Forwards and Futures

Introduction

Forward and futures contracts allow firms and investors to fix—or substantially stabilize—a future purchase or sale price. They are widely used to manage foreign-exchange, interest-rate, equity-market and commodity risks. They also support speculation, arbitrage and price discovery.

A derivative hedge does not usually eliminate every source of uncertainty. Differences between the exposure and the chosen contract, imperfect timing, changing quantities, liquidity constraints and daily settlement can create residual risk. Effective risk management therefore requires both correct valuation and careful hedge design.

Learning Objectives

After studying this chapter, students should be able to:

• distinguish forward contracts from futures contracts;
• calculate contract payoffs and interpret long and short positions;
• derive forward prices using no-arbitrage and cost-of-carry principles;
• value an existing forward contract;
• explain basis, convergence and basis risk;
• determine minimum-variance hedge ratios and contract quantities;
• hedge foreign-exchange, interest-rate, equity-index and commodity exposures; and
• evaluate operational, accounting, governance and regulatory considerations.

1. Forward and Futures Contracts

A forward contract is an over-the-counter agreement under which two parties commit today to exchange an asset at a specified future date for a price fixed at contract initiation. The agreed delivery price is commonly denoted by K.

A futures contract is a standardized, exchange-traded commitment to buy or sell an underlying asset. Futures are marked to market daily through a clearinghouse, and traders must maintain margin accounts.

The long party agrees to buy the underlying asset. The short party agrees to sell it. Futures contracts are commonly closed before delivery by taking an equal and opposite position.

1.1 Payoffs at maturity

For a contract on one unit of an asset, the long forward payoff at maturity T is:

The short forward payoff is:

Here, S_T is the spot price at maturity and K is the contractual delivery price. Payoffs have currency units per underlying unit. The two payoffs sum to zero before transaction costs and counterparty losses.

A payoff is not the same as profit if a contract has been acquired after initiation for a nonzero price. At inception, however, a standard forward is normally structured to have zero value, so its delivery price is set equal to the prevailing forward price.

2. No-Arbitrage and Cost of Carry

The forward price is linked to the spot price by the cost of carrying the asset from today until delivery. If equivalent future cash flows have different prices, an arbitrageur may buy the cheaper strategy and sell the more expensive one.

2.1 Investment asset with no income

Suppose an asset can be purchased and stored without cost, pays no income, can be sold short, and markets have no taxes, transaction costs or binding funding constraints. Let:

• S₀ = current spot price in currency per unit;
• F₀ = current forward price for delivery at time T;
• r = continuously compounded annual risk-free rate;
• T = years until delivery.

Buying the asset today and financing it until maturity costs S₀e^rT at time T. No-arbitrage therefore requires:

Derivation. A cash-and-carry strategy consists of borrowing S₀, buying the asset and shorting a forward. At maturity, the asset is delivered into the forward for F₀, while the loan repayment is S₀e^rT. A riskless positive profit would arise if F₀>S₀e^rT. Reverse cash-and-carry gives the opposite restriction when short selling is feasible.

With annual discrete compounding at rate r for T years, the corresponding special-case formula is:

The compounding convention used for r must match the formula.

2.2 Known cash income

If the asset pays known cash income with present value I₀ before delivery, the investor carrying the asset receives that income. Under the preceding assumptions:

For example, if a share index has a spot value of 4,000, dividends with present value 60, a continuously compounded annual risk-free rate of 5% and a one-year maturity, then:

2.3 Known proportional yield

If an asset provides a continuous income yield q, such as a dividend yield, the forward price is:

Both r and q are continuously compounded annual rates. This model is frequently used for broad equity-index futures.

2.4 Foreign-exchange forwards

A foreign currency can be viewed as an asset yielding the foreign risk-free interest rate. Let:

• S₀ = domestic-currency price of one unit of foreign currency;
• r_d = continuously compounded domestic interest rate;
• r_f = continuously compounded foreign interest rate;
• T = maturity in years.

Covered interest parity gives the dedicated FX cost-of-carry relation:

This assumes comparable default-free borrowing and lending opportunities in both currencies and no material capital controls or transaction costs.

Worked example: FX forward price. Suppose EUR 1 costs USD 1.1000, the one-year continuously compounded US rate is 4% and the euro rate is 2%:

The euro trades at a forward premium against the dollar because the US interest rate exceeds the euro interest rate. This conclusion follows from the quotation convention; reversing the currency quote reverses the interpretation.

