Forward contract
Encyclopedia
In finance, a forward contract or simply a forward is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed today. This is in contrast to a spot contract, which is an agreement to buy or sell an asset today. It costs nothing to enter a forward contract. The party agreeing to buy the underlying asset in the future assumes a long position, and the party agreeing to sell the asset in the future assumes a short position. The price agreed upon is called the delivery price, which is equal to the forward price
at the time the contract is entered into.
The price of the underlying instrument, in whatever form, is paid before control of the instrument changes. This is one of the many forms of buy/sell orders where the time of trade is not the time where the securities themselves are exchanged.
The forward price
of such a contract is commonly contrasted with the spot price
, which is the price at which the asset changes hands on the spot date
. The difference between the spot and the forward price is the forward premium or forward discount, generally considered in the form of a profit
, or loss, by the purchasing party.
Forwards, like other derivative securities, can be used to hedge
risk (typically currency or exchange rate risk), as a means of speculation
, or to allow a party to take advantage of a quality of the underlying instrument which is time-sensitive.
A closely related contract is a futures contract
; they differ in certain respects. Forward contracts are very similar to futures contracts, except they are not exchange-traded, or defined on standardized assets. Forwards also typically have no interim partial settlements or "true-ups" in margin requirements like futures – such that the parties do not exchange additional property securing the party at gain and the entire unrealized gain or loss builds up while the contract is open. However, being traded OTC, forward contracts specification can be customized and may include mark-to-market and daily margining. Hence, a forward contract arrangement might call for the loss party to pledge collateral or additional collateral to better secure the party at gain.
) and the underlying price (
) at that time.
At the end of one year, suppose that the current market valuation of Andy's house is $110,000. Then, because Andy is obliged to sell to Bob for only $104,000, Bob will make a profit of $6,000. To see why this is so, one needs only to recognize that Bob can buy from Andy for $104,000 and immediately sell to the market for $110,000. Bob has made the difference in profit. In contrast, Andy has made a potential loss of $6,000, and an actual profit of $4,000.
The similar situation works among currency forwards, where one party opens a forward contract to buy or sell a currency (ex. a contract to buy Canadian dollars) to expire/settle at a future date, as they do not wish to be exposed to exchange rate/currency risk over a period of time. As the exchange rate between U.S. dollars and Canadian dollars fluctuates between the trade date and the earlier of the date at which the contract is closed or the expiration date, one party gains and the counterparty loses as one currency strengthens against the other. Sometimes, the buy forward is opened because the investor will actually need Canadian dollars at a future date such as to pay a debt owed that is denominated in Canadian dollars. Other times, the party opening a forward does so, not because they need Canadian dollars nor because they are hedging currency risk, but because they are speculating on the currency, expecting the exchange rate to move favorably to generate a gain on closing the contract.
In a currency forward, the notional amount
s of currencies are specified (ex: a contract to buy $100 million Canadian dollars equivalent to, say $114.4 million USD at the current rate—these two amounts are called the notional amount(s)). While the notional amount or reference amount may be a large number, the cost or margin requirement to command or open such a contract is considerably less than that amount, which refers to the leverage
created, which is typical in derivative
contracts.
("tradeables"), spot–forward parity provides the link between the spot market and the forward market. It describes the relationship between the spot and forward price of the underlying asset in a forward contract. While the overall effect can be described as the cost of carry, this effect can be broken down into different components, specifically whether the asset:
) and spot (
) prices is
where
is the continuously compounded risk free rate of return, and T is the time to maturity. The intuition behind this result is that given you want to own the asset at time T, there should be no difference in a perfect capital market between buying the asset today and holding it and buying the forward contract and taking delivery. Thus, both approaches must cost the same in present value terms. For an arbitrage proof of why this is the case, see Rational pricing below.
For an asset that pays known income, the relationship becomes:
where
is the present value of the discrete income at time 
, and 
is the continuous dividend yield over the life of the contract. The intuition is that when an asset pays income, there is a benefit to holding the asset rather than the forward because you get to receive this income. Hence the income (
or 
) must be subtracted to reflect this benefit. An example of an asset which pays discrete income might be a stock
, and example of an asset which pays a continuous yield might be a foreign currency or a stock index.
