# Present Value of a Growing Annuity

PV of a Growing Annuity Calculator (Click Here or Scroll Down)

The present value of a growing annuity formula calculates the present day value of a series of future periodic payments
that grow at a proportionate rate. A growing annuity may sometimes be referred to as an increasing annuity. A simple example
of a growing annuity would be an individual who receives $100 the first year and successive payments increase by 10% per year
for a total of three years. This would be a receipt of $100, $110, and $121, respectively.

The present value of a growing annuity formula relies on the concept of time value of money. The premise to this concept is
that a specific quantity of money is worth more today than at a future time.

Like all financial formulas that involve a rate, it is important to correlate the rate per period to the number of periods
in the present value of a growing annuity formula. If the payments are monthly, then the rate would need to be the monthly
rate.

### How is the Present Value of a Growing Annuity Derived?

The present value of a growing annuity is the sum of future cash flows. For a growing annuity, each cash flow increases at
a certain rate. The formula for the present value of a growing annuity can be written as

This formula is the general formula for summing the discounted future cash flows along with using *1 + g*
to factor in that each future cash flow will increase at a specific rate.

This present value of a growing annuity formula can then be rewritten as

This would be considered a geometric series where **(1+g)/(1+r)** is the common ratio. By using the
geometric series formula, the present value of a growing annuity will be shown as

This formula can be simplified by multiplying it by *(1+r)/(1+r)*, which is to multiply it by 1.
This cancels out many of these throughout the formula, which leaves

In the denominator, *(1+r) - (1+g)* will return *r-g*. At this point,
*P* and *r-g* can be factored out, which will lead to the present value of a
growing annuity formula shown at the top of the page.

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