FRM Exam Part I · Interest Rates

Interest Rate Hedging and Futures for FRM Part I

Updated 11 October 2026 · Fact-checked

Interest rate hedging uses futures or forward rate agreements to offset the change in a portfolio's value when yields move. For duration-based hedging, you match the portfolio's DV01 to the futures DV01: N = (DV01 of portfolio ÷ DV01 of one futures contract), and go short futures to hedge a long bond position.

Understand Interest Rate Hedging and Futures

A bond portfolio loses value when yields rise. A hedge adds a position that gains when yields rise, so the total value barely moves. The usual tool is a short position in an interest rate futures contract.

The key measure is DV01, the dollar change in value for a one basis point change in yield. If you know the DV01 of your portfolio and the DV01 of one futures contract, you can find how many contracts cancel the exposure. This is duration-based hedging. It assumes a small, parallel shift in yields, so it does not protect against twists in the curve.

Treasury bond and note futures are not tied to one bond. The short party can deliver any bond from an eligible basket. Each bond has a conversion factor that adjusts its price to a standard 6% yield. The short will choose the cheapest-to-deliver (CTD) bond, the one that minimises: quoted bond price − (settlement futures price × conversion factor). The futures price therefore tracks the CTD bond, and the CTD's duration is what you use for hedging.

Eurodollar-style short-term rate futures settle on a three-month rate. The contract price is 100 − the rate. A one basis point move is worth a fixed amount per contract ($25 per basis point for a $1 million three-month contract). Futures are marked to market daily, while forward rate agreements (FRAs) settle once. Because of this, the futures rate is higher than the forward rate, and you subtract a convexity adjustment to get the forward rate.

An FRA locks in a rate for a future period. The buyer of an FRA pays the fixed rate and receives the floating rate, so a borrower uses it to fix a future borrowing cost.

Key formulas to remember

Duration-based hedge ratio (DV01 form): N = − DV01 of portfolio ÷ DV01 of one futures contract; Negative means short futures. Use the CTD bond's DV01 for Treasury futures.
Duration-based hedge ratio (duration form): N = − (P × D_P) ÷ (F × D_F); P is portfolio value, F is the futures contract price (value per contract), D_P and D_F are the durations. Use consistent duration measures.
Futures DV01 from CTD: DV01 of futures ≈ DV01 of CTD bond ÷ conversion factor; Futures price ≈ CTD price ÷ conversion factor, so the futures DV01 per contract equals the CTD DV01 divided by the conversion factor (per unit of bond face value).
Cheapest to deliver: Cost = quoted bond price − (futures settlement price × conversion factor); The bond with the smallest cost is the CTD.
Cash price of a delivered bond: Cash received = (settlement price × conversion factor) + accrued interest; Used in Treasury futures delivery.
Eurodollar futures price: Quote = 100 − futures rate (%); A 1 basis point move equals 25 dollars on a 1 million dollar, three-month contract.
Convexity adjustment (approximation): Forward rate = futures rate − ½ σ² T₁ T₂; σ is the annual standard deviation of the short rate change, T₁ is the futures maturity and T₂ is the maturity of the underlying rate, in years.
FRA settlement: Payment = L × (R_floating − R_fixed) × τ ÷ (1 + R_floating × τ); L is notional, τ is the accrual fraction. Paid at the start of the period, so it is discounted. Positive means the fixed-rate payer receives.
Target duration hedge: N = (D_T − D_P) × P ÷ (D_F × F); Changes duration from D_P to a target D_T. Positive N means buy futures to lengthen duration.

How to solve Interest Rate Hedging and Futures questions

Use this order for any hedging question on interest rate futures or FRAs.

1Identify the exposure: are you long bonds (hurt by rising yields) or a borrower (hurt by rising rates)? This sets the direction of the hedge.
2Collect the portfolio value, duration or DV01, and the futures price, duration or DV01. Check that durations are the same type (modified or Macaulay) and yields are on the same basis.
3For Treasury futures, find the CTD and its conversion factor if the question asks. The CTD bond drives the futures duration.
4Compute the hedge ratio: N = −(P × D_P) ÷ (F × D_F), or the DV01 form. Keep the sign.
5Round to the nearest whole contract only at the end, and state long or short.
6For FRAs or Eurodollar futures, compute the rate or payment with the accrual fraction, and discount the FRA payment if it settles at the start of the period.
7If the question mentions futures versus forward rates, subtract the convexity adjustment from the futures rate.
8Sanity-check: does the hedge gain when rates move against the underlying exposure?

