We can discount cash flows either by using spot rates or forward rates, because a spot rate is simply a package of short-term forward rates. Assume that the cash flow of period T is $1; then, the present value of the cash flow using the spot rate for period T will be as follows:
PV of $1 in M periods =
We know that,
y_{T}^{ }= [(1 + y_{1}) (1 + _{1}f_{1}) (1 + _{1}f_{2}) (1 + _{1}f_{3}) ..... (1 + _{1}f_{T-1})]^{1/ T} - 1
Adding 1 on both sides of the equations,
(1 + y_{T}) ^{ }= [(1 + y_{1}) (1 + _{1}f_{1}) (1 + _{1}f_{2}) (1 + _{1}f_{3}) ..... (1 + _{1}f_{T - 1})]^{1/ T}
Raising both sides of the equations to the T-th power, we get:
(1 + y_{T})^{T} = [(1 + y_{1}) (1 + _{1}f_{1}) (1 + _{1}f_{2}) (1 + _{1}f_{3}) ..... (1 + _{1}f_{T - 1})]
The present value of X1 in M periods can be determined by substituting the value calculated in the above step into the present value formula.
PV of X1 M periods =
The present value of Rs.1 in T period is called the forward discount factor for period T.