Pricing and Valuation of Forward Contracts at Initiation Date
When a buyer and seller come together to create a forward contract, no money changes hands, since neither party initially has a liability to the other. At initiation, the contract has zero value:
$$\displaystyle V_{0}\left ( T \right )= 0$$
But there needs to be a way to determine the forward __price at initiation__. Assume that you own an asset and would like to sell the asset in the future through a forward contract. By entering into the contract, you are creating a risk-free position and are entitled to earn the risk-free return:
$$\displaystyle \frac{F_{0}\left ( T \right )}{S_{0}}= \left ( 1+r \right )^{T}$$
Rearranging this expression produces:
$$\displaystyle F_{0}\left ( T \right )= S_{0}\left ( 1+r \right )^{T}$$
So the forward price at initiation is simply the spot price compounded at the risk-free rate over the term of the contract.
Suppose you own an asset that you feel is priced too high, but since you do not want to sell it just yet, you decide to sell the asset by engaging in a forward contract. If the risk-free rate is currently 2% and the asset is trading at USD 25, what do you think the forward price should be on a six-month contract?
Incorrect.
The price of a forward contract at initiation cannot simply be the prevailing spot price for the underlying asset. This would ignore the fact that the asset's sale will take place at a point in the future, suggesting that the spot price needs to be adjusted to account for the time value of money.
That's right!
Using the formula above, you can see that the forward price at initiation is the spot price compounded at the risk-free return:
$$\displaystyle F_{0}\left ( T \right )= S_{0}\left ( 1+r \right )^{T}$$
$$\displaystyle F_{0}\left ( T \right )=25\left ( 1 + 0.02 \right )^{\frac{6}{12}} = 25.25$$
Incorrect.
The forward price at initiation is simply the asset's spot price compounded at the risk-free rate:
$$\displaystyle F_{0}\left ( T \right )= S_{0}\left ( 1+r \right )^{T}$$
But the forward contract will run for only half the year, so the value for _T_ must be one-half as well, otherwise the expression would yield:
$$\displaystyle F_{0}\left ( T \right ) = 25 \left ( 1 + 0.02 \right )$$
$$\displaystyle F_{0}\left ( T \right ) = 25.50$$
which in this case is not the correct price.
For assets that produce cash payments or other benefits and that incur costs, these variables need to be considered when pricing a forward contract. An asset's dividends, interest payments, storage costs, and convenience yield, or its benefits, represent the _net cost of carry_ that will also need to be compounded, since it is in present value form.
Adding these variables representing benefits or income ($$I$$) and any costs ($$C$$) to the earlier pricing formula produces:
$$\displaystyle F_{0}(T) = [S_{0} - PV_0(I) + PV_0(C)](1+r)^{T}$$
So the future value of the asset's benefits will reduce the forward price, and the costs will increase the price. This makes sense, because if you acquire an asset at a point in the future, you are forgoing the benefits but avoiding the costs.
Suppose that you would still like to sell your asset six months from now. It is trading at USD 25, and the risk-free rate remains unchanged at 2%. If the present value of the asset's benefits are expected to total USD 3 and its costs in present value will be USD 1, what do you think the forward price should be?
Exactly!
This time around, the forward price needs to factor in the net cost of carry, so the following expression is used:
$$\displaystyle F_{0}(T) = [S_{0} - PV_0(I) + PV_0(C)](1+r)^{T}$$
Plugging in the values, as you did earlier, and the benefits and costs produces:
$$\displaystyle F_{0}(T) = [25 - 3 + 1](1+0.02)^{0.5} \approx 23.23$$
The forward price for your asset is now lower than it was before because it accounts for the net cost of carry. By selling the asset six months from now, you would be entitled to the net benefits of continuing to hold it, which would not be the case if you simply sold the asset in the spot market today.
Incorrect.
Be sure to use the formula for calculating the forward price with the net cost of carry:
$$\displaystyle F_{0}(T) = [S_{0} - PV_0(I) + PV_0(C)](1+r)^{T}$$
Clearly, the forward price cannot be higher than it was without the net cost of carry, so the price cannot be USD 27.27.
Incorrect.
The forward price, after factoring in the net cost of carry, cannot be the same as the price without considering this variable. By continuing to hold the asset, as opposed to selling it today in the spot market, you would be entitled to retain its net benefits, so this needs to be included when determining the forward price at the initiation date.
To summarize:
[[summary]]
USD 25.00
USD 25.25
USD 25.50
USD 23.23
USD 25.25
USD 27.27
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