For investment assets which are commodities, such as gold
and silver
, storage costs must also be considered. Storage costs can be treated as 'negative income', and like income can be discrete or continuous. Hence with storage costs, the relationship becomes:
where
is the present value of the discrete storage cost at time 
, and 
is the storage cost where it is proportional to the price of the commodity, and is hence a 'negative yield'. The intuition here is that because storage costs make the final price higher, we have to add them to the spot price.
where
is the convenience yield over the life of the contract. Since the convenience yield provides a benefit to the holder of the asset but not the holder of the forward, it can be modelled as a type of 'dividend yield'. However, it is important to note that the convenience yield is a non cash item, but rather reflects the market's expectations concerning future availability of the commodity. If users have low inventories of the commodity, this implies a greater chance of shortage, which means a higher convenience yield. The opposite is true when high inventories exist.
. Hence,
and the expected future spot price, 
.
The economists John Maynard Keynes
and John Hicks
argued that in general, the natural hedgers of a commodity are those who wish to sell the commodity at a future point in time. Thus, hedgers will collectively hold a net short position in the forward market. The other side of these contracts are held by speculators, who must therefore hold a net long position. Hedgers are interested in reducing risk, and thus will accept losing money on their forward contracts. Speculators on the other hand, are interested in making a profit, and will hence only enter the contracts if they expect to make money. Thus, if speculators are holding a net long position, it must be the case that the expected future spot price is greater than the forward price.
In other words, the expected payoff to the speculator at maturity is:
, where 
is the delivery price at maturity
Thus, if the speculators expect to profit,
, as 
when they enter the contract
This market situation, where
, is referred to as normal backwardation. Since forward/futures prices converge with the spot price at maturity (see basis), normal backwardation implies that futures prices for a certain maturity are increasing over time. The opposite situation, where 
, is referred to as contango
. Likewise, contango implies that futures prices for a certain maturity are falling over time.
is the spot price
of an asset at time
, and 
is the continuously compounded rate, then the forward price at a future time 
must satisfy 
.
To prove this, suppose not. Then we have two possible cases.
Case 1: Suppose that
. Then an investor can execute the following trades at time 
:
The initial cost of the trades at the initial time sum to zero.
At time
the investor can reverse the trades that were executed at time 
. Specifically, and mirroring the trades 1., 2. and 3. the investor
The sum of the inflows in 1.' and 2.' equals
, which by hypothesis, is positive. This is an arbitrage profit. Consequently, and assuming that the non-arbitrage condition holds, we have a contradiction. This is called a cash and carry arbitrage because you "carry" the stock until maturity.
Case 2: Suppose that
. Then an investor can do the reverse of what he has done above in case 1. But if you look at the convenience yield
page, you will see that if there are finite stocks/inventory, the reverse cash and carry arbitrage is not always possible. It would depend on the elasticity of demand for forward contracts and such like.
is the time value of cash flows X at the contract expiration time 
. The forward price
is then given by the formula:
The cash flows can be in the form of dividend
s from the asset, or costs of maintaining the asset.
If these price relationships do not hold, there is an arbitrage
opportunity for a riskless profit similar to that discussed above. One implication of this is that the presence of a forward market will force spot prices to reflect current expectations of future prices. As a result, the forward price for nonperishable commodities, securities or currency is no more a predictor of future price than the spot price is - the relationship between forward and spot prices is driven by interest rates. For perishable commodities, arbitrage does not have this
The above forward pricing formula can also be written as:
Where
is the time t value of all cash flows over the life of the contract.
For more details about pricing, see forward price
.
incentives to anticipate their production through forward contracts.
Other types of trade contracts:
Forward price
The forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...
at the time the contract is entered into.
The price of the underlying instrument, in whatever form, is paid before control of the instrument changes. This is one of the many forms of buy/sell orders where the time of trade is not the time where the securities themselves are exchanged.
The forward price
Forward price
The forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...
of such a contract is commonly contrasted with the spot price
Spot price
The spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...
, which is the price at which the asset changes hands on the spot date
Spot date
In finance, the spot date of a transaction is the normal settlement day when the transaction is done today. This kind of transaction is referred to as a spot transaction or simply spot....
. The difference between the spot and the forward price is the forward premium or forward discount, generally considered in the form of a profit
Profit (accounting)
In accounting, profit can be considered to be the difference between the purchase price and the costs of bringing to market whatever it is that is accounted as an enterprise in terms of the component costs of delivered goods and/or services and any operating or other expenses.-Definition:There are...
, or loss, by the purchasing party.
Forwards, like other derivative securities, can be used to hedge
Hedge (finance)
A hedge is an investment position intended to offset potential losses that may be incurred by a companion investment.A hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance, forward contracts, swaps, options, many types of...
risk (typically currency or exchange rate risk), as a means of speculation
Speculation
In finance, speculation is a financial action that does not promise safety of the initial investment along with the return on the principal sum...