2.5 Commodities, storage costs and convenience yield

Physical commodities may involve storage, insurance, spoilage and financing costs. Holding inventory may also provide a convenience yield, representing noncash operational benefits such as avoiding production interruptions.

If u is a continuous proportional storage-cost rate and y is a continuous convenience yield, a common cost-of-carry model is:

The formula is most useful when storage costs can be represented proportionally and the convenience yield can be estimated. If storage costs are known fixed cash amounts with present value U₀, the appropriate treatment is instead based on (S₀+U₀)e^rT, adjusted for any benefits of ownership.

When F₀>S₀, the market is commonly described as contango. When F₀<S₀, it is commonly described as backwardation. These labels describe the term-price relationship; they do not, by themselves, prove that expected spot prices will rise or fall.

3. Valuing an Existing Forward Contract

The forward price is the delivery price that would make a new contract worth zero. The value of an existing contract is different because its contractual delivery price K is fixed while market conditions change.

For a long forward on a non-income-producing investment asset, at time t before maturity:

Equivalently, if F_t is the current forward price for the remaining maturity:

The short position has value -V_t. These equations assume the same underlying, maturity and settlement terms, and use the relevant continuously compounded discount rate.

Worked example. A six-month-old long forward has six months remaining, K=105, S_t=110 and r=4\% continuously compounded:

Thus, the long contract is worth approximately 7.08 currency units per underlying unit.

For a currency forward, the long value in domestic currency is:

where \tau=T-t is remaining maturity. The first term is the domestic value of investing one foreign-currency unit for delivery, while the second is the present value of the domestic payment.

4. Why Futures and Forward Prices May Differ

If interest rates are deterministic and contract terms are otherwise comparable, forward and futures prices are theoretically equal. With stochastic interest rates, daily settlement can create differences.

A futures gain received today can be invested, while a loss must be funded immediately. If the underlying price tends to rise when interest rates rise, long futures positions receive gains when reinvestment rates are relatively high. This can make futures prices exceed otherwise comparable forward prices. The direction reverses when the correlation is negative.

For short maturities and stable interest rates, the difference is often small. For long-dated interest-rate-sensitive contracts, the distinction can be material. Credit support, liquidity, taxes and funding arrangements may also produce observed differences.

5. Basis, Convergence and Hedge Outcomes

For a futures contract maturing at T, define the basis at time t as:

Here, S_t is the spot price and F_t is the futures price for the selected contract. Some markets use the opposite convention, so analysts must always state their definition.

At delivery, futures and spot prices should converge for a deliverable contract after allowing for contractual grade, location and timing rules:

Without convergence, traders could potentially exploit a price discrepancy through delivery or offsetting transactions, subject to contract rules and costs.

Figure: Spot and futures prices converge as calendar time advances from initiation to contract maturity.

5.1 Effective price in a short hedge

A producer expecting to sell an asset may short futures at F₀ and close the position at F_T. The futures gain is F₀-F_T. If the asset is sold in the spot market for S_T, the effective sale price is:

5.2 Effective price in a long hedge

A purchaser may buy futures at F₀. The futures gain is F_T-F₀, which reduces the net cost of buying the asset at S_T:

Both hedges therefore produce F₀+b_T under the stated basis convention. If the hedge is held to delivery and the underlying exactly matches the contract, b_T=0 and the effective price is F₀, apart from transaction costs and financing effects.

Basis risk arises when the basis at hedge termination is uncertain. It is particularly important in cross-hedges, early closures and contracts with delivery-grade or location differences.

6. Selecting the Hedge Ratio

A one-for-one hedge may not minimize risk when the exposure and futures contract differ. Let \Delta S be the change in the exposure's spot price and \Delta F the change in the futures price. The minimum-variance hedge ratio is:

Here, \rho_SF is the correlation between price changes, while \sigma_S and \sigma_F are their standard deviations measured over matching intervals. The formula results from minimizing the variance of \Delta S-h\Delta F. It assumes a stable linear relationship and estimates representative of the hedge horizon.

The required number of futures contracts is:

where Q_A is the quantity exposed and Q_F is the quantity represented by one futures contract. Contract numbers must normally be rounded, creating a small overhedge or underhedge.

A short hedge uses short futures when the firm is exposed to falling prices; a long hedge uses long futures when exposed to rising prices.

