Level III Core · Swaps, Forwards, and Futures Strategies

Bond Futures and Duration Adjustment for CFA Level III

Updated 8 October 2026 · Fact-checked

Bond futures change a portfolio's duration without trading the bonds. Compute the portfolio's basis point value (BPV), set a target BPV, then divide the BPV gap by the futures BPV, which comes from the cheapest-to-deliver bond and its conversion factor. A negative result means sell contracts; a positive result means buy.

Understand Futures for Fixed Income and Duration Adjustment

Duration tells you how much a bond portfolio's value changes when yields move. Sometimes a manager wants more or less of that sensitivity but does not want to sell bonds, because of trading costs, taxes, or a short-term view. Bond futures let you shift duration quickly and cheaply.

Basis point value (BPV) is the change in value for a 1 basis point change in yield. For a portfolio, BPV ≈ modified duration × market value × 0.0001. If you want less duration, you sell futures. A short futures position gains when yields rise, which offsets losses on the bonds. If you want more duration, you buy futures.

A bond futures contract does not reference one bond. The short can deliver any bond from an eligible basket. Each bond has a conversion factor (CF) that puts it on a common footing with the notional contract. The short will deliver the cheapest-to-deliver (CTD) bond, which is the one with the lowest net cost of delivery. The futures price therefore behaves like the CTD bond's price divided by its CF.

That is why futures BPV is based on the CTD: BPV of futures ≈ BPV of the CTD per contract ÷ CF. Once you have that number, the hedge ratio is a simple division. The result is an approximation. It assumes parallel yield shifts and a stable CTD. If yield changes alter which bond is CTD, the futures BPV changes and you must re-estimate it.

Compared with interest rate swaps, futures are standardized, exchange-traded, liquid and margined daily, with little credit risk. They have limited maturities and contract sizes, so they cannot be tailored exactly. They also carry basis risk and CTD or delivery-option risk. Swaps can be customized to match exact dates and exposures, but they bring counterparty exposure and are less flexible to unwind.

Key rules to remember

Portfolio BPV: BPV = modified duration × market value × 0.0001; Use the same unit for market value throughout. The result is the value change for a 1 basis point yield change.
Futures BPV: BPV of futures ≈ BPV of CTD bond (per contract notional) ÷ conversion factor; Use the CTD's price and modified duration, and apply them to the contract's par amount.
Number of contracts: N = (BPV target − BPV portfolio) ÷ BPV futures; Negative means sell contracts to cut duration. Positive means buy contracts to add duration.
Target BPV: BPV target = target modified duration × market value × 0.0001; For a full hedge, set target BPV to zero.
Cost to deliver (CTD test): Cost = bond quoted price − (futures price × conversion factor); Choose the bond with the lowest cost. Add accrued interest consistently if the exam gives it.
Futures price from CTD: Futures price ≈ [CTD full price × (1 + r)^T − accrued interest at expiry − FV of coupons] ÷ CF; The usual no-arbitrage relationship using the CTD. Use the exam's inputs exactly.

How to solve Futures for Fixed Income and Duration Adjustment questions

Use this order for any question on adjusting duration with bond futures. Write each number so partial work earns credit.

1Identify the goal: reduce duration, increase duration, or hedge fully. This sets the sign of your answer.
2Compute portfolio BPV = modified duration × market value × 0.0001. If the question gives a BPV directly, use it.
3Compute target BPV from the target duration, or use zero for a full hedge.
4Identify the CTD bond. If the question gives the costs to deliver, pick the lowest. Otherwise compute price − futures price × CF for each bond.
5Compute BPV of the CTD for the contract notional, then divide by the CF to get futures BPV.
6Compute N = (target BPV − portfolio BPV) ÷ futures BPV. State buy or sell and round to a whole contract.
7Check that the answer makes sense: a lower target duration should give a negative N. Mention limits if asked, such as non-parallel shifts or a change in CTD.

Quickest way: Three-line BPV gap method

When to use it: Use it when the vignette gives the portfolio, target duration, and CTD data and asks only for the number of contracts.

Line 1: BPV gap = (target duration − current duration) × market value × 0.0001.
Line 2: futures BPV = CTD price × contract par × CTD modified duration × 0.0001 ÷ CF (price as a decimal fraction of par).
Line 3: N = gap ÷ futures BPV. A negative sign means sell. Then answer in whole contracts.

