Present Value of a Perpetuity Due

The present value of a perpetuity (an infinite stream of cash flows) is calculated by dividing the periodic cash flow by the discount rate:

present value of perpetuity = cash flow / discount rate

But this assumes that the first cash flow occurs at the end of this period. What if the first cash flow occurs today?

When the first payment in an infinite stream of payments occurs today, this is called a perpetuity due. To find the present value of a perpetuity, you simply add the first cash flow (the one that occurs today) to the present value of the regular perpetuity.

present value of perpetuity due = (cash flow / discount rate) + cash flow

Thus, if I promise to give you $100/year forever with the first payment starting today, and your discount rate is 5%, the present value of this infinite stream of cash flows would be $2,100.

present value of perpetuity due = ($100 / 0.05) + $100
present value of perpetuity due = $2,100

You could also calculate the present value of a perpetuity as follows:

present value of perpetuity due = (cash flow / discount rate) * (1 + r)

The above formula shows that the present value of a perpetuity due is simply the present value of a regular perpetuity times one plus the discount rate.

0:00 Introduction
0:10 Example of regular perpetuity
0:36 Critical assumption
0:46 Perpetuity due
1:11 Example of perpetuity due
2:00 Alternative formula
2:26 Relationship between perpetuity and perpetuity due
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