.

Write the rate change at each maturity point in a line, with its sign.
Mark where each portfolio has its largest key rate durations.
The portfolio with more duration where yields fell most (or rose least) wins.
For a twist or butterfly, compare short, middle and long exposures only. Equal total duration does not mean equal result.
Only calculate if the options are close in value.

Common mistakes in Yield Curve Factor Models and Duration

Using effective duration for a non-parallel shift
Effective duration is the duration number students know best, so they apply it by habit.
Fix: Effective duration assumes a parallel shift. For any twist or butterfly, use key rate durations point by point.
Forgetting the negative sign
Students multiply duration by yield change and stop.
Fix: Price change is −KRD × Δy. State the direction of price first, then calculate.
Confusing steepness with level
A rising long rate looks like a rate increase overall.
Fix: Level means all yields move together. Steepness means short and long move differently. Check the short end before you label it.
Mixing basis points and percent
Vignettes give changes in bp but durations are per 1%.
Fix: Convert first. A 25 bp change is 0.25%, so a KRD of 4 gives a 1.00% price effect, not 100%.
Thinking equal duration means equal risk
Students compare a barbell and a bullet with the same effective duration and call them the same.
Fix: Compare KRD profiles. The barbell has more end exposure and less middle exposure, so it reacts differently to steepness and curvature changes.
Assuming all KRDs of a bond are non-zero
Students spread duration evenly across maturities.
Fix: A bond's KRD sits where its cash flows are. A zero-coupon bond has nearly all its KRD at its maturity point.

Worked examples

Example 1

A portfolio has key rate durations of 0.8 at 2 years, 2.5 at 5 years and 4.7 at 10 years. The curve moves: 2-year yield +20 bp, 5-year yield 0 bp, 10-year yield −30 bp. (1) Classify the move. (2) Estimate the percentage price change. (3) Does the sum of KRDs match effective duration?

Show the solution

Classify: the short rate rose and the long rate fell, so the curve flattened. This is a steepness (twist) move, not a parallel shift.
Convert to percent: +0.20%, 0%, −0.30%.
Contribution at 2 years: −0.8 × 0.20 = −0.16%.
Contribution at 5 years: −2.5 × 0 = 0%.
Contribution at 10 years: −4.7 × (−0.30) = +1.41%.
Total: −0.16 + 0 + 1.41 = +1.25%.
Sum of KRDs: 0.8 + 2.5 + 4.7 = 8.0, which approximately equals effective duration. For a parallel +20 bp shift, price change ≈ −8.0 × 0.20 = −1.60%, so the twist matters.

Answer: (1) Flattening twist. (2) About +1.25%. (3) Sum of KRDs is 8.0, approximately the effective duration.

Example 2

Portfolio A (bullet) has key rate durations of 0, 6.0 and 0 at 2, 7 and 20 years. Portfolio B (barbell) has 3.0, 0 and 3.0 at the same points. Yields change: 2-year +10 bp, 7-year −20 bp, 20-year +10 bp. (1) Which factor moved? (2) Which portfolio performs better and by how much?

Show the solution

Classify: the middle fell while both ends rose. The middle moved differently from the ends, so this is a curvature (butterfly) move, with a more humped curve.
Convert to percent: +0.10%, −0.20%, +0.10%.
Portfolio A: −6.0 × (−0.20) = +1.20%.
Portfolio B: −3.0 × 0.10 + (−3.0 × 0.10) = −0.30 − 0.30 = −0.60%.
Difference: 1.20 − (−0.60) = 1.80 percentage points.
Both portfolios have total duration 6.0, so effective duration would not have separated them.

Answer: (1) Curvature (butterfly). (2) The bullet, Portfolio A, outperforms: about +1.20% versus −0.60%, a gap of about 1.80 percentage points.

Exam tips

Label the move first. Most wrong answers start with a wrong factor label, so do it before any arithmetic.
When two portfolios share the same duration, the question is testing key rate exposure. Go straight to the exhibit's KRD columns.
Watch units: bp versus percent is the most common trap in the answer options.
Know the qualitative claims: level explains most variation, and the sum of KRDs approximates effective duration.
If the answer options differ in sign, decide the direction by logic and eliminate before you calculate.

Yield Curve Factor Models and Duration in other exams

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

Yield Curve Factor Models and Duration: frequently asked questions

What are the three yield curve factors in CFA Level II?

They are level, steepness and curvature. Level is a parallel shift, steepness is a twist where short and long rates move differently, and curvature is a butterfly where the middle moves against the ends. Level explains the largest share of curve changes.

What is the difference between effective duration and key rate duration?

Effective duration measures price sensitivity to a parallel shift of the whole curve and gives one number. Key rate duration measures sensitivity to a change at one maturity point only. The sum of key rate durations approximates effective duration.

How do you calculate price change using key rate durations?

Multiply each key rate duration by the yield change at that maturity point, add the results, and put a negative sign in front. Make sure yield changes are in percent when durations are per 1%.

What is a butterfly movement in the yield curve?

A butterfly is a curvature change where the middle maturities move by a different amount, or in a different direction, from the short and long maturities. It changes how humped the curve is. Barbell and bullet portfolios react differently to it.