Let us take an example. Assume Leo owns a share of GE, whose spot price today (S0) is $100. Rachel enters into a forward contract with Leo today to buy a share of GE at $101 after 1 month.
The terms, parties and ways to settle the contract are described below:
The above example is shown in a generic way in the diagram below. It shows the transactions from a buyer’s perspective at times t = 0 when the contract is initiated and time t = T, when the contract expires.
Interpreting the notation and key features of a forward contract:
The forward price agreed at the initiation date of the contract is the spot rate compounded at the risk-free rate over the life of the contract.
Assuming a risk-free rate of 10%, spot price of , and time period of 1 month, we calculate the forward price as:
Why is the forward price equal to the spot rate compounded at the risk-free rate over the contract period?
In theory, if Rachel did not enter into the forward contract, then she could invest $100 for period T in a risk-free asset and earn 10% or $101 after 1 month. So, the forward contract must earn the risk-free rate as an excess return will result in an arbitrage opportunity.
Value of a forward contract at initiation
Value of a forward contract at initiation = 0
The value of a forward contract at initiation is zero because neither party pays any money to the other. There is no value to either party.
Value at expiration for the long party:
ST= spot price of the underlying
F0(T)= forward price agreed in the contract
The value of a forward contract is positive to the long party if ST > F0(T) and negative if ST<F0(T) at expiration.
Value at expiration for the short party:
The value of a forward contract is positive to the short party if F0(T) > ST and negative if F0(T) < ST at expiration.
If the spot price at time T is 105, then Rachel (long) gets the asset by paying 101 and can immediately sell it for 105 at the then market price. Value to the long = = 105 – 101 = 4.
Value of a forward contract during the life of the contract
This is the difference between the spot price at time t, and the present value of forward price for the remaining life of the contract. In our example, at t = 15 days, if St= 106, then Vt =
Since the price has gone up, the value to the long party is positive.
As market conditions change, only the value of a forward contract changes. Its price does not change as it is fixed at contract initiation.
Forward Price with Benefits and Costs
Now, let us consider an asset with benefits and costs. How is the forward price for such an asset calculated? The forward price of an asset with benefits and/or costs is the spot price compounded at the risk-free rate over the life of the contract minus the future value of those benefits and costs.
It can be rewritten as:
γ = present value of the benefit. It is subtracted from the spot price because if you own the asset, you receive any benefits associated with the asset, during the life of the contract.
θ = present value of costs incurred on the asset during the life of the contract. These costs make it more expensive to hold the asset and hence increase the forward price.
In the previous example, assume $10 is the present value of benefits from the asset and $20 is the present value of costs associated with holding the asset. The forward price at time 0 when Rachel enters into the contract is:
The value of a forward contract is the spot price of the underlying asset minus the present value of the forward price. The value of the contract at time t is given by the expression below:
(with benefits and costs)
In words, the value of the contract is the spot price minus the net benefit for the remaining period minus the present value of the forward price. Note that (γ – θ) is the net benefit.
The forward price can be lower than the spot price though it is not common.
When the future value of benefits is higher than the future value of costs and the compounded spot price, then the forward price is lower than the spot price. If a commodity is in short supply, then its non-monetary benefits/convenience yield may be higher.
So far, the forward contracts we have seen had an asset as the underlying. But, some forward contracts which have an interest rate as the underlying are called forward rate agreements. FRAs allow us to lock in a rate today for a loan in the future and act as a hedge against interest rate risk. They are forward contracts that allow participants to make a known interest payment at a later date and receive in return an unknown interest payment.
The payoffs on an FRA are determined by market interest rates at expiration:
Example of an FRA
The exhibit below shows going long a 30-day FRA in which the underlying is 90-day Libor.
Source: CFA Program Curriculum, Basics of Derivative Pricing and Valuation
Synthetic FRA: Instead of engaging in a real FRA, a synthetic FRA can be constructed by lending a 30-day Eurodollar time deposit and buying a 120-day Eurodollar time deposit.
A quick summary of the concepts discussed in earlier readings.