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FRM Part I 2025 Market Risk: Yield-Curve Shifts—Key-Rate Walkthrough

FRM Part I 2025 Market Risk: Yield-Curve Shifts—Key-Rate Walkthrough
FRM Part I 2025 Market Risk: Yield-Curve Shifts—Key-Rate Walkthrough


Interest rate risk—the potential for financial loss due to changes in interest rates—is a fundamental component of market risk in the FRM® Part I syllabus. Traditional duration measures assume a parallel shift in the yield curve, but real-world movements often involve twists and curvature. Key-rate duration (KRD) decomposes interest rate risk into sensitivities at specific maturities, enabling more granular risk management when the curve shifts non-parallel.


1. Traditional Measures & Their Limitations


Modified duration (or DV01) estimates a bond’s price sensitivity to a small, parallel change in yield:

Modified Duration ≈ – (1 / P₀) × (ΔP / Δy)

  • Limitation: Assumes all maturities move by the same basis point amount, which rarely holds in practice.

  • Consequence: Risk can be mis-measured if the curve steepens, flattens, or exhibits a butterfly move.


2. Yield-Curve Shifts: Types of Moves


Yield-curve movements can be decomposed into three principal components:

  1. Level (Parallel Shift): All maturities shift by the same amount.

  2. Twist (Steepener/Flattener): Short and long ends move in opposite directions.

  3. Curvature (Butterfly): Intermediate maturities move relative to both ends.

Using only modified duration ignores twists and curvature, potentially underestimating risk in portfolios concentrated in certain tenors. FRM Part I 2025 Market Risk


3. Key-Rate Duration: Concept & Formula FRM Part I 2025 Market Risk


Key-rate duration measures the sensitivity of a bond or portfolio price to a small change in yield at a single maturity point, holding other key rates constant.


Definition

For a key rate at maturity i:

KRDᵢ = – [V(yᵢ + Δy) – V(yᵢ – Δy)] / (2 × Δy × V₀)

  • V(yᵢ ± Δy) = portfolio value when the yield at tenor i is bumped up/down by Δy

  • V₀ = initial portfolio value

  • Δy typically = 1 basis point (0.01%) for FRM calculations

Aggregating across all key rates, the approximate percent price change for arbitrary curve shifts is:

ΔP / P₀ ≈ – ∑₍ᵢ₌₁₎ᵐ (KRDᵢ × Δyᵢ)

where m = number of key maturities.


4. Step-by-Step KRD Calculation


FRM candidates should follow these steps when computing key-rate durations:

  1. Select Key Maturities: Common tenors include 2-year, 5-year, 7-year, 10-year, 20-year, and 30-year.

  2. Construct Zero-Coupon Curve: Build or obtain spot rates for each key maturity.

  3. Bump a Single Key Rate by +1 bp (Δy = 0.0001), keeping all other spot rates unchanged.

  4. Reprice the portfolio using the bumped curve to get V⁺.

  5. Repeat with a –1 bp bump for the same key rate to get V⁻.

  6. Compute KRD using the two prices and the formula above.

  7. Repeat for each key maturity to obtain the full KRD vector.


5. 2025 U.S. Treasury Yields as of July 11, 2025

Maturity

Yield (%)

Source

2-year

3.90 %

(ETF Database)

5-year

3.99 %

(YCharts)

7-year

4.12 %

(FRED)

10-year

4.43 %

(YCharts)

30-year

4.96 %

(ETF Database)

These yields illustrate a modest steepening from short to long maturities in mid-2025.


6. Numerical Example: Two-Bond Portfolio


Portfolio: 50 % in a 5-year zero-coupon bond, 50 % in a 10-year zero-coupon bond.

  • Face value of each bond: $100

  • Spot yields:

    • 5-year: y₅ = 3.99 %

    • 10-year: y₁₀ = 4.43 %


6.1 Un-bumped Prices

P₅,₀ = 100 / (1 + y₅)⁵ = 100 / 1.0399⁵ ≈ 82.2322  
P₁₀,₀ = 100 / (1 + y₁₀)¹⁰ = 100 / 1.0443¹⁰ ≈ 64.8257  
Portfolio Value V₀ = 0.5×82.2322 + 0.5×64.8257 ≈ 73.5290

6.2 Price after ±1 bp Bumps

  • 5-year bump Δy = ±0.0001:

    P₅,⁺ = 100 / (1.0400)⁵ ≈ 82.2077 P₅,⁻ = 100 / (1.0398)⁵ ≈ 82.2563

  • 10-year bump Δy = ±0.0001:

    P₁₀,⁺ = 100 / (1.0444)¹⁰ ≈ 64.7927 P₁₀,⁻ = 100 / (1.0442)¹⁰ ≈ 64.8587


6.3 Key-Rate Durations

KRD₅  = – [0.5×(P₅,⁺ – P₅,⁻)] / (2×0.0001×V₀) ≈ 2.69  
KRD₁₀ = – [0.5×(P₁₀,⁺ – P₁₀,⁻)] / (2×0.0001×V₀) ≈ 4.22

  • The portfolio is more sensitive to 10-year yield moves than to 5-year moves.

  • Sum of KRDs (2.69 + 4.22 = 6.91) approximates the portfolio’s modified duration under a parallel shift assumption.


7. Aggregating Risks & Curve Scenarios


Once all KRDs are calculated, you can estimate price changes for any non-parallel curve shift. For example, if the 5-year yield steepens by +10 bp while the 10-year flattens by –5 bp:

ΔP/P₀ ≈ – [ KRD₅×(+0.0010) + KRD₁₀×(–0.0005) ]  
       = – [2.69×0.0010 + 4.22×(–0.0005)]  
       ≈ – (0.00269 – 0.00211)  
       ≈ – 0.00058  ⇒ –0.058 %

This level of granularity is critical for accurate risk attribution and hedging in FRM-level market risk management.


8. Common Pitfalls & Exam Tips


  1. Bump Size Mis-Specification: FRM uses 1 bp (0.0001), not 1 % (0.01).

  2. Curve Construction Errors: Ensure the spot rate curve corresponds to zero-coupon yields.

  3. Ignoring Interpolation: Real yield curves require interpolation between auction maturities.

  4. Neglecting Convexity: KRD is a first-order measure; large Δy may need convexity adjustments.

  5. Summation Check: The sum of all KRDs should approximate the parallel-shift duration.

  6. Unit Consistency: Always convert yields to decimals (e.g., 4.43 % = 0.0443) in calculations.


In FRM Part I, Key-Rate Duration is an indispensable tool for measuring interest rate risk under non-parallel yield-curve scenarios. By breaking down exposure into individual tenors and applying the simple KRD formula, candidates can accurately estimate price impacts of steepeners, flatteners, and butterfly shifts. Mastery of this technique not only secures exam success but also lays the groundwork for sophisticated risk management in a volatile 2025 market environment.



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