## Present Value of Perpetuity

A perpetuity is a type of annuity that provides payments indefinitely. The present value of a perpetuity is the value of these infinite cash flows today.

Because perpetuities continue indefinitely, we cannot calculate their present value using the ordinary annuity formula. Instead, we use a much simpler formula:

PV = Pmt / r

Where:

- PV is the present value of the perpetuity
- Pmt is the amount of each equal payment
- r is the discount rate per period

This formula assumes that the first payment isn’t received until one period from now (like an ordinary annuity). If the first payment is received immediately, you would add the payment amount to the calculated present value.

It’s important to note that perpetuities are largely theoretical constructs. In real life, very few, if any, financial instruments guarantee payments forever. Still, the concept is useful for understanding the time value of money and for estimating the value of some types of financial instruments.

## Example of the Present Value of Perpetuity

Let’s say you are being offered an investment that will pay you $5,000 per year indefinitely, and you want to know how much you should be willing to pay for this investment today. Suppose the appropriate discount rate for similar investments is 7%.

We can calculate the present value of this perpetuity using the formula:

PV = Pmt / r

Substituting the given values into the formula:

PV = $5,000 / 0.07

Therefore:

PV = $71,428.57

So, the present value of a perpetuity that pays $5,000 per year, with a discount rate of 7%, is approximately $71,428.57. This means you should be willing to pay around $71,428.57 today for the right to receive $5,000 per year indefinitely, assuming a 7% annual return on comparable investments.

Again, remember that this is a theoretical example. In practice, very few investments can guarantee payments indefinitely.