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

Immunization and Hedging Interest Rate Exposure

Updated 11 October 2026 · Fact-checked

Immunization protects a portfolio or balance sheet from small parallel rate shifts by matching the duration of assets to liabilities, with equal present values. Hedging adds futures or swaps sized by DV01 or duration so that the value change from a rate move is offset. Hedge ratio = target DV01 ÷ hedge instrument DV01.

Understand Immunization and Hedging Interest Rate Exposure

Interest rate risk arises because assets and liabilities reprice and change value differently when rates move. A bank with long-dated fixed-rate loans funded by short deposits loses economic value when rates rise. Immunization and hedging are the two main ways to remove that exposure.

Immunization sets up a portfolio so that a small, one-time, parallel shift in yields leaves the value of the net position, or the funded future liability, unchanged. The idea is that a rate rise lowers bond prices but raises reinvestment income. When you fund a liability, as a pension or insurer does, the duration of the assets equals the duration of the liabilities and their present values match. The two effects then offset.

A leveraged bank is a different setting. Its assets are larger than its liabilities, so matching durations does not protect equity. Equity is immunized when D_A = (L ÷ A) × D_L. This makes D_A lower than D_L. The condition D_A = D_L with PV_A = PV_L applies to liability-funding immunization, not to a leveraged bank's equity.

For a bank, you work with the duration gap. Equity value change is approximately -(D_A - (L ÷ A) × D_L) × A × Δy ÷ (1 + y). If the gap is zero, equity value is protected against small parallel moves. For a pension or insurer with a single liability date, you set asset duration equal to the liability horizon.

Hedging with derivatives is faster than restructuring the balance sheet. You choose an instrument whose value moves opposite to your exposure. A pay-fixed swap or a short bond futures position gains when rates rise. A receive-fixed swap or long futures gains when rates fall. You size the position using DV01 (dollar value of a basis point) or duration so the hedge DV01 equals the exposure DV01.

Immunization is only approximate. It holds for small, instantaneous, parallel shifts. It must be rebalanced as time passes and yields change because duration drifts. Convexity, non-parallel moves and credit or option features (prepayments, deposit withdrawals) break the result. Matching convexity, or using key rate durations, improves the hedge.

Key formulas to remember

Price change from duration: ΔP ÷ P ≈ -D_mod × Δy; D_mod = Macaulay duration ÷ (1 + y) for annual compounding. Valid for small moves.
DV01: DV01 ≈ D_mod × P × 0.0001; Value change for a one basis point rise in yield. Quote as a positive number.
DV01 hedge ratio: N = DV01_portfolio ÷ DV01_hedge instrument; Take the opposite position to the exposure. Hedge DV01 must equal portfolio DV01.
Duration-based futures hedge: N = (D_P × P) ÷ (D_F × F); D_P and D_F are modified durations of portfolio and futures (or cheapest-to-deliver). F is the futures contract value.
Bank duration gap: DGAP = D_A - (L ÷ A) × D_L; Equity change ≈ -DGAP × A × Δy ÷ (1 + y). Zero gap means equity is immunized for small parallel shifts.
Classical immunization conditions: PV(assets) = PV(liabilities) and D_A = D_L; This applies to funding a liability, not to a leveraged bank's equity, where D_A = (L ÷ A) × D_L. Also want asset convexity at least as large as liability convexity. Assumes a parallel shift, once.
Swap value to fixed payer: V_pay-fixed = PV(floating leg) - PV(fixed leg); Pay-fixed swap behaves like long a floating bond and short a fixed bond, so it has negative duration. Sign convention: a receive-fixed swap has duration = fixed-leg duration minus floating-leg duration. A pay-fixed swap has the negative of that.

How to solve Immunization and Hedging Interest Rate Exposure questions

Use this sequence for any immunization or hedge-sizing question. It keeps the sign and the units straight.

1Identify the exposure: does the position lose when rates rise (long fixed assets, funded short) or lose when rates fall (long fixed liabilities)?
2Collect values and durations. Convert to modified duration if the question gives Macaulay duration.
3Compute the exposure DV01 or the duration gap. For a bank, include liabilities and the L ÷ A weight.
4Compute the DV01 or duration of the hedge instrument. For a swap, a receive-fixed swap has duration equal to fixed-leg duration minus floating-leg duration. A pay-fixed swap has the negative of that.
5Compute the hedge ratio N = exposure DV01 ÷ hedge DV01 and choose the direction opposite to the exposure.
6Check the result: after the hedge, net DV01 should be zero. State the sign and the number of contracts, rounded sensibly.
7Note the limits if asked: parallel shifts, convexity, basis risk, rebalancing.

