## Perpetuity Definition:

A Perpetuity a type of annuity with payments that last forever.

## Perpetuity Immediate:

This is a perpetuity with payments at the end of the interest crediting period.

## Perpetuity Immediate PV Proof

Recall the interest discounting formula v where:
 v  = 1 1 + i

The payments continue forever
Write the present value of a payment of 1 each period with notation a∞|i
with interest rate (i) as follows
a∞|i = v + v2 + v3 + · · ·
 a∞|i  = v 1 - v

Since 1 - v = iv, we have
 a∞|i  = v iv

## Cancel the v on the top and bottom:

 a∞|i  = v iv

 a∞|i  = 1 i

## Plug in a payment of P each period:

 a∞|i  = P i

## Perpetuity Due:

This is a perpetuity with payments at the beginning of the interest crediting period.

## Perpetuity Due PV Formula Proof

Recall the interest discounting formula d = iv where:
The payments continue forever
Write the present value of a payment of 1 each period with notation ä∞|i
with interest rate (i) as follows
a∞|i = 1 + v + v2 + · · ·
 ä∞|i  = 1 1 - v

Since d = 1 - v, we have
 ä∞|i  = 1 d

Since d = i/(1 + i), we take the reciprocal of this since it's in the denominator and we have
 ä∞|i  = 1 + i i

## Plug in a payment of P each period:

 PV  = P(1 + i) i

## Perpetuity Calculator:

For more help with perpetuity calculations, check out our perpetuity calculator

##### How does the Perpetuity Calculator work?
Free Perpetuity Calculator - Walks you through the definition of a perpetuity, the present value of a perpetuity immediate, and the present value of a perpetuity due.

### What 2 formulas are used for the Perpetuity Calculator?

P = 1/i
v = 1/(1 + i)

For more math formulas, check out our Formula Dossier

### What 4 concepts are covered in the Perpetuity Calculator?

discount rate
the interest rate used in discounted cash flow (DCF) analysis to determine the present value of future cash flows. Key Takeaways.
interest rate
the proportion of a loan that is charged as interest to the borrower or proportion of principal credit given to a depositor
perpetuity
An infinite stream of payments
present value
the value in the present of a sum of money, in contrast to some future value it will have when it has been invested at compound interest.
PV = FV/(1 + i)n
where I is the interest rate per period, PV = Present Value, and FV = Future Value