7. Applications Across Major Risk Classes

7.1 Foreign-exchange exposure

An importer that must pay foreign currency has a long economic exposure to that currency and can buy it forward. An exporter expecting a foreign-currency receipt can sell the currency forward.

Suppose a US company must pay EUR 2 million in six months. A bank quotes USD 1.1150 per euro for a six-month forward. By buying euros forward, the company fixes the dollar payment at:

The hedge removes uncertainty about the dollar payment but also prevents the firm from benefiting if the euro depreciates. Counterparty limits, collateral and forecast accuracy remain relevant.

7.2 Equity-index exposure

Equity-index futures can alter a portfolio's market beta without immediately selling its securities. An approximate hedge contract count is:

where \beta_P is the current portfolio beta, \beta_T the target beta, V_P the portfolio value, F₀ the futures index level and M the currency multiplier per index point. A positive result indicates the number of contracts to short when reducing beta.

Example. A USD 10 million portfolio has beta 1.2. The index futures level is 5,000 and the multiplier is USD 50. To reduce beta to zero:

The manager should short approximately 48 contracts. The hedge may be imperfect because the portfolio may not track the index consistently.

7.3 Interest-rate exposure and duration hedging

Bond prices generally fall when yields rise. A holder of fixed-rate bonds can therefore short interest-rate futures. For small parallel yield changes, modified duration gives:

Here, D_mod is modified duration in years, V is market value in currency and \Delta y is the yield change in decimal form. The result has currency units.

A duration-based futures hedge is commonly approximated by:

where D_P and V_P are portfolio duration and value, while D_F and V_F are the duration and value associated with one futures contract. Conversion factors, cheapest-to-deliver options and nonparallel yield-curve movements may require more detailed models.

Verified duration calculation. Suppose 169 short futures contracts each have a value of USD 98,000 and duration 7.5 years. If the relevant yield rises by 0.005, the approximate gain is:

The approximation assumes a parallel yield increase and ignores convexity, daily settlement and changes in the delivery option.

Short-term interest-rate futures may instead be interpreted through their quotation conventions and changes in implied rates. Contract specifications must be checked carefully because price changes and rate changes move in opposite directions.

7.4 Commodity exposure

Producers often use short hedges to protect selling prices, while consumers use long hedges to protect purchasing costs. Commodity hedging requires attention to grade, location, seasonality, storage availability and contract delivery rules.

8. Illustrative Case: Airline Fuel Hedge

Background

A fictional airline, Northstar Air, expects to purchase 1.26 million gallons of jet-fuel-equivalent product in three months. Because liquid jet-fuel futures are unavailable in its region, it uses a closely related fuel futures contract containing 42,000 gallons per contract.

Problem

The airline fears a price increase. Current physical fuel costs USD 2.00 per gallon, while three-month futures trade at USD 2.05 per gallon. A cross-hedge creates basis risk because the futures product and purchased fuel are not identical.

Approach

For a full nominal hedge, the airline buys:

At hedge termination, the spot purchase price is USD 2.03 per gallon and futures are USD 2.04 per gallon. The ending spot price of USD 2.03 is USD 0.03 above the initial spot price of USD 2.00 and USD 0.02 below the initial futures price of USD 2.05.

The long futures loss per gallon is:

The physical purchase costs:

The futures loss is:

Therefore, total effective expenditure is USD 2,570,400, or:

The basis at termination is b_T=S_T-F_T=2.03-2.04=-0.01. Accordingly:

Outcome and interpretation

The hedge did not guarantee USD 2.05 because it was closed before delivery and retained a negative basis. It nevertheless reduced uncertainty relative to remaining unhedged. Interestingly, because prices fell relative to the initial futures price, the futures position generated a loss; this is economically offset by purchasing physical fuel below that initial futures price.

Learning points

• Hedge success should be assessed using the combined physical and derivative outcome.
• A futures loss does not necessarily mean that a hedge failed.
• Cross-commodity relationships and basis behavior must be monitored.
• Forecast-volume errors can create overhedging or underhedging.
• Transaction costs, margin funding and accounting effects should be included in a complete evaluation.

9. Implementation, Governance and Limitations

9.1 Operational and model risks

Forward and futures models rely on assumptions that may not hold exactly. Borrowing rates can exceed lending rates, short selling may be restricted, and storage may be scarce. Historical hedge ratios can become unstable during market stress. Contract data, settlement calendars and currency quotation errors can also cause material losses.

