Forward contract

A forward contract is an agreement between two parties to buy or sell an asset (which can be of any kind) at a pre-agreed future point in time. Therefore, the trade date and delivery date are separated. It is used to control and hedge risk, for example currency exposure risk (e.g. forward contracts on USD or EUR) or commodity prices (e.g. forward contracts on oil). 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 Stackelberg incentives to anticipate its production through forward contracts.

One party agrees to buy, the other to sell, for a forward price agreed in advance. In a forward transaction, no actual cash changes hands. If the transaction is collaterised, exchange of margin will take place according to a pre-agreed rule or schedule. Otherwise no asset of any kind actually changes hands, until the maturity of the contract.

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, usually next business day). The difference between the spot and the forward price is the forward premium or forward discount.

A standardized forward contract that is traded on an exchange is called a futures contract.

Example of how the payoff of a forward contract works

Suppose that Bob wants to buy a house in one year's time. At the same time, suppose that Andy currently owns a house that he wishes to sell in one year's time. 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 recognise 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 loss of $6,000. To see why this is so, one needs only recognise that Andy could have sold to the open market $110,000 rather than Bob for $104,000. Unfortunately for Andy, he is legally obliged to sell to Bob at the lower price.

Mathematical Definition of Forward Contract Payoff

To generalize, if we enter into a long forward contract at time $t$ with maturity of time $T$ and with forward price given by $F_{t,T}$ , then the payoff of the contract at time will be given by $S_{T}-F_{t,T}$ . Here, the price of the underlying is given by $S_{T}$ . In contrast, the short forward contract will have a payoff given by $F_{t,T}-S_{T}$ .

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. But Bob knows that if he buys the home now, a loan of $100,000, he will be required to pay back $100,000 plus 4% interest. Thus, entering into a forward contract now means that he doesn't have to pay this interest. Bob will be willing to go at least as high $104,000 for a purchase in one year's time.

This example can be made more complicated by including in rent payments that the current owner accrues but is not seen by the holder of the forward contract.

Rational Pricing

If $S_{t}$ is the spot price of an asset at time $t$ , and $r$ is the continuously compounded rate, then the forward price must satisfy $F_{t,T}=S_{t}e^{r(T-t)}$ .

To prove this, suppose not. Then we have two possible cases.

Case 1: Suppose that $F_{t,T}>S_{t}e^{r(T-t)}$ . Then an investor can execute the following trades at time $t$ :

go to the bank and get a loan for $S_{t}$ at the continuously compounded rate r;
with this money from the bank, buy one unit of stock for $S_{t}$ ;
enter into one short forward contract costing 0.

The initial cost of the trades at the initial time sum to zero.

At time $T$ the investor can reverse the trades that was executed at time $t$ . Specifically,

' repay the loan to the bank. The investor owes the bank $S_{t}e^{r(T-t)}$ ;
' sell the stock for $S_{T}$ , which was an uncertain amount at time $t$ ;
' settle the short forward contract by paying $S_{T}-F_{t,T}$ , (that is there is an inflow of funds to the investor of $F_{t,T}-S_{T}$ ).

The sum of 1.', 2.' and 3.' equals $F_{t,T}-S_{t}e^{r(T-t)}$ , which by hypothesis, is positive. This is an arbitrage profit. Consequently, 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 $F_{t,T}<S_{t}e^{r(T-t)}$ . 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.

Extensions to the Forward Pricing Formula

Suppose that $PV_{t}(X)$ is the time value of cash flows X at time $t$ . The forward price is then given by the formula:

F_{t,T}=S_{t}e^{r(T-t)}-PV_{T}(\mathrm {All\ cash\ flows\ over\ the\ life\ of\ the\ contract} )

The cash flows can be in the form of dividends 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:

F_{t,T}=(S_{t}-I_{t})e^{r(T-t)}

Where $I_{t}$ is the time t value of all cash flows over the life of the contract.