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

Duration Gap and Economic Value of Equity Explained

Updated 11 October 2026 · Fact-checked

Duration gap measures how far a bank's asset duration is from its liability duration, scaled by leverage: DGAP = D_A − (L ÷ A) × D_L. The change in economic value of equity is approximately −DGAP × A × Δy ÷ (1 + y). A positive gap means equity loses value when rates rise.

Understand Duration Gap and Economic Value of Equity

A bank holds assets (loans, bonds) and owes liabilities (deposits, borrowings). Both have market values that move when interest rates change. Economic value of equity (EVE) is the market value of assets minus the market value of liabilities. It is a value measure, not an accounting earnings measure.

Duration tells you the approximate percentage change in a price for a small parallel change in yield. If assets have a longer duration than liabilities, a rate rise cuts asset values by more than it cuts liability values. Equity absorbs the difference, so EVE falls.

The catch is size. Assets are larger than liabilities because equity funds part of them. So you cannot compare the two durations directly. You must weight the liability duration by the ratio L ÷ A. The result is the duration gap. It is the effective duration of equity measured against assets.

A positive duration gap means EVE falls when rates rise and rises when rates fall. A negative gap means the reverse. A gap of zero means the bank is immunized against small parallel shifts: the change in asset value equals the change in liability value.

The repricing gap is a different tool. It compares rate-sensitive assets and liabilities that reprice within time buckets and mainly measures the effect on net interest income (earnings). Duration gap measures the effect on market value of equity over the whole life of the balance sheet. Duration gap uses all cash flows. Repricing gap ignores cash flows beyond the bucket.

Both rely on limits. Duration is a first-order measure and assumes a small parallel shift. Large moves, non-parallel shifts, options such as prepayments and non-maturity deposit behaviour all reduce its accuracy.

Key formulas to remember

Equity identity
E = A − L
Use market values, not book values, for EVE.
Duration gap
DGAP = D_A − (L ÷ A) × D_L
D_A and D_L are the modified or Macaulay durations, but use the same type for both. L ÷ A is the liability-to-asset ratio.
Change in asset value
ΔA ≈ −D_A × A × Δy ÷ (1 + y)
With modified duration, drop the division by (1 + y) and use ΔA ≈ −D_mod × A × Δy.
Change in EVE
ΔE ≈ −DGAP × A × Δy ÷ (1 + y)
Valid for small parallel shifts. Equivalent to ΔA − ΔL.
Percentage change in EVE
ΔE ÷ E ≈ −DGAP × (A ÷ E) × Δy ÷ (1 + y)
A ÷ E is the leverage multiplier. It amplifies the effect on equity.
Immunization condition
D_A = (L ÷ A) × D_L
DGAP = 0. Equity value is protected against small parallel yield shifts.

How to solve Duration Gap and Economic Value of Equity questions

Use this order for any duration gap or EVE sensitivity question.

  1. 1Write down market values of assets A and liabilities L. Find equity E = A − L if it is not given.
  2. 2Note the duration of assets D_A and of liabilities D_L, and check they are the same type (Macaulay or modified).
  3. 3Compute L ÷ A and then DGAP = D_A − (L ÷ A) × D_L.
  4. 4Read the sign: positive gap means EVE falls when rates rise; negative gap means EVE rises when rates rise.
  5. 5Find the rate change Δy in decimals (100 bp = 0.01) and the starting yield y if Macaulay durations are used.
  6. 6Compute ΔE = −DGAP × A × Δy ÷ (1 + y), or without the (1 + y) term for modified durations.
  7. 7If asked for the percentage change in equity, divide ΔE by E.
  8. 8State the answer in currency with a sign and a one-line interpretation, including the parallel-shift assumption.

Quickest way: Direct dollar-duration shortcut

When to use it: Use when the question gives durations and balance sheet values and asks only for the change in equity value.

  1. Compute the asset dollar change: D_A × A × Δy.
  2. Compute the liability dollar change: D_L × L × Δy.
  3. Subtract: ΔE = −(asset term − liability term). This avoids computing DGAP at all.
  4. Check the sign: if the asset term is larger and rates rise, equity falls.
  5. Scan the options. Look for the one with the right sign and size before doing extra decimals.

Common mistakes in Duration Gap and Economic Value of Equity

  • Subtracting D_L from D_A without weighting by L ÷ A.

    The idea that a gap is a simple difference of durations feels natural.