, or to allow a party to take advantage of a quality of the underlying instrument which is time-sensitive.
A closely related contract is a futures contract
Futures contract
In finance, a futures contract is a standardized contract between two parties to exchange a specified asset of standardized quantity and quality for a price agreed today with delivery occurring at a specified future date, the delivery date. The contracts are traded on a futures exchange...
; they differ in certain respects. Forward contracts are very similar to futures contracts, except they are not exchange-traded, or defined on standardized assets. Forwards also typically have no interim partial settlements or "true-ups" in margin requirements like futures – such that the parties do not exchange additional property securing the party at gain and the entire unrealized gain or loss builds up while the contract is open. However, being traded OTC, forward contracts specification can be customized and may include mark-to-market and daily margining. Hence, a forward contract arrangement might call for the loss party to pledge collateral or additional collateral to better secure the party at gain.
Payoffs
The value of a forward position at maturity depends on the relationship between the delivery price () and the underlying price () at that time.
- For a long position this payoff is:
- For a short position, it is:
How a forward contract works
Suppose that Bob wants to buy a house a year from now. At the same time, suppose that Andy currently owns a $100,000 house that he wishes to sell a year from now. Both parties could enter into a forward contract with each other. Suppose that they both agree on the sale price in one year's time of $104,000 (more below on why the sale price should be this amount). Andy and Bob have entered into a forward contract. Bob, because he is buying the underlying, is said to have entered a long forward contract. Conversely, Andy will have the short forward contract.At the end of one year, suppose that the current market valuation of Andy's house is $110,000. Then, because Andy is obliged to sell to Bob for only $104,000, Bob will make a profit of $6,000. To see why this is so, one needs only to recognize that Bob can buy from Andy for $104,000 and immediately sell to the market for $110,000. Bob has made the difference in profit. In contrast, Andy has made a potential loss of $6,000, and an actual profit of $4,000.
The similar situation works among currency forwards, where one party opens a forward contract to buy or sell a currency (ex. a contract to buy Canadian dollars) to expire/settle at a future date, as they do not wish to be exposed to exchange rate/currency risk over a period of time. As the exchange rate between U.S. dollars and Canadian dollars fluctuates between the trade date and the earlier of the date at which the contract is closed or the expiration date, one party gains and the counterparty loses as one currency strengthens against the other. Sometimes, the buy forward is opened because the investor will actually need Canadian dollars at a future date such as to pay a debt owed that is denominated in Canadian dollars. Other times, the party opening a forward does so, not because they need Canadian dollars nor because they are hedging currency risk, but because they are speculating on the currency, expecting the exchange rate to move favorably to generate a gain on closing the contract.
In a currency forward, the notional amount
Notional amount
The notional amount on a financial instrument is the nominal or face amount that is used to calculate payments made on that instrument...
s of currencies are specified (ex: a contract to buy $100 million Canadian dollars equivalent to, say $114.4 million USD at the current rate—these two amounts are called the notional amount(s)). While the notional amount or reference amount may be a large number, the cost or margin requirement to command or open such a contract is considerably less than that amount, which refers to the leverage
Leverage (finance)
In finance, leverage is a general term for any technique to multiply gains and losses. Common ways to attain leverage are borrowing money, buying fixed assets and using derivatives. Important examples are:* A public corporation may leverage its equity by borrowing money...
created, which is typical in derivative
Derivative (finance)
A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...
contracts.
Example of how forward prices should be agreed upon
Continuing on the example above, suppose now that the initial price of Andy's house is $100,000 and that Bob enters into a forward contract to buy the house one year from today. But since Andy knows that he can immediately sell for $100,000 and place the proceeds in the bank, he wants to be compensated for the delayed sale. Suppose that the risk free rate of return R (the bank rate) for one year is 4%. Then the money in the bank would grow to $104,000, risk free. So Andy would want at least $104,000 one year from now for the contract to be worthwhile for him – the opportunity cost will be covered.Spot–forward parity
For liquid assetsMarket liquidity
In business, economics or investment, market liquidity is an asset's ability to be sold without causing a significant movement in the price and with minimum loss of value...
("tradeables"), spot–forward parity provides the link between the spot market and the forward market. It describes the relationship between the spot and forward price of the underlying asset in a forward contract. While the overall effect can be described as the cost of carry, this effect can be broken down into different components, specifically whether the asset:
- pays income, and if so whether this is on a discrete or continuous basis
- incurs storage costs
- is regarded as
- an investment asset, i.e. an asset held primarily for investment purposes (e.g. gold, financial securities);
- or a consumption asset, i.e. an asset held primarily for consumption (e.g. oil, iron ore etc.)
