# Forward price, price of a forward contract, value of a forward contract

I am confused about the following terms:

1. forward price
2. price of a forward contract
3. value of a forward contract

The Hull textbook says that the forward price `F0` for an investment that pays no income (such as a non-dividend paying stock) is given by:

`F0 = S0 * e ^ (rT)`

where `S0` is the current price of the stock, `r` is the risk-free rate and `T` is time till maturity.

Later in the text, it says that the value of a forward contract `f` is given by:

`f = (F0 - K) * e ^ (-rT)`

where `K` is the delivery price.

When Person A and Person B create a forward contract, I believe there is no exchange of money. The price/value of a forward contract is 0. When a Person C wants to get the contract from Person A, does he pay the value of the contract, which I guess is the same as the price of the forward contract?

Is what I am saying correct? Can anyone help me distinguish the terms?

No you are wrong.

The definition of a Forward contract is "an agreement to buy/sell an underlying at a later time, at a fixed price agreed today".

You missed the part where if the buyer of the forward contract holds the contract until maturity, over the course of the contract (including the last day), he would need to pay exactly `F0` (the price initially), regardless of what the spot price on the delivery date is. The seller would have to deliver the underlying to the buyer.

So "no exchange of money" initially doesn't mean that they did not agree to a "price" that could be paid in full on maturity.

Your textbook is entirely correct. Price of a forward contract is the stock price agreed between two parties initially, and value of a forward contract is \$0 initially, but fluctuates as the new forward price in the market changes.

A forward contract is similar to a Future. Comparing it to a stock, is less than ideal since there are significant differences between the two... although you could very well write a forward on a stock price, this would be a specific instance of a forward and not the general rule.

The forward is like a future. It is a contract between 2 parties where one agrees to buy a given amount of an underlying asset at some point in the future, at a specific price that is set today. But unlike the future, it is "custom made" and the amount of underlying asset represented by the contract, the expiration date, the margin requirement, the inclusion of a third party (clearing firm) that makes sure each of the other 2 keep to their word and the price are negotiable. In a future contract, the only part that is negotiable is the price. Everything else is kept standard. Because of this, it is very easy to get in/out of a future contract, you buy it and sell it in the market where it is traded. The forward in the other hand is very iliquid and it is very difficult to get out of such a contract before expiration.

e^(rt) is the formula for continuously compounded interest rate. It measures the value of money in time. The positive exponent brings the value into the future, while the negative exponent brings the value into the present (or past, depending on where you're standing)

The other part of the formula is the difference in price for the contract between the day when it is written and expiration date (on the second formula) or the spot price (in the first formula).

This is basically following the form: FutureValue = SpotPrice + CostOfCarry where CostOfCarry is simplified to the interest rate (this is true for a financial asset, not so much for physical commodities).

The FwdPrice is the price of the Fwd in the present taken into a future value adding the cost of carry.

The Fwd Value is the price of that Fwd in the future once the expiration price (K) is known, brought to present value.

At the time when the contract is written there is an exchange of money (margin), depending on how the contract is negotiated this can be the full price of the contract or a fraction of it.

Two very good texts, for your reference, are by: (I) Robert L. McDonald, titled "Derivatives Markets" and (II) Matthew J. Hassett / Michael I. Ratliff / Toni Coombs Garcia / Amy C. Steeby, named "ACTEX SOA FM Study Manual". Like you, I was confused about the three terms that you are asking about, as they pertain to forward prices.

Reference (II) above is a study manual I am using to prepare for a Financial Mathematics exam for the United States Society of Actuaries (SOA) -- located in Schaumburg, Illinois.

To sort out the differences in the  forward price;  price of a forward contract and  value of a forward contract, I base my explanations from my first reference above (by McDonald):

1. The forward price would be a premium, if any, that would exist at the time the forward contract is created -->> that being time t=0 (right now). As the others above said, forward price (or premium) is zero (0.00). In case you are curious as to why there is no premium paid up front with the forward contracts, both parties (the buyer and seller) are legally obligated to exchange cash or the asset at the time of expiration. Since neither party has the option, neither the buyer, nor the seller, is at the mercy of the other party.

2. Your wording of "price of a forward" implies the contractual price of any forward contract. Depending on whether the market price (a.k.a. index or spot price) at the time of expiration (which is also stated in the forward contract-"how long until the contract is 'over'?") is higher or lower than the agreed-upon price in the contract, the buyer owes the seller, or vice versa, the difference in the spot price and forward price (specified in the contract at the time the contract was created). Think of "price of a forward" as the price that the buyer and seller of the contract "shook hands on" at contract inception (again, that being at time t=0), and, was written in the contract. (It is not the premium that I am describing in "1." above.)

3. The "value of a forward contract" refers to how much of a difference there is between the spot price at expiration and the contractual forward price. If {A} the spot price is greater than the contractual forward price, the buyer profits and gets the difference as cash from the seller and if: {B} the spot price at expiration is less than the forward price, the seller has gained, so the buyer owes the seller the difference in the two prices.

Hope this helps. :)