    Fix: Always scale liability duration by L ÷ A. Equity funding makes assets larger than liabilities.

  • Getting the sign of the equity change wrong.

    Students forget that a positive gap means asset values fall more than liability values when rates rise.

    Fix: Put the minus sign in the formula: ΔE = −DGAP × A × Δy. Positive gap and rising rates give a loss.

  • Using book values instead of market values.

    Balance sheet figures in a question are often called assets and liabilities without comment.

    Fix: EVE is a market value concept. Use the values given for the calculation and do not substitute accounting equity.

  • Confusing duration gap with repricing gap.

    Both are called gaps and both link to rate risk.

    Fix: Repricing gap targets net interest income over a horizon. Duration gap targets market value of equity using all cash flows.

  • Mixing Macaulay and modified duration, or forgetting the (1 + y) term.

    Questions use either duration type, and the formula differs.

    Fix: With Macaulay duration, divide by (1 + y). With modified duration, do not. Do not apply both.

  • Treating a zero duration gap as zero rate risk.

    Immunization sounds complete.

    Fix: A zero gap protects only against small parallel shifts. Convexity, non-parallel moves and embedded options still create risk.

Worked examples

Example 1

A bank has assets of $1,000 million with duration 4.0 years and liabilities of $900 million with duration 3.0 years. Durations are modified durations. Rates rise by 50 bp in a parallel shift. Find the duration gap and the change in the market value of equity.

Show the solution
  1. L ÷ A = 900 ÷ 1,000 = 0.90.
  2. DGAP = 4.0 − 0.90 × 3.0 = 4.0 − 2.7 = 1.3 years.
  3. Δy = 0.005.
  4. ΔE = −DGAP × A × Δy = −1.3 × 1,000 × 0.005 = −$6.5 million.
  5. Check by dollar terms: asset change = −4.0 × 1,000 × 0.005 = −20; liability change = −3.0 × 900 × 0.005 = −13.5; ΔE = −20 − (−13.5) = −6.5.

Answer: DGAP = 1.3 years. EVE falls by about $6.5 million, as the gap is positive and rates rose.

Example 2

A bank has assets of €2,000 million with modified duration 3.5 years and equity of €200 million. Liabilities have modified duration 2.8 years. Rates fall by 100 bp in a parallel shift. Find the percentage change in equity value.

Show the solution
  1. L = 2,000 − 200 = €1,800 million. L ÷ A = 0.90.
  2. DGAP = 3.5 − 0.90 × 2.8 = 3.5 − 2.52 = 0.98 years.
  3. Δy = −0.01.
  4. ΔE = −0.98 × 2,000 × (−0.01) = +€19.6 million.
  5. Percentage change = 19.6 ÷ 200 = 9.8%.
  6. Check by dollar terms: assets rise 3.5 × 2,000 × 0.01 = 70; liabilities rise 2.8 × 1,800 × 0.01 = 50.4; difference = 19.6.

Answer: EVE rises by about €19.6 million, which is 9.8% of equity.

Exam tips

  • Questions often hide the liability value. Derive L = A − E before you start.
  • Check whether durations are Macaulay or modified. The wording decides whether you divide by (1 + y).
  • Questions test sign and interpretation as often as arithmetic. State whether EVE rises or falls and why.
  • When asked to compare with repricing gap, remember: earnings versus value, partial versus full cash flows.
  • Look for options that name the right limitation: parallel shift, convexity, embedded options, deposit behaviour.

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

Duration Gap and Economic Value of Equity in other exams

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

Duration Gap and Economic Value of Equity: frequently asked questions

What is the duration gap formula for a bank?

DGAP = D_A − (L ÷ A) × D_L. It weights liability duration by the liability-to-asset ratio. A positive result means equity value falls when rates rise.

How is duration gap different from repricing gap?

Repricing gap compares assets and liabilities that reprice in time buckets and mostly shows the effect on net interest income. Duration gap uses the timing of all cash flows and shows the effect on market value of equity.

What does economic value of equity sensitivity tell you?

It shows how the market value of assets minus liabilities changes under a rate shock. It is a long-run value view, not a one-year earnings view. Supervisors use it to assess interest rate risk in the banking book.

Does a zero duration gap remove all interest rate risk?

No. It protects only against small parallel yield shifts. Convexity differences, non-parallel moves and embedded options such as prepayments can still change equity value.