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

Key Rate Exposures and Key Rate Shifts Explained

Updated 11 October 2026 · Fact-checked

A key rate exposure measures how a bond or portfolio value changes when one chosen point on the curve moves by a small amount while the other key rates stay fixed. Key rate 01 is the dollar change per 1 bp shift. Key rate duration is that change as a percentage of value. Summing them gives total DV01 or duration.

Understand Key Rate Exposures and Key Rate Shifts

Ordinary duration and DV01 assume a parallel shift: every rate moves by the same amount. Real curves twist and steepen. A 2-year rate can fall while the 30-year rate rises. One duration number cannot describe this.

Key rate exposures fix that. You pick a few key rates on the curve, for example the 2-year, 5-year, 10-year and 30-year. You then shift one key rate by a small amount, usually 1 bp, and hold the others unchanged. The rates between key points move linearly, forming a triangle-shaped shift that peaks at the chosen key rate and falls to zero at the neighbouring key rates. You reprice the bond and record the change.

The result is a key rate 01 (KR01): the change in value for a 1 bp shift at that key rate. If you divide the change by the bond value, and scale it, you get a key rate duration (KRD). A bond has one KR01 for each key rate, so you see where on the curve it is exposed. A zero-coupon bond maturing in 10 years has almost all its exposure at the 10-year key rate. A coupon bond spreads exposure across several key rates, with the largest at the maturity point because the principal is paid there.

The key rate shifts are built so that moving all key rates by the same amount gives a parallel shift. So the key rate exposures add up to the total exposure: the sum of the KR01s equals the DV01, and the sum of the KRDs equals the effective duration. This is the main check you should use.

Key rate duration differs from effective duration. Effective duration is one number for a parallel shift of the whole curve. Key rate duration is a set of numbers, one for each point. Key rates can be defined on the spot curve or the par curve. Spot-based shifts change the discount rates directly. Par-based shifts change par yields and the spot curve is then re-derived by bootstrapping, so the exposure is spread differently. The method is the same either way.

Key formulas to remember

Key rate 01: KR01(k) = −[V(+1 bp at key rate k) − V(−1 bp at key rate k)] ÷ 2, or −[V(+1 bp) − V₀] for a one-sided shift; Only key rate k moves, with linear interpolation to neighbouring key rates. A positive KR01 means value falls when that rate rises.
Key rate duration: KRD(k) = −[V(+Δ) − V(−Δ)] ÷ (2 × V₀ × Δ); Δ is the shift in decimal form, so 1 bp = 0.0001. Equivalent to KR01 ÷ (V₀ × 0.0001).
Relation of KR01 to KRD: KR01(k) = KRD(k) × V₀ × 0.0001; Use this to switch between the two measures.
Sum rule: Σ KR01(k) = DV01 and Σ KRD(k) = effective duration; Holds when the key rate shifts together make a parallel shift of 1 bp. For a portfolio, add KR01s across positions at each key rate.
Approximate price change: ΔV ≈ −Σ KR01(k) × Δy(k), with Δy in bp; Use this to estimate the value change for a non-parallel move.

How to solve Key Rate Exposures and Key Rate Shifts questions

Use this method for any question that gives key rate exposures or asks you to compute or apply them.

1Identify the key rates and the shift sizes in bp for each one. Note whether the question uses KR01 (currency per bp) or KRD (percent per unit of yield).
2Check the units. Convert KRD to KR01 with KR01 = KRD × V₀ × 0.0001, or the other way if needed.
3If you must compute an exposure, reprice the bond with only that key rate shifted. Keep the other key rates fixed.
4Take the value change per bp. Use the two-sided formula if both up and down prices are given.
5For a non-parallel move, multiply each KR01 by the bp shift at that key rate and add them up. Put a minus sign in front.
6For a portfolio, add the KR01s at each key rate across positions. Short positions carry negative KR01s.
7Check with the sum rule: the KR01s should add to the DV01, and the KRDs to the duration.
8State the answer with the correct sign and unit, and say whether value rises or falls.

Quickest way: KR01 table and dot-product shortcut

When to use it: Use this when the question gives a list of key rate exposures and a curve move, or asks how to hedge or compare exposures.

Write the KR01s in a row and the bp shifts below them.
Multiply each pair mentally and add. Change the sign of the total.
Positive KR01 and rising rate means a loss. Negative KR01 and rising rate means a gain.
To hedge a key rate, find the position whose KR01 at that point is equal and opposite.
Use the sum rule as a fast check on any option that gives a total DV01 or duration.

