FRM Exam Part I · Using Futures for Hedging

Tailing the Hedge and Rolling Hedges Forward Explained

Updated 11 October 2026

Tailing the hedge reduces the number of futures contracts to allow for daily settlement: multiply the usual hedge position by the discount factor, roughly 1 ÷ (1 + r × T). Rolling a hedge forward closes a near contract and opens a later one when the hedge horizon exceeds contract maturity. It adds basis and cash-flow risk.

Understand Tailing the Hedge and Rolling Hedges Forward

A futures contract is settled every day. A forward contract is settled once, at maturity. So gains and losses on a futures hedge arrive early. Because they are settled daily and then compounded at the interest rate up to the horizon, the futures payoff is magnified by roughly (1 + rT).

The tail applies to a hedge position sized by value: N* = h* × V_A ÷ V_F, where V_F is the futures price times the contract size. You then multiply N* by the discount factor to the hedge horizon. If you skip this step, the compounding makes the futures payoff at the horizon about (1 + rT) times larger than the price change you want to offset. Tailing the hedge fixes this by scaling the futures position down by 1 ÷ (1 + rT). The tail factor is the discount factor from the hedge date to the hedge horizon. For a one-year hedge at 5% it is about 1 ÷ 1.05 = 0.952. Tailing matters most for long horizons and high rates. For short horizons it is small.

A second problem is maturity. Liquid futures often expire before your exposure ends. In a stack and roll strategy you buy or sell a nearby contract, then close it near expiry and open the same position in the next contract. You repeat this until the horizon. Each roll is a new trade at a new futures price. Your final result depends on the sequence of basis moves, not on one fixed price.

Rolling therefore leaves rollover basis risk. It also creates liquidity risk. The Metallgesellschaft case shows this. Its US subsidiary sold long-dated fixed-price oil contracts to customers and hedged with short-dated long futures, rolled monthly. MGRM used a roughly one-for-one stack-and-roll hedge in barrels, which analysts have criticised as not optimal. Hedging long-dated exposure with short-dated futures created a maturity mismatch and large interim margin flows. When oil prices fell, the futures lost heavily and margin calls were large. The customer contracts gained value, but those gains were not cash. The firm faced a funding squeeze. During 1993 the market also moved from backwardation to contango, which turned the roll yield negative for the long futures. Even a sound economic hedge can fail if you cannot fund its interim cash flows.

Key formulas to remember

Hedge ratio before tailing: N* = h* × Q_A ÷ Q_F; h* is the minimum variance hedge ratio, Q_A is the exposure size and Q_F is the size of one futures contract. This gives the number of contracts. When the tail is applied, size the position by value (V_A ÷ V_F, with V_F the futures price times contract size) and then discount.
Tailing factor (discrete): Tail factor = 1 ÷ (1 + r × T); T is the time to the hedge horizon in years. Tailed contracts = N* × tail factor, where N* is the value-based hedge position. Use the discounting convention given in the question.
Tailing factor (continuous): Tail factor = e^(−rT); Use when the question gives a continuously compounded rate. It is the discount factor to the hedge horizon.
Tailed contract count: N_tailed = N* × (1 ÷ (1 + r × T)); Always fewer contracts than the untailed number for positive rates.
Rolling hedge result: Long (purchase) hedge: effective cost ≈ S_end − Σ(futures gains over each roll). Short (sale) hedge: effective proceeds ≈ S_end + Σ(futures gains over each roll); S_end is the spot price at the horizon, adjusted for the cumulative futures gains or losses. Each roll opens a new futures position at a new futures price, so the price is not locked in. The final outcome depends on the basis at each roll, that is, the spread between spot and futures and between the near and far contracts. Check the direction: a purchase hedge benefits from futures gains, so they reduce the net cost.

How to solve Tailing the Hedge and Rolling Hedges Forward questions

Use this method for any question on tailing or rolling hedges.

1Identify the exposure: its size, its direction (long or short) and the hedge horizon.
2Compute the untailed hedge: N* = h* × Q_A ÷ Q_F. If no h* is given, use 1.
3Find the tailing factor from the rate and horizon. Use the compounding the question specifies.
4Multiply N* by the tailing factor and round to a whole number of contracts only at the end, if asked.
5If the hedge needs rolling, list each contract period and note that each roll sets a new futures price.
6Identify the risks: rollover basis risk, margin funding risk and, for commodities, contango or backwardation effects.
7Check the direction: a long exposure needs short futures, a short exposure needs long futures. Then state the final answer with units.

Quickest way: Discount and sanity-check

When to use it: Use this when the question gives a rate, a horizon and a hedge size, and asks for the tailed number of contracts.

Compute the untailed contracts first.
Divide by (1 + r × T), or multiply by e^(−rT) if continuous.
Check that the answer is smaller than the untailed figure but close to it.
For a conceptual option, pick the answer that says the tailed position is smaller because of daily settlement and interest on margin flows.

