FRM Exam Part I · Interest Rate Futures

Duration-Based Hedging with Interest Rate Futures

Updated 11 October 2026 · Fact-checked

Duration-based hedging uses a futures contract to offset the interest rate sensitivity of a bond portfolio. The number of contracts is N = −(P × D_P) ÷ (F × D_F), or equivalently the portfolio DV01 divided by the futures DV01. A negative sign means you sell (short) futures.

Understand Duration-Based Hedging with Interest Rate Futures

A bond's price falls when yields rise. Duration measures how much. For a small parallel yield shift, the percentage price change is about −D × Δy. So a portfolio with a larger duration loses more in dollar terms when rates rise.

A futures contract on a Treasury bond also changes in value when yields move. If you want to cancel the portfolio's rate risk, you take a futures position whose value change is equal and opposite. You match the dollar sensitivity of the two sides, not just the durations.

The dollar sensitivity of the portfolio is its value times its duration (P × D_P). The dollar sensitivity of one futures contract is its price times the duration of the underlying bond (F × D_F). Dividing one by the other gives the number of contracts. Since you want the futures to gain when the portfolio loses, you sell futures to hedge a long bond portfolio.

The same formula changes duration. If you want to move duration from D_P to a target D_T, replace D_P with the difference (D_T − D_P) in the numerator. A positive result means you buy futures to lengthen duration. A negative result means you sell to shorten it.

The hedge is only approximate. It assumes small, parallel yield shifts, and it ignores convexity. It also ignores basis risk and the fact that the futures yield may not move one-for-one with the portfolio yield. Duration of the futures is usually taken from the cheapest-to-deliver bond.

Key formulas to remember

Number of futures contracts to hedge: N* = −(P × D_P) ÷ (F × D_F); P = portfolio value, D_P = portfolio duration, F = contract price (value of one contract), D_F = duration of the futures underlying (cheapest-to-deliver). Negative means short.
DV01 form: N* = −DV01_P ÷ DV01_F; DV01 is the value change for a one basis point yield change. Use the same sign convention and units on both sides.
Change duration to a target: N = (D_T − D_P) × P ÷ (F × D_F); D_T is the target duration. Positive N means buy futures. Negative N means sell.
Duration price approximation: ΔP ≈ −D × P × Δy; D is modified duration when Δy is the change in the periodic yield. Valid for small changes.
Futures contract value (U.S. Treasury bond): F = quoted price ÷ 100 × ₹ or $ contract size; A US T-bond contract has a face value of $100,000, so a quote of 120 means F = $120,000. Convert 32nds carefully.

How to solve Duration-Based Hedging with Interest Rate Futures questions

Use this order for any question on hedging or changing duration with interest rate futures.

1Identify the goal: fully hedge the rate risk, or move duration to a target.
2Write down P and D_P for the portfolio. If the goal is a target duration, also write D_T.
3Find the futures contract value F. Multiply the quoted price by the contract size and divide by 100. Note the futures duration D_F.
4Choose the formula: N = −(P × D_P) ÷ (F × D_F) for a full hedge, or N = (D_T − D_P) × P ÷ (F × D_F) for a target.
5Substitute the numbers and compute. Keep the sign.
6Read the sign: negative means sell futures, positive means buy. Round to a whole number of contracts.
7Sanity check: the hedge should reduce risk. A long bond portfolio needs short futures. Lengthening duration needs long futures.

Quickest way: Dollar-sensitivity shortcut

When to use it: Use it when the question gives you DV01 or when numbers are large and you want to avoid slips.

Compute the portfolio sensitivity P × D_P and the contract sensitivity F × D_F.
Divide the first by the second and attach the sign from the goal.
For a target duration, compute (D_T − D_P) × P first, then divide.
Check the sign in one line: long bonds need short futures to hedge.
Scan the options. Often two choices have the wrong sign. Remove them first.