Common mistakes in Key Rate Exposures and Key Rate Shifts

Treating key rate duration as the same as effective duration.
Both are called duration and both are in years.
Fix: Effective duration is for a parallel shift. A KRD is for one key rate only. Only the sum of KRDs equals effective duration.
Shifting all rates when computing one key rate exposure.
Students reuse the parallel-shift routine.
Fix: Move only the chosen key rate, with linear interpolation to its neighbours, and leave the other key rates unchanged.
Forgetting the minus sign when finding the price change.
KR01 is often quoted as a positive number.
Fix: Bond prices fall when rates rise. Use ΔV ≈ −Σ KR01 × Δy, or check the sign convention in the question.
Mixing KRD and KR01 units.
One is a percent measure and the other is a currency amount per bp.
Fix: Convert with KR01 = KRD × V₀ × 0.0001 before you add anything.
Expecting all exposure of a coupon bond at the maturity key rate.
Zero-coupon bonds behave that way and students generalise.
Fix: Only a zero-coupon bond has nearly all exposure at one point. A coupon bond has exposure at each key rate near its cash flow dates, with most at maturity.
Using the sum rule when the shifts do not form a parallel shift.
The rule is remembered without its condition.
Fix: The sum equals DV01 only when all the key rate shifts together equal a uniform 1 bp move.

Worked examples

Example 1

A bond portfolio worth $50 million has these key rate 01s: 2-year $1,200; 5-year $3,500; 10-year $6,800; 30-year $2,500 (value falls as the rate rises). The curve moves: 2-year +10 bp, 5-year +5 bp, 10-year −5 bp, 30-year −10 bp. Estimate the change in value.

Show the solution

Formula: ΔV ≈ −Σ KR01(k) × Δy(k).
2-year: 1,200 × 10 = 12,000.
5-year: 3,500 × 5 = 17,500.
10-year: 6,800 × (−5) = −34,000.
30-year: 2,500 × (−10) = −25,000.
Sum: 12,000 + 17,500 − 34,000 − 25,000 = −29,500.
Apply the minus sign: ΔV ≈ −(−29,500) = +29,500.

Answer: The portfolio gains about $29,500.

Example 2

A bond worth $200,000 has a 10-year key rate duration of 4.50. What is its 10-year KR01? If all the key rate durations sum to 6.20, what is the bond's DV01?

Show the solution

Formula: KR01 = KRD × V₀ × 0.0001.
KR01(10-year) = 4.50 × 200,000 × 0.0001.
200,000 × 0.0001 = 20, so KR01 = 4.50 × 20 = $90.
Sum rule: the KRDs add to the effective duration of 6.20.
DV01 = 6.20 × 200,000 × 0.0001 = 6.20 × 20 = $124.

Answer: The 10-year KR01 is $90 per bp, and the DV01 is $124 per bp.

Exam tips

Questions often give KR01s and a non-parallel move and ask for the value change. Do the dot product and watch the sign.
Check units first. Many wrong options come from using KRD as if it were KR01.
Use the sum rule to eliminate options: the key rate figures must add up to the total DV01 or duration.
Know why a zero-coupon bond has a single large exposure and a coupon bond has several.
Expect a conceptual question on why key rate exposures are better than one duration for curve twists and for hedging.

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

Key Rate Exposures and Key Rate Shifts: frequently asked questions

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

Effective duration measures sensitivity to a parallel shift of the whole curve and is one number. Key rate duration measures sensitivity to a shift at one point and you get one for each key rate. The key rate durations add up to the effective duration.

How do I calculate a key rate 01?

Shift one key rate by 1 bp, keep the other key rates fixed, reprice the bond, and take the change in value. With up and down prices, use −(V up − V down) ÷ 2. The result is the KR01 in currency per bp.

Do key rate exposures add up to DV01?

Yes, when the key rate shifts together make a parallel 1 bp move. The sum of KR01s equals DV01 and the sum of KRDs equals effective duration. Use this to check your work.

Are key rates defined on the spot curve or the par curve?

Either can be used. Spot-based shifts move discount rates directly. Par-based shifts move par yields and the spot curve is re-derived. The calculation method is the same, but the exposures can differ in how they are spread.