# Present Value of a Growing Annuity

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.

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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Category: Annuity

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