Sound practice includes independent price verification, position reconciliation, stress testing, documented limits and separation of trading, confirmation and settlement responsibilities.

9.2 Liquidity and margin risk

A futures hedge can be economically effective yet produce large variation-margin calls before the underlying cash flow occurs. Treasury managers must maintain sufficient liquidity and collateral. Closing a large position in an illiquid market may also move prices or incur substantial bid–ask costs.

9.3 Counterparty and clearing risk

OTC forwards expose each party to replacement-cost risk if the counterparty defaults. Netting, collateral agreements, credit limits and central clearing can reduce—but not eliminate—this risk. Futures clearinghouses mutualize and manage default risk through margin, default funds and position controls.

9.4 Accounting and ethical considerations

Under applicable financial-reporting standards, derivatives are generally recognized at fair value. Hedge-accounting treatment may align some derivative gains and losses with the hedged exposure, but it requires eligibility, formal designation and appropriate documentation and assessment. Rules differ by jurisdiction and hedge type.

Managers should not label speculative positions as hedges to conceal risk-taking. Boards should define permitted instruments, authorized counterparties, hedge objectives and reporting responsibilities. Compensation arrangements should not encourage excessive contract volume or manipulation of valuation assumptions.

9.5 Technology and contemporary developments

Electronic trading, automated margining and real-time risk dashboards have improved execution and monitoring. Machine-learning methods may help forecast basis relationships, liquidity needs and hedge effectiveness, but unstable regimes and opaque models can create false confidence. Human oversight, explainability, data governance and independent validation remain necessary.

Current developments include increased central clearing, heightened collateral requirements, benchmark-rate transitions, climate-related commodity volatility and growing use of exchange-traded contracts linked to electricity, emissions and digital assets. Regardless of the underlying market, no-arbitrage reasoning, contract specifications and liquidity management remain foundational.

Chapter Summary

Forwards and futures commit parties to transact at future dates. Forwards are customizable OTC agreements, while futures are standardized, exchange-traded and marked to market daily. Long positions benefit from price increases; short positions benefit from price decreases.

No-arbitrage connects forward prices to spot prices and carrying costs. For an asset without income, F₀=S₀e^rT. Income reduces the forward price, storage costs increase it, and convenience yield reduces it. For currencies, covered interest parity gives F₀=S₀e^(r_d-r_f)T. Existing forwards are valued by comparing their contractual delivery prices with current market terms.

With basis defined as b_t=S_t-F_t, both the effective short-hedge sale price and long-hedge purchase price equal F₀+b_T. Convergence drives the basis toward zero at delivery, although early closure and cross-hedging create basis risk. Minimum-variance hedge ratios account for imperfect price correlation.

Forward and futures contracts can manage FX, equity-index, interest-rate and commodity exposures. Effective implementation also requires attention to liquidity, margin, counterparty risk, contract specifications, accounting, governance and model limitations.

Key Terms

Arbitrage: A strategy designed to exploit inconsistent prices while avoiding net market risk under stated assumptions.

Backwardation: A market condition in which a futures price is below the current spot price.

Basis: The difference between spot and futures prices, defined in this chapter as S_t-F_t.

Basis risk: Uncertainty arising because the basis may change before a hedge is closed.

Cash-and-carry arbitrage: Buying an asset, financing and carrying it, and selling it forward when the forward price is too high.

Contango: A market condition in which a futures price is above the current spot price.

Convenience yield: The noncash operational benefit obtained from holding a physical commodity.

Convergence: The tendency of spot and futures prices to become equal as delivery approaches.

Cost of carry: The net cost of holding an asset, including financing and storage less income or convenience benefits.

Cross-hedge: A hedge using a derivative whose underlying differs from the exposure but has correlated price movements.

Forward contract: A customized OTC agreement to transact at a specified future date and price.

Forward price: The delivery price that gives a newly initiated forward contract zero value.

Futures contract: A standardized exchange-traded contract subject to daily marking to market.

Hedge ratio: The size of a derivative position relative to the underlying exposure.

Long hedge: A long derivative position used to manage the risk of rising purchase prices.

Marking to market: Daily settlement of futures gains and losses.

Minimum-variance hedge ratio: The hedge ratio that minimizes the variance of the combined spot and futures position.

Short hedge: A short derivative position used to manage the risk of falling selling prices.