Common mistakes in Duration-Based Hedging with Interest Rate Futures

Using the wrong sign and buying futures to hedge a long bond portfolio
Students forget that bond values and futures values must move in opposite directions after hedging.
Fix: Ask who loses when rates rise. A long bond holder loses, so you need a position that gains: short futures.
Using the quoted futures price instead of the contract value
The quote is per 100 of face value, not per contract.
Fix: Multiply the quote ÷ 100 by the contract face value. A quote of 120 on a $100,000 contract is $120,000.
Dividing durations only, ignoring portfolio and contract size
Students recall the ratio D_P ÷ D_F and drop the values.
Fix: Always match dollar sensitivity: (P × D_P) ÷ (F × D_F).
For a target duration, using the full portfolio duration instead of the difference
The hedge formula is memorised without understanding.
Fix: Use (D_T − D_P). Setting D_T = 0 gives the full hedge formula back.
Using the portfolio duration for the futures duration
Students overlook that the futures duration comes from the cheapest-to-deliver bond.
Fix: Use the D_F given in the question. It belongs to the underlying of the contract.
Claiming the hedge is perfect
The formula looks exact.
Fix: State the limits: it assumes small parallel shifts, ignores convexity, and carries basis risk and risk of non-parallel yield moves.

Worked examples

Example 1

A portfolio manager holds a $10 million bond portfolio with a modified duration of 6.8. Treasury bond futures trade at 110 per 100 of face value, with a contract size of $100,000 and a cheapest-to-deliver duration of 9.0. How many contracts should be traded to fully hedge the duration exposure?

Show the solution

P = 10,000,000 and D_P = 6.8, so P × D_P = 68,000,000.
F = 110 ÷ 100 × 100,000 = 110,000.
F × D_F = 110,000 × 9.0 = 990,000.
N = −68,000,000 ÷ 990,000 = −68.69.
Round to the nearest whole contract: −69.

Answer: Sell about 69 Treasury bond futures contracts.

Example 2

A fund has $50 million of bonds with duration 4.0. It wants to raise duration to 7.0 using futures priced at $120,000 per contract with an underlying duration of 8.0. How many contracts should it trade?

Show the solution

D_T − D_P = 7.0 − 4.0 = 3.0.
Numerator: 3.0 × 50,000,000 = 150,000,000.
Denominator: F × D_F = 120,000 × 8.0 = 960,000.
N = 150,000,000 ÷ 960,000 = 156.25.
Round to 156. The sign is positive because duration must increase.

Answer: Buy about 156 futures contracts.

Exam tips

Check the sign first. Many wrong options differ from the right one only by sign.
Read whether the question asks for a full hedge or a target duration. The formulas differ by the (D_T − D_P) term.
Compute the contract value from the quote before anything else. A common trap is a quote like 110-16, which is 110 and 16/32.
If DV01 is given, use N = −DV01_P ÷ DV01_F and skip duration entirely.
Expect conceptual questions on limitations: parallel shifts, convexity, basis risk and the cheapest-to-deliver option.

Practice questions from Interest Rate Futures

Duration-Based Hedging with Interest Rate Futures in other exams

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

Duration-Based Hedging with Interest Rate Futures: frequently asked questions

What is the duration-based hedge ratio formula?

N* = −(P × D_P) ÷ (F × D_F). P and D_P are the portfolio value and duration. F and D_F are the futures contract value and the duration of its underlying bond. The negative sign means you short futures to hedge a long portfolio.

How do I change portfolio duration using futures?

Use N = (D_T − D_P) × P ÷ (F × D_F). A positive answer means buy futures to raise duration. A negative answer means sell futures to cut duration.

What are the limitations of duration-based hedging?

It assumes small, parallel shifts in the yield curve and ignores convexity. It also carries basis risk and depends on the cheapest-to-deliver bond being a good proxy. Large or non-parallel moves leave a residual exposure.

Should I use modified or Macaulay duration?

Use the measure the question supplies, and use the same type for both portfolio and futures. Modified duration is the usual choice because it links price change directly to the yield change.