Perpetuity

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:

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