Variation margin: Funds transferred because of daily changes in a futures position's value.

Review Questions

Short Answer Questions

1. Distinguish a forward contract from a futures contract.
2. What is meant by convergence in a futures market?
3. Explain why convenience yield reduces a commodity's theoretical futures price.
4. Distinguish a forward price from the value of an existing forward contract.
5. Why can a successful hedge generate a loss on its derivative component?

Long Answer Questions

1. Derive the no-arbitrage forward price for a non-income-producing investment asset and explain the assumptions required.
2. Explain covered interest parity and show how domestic and foreign interest rates affect an FX forward quotation.
3. Analyze the causes and consequences of basis risk in commodity cross-hedging.
4. Compare forward and futures contracts from valuation, credit-risk, liquidity and cash-flow perspectives.
5. Discuss the governance, accounting and liquidity-management requirements of a corporate derivatives program.

Analytical / Application Questions

1. A commodity has a spot price of USD 80, a continuous financing rate of 5%, storage costs of 2% and convenience yield of 1%. Calculate its theoretical one-year futures price and interpret the result.
2. A firm will sell 500,000 units of a commodity. The estimated minimum-variance hedge ratio is 0.82 and each futures contract covers 10,000 units. Determine the contract position and discuss the effect of rounding.
3. A portfolio worth USD 25 million has beta 1.3. Index futures stand at 4,000 with a USD 50 multiplier. Calculate the number of contracts required to reduce the portfolio beta to 0.5 and identify the required position.

Multiple Choice Questions with Answers

Question 1

Which feature most directly distinguishes exchange-traded futures from conventional OTC forwards?

A. Futures always require physical delivery

B. Futures are standardized and marked to market daily

C. Forwards cannot hedge currency exposure

D. Forwards always have positive value at inception

Correct Answer: B. Futures are standardized and marked to market daily

Question 2

A non-income-producing asset has a spot price of 100. The continuously compounded risk-free rate is 5% and maturity is one year. What is the no-arbitrage forward price?

A. 95.12

B. 100.00

C. 105.13

D. 110.25

Correct Answer: C. 105.13

Question 3

Using b_T=S_T-F_T, what is the effective purchase price for a long hedge initiated at F₀ and closed at time T?

A. S_T+(F_T-F₀)

B. F₀-b_T

C. S_T-(F_T-F₀)=F₀+b_T

D. F_T+b_T

Correct Answer: C. S_T-(F_T-F₀)=F₀+b_T

Question 4

The spot exchange rate is USD 1.20 per euro, the continuously compounded US rate is 5%, the euro rate is 2%, and maturity is one year. What is the theoretical forward rate?

A. USD 1.1645 per euro

B. USD 1.2000 per euro

C. USD 1.2365 per euro

D. USD 1.2840 per euro

Correct Answer: C. USD 1.2365 per euro

Question 5

Which change, considered alone, increases a commodity's theoretical futures price under the cost-of-carry model?

A. A higher convenience yield

B. A higher storage-cost rate

C. A higher income yield

D. A shorter maturity when net carry is positive

Correct Answer: B. A higher storage-cost rate

Question 6

A long forward has delivery price 102. The current forward price for the same remaining maturity is 108, and the discount factor is 0.98. What is its value per unit?

A. 5.88

B. 6.00

C. 10.00

D. 105.84

Correct Answer: A. 5.88

Question 7

If \rho_SF=0.8, \sigma_S=12\% and \sigma_F=10\%, what is the minimum-variance hedge ratio?

A. 0.67

B. 0.80

C. 0.96

D. 1.20

Correct Answer: C. 0.96

Question 8

A USD 10 million equity portfolio has beta 1.2. Index futures are 5,000 with a USD 50 multiplier. How many contracts should be shorted to target a zero beta?

A. 24

B. 40

C. 48

D. 60

Correct Answer: C. 48

Question 9

What is the approximate gain on 169 short futures contracts, each worth USD 98,000 with duration 7.5, when yields rise by 0.005?

A. USD 124,215

B. USD 490,000

C. USD 621,075

D. USD 735,000

Correct Answer: C. USD 621,075

Question 10

Which risk can require cash funding even when a futures hedge is economically effective?

A. Variation-margin liquidity risk

B. Only physical-delivery risk

C. Dividend-voting risk

D. Inventory ownership risk in every futures position

Correct Answer: A. Variation-margin liquidity risk