FRM Exam Part II · Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Limitations of Duration and Non-Parallel Rate Shifts in FRM Part 2

Updated 11 October 2026 · Fact-checked

Duration estimates a price change for a small, parallel shift in yields, using a linear approximation. It fails for large moves, non-parallel curve shifts, and cash flows that change with rates, such as callable bonds and mortgages. Key rate durations fix the shape problem; effective duration and convexity handle options.

Understand Limitations of Duration and Non-Parallel Rate Shifts

Duration tells you how much a bond or portfolio changes in value for a small change in yield. It is a first-order, linear estimate. The price-yield curve is curved, so the estimate gets worse as the yield move gets bigger.

The standard formula also assumes one yield for all maturities, and that every maturity moves by the same amount. This is a parallel shift. Real yield curves twist and bend. Short rates can rise while long rates fall. Two portfolios with the same duration can then lose or gain very different amounts.

Key rate durations (partial durations) solve this. You pick key maturities, such as 2, 5, 10 and 30 years. You shift one key rate by 1 basis point, keep the others fixed, and record the price change. The sum of the key rate durations is approximately the effective duration for a parallel shift. The profile shows where on the curve the risk sits.

Duration also assumes cash flows are fixed. With a callable bond or a mortgage, they are not. When rates fall, borrowers prepay or issuers call, so the asset's life shortens and its price rises less than a plain bond. This is negative convexity. When rates rise, prepayments slow, the life extends and the price falls more. This is extension risk. Modified duration cannot capture this. You need effective duration, which reprices the asset after shifting the curve up and down, using a model that lets cash flows change.

For asset-liability management, a duration-matched balance sheet is protected only against small parallel moves. It stays exposed to curve reshaping, large shocks, optionality and behavioural cash flows such as deposit runoff.

Key formulas to remember

Duration price approximation: ΔP ÷ P ≈ -D × Δy; Valid for small, parallel yield changes. D is modified duration for fixed cash flows.
Duration plus convexity: ΔP ÷ P ≈ -D × Δy + ½ × C × (Δy)²; Convexity improves the estimate for larger moves. Negative convexity means the second term hurts you.
Effective duration: D_eff = (P₋ - P₊) ÷ (2 × P₀ × Δy); P₋ and P₊ are prices after yields fall and rise by Δy, using a model that lets cash flows change. Use it for bonds with options and for MBS.
Key rate duration: KRD_i = -(1 ÷ P) × (ΔP_i ÷ Δy_i); Only key rate i is shifted. Other key rates are held constant.
Sum of key rate durations: Σ KRD_i ≈ effective duration; Holds when all key rates move by the same amount (a parallel shift).
Portfolio price change from curve shifts: ΔP ÷ P ≈ -Σ (KRD_i × Δy_i); Use the actual yield change at each key rate.

How to solve Limitations of Duration and Non-Parallel Rate Shifts questions

Use this order for any question on the limits of duration or non-parallel shifts.

1Identify the shift in the question: small or large, parallel or non-parallel.
2Check whether cash flows are fixed. If there is a call, prepayment or put, plain modified duration is unsuitable; use effective duration.
3For a parallel small shift with fixed cash flows, apply ΔP ÷ P ≈ -D × Δy.
4For a large shift, add the convexity term.
5For a non-parallel shift, multiply each key rate duration by its own yield change and sum with a negative sign.
6Convert to currency by multiplying the percentage change by the portfolio value.
7State the interpretation: what risk remains (curve, option or extension risk) after the hedge.

Quickest way: Match the shift to the tool

When to use it: Use when you have about a minute per question and the options are close.

Parallel and small: duration is enough.
Large: add convexity.
Twist or butterfly: key rate durations.
Callable or MBS: effective duration, expect negative convexity.
Compute the signed sum Σ KRD × Δy, then check the sign: rising yields mean a loss for a positive duration.