Quickest way: DV01 matching in three lines

When to use it: Use when the question gives or lets you compute DV01 or modified duration for both the portfolio and the futures contract.

Portfolio DV01 = P × modified duration × 0.0001.
Futures DV01 = F × futures duration × 0.0001 (or given directly).
Contracts = portfolio DV01 ÷ futures DV01. Short if you hold bonds, long if you need more duration.

Common mistakes in Interest Rate Hedging and Futures

Going long futures to hedge a long bond portfolio
Students think hedge means same direction as the position.
Fix: A long bond loses when yields rise. A short futures position gains when yields rise. Short futures to hedge long bonds.
Using the portfolio's duration for the futures contract
The futures looks like a bond, so students reuse the bond's duration.
Fix: Use the duration of the CTD bond (or the futures DV01 given). The futures duration is not the portfolio's.
Forgetting the conversion factor when finding the CTD
Students compare quoted bond prices only.
Fix: Compute quoted price − futures price × conversion factor for each bond. The lowest value is CTD.
Adding the convexity adjustment instead of subtracting it
Students remember that futures and forwards differ but not the direction.
Fix: Daily settlement makes the futures rate higher than the forward rate. Forward = futures − convexity adjustment.
Mixing up the Eurodollar tick value with the notional
Students apply 0.01% to 1 million dollars without the three-month fraction.
Fix: 0.0001 × 1,000,000 × 0.25 = 25 dollars per basis point.
Assuming the duration hedge removes all rate risk
The formula gives one clean number, so it feels complete.
Fix: It hedges small parallel shifts only. Curve twists, basis risk, and convexity remain, and the CTD can change.

Worked examples

Example 1

A fund holds $50 million of Treasury bonds with modified duration 7.2. A Treasury futures contract is priced at $120,000 per contract with duration 6.0 (the CTD's). How many contracts does the fund need to hedge against a small parallel yield shift?

Show the solution

Formula: N = −(P × D_P) ÷ (F × D_F).
P × D_P = 50,000,000 × 7.2 = 360,000,000.
F × D_F = 120,000 × 6.0 = 720,000.
N = −360,000,000 ÷ 720,000 = −500.
Negative sign means short.

Answer: Short 500 futures contracts.

Example 2

A treasurer expects to borrow $10 million for 3 months starting in 6 months. The 6-month rate is 4.00% and the 9-month rate is 4.40%, both quoted as simple (money-market) rates. What is the forward rate for the 6x9 period, which the treasurer could lock in with an FRA?

Show the solution

Using simple interest, (1 + 0.044 × 0.75) = (1 + 0.04 × 0.5) × (1 + f × 0.25).
Left side: 1 + 0.033 = 1.033.
Right first factor: 1 + 0.02 = 1.02.
1.033 ÷ 1.02 = 1.012745.
So f × 0.25 = 0.012745, hence f = 0.05098, or about 5.10%.

Answer: The 6x9 forward rate is about 5.10% per annum. The treasurer buys an FRA at this rate to fix borrowing cost.

Exam tips

Expect a numeric hedge-ratio question. Write the formula, plug numbers, and check the sign before choosing an answer.
Know the direction rules cold: short futures protects a long bond, and rising yields mean falling futures prices.
For CTD questions, compute the cost for each bond. Two options are often close, so check your arithmetic.
Eurodollar questions test the 100 minus rate quote, the 25 dollar tick, and the convexity adjustment direction.
Read whether the question gives DV01 or duration. Do not mix the two without converting.

Practice questions from Interest Rates

Interest Rate Hedging and Futures in other exams

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

Interest Rate Hedging and Futures: frequently asked questions

What is the duration-based hedge ratio formula for FRM?

N = −(P × D_P) ÷ (F × D_F), where P is portfolio value, F is futures contract value, and D is duration. The DV01 form is simply portfolio DV01 divided by futures DV01. A negative result means short futures.

How do you hedge a bond portfolio with Treasury futures?

Compute the portfolio DV01 and the DV01 of one futures contract based on the CTD bond. Divide to get the number of contracts. Short that many contracts for a long bond portfolio.

What is the cheapest-to-deliver bond?

It is the deliverable bond that costs the short the least relative to what it receives. You find it by minimising quoted price minus futures price times conversion factor. The futures price tracks this bond.

Why is there a convexity adjustment for Eurodollar futures?

Futures are settled daily, so gains and losses are reinvested or financed at changing rates. This makes the futures rate higher than the equivalent forward rate. You subtract the adjustment to get the forward rate.