Forward price
The Forward price is the agreed upon price of an asset in a Forward contract. The forward price is calculated by assuming that the long position / short position ) will not have have an arbitrage opportunity via this transaction. There are equivalent ways of expressing this (simply put, neither the short position nor the long position should be able to make any money off the contract alone).
Notation
F is the forward price to be paid at time T C is any rent/dividend/coupon that the asset procures (with 100% certainty) during (0, T) exp(x) is the exponent e^x S is the spot price of the asset (i.e. what it would sell for at time 0)
The Forward price is given by
F = (S - C * exp(-T)) exp(T)
Proof of the Forward price formula
The main dilemma here is what price should the short position (the seller of the asset) offer to maximize his gain; what price should the long position (the buyer of the asset) accept to maximize his gain?
At the very least we know that both do not want to lose any money in the deal.
The short position knows as much as the long position knows: the short/long positions are both aware of any schemes that they could partake on to gain a profit given some forward price.
So of course they will have to settle on a fair price or else the transaction cannot occur.
An economic articulation would be:
(fair price + future value of asset's dividends) - spot price of asset = cost of capital
The future value of thet asset's dividens (this could also be coupons from bonds, monthly rent from a house, fruit from a crop, etc.) is calculated using the risk-free force of interest. This is because we are in a risk-free situation (the whole point of the forward contract is to get rid of risk or to at least reduce it) so why would the owner of the asset take any chances? He would reinvest at the risk-free rate (i.e. U.S. T-bills which are considered risk-free). The spot price of the asset is simply the market value at the instant in time when the forward contract is entered into. So OUT - IN = NET GAIN and his net gain can only come from the opportunity cost of keeping the asset for that time period (he could have sold it and invested the money at the risk-free rate).
let:
K = fair price
C = cost of capital
S = spot price of asset
F = future value of asset's dividens
I = present value of F (discounted using r)
r = risk-free interest rate compounded continuously
T = length of time from when the contract was entered into
Solving for fair price and substituting mathematics we get: K = C + S - F
where:
C = S(e^rT - 1)
(since [ e^rT = 1 + j ] where j is the effective rate of interest per time period of T)
F = c_1 * e^(r(T - t_1)) + ... + c_n * e^(r(T - t_n))
(where c_i is the i'th dividend paid at time t_i)
doing some reduction we end up with:
K = (S - I)e^(rT)