Common mistakes in Limitations of Duration and Non-Parallel Rate Shifts

Assuming equal duration means equal risk.
Duration is a single number, so it feels complete.
Fix: Two portfolios can match in duration yet have different key rate profiles. A curve twist then affects them differently.
Using modified duration for a callable bond or MBS.
Modified duration assumes fixed cash flows.
Fix: Use effective duration from repricing under up and down shifts with a model for the option or prepayments.
Forgetting the sign.
Duration is quoted as a positive number.
Fix: Apply the minus sign: higher yields mean lower prices for positive duration.
Applying each key rate duration to the same yield change.
The parallel habit carries over.
Fix: Use each key rate's own change, for example +10 bp at 2 years and -20 bp at 10 years.
Thinking MBS duration is constant when rates move.
Bond duration seems stable.
Fix: MBS duration shortens when rates fall (faster prepayment) and lengthens when rates rise (extension risk).

Worked examples

Example 1

A ₹100 crore bond portfolio has key rate durations of 0.5 at 2 years, 2.0 at 5 years and 4.5 at 10 years. The 2-year rate rises 20 bp, the 5-year rate is unchanged, and the 10-year rate falls 10 bp. Estimate the change in portfolio value.

Show the solution

Total effective duration ≈ 0.5 + 2.0 + 4.5 = 7.0.
Convert the changes: +0.20% at 2 years, 0% at 5 years, -0.10% at 10 years.
ΔP ÷ P ≈ -(0.5 × 0.20 + 2.0 × 0 + 4.5 × (-0.10)) %.
= -(0.10 + 0 - 0.45) % = +0.35%.
Value change = 0.35% × ₹100 crore = ₹0.35 crore.

Answer: The portfolio gains about 0.35%, or ₹0.35 crore (₹35,00,000). A parallel-shift duration approach would miss this.

Example 2

A mortgage portfolio worth USD 200 million is priced at USD 200 million. If yields fall 50 bp, a prepayment model gives USD 201.6 million; if yields rise 50 bp, it gives USD 196.8 million. Compute effective duration and explain why a plain bond with that duration would behave differently.

Show the solution

D_eff = (P₋ - P₊) ÷ (2 × P₀ × Δy).
P₋ - P₊ = 201.6 - 196.8 = 4.8.
Denominator = 2 × 200 × 0.005 = 2.0.
D_eff = 4.8 ÷ 2.0 = 2.4.
Gain on fall = 0.8%; loss on rise = 1.6%. The loss exceeds the gain, so the portfolio has negative convexity.
A plain bond with duration 2.4 would gain and lose similar amounts, with a gain slightly larger than the loss.

Answer: Effective duration is 2.4. The portfolio gains 0.8% if yields fall but loses 1.6% if they rise, showing negative convexity from prepayment and extension risk.

Exam tips

Look for the words twist, steepening, butterfly or non-parallel: they point to key rate durations.
If a question mentions a call, prepayment or an option, choose effective duration.
Check the sign and direction of each yield change before multiplying.
Know that key rate durations sum to approximately effective duration.
Expect interpretation: say which risk the hedge leaves open.

Practice questions from Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Limitations of Duration and Non-Parallel Rate Shifts in other exams

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

FRM Part ILimitations of Duration and Key Rate Measures

Limitations of Duration and Non-Parallel Rate Shifts: frequently asked questions

What is key rate duration in simple words?

It is the sensitivity of a portfolio's value to a 1 bp move in one point on the yield curve while all others stay fixed. A set of them shows where on the curve your risk sits.

Why does prepayment risk change the duration of mortgage assets?

Borrowers prepay more when rates fall and less when rates rise. This shortens the asset's life in falling rates and lengthens it in rising rates, causing negative convexity.

Does a duration-matched portfolio remove interest rate risk?

No. It protects against small parallel shifts only. Curve reshaping, large moves and embedded options still create exposure.

How is effective duration different from modified duration?

Modified duration comes from a formula with fixed cash flows. Effective duration comes from repricing under shifted curves with a model that lets cash flows change, so it works for options and MBS.