Investment assets
For an asset that provides no income, the relationship between the current forward () and spot () prices iswhere
is the continuously compounded risk free rate of return, and T is the time to maturity. The intuition behind this result is that given you want to own the asset at time T, there should be no difference in a perfect capital market between buying the asset today and holding it and buying the forward contract and taking delivery. Thus, both approaches must cost the same in present value terms. For an arbitrage proof of why this is the case, see Rational pricing below.For an asset that pays known income, the relationship becomes:
- Discrete:
- Continuous:
where
is the present value of the discrete income at time , and is the continuous dividend yield over the life of the contract. The intuition is that when an asset pays income, there is a benefit to holding the asset rather than the forward because you get to receive this income. Hence the income ( or ) must be subtracted to reflect this benefit. An example of an asset which pays discrete income might be a stockStock
The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
, and example of an asset which pays a continuous yield might be a foreign currency or a stock index.
For investment assets which are commodities, such as gold
Gold
Gold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...
and silver
Silver
Silver is a metallic chemical element with the chemical symbol Ag and atomic number 47. A soft, white, lustrous transition metal, it has the highest electrical conductivity of any element and the highest thermal conductivity of any metal...
, storage costs must also be considered. Storage costs can be treated as 'negative income', and like income can be discrete or continuous. Hence with storage costs, the relationship becomes:
- Discrete:
- Continuous:
where
is the present value of the discrete storage cost at time , and is the storage cost where it is proportional to the price of the commodity, and is hence a 'negative yield'. The intuition here is that because storage costs make the final price higher, we have to add them to the spot price.Consumption assets
Consumption assets are typically raw material commodities which are used as a source of energy or in a production process, for example crude oil or iron ore. Users of these consumption commodities may feel that there is a benefit from physically holding the asset in inventory as opposed to holding a forward on the asset. These benefits include the ability to profit from temporary shortages and the ability to keep a production process running, and are referred to as the convenience yield. Thus, for consumption assets, the spot-forward relationship is:- Discrete storage costs:
- Continuous storage costs:
where
is the convenience yield over the life of the contract. Since the convenience yield provides a benefit to the holder of the asset but not the holder of the forward, it can be modelled as a type of 'dividend yield'. However, it is important to note that the convenience yield is a non cash item, but rather reflects the market's expectations concerning future availability of the commodity. If users have low inventories of the commodity, this implies a greater chance of shortage, which means a higher convenience yield. The opposite is true when high inventories exist.Cost of carry
The relationship between the spot and forward price of an asset reflects the net cost of holding (or carrying) that asset relative to holding the forward. Thus, all of the costs and benefits above can be summarised as the cost of carry,. Hence,- Discrete:
- Continuous:
Relationship between the forward price and the expected future spot price
The market's opinion about what the spot price of an asset will be in the future is the expected future spot price. Hence, a key question is whether or not the current forward price actually predicts the respective spot price in the future. There are a number of different hypotheses which try to explain the relationship between the current forward price, and the expected future spot price, .The economists John Maynard Keynes
John Maynard Keynes
John Maynard Keynes, Baron Keynes of Tilton, CB FBA , was a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments...
and John Hicks
John Hicks
Sir John Richard Hicks was a British economist and one of the most important and influential economists of the twentieth century. The most familiar of his many contributions in the field of economics were his statement of consumer demand theory in microeconomics, and the IS/LM model , which...
argued that in general, the natural hedgers of a commodity are those who wish to sell the commodity at a future point in time. Thus, hedgers will collectively hold a net short position in the forward market. The other side of these contracts are held by speculators, who must therefore hold a net long position. Hedgers are interested in reducing risk, and thus will accept losing money on their forward contracts. Speculators on the other hand, are interested in making a profit, and will hence only enter the contracts if they expect to make money. Thus, if speculators are holding a net long position, it must be the case that the expected future spot price is greater than the forward price.
In other words, the expected payoff to the speculator at maturity is:
, where is the delivery price at maturityThus, if the speculators expect to profit,
, as when they enter the contractThis market situation, where
, is referred to as normal backwardation. Since forward/futures prices converge with the spot price at maturity (see basis), normal backwardation implies that futures prices for a certain maturity are increasing over time. The opposite situation, where , is referred to as contangoContango
Contango is the market condition wherein the price of a forward or futures contract is trading above the expected spot price at contract maturity. The resulting futures or forward curve would typically be upward sloping , since contracts for further dates would typically trade at even higher prices...
