In the forward commitment example from the prior lesson, where no costs or benefits were associated with the underlying asset, the following relationship between the spot and forward prices was established:
F0(T) = S0(1 + r)T.
This relationship is shown under continuous compounding
F0(T) = S0erT.
The risk-free rate, r, denotes the opportunity cost of holding (“carrying”) the asset, whether or not the long investor borrows to finance the asset.
For underlying assets with ownership benefits or income (I) or costs (C) expressed as a known amount in present value terms at t = 0—shown as PV0()—the relationship between spot and forward prices in discrete compounding terms can be shown as
F0(T) = [S0 – PV0(I) + PV0(C)](1 + r)T.
In other instances, the additional costs or benefits are expressed as a rate of return over the life of the contract. For income (i) and cost (c) expressed as rates of return, the relationship between spot and forward prices under continuous compounding is
F0(T) = S0e(r + c–i)T.
The relevant discount rate here involves the difference between the foreign and domestic risk-free rates, as shown in the following modified
F0,f/d(T) = S0,f/de(rf–rd)T.
