Pricing and Valuation of Forward Contracts During the Life of the Contract
The value of a forward contract can easily be determined during the life of the contract. It is simply the spot price of the underlying asset less the present value of the forward price:
$$\displaystyle V_t(T) = S_t - F_{0}(T)(1 + r)^{-(T-t)}$$
where:
$$V_t(T) = $$ Value at time $$t$$
$$S_t = $$ Asset price at time $$t$$
$$F_{0}(T) = $$ Forward price set at initiation
$$r =$$ Risk free rate
$$T-t =$$ Remaining time to expiration
The time to expiration is expressed as a fraction of one year.
Suppose that you have identified an undervalued asset, but since you recently spent most of your savings downloading music to your mobile phone, you commit to buying the asset in the future, once you replenish your savings, using a forward contract.
The asset is currently trading for USD 40, and your forward contract will allow you to buy the asset for USD 40.15 three months from now. If the risk-free interest rate is 1.5%, what do you think the value of the forward contract to you should be one month from now if the asset's price increased to USD 43?
Incorrect.
The value of your forward contract after one month is not simply the difference between the asset's spot price at time _t_, USD 43, and the forward price, USD 40.15. Instead, it is calculated using the formula below:
$$\displaystyle V_{t}\left ( T \right )= S_{t}-F_{0}\left ( T \right )\left ( 1+r \right )^{-\left ( T-t \right )}$$
Correct!
Using the formula for calculating a forward contract's value after initiation, you get:
$$\displaystyle V_t(T) = S_t - F_{0}(T)(1 + r)^{-(T-t)}$$
$$\displaystyle V_t(T) = 43 - 40.15(1 + 0.015)^{-(2/12)} \approx 2.95$$
The value of the forward contract after one month is positive from your perspective, since the asset's price has increased, relative to the price at which you are committed to buying it at expiration.
Incorrect.
The value of the forward contract has indeed changed since it was initiated, in light of the increased asset price. Because you are committed to buying the asset at the forward price, USD 40.15, the value of the position from your perspective cannot be negative. The asset's price has moved in your favor, suggesting that it now has positive value for you.
If the underlying asset in a forward contract has a net cost of carry, which simply means that the holder is entitled to income or benefits ($$I$$) such as dividends, interest, and a convenience yield, after factoring in the storage cost ($$C$$), if any, then an adjustment will need to be made to determine the contract's value at time _t_:
$$\displaystyle V_t(T) = [S_t - PV_t(I) + PV_t(C)] - F_{0}(T)(1 + r)^{-(T-t)}$$
As the buyer in the forward contract, you are forgoing the benefits of owning the asset prior to expiration, but you are also avoiding the related costs. So the forward contract's value will be adjusted downward to account for the foregone benefits but adjusted upward for the avoided costs.
The benefits and costs are expressed in present value terms, as of the initiation date, so they will need to be compounded to time _t_, with _t_ being the fraction of a year that has passed.
Assume that your earlier commitment to buy the asset through a three-month forward contract remains unchanged. The forward price is USD 40.15, and the risk-free interest rate is still 1.5%. If the PV of benefits associated with owning the asset are USD 2.50 and the PV of costs are USD 0.75, what do you think the value of the forward contract to you should be one month from now if the asset's price were to increase to USD 43?
Incorrect.
Your forward contract still has value to you, even after factoring in the net cost of carry for the asset. The net cost of carry, although reducing the USD 2.95 value of the forward contract calculated earlier, does not completely offset this value, as determined using the formula below:
$$\displaystyle V_t(T) = [S_t - PV_t(I) + PV_t(C) - F_{0}(T)](1 + r)^{-(T-t)}$$
That's right!
Using the expression below, the value at time _t_ is:
$$\displaystyle V_t(T) = [S_t - PV_t(I) + PV_t(C)] - F_{0}(T)(1 + r)^{-(T-t)}$$
$$\displaystyle V_t(T) = [43 - 2.50(1+0.015)^{1/12} + 0.75(1+0.015)^{1/12}] - 40.15(1 + 0.015)^{-(2/12)}$$
$$\displaystyle V_t(T) \approx [43 - 2.5031 + 0.7509] - 40.0505 \approx 1.20$$
Since you do not yet own the asset, you do not reap the benefits of ownership, but you also do not incur the costs either. The net result is that your forward contract's value at time _t_ is reduced to reflect these variables, but the value is still positive for you.
Incorrect.
Your role as the buyer in this forward contract has not changed, so even after considering the net cost of carry, which reduces the contract value from the USD 2.95 figure calculated earlier, the value at time _t_ is still positive from your point of view.
To summarize:
[[summary]]
USD 2.85
USD 2.95
USD –2.95
USD 0
USD 1.20
USD –1.20
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