Common mistakes in Tailing the Hedge and Rolling Hedges Forward

Increasing the number of contracts when tailing.
Students multiply by (1 + rT) instead of dividing.
Fix: Tailing discounts. For positive rates the tailed position is always smaller than the untailed one.
Using the futures maturity as T instead of the hedge horizon.
The contract date is easy to see in the question.
Fix: The tail factor discounts to the hedge horizon, the date the exposure is settled or hedged to, unless the question says otherwise.
Believing tailing is needed because futures are riskier than forwards.
Confusing daily settlement with credit or price risk.
Fix: Tailing exists only because gains and losses are realized and reinvested or funded before maturity.
Thinking a roll eliminates basis risk.
Students assume the same position means the same price.
Fix: Each roll resets the futures price. Rollover basis risk remains, and so does cash-flow risk.
Blaming Metallgesellschaft's loss only on a bad price view.
Focusing on the falling oil price.
Fix: The key problem was funding. Futures losses were paid in cash while offsetting gains on customer contracts were not, and the position was large and rolled short-dated against long-dated exposure. The market also moved from backwardation to contango, which made the roll yield negative for the long futures.
Rounding contracts too early.
Wanting a whole number at each step.
Fix: Keep full precision until the final step, then round to the nearest whole contract.

Worked examples

Example 1

A firm holds a USD 20 million equity portfolio with beta 1.0 and wants to hedge it over 2 years. The futures contract value is USD 250,000 per contract, the optimal hedge ratio is 1.0, and the rate is 4% per year (simple discounting, 1 ÷ (1 + rT)). How many futures contracts should it short after tailing?

Show the solution

Untailed contracts: N* = 1.0 × 20,000,000 ÷ 250,000 = 80.
Tail factor = 1 ÷ (1 + 0.04 × 2) = 1 ÷ 1.08 = 0.9259.
Tailed contracts = 80 × 0.9259 = 74.07.
Round to the nearest whole contract: 74. The firm shorts futures because it is long the portfolio.

Answer: Short about 74 contracts.

Example 2

A refiner must hedge a purchase of 500,000 barrels of oil in 9 months. Each futures contract covers 1,000 barrels, h* = 0.90, and the continuously compounded rate is 6%. The available futures expire in 3 months and are rolled. Apply a single tail factor to the 9-month hedge horizon. How many contracts does the refiner buy after tailing, and what is the main risk of the roll?

Show the solution

Untailed contracts: N* = 0.90 × 500,000 ÷ 1,000 = 450.
Tail factor = e^(−0.06 × 0.75) = e^(−0.045) = 0.9560. Using one factor for the whole 9-month horizon is a simplification. It follows the convention of discounting to the hedge horizon. A period-by-period tail on each rolled contract would differ slightly.
Tailed contracts = 450 × 0.9560 = 430.2, so about 430.
The refiner is buying oil later, so it goes long futures.
The main risk of rolling is rollover basis risk: the spread between the near and far contracts at each roll is unknown in advance. It also faces margin funding risk if prices fall.

Answer: Buy about 430 contracts (long), using a single tail factor to the 9-month horizon. The main risk is rollover basis risk, along with margin funding risk.

Exam tips

Expect a numeric tailing question: compute N* first, then discount. Watch whether the rate is simple or continuous.
Expect conceptual questions on why tailing is needed. The answer is daily settlement, not credit risk.
For Metallgesellschaft, link the loss to funding liquidity and margin calls, not only to price direction.
Remember that the tail reduces the position, so eliminate any option showing more contracts than the untailed number.
Check direction: long exposure needs short futures, short exposure needs long futures.

Practice questions from Using Futures for Hedging

Tailing the Hedge and Rolling Hedges Forward: frequently asked questions

What is tailing the hedge?

It is scaling down the futures position to allow for daily settlement. The tail factor is the discount factor to the hedge horizon, so you hold slightly fewer contracts than the untailed number.

Why does daily settlement change the hedge ratio?

Futures gains and losses are paid each day, before the hedge horizon, and are then compounded at the interest rate up to the horizon. This magnifies the futures payoff by roughly (1 + rT). So you size the hedge position by value, N* = h* × V_A ÷ V_F, and scale it down by 1 ÷ (1 + rT), the discount factor to the horizon. That is the tail.

What is a stack and roll hedge?

You hold futures in a near contract, close it near expiry and open the same position in the next contract. You repeat until the exposure ends. The strategy is used when liquid futures mature before the hedge horizon.

What did the Metallgesellschaft case show about rolling hedges?

It showed that a hedge can be sound economically but fail on funding. Futures losses created large cash margin calls, while the gains on the offsetting customer contracts were not received in cash. Rolling also became costly when the market moved from backwardation to contango, which turned the roll yield negative for the long futures.