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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,
yT = [(1 + y1) (1 + 1f1) (1 + 1f2) (1 + 1f3) ..... (1 + 1fT-1)]1/ T - 1
Adding 1 on both sides of the equations,
(1 + yT) = [(1 + y1) (1 + 1f1) (1 + 1f2) (1 + 1f3) ..... (1 + 1fT - 1)]1/ T
Raising both sides of the equations to the T-th power, we get:
(1 + yT)T = [(1 + y1) (1 + 1f1) (1 + 1f2) (1 + 1f3) ..... (1 + 1fT - 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.
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