Quickest way: DV01 match in three lines

When to use it: Use for any numerical hedge question with futures or swaps when you have price and duration data.

Exposure DV01 = D_mod × value × 0.0001.
Hedge DV01 = per-contract DV01 from the question or D_mod × contract value × 0.0001.
Contracts = exposure DV01 ÷ hedge DV01. Short futures if you hold bonds; pay-fixed if assets are fixed-rate.

Common mistakes in Immunization and Hedging Interest Rate Exposure

Matching asset and liability duration but not present values.
Students remember 'match duration' and forget the funding level.
Fix: Check D_A = D_L and PV_A = PV_L. For a leveraged bank, use the duration gap with L ÷ A.
Using the wrong direction for the hedge.
Confusion between price and rate moves.
Fix: Ask who loses if rates rise. Long bonds lose, so short futures or pay-fixed swap. Say the direction before computing.
Mixing Macaulay and modified duration.
Questions give one and the formula needs the other.
Fix: Convert: D_mod = D_Mac ÷ (1 + y ÷ m). Use modified duration for price sensitivity.
Claiming immunization works for any rate change.
Overstating the rule.
Fix: State the conditions: small, instantaneous, parallel shift, rebalancing over time. Large or twisting moves leave residual risk.
Ignoring basis risk and cheapest-to-deliver in futures hedges.
Treating the futures as the same instrument as the bond.
Fix: Use the futures' own duration (cheapest-to-deliver) and note that yield changes may not move equally.
Treating a pay-fixed swap as having positive duration.
Focus on paying a fixed bond coupon.
Fix: Pay-fixed is short a fixed bond and long a floater, so value rises when rates rise.

Worked examples

Example 1

A bank holds a bond portfolio worth $200 million with modified duration 6.0. It hedges with Treasury futures whose contract value is $120,000 and modified duration 8.0 (cheapest-to-deliver basis). How many contracts, and in which direction, are needed to make DV01 zero?

Show the solution

Exposure DV01 = 6.0 × 200,000,000 × 0.0001 = $120,000 per basis point.
Futures DV01 per contract = 8.0 × 120,000 × 0.0001 = $96.
N = 120,000 ÷ 96 = 1,250 contracts.
The bank loses when rates rise, so it must be short futures.

Answer: Short 1,250 futures contracts.

Example 2

A bank has assets of $1,000 million with duration 5.0 and liabilities of $900 million with duration 2.0. A single yield of 5% applies to both assets and liabilities, and it rises by 50 basis points in a parallel shift. Estimate the change in equity value and state what asset duration would immunize equity.

Show the solution

DGAP = 5.0 - (900 ÷ 1,000) × 2.0 = 5.0 - 1.8 = 3.2 years.
Use the single 5% yield for both assets and liabilities, so the divisor is 1 + y = 1.05.
Equity change ≈ -3.2 × 1,000 × 0.005 ÷ 1.05.
= -16 ÷ 1.05 = -15.24 million.
For immunization DGAP = 0, so D_A = (900 ÷ 1,000) × 2.0 = 1.8 years.

Answer: Equity falls by about $15.2 million. Asset duration of 1.8 years would immunize equity.

Exam tips

Always state the direction first: who loses if rates rise. Many wrong options differ only by sign.
Check whether duration is Macaulay or modified before using the price formula.
For bank questions, look for the L ÷ A leverage term. Equal asset and liability duration does not give a zero gap when leverage exists.
When asked about limitations, name parallel shift, convexity, rebalancing and embedded options, and mention key rate durations as the fix.
On phones, write DV01 numbers with units per basis point to avoid scale errors of 100 or 10,000.

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

Immunization and Hedging Interest Rate Exposure: frequently asked questions

What are the conditions for portfolio immunization?

Asset present value equals liability present value, and asset duration equals liability duration. Asset convexity should be at least liability convexity. The result holds for a small, one-time parallel yield shift and needs rebalancing as time passes.

How do I calculate a hedge ratio using duration?

Divide the exposure's DV01 by the hedge instrument's DV01, and take the opposite position. DV01 is modified duration × value × 0.0001. For futures you can also use (D_P × P) ÷ (D_F × F).

Why does immunization fail in practice?

Real yield curve moves are not parallel, duration changes as rates and time change, and embedded options such as prepayments alter cash flows. Basis risk adds further error. Key rate durations and regular rebalancing reduce the gap.

Should a bank use futures or swaps to hedge?

Futures are liquid, margined and good for short-term or tactical hedges. Swaps can be customised to match cash flow dates and tenors but bring counterparty risk and collateral needs. The best choice depends on the exposure's size, horizon and funding.