Common mistakes in Futures for Fixed Income and Duration Adjustment

Forgetting to divide by the conversion factor when computing futures BPV
Students treat the CTD bond itself as the futures contract.
Fix: Always write futures BPV = CTD BPV ÷ CF. A CF below 1 makes the futures BPV larger than the CTD's BPV per contract.
Getting the sign wrong and buying futures to cut duration
Students focus on the size of the number and forget the direction.
Fix: Lower duration needs a short position. Check that target BPV minus portfolio BPV is negative and then sell.
Choosing the CTD by highest yield or lowest price
Price alone ignores the conversion factor.
Fix: Compare price − (futures price × CF) across bonds. The lowest value is the CTD.
Using the CTD's duration for the portfolio or the portfolio's duration for the futures
Both are called duration, and the vignette lists several.
Fix: Label each number: portfolio duration for portfolio BPV, CTD duration for futures BPV.
Treating a full hedge as exact
The formula gives a precise-looking number.
Fix: Remember the result assumes parallel shifts and a fixed CTD. Credit spread moves, yield curve twists, and a change in CTD all leave residual risk.
Mixing units, such as using millions for value and a single contract's par in thousands
Rushing through the BPV calculation.
Fix: Write the full number, for example 200,000,000, before multiplying by 0.0001.

Worked examples

Example 1

A portfolio worth $200 million has a modified duration of 6.2. The manager wants to cut modified duration to 4.0 using bond futures. The CTD bond has a price of 98.50 (per 100 of par) and a modified duration of 7.5. The CF is 0.9000 and each contract has a par value of $100,000. How many contracts should the manager trade?

Show the solution

Portfolio BPV = 200,000,000 × 6.2 × 0.0001 = $124,000.
Target BPV = 200,000,000 × 4.0 × 0.0001 = $80,000.
BPV gap = 80,000 − 124,000 = −$44,000, so the manager must sell.
CTD value per contract = 98.50% × 100,000 = $98,500. CTD BPV = 98,500 × 7.5 × 0.0001 = $73.875.
Futures BPV = 73.875 ÷ 0.9000 = $82.0833.
N = −44,000 ÷ 82.0833 = −536.04.

Answer: Sell about 536 futures contracts.

Example 2

The futures price is 120.00. Three bonds are deliverable, with quoted price and conversion factor: Bond A 104.50 and 0.8600; Bond B 118.40 and 0.9800; Bond C 96.30 and 0.8000. Ignore accrued interest. (a) Which bond is the CTD? (b) Bond C has a modified duration of 5.0 and the contract par is $100,000. How many contracts must be bought to add $30,000 of BPV?

Show the solution

Cost for A = 104.50 − (120.00 × 0.8600) = 104.50 − 103.20 = 1.30.
Cost for B = 118.40 − (120.00 × 0.9800) = 118.40 − 117.60 = 0.80.
Cost for C = 96.30 − (120.00 × 0.8000) = 96.30 − 96.00 = 0.30. The lowest cost is Bond C, so C is the CTD.
CTD value per contract = 96.30% × 100,000 = $96,300. CTD BPV = 96,300 × 5.0 × 0.0001 = $48.15.
Futures BPV = 48.15 ÷ 0.8000 = $60.1875.
N = 30,000 ÷ 60.1875 = 498.44, so buy about 498 contracts.

Answer: (a) Bond C is the CTD. (b) Buy about 498 contracts.

Exam tips

Read the command word. If asked to "calculate", show the BPV steps and type the final number clearly, because a correct number earns full credit. If asked to "justify" or "explain", give one reason tied to the client's goal.
Underline whether the target is a duration, a BPV, or a full hedge before you start. Many errors come from using the wrong target.
In item sets, check whether the vignette already gives futures BPV or the CTD's BPV before you recompute it. This saves time.
Expect a short qualitative follow-up on futures versus swaps. Use clear contrasts: standardized versus customized, daily margin versus counterparty exposure, CTD basis risk versus exact matching.
Round the number of contracts only at the end, and state buy or sell. Only the number of responses asked for is evaluated, so do not add extra answers.

Futures for Fixed Income and Duration Adjustment: frequently asked questions

How do I calculate the BPV hedge ratio for bond futures?

Find the portfolio BPV and the target BPV, then take the difference. Divide it by the futures BPV, which is the CTD's BPV per contract divided by its conversion factor. A negative result means sell contracts.

How do I find the cheapest-to-deliver bond using conversion factors?

For each deliverable bond, compute quoted price minus futures price times the conversion factor. The bond with the lowest result is the CTD. If accrued interest differs, include it on both sides consistently.

Why is futures BPV divided by the conversion factor?

The futures price tracks the CTD price scaled by the conversion factor. A move in the CTD's price therefore moves the futures price by roughly that move divided by the CF. Dividing the CTD's BPV by the CF reflects this.

What is the difference between futures and swaps for duration management?

Futures are standardized, liquid and margined daily, so credit risk is low, but they have fixed maturities and CTD basis risk. Swaps can be tailored to exact dates and exposures but carry counterparty exposure and are less easy to unwind.