Perpetuity refers to an infinite amount of time (). In finance, perpetuity is a constant stream of identical cash flows, (), with no end.
The present value () of a security with perpetual cash flows can be determined as:
with being the discount rate or cost of capital. Present value just states:
How much money would you need to deposit into an interest earning account (with rate ) or investment today, in order to get amount of money in years.
In other words, present value is the result of interest being deducted or discounted from a future amount (compounding in reverse).
So back to our original formula. Why can we rewrite it as follows?
If we look at the original formula we can see that it is a geometric series:
with and .
Since for :
we can easily see that:
Proof that for
Multiplying with we get:
Solving this for we get:
Using this we can :
Above we used simply because our formula is for .
For , which is our case because we get:
Similarly we can derive the Present Value of Growing Perpetuity where periodic payments grow at a proportionate rate :
which can be rewritten as:
It is simply a geometric series:
with and .