. Likewise, contango implies that futures prices for a certain maturity are falling over time.
Rational pricing
If is the spot priceSpot price
The spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...
of an asset at time
, and is the continuously compounded rate, then the forward price at a future time must satisfy .To prove this, suppose not. Then we have two possible cases.
Case 1: Suppose that
. Then an investor can execute the following trades at time :- go to the bank and get a loan with amount
at the continuously compounded rate r; - with this money from the bank, buy one unit of stock for
; - enter into one short forward contract costing 0. A short forward contract means that the investor owes the counterpartyCounterpartyA counterparty is a legal and financial term. It means a party to a contract. A counterparty is usually the entity with whom one negotiates on a given agreement, and the term can refer to either party or both, depending on context....
the stock at time
.
The initial cost of the trades at the initial time sum to zero.
At time
the investor can reverse the trades that were executed at time . Specifically, and mirroring the trades 1., 2. and 3. the investor- ' repays the loan to the bank. The inflow to the investor is
; - ' settles the short forward contract by selling the stock for
. The cash inflow to the investor is now 
because the buyer receives 
from the investor.
The sum of the inflows in 1.' and 2.' equals
, which by hypothesis, is positive. This is an arbitrage profit. Consequently, and assuming that the non-arbitrage condition holds, we have a contradiction. This is called a cash and carry arbitrage because you "carry" the stock until maturity.Case 2: Suppose that
. Then an investor can do the reverse of what he has done above in case 1. But if you look at the convenience yieldConvenience yield
A convenience yield is an adjustment to the cost of carry in the non-arbitrage pricing formula for forward prices in markets with trading constraints....
page, you will see that if there are finite stocks/inventory, the reverse cash and carry arbitrage is not always possible. It would depend on the elasticity of demand for forward contracts and such like.
Extensions to the forward pricing formula
Suppose that is the time value of cash flows X at the contract expiration time . The forward priceForward price
The forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...
is then given by the formula:
The cash flows can be in the form of dividend
Dividend
Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...
s from the asset, or costs of maintaining the asset.
If these price relationships do not hold, there is an arbitrage
Arbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
opportunity for a riskless profit similar to that discussed above. One implication of this is that the presence of a forward market will force spot prices to reflect current expectations of future prices. As a result, the forward price for nonperishable commodities, securities or currency is no more a predictor of future price than the spot price is - the relationship between forward and spot prices is driven by interest rates. For perishable commodities, arbitrage does not have this
The above forward pricing formula can also be written as:
Where
is the time t value of all cash flows over the life of the contract.For more details about pricing, see forward price
Forward price
The forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...
.
Theories of why a forward contract exists
Allaz and Vila (1993) suggest that there is also a strategic reason (in an imperfect competitive environment) for the existence of forward trading, that is, forward trading can be used even in a world without uncertainty. This is due to firms having StackelbergStackelberg competition
The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially...
incentives to anticipate their production through forward contracts.
See also
- Forward to forward contractForward to forward contractA forward to forward contract is a swap transaction that involves the simultaneous sale and purchase of one currency for another, where both transactions are forward contracts. It allows the company to take advantage of the forward premium without locking on to the spot rate. The spot rate has to...
- Futures contractFutures contractIn finance, a futures contract is a standardized contract between two parties to exchange a specified asset of standardized quantity and quality for a price agreed today with delivery occurring at a specified future date, the delivery date. The contracts are traded on a futures exchange...
- Derivative (finance)Derivative (finance)A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...
- Forward marketForward marketThe forward market is the over-the-counter financial market in contracts for future delivery, so called forward contracts. Forward contracts are personalized between parties The forward market is the over-the-counter financial market in contracts for future delivery, so called forward contracts. ...
- Forward priceForward priceThe forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...
- HedgingHedge (finance)A hedge is an investment position intended to offset potential losses that may be incurred by a companion investment.A hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance, forward contracts, swaps, options, many types of...
- OptionOption (finance)In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...
- Swap (finance)Swap (finance)In finance, a swap is a derivative in which counterparties exchange certain benefits of one party's financial instrument for those of the other party's financial instrument. The benefits in question depend on the type of financial instruments involved...
- 988 transaction
- Non-deliverable forward
Other types of trade contracts:
- Spot priceSpot priceThe spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...
- Spot marketSpot marketThe spot market or cash market is a public financial market, in which financial instruments or commodities are traded for immediate delivery. It contrasts with a futures market in which delivery is due at a later date...
Further reading
- Allaz, B. and Vila, J.-L., Cournot competition, futures markets and efficiency, Journal of Economic Theory 59,297-308.