How to Calculate the Present Value of a Growing Annuity Using the Future Value

by Bryan Keythman, studioD

A growing annuity is a series of increasing, periodic cash flows that grow at a fixed percentage. For example, you might deposit money into an account each year for five years. If each deposit is 5 percent greater than the last, the series of deposits is a growing annuity. The future value of the growing annuity is the accumulated value of the payments at the end of the last payment period. The present value is the value of the payments in today’s dollars. If you know the future value, you can use two growing annuity formulas to calculate the present value.

Step 1

Substitute the future value of the growing annuity into the formula FV/[((1 + R)^N - (1 + G)^N)/(R - G)], in which FV represents future value, R represents the interest rate per period, G represents the payment growth rate per period and N represents the number of payment periods. For example, assume a five-year growing annuity has a 6 percent annual interest rate, a future value of $250,000 and payments that grow 2 percent per year. The formula is $250,000/[((1 + 0.06)^5 - (1 + 0.02)^5)/(0.06 - 0.02)].

Step 2

Calculate the numbers in each set of parentheses. In this example, add 1 to 0.06 to get 1.06. Add 1 to 0.02 to get 1.02. Subtract 0.02 from 0.06 to get 0.04. This leaves $250,000/[(1.06^5 - 1.02^5)/(0.04)].

Step 3

Calculate the numbers in brackets. In this example, raise 1.06 to the fifth power to get 1.3382. Raise 1.02 to the fifth power to get 1.1041. Subtract 1.1041 from 1.3382 to get 0.2341. Divide 0.2341 by 0.04 to get 5.8525. This leaves $250,000/5.8525.

Step 4

Divide the remaining numbers to calculate the growing annuity’s first payment. In this example, divide $250,000 by 5.8525 to get an initial payment of $42,717.

Step 5

Plug the initial payment and other variables into the formula [P/(R - G)] x [1 - ((1 + G)/(1 + R))^N], in which P represents the initial payment. The other variables are the same as the previous formula. Continuing the example, the formula is [$42,717/(0.06 - 0.02)] x [1 - ((1 + 0.02)/(1 + 0.06))^5].

Step 6

Solve the numbers in the first set of brackets. In this example, subtract 0.02 from 0.06 to get 0.04. Divide $42,717 by 0.04 to get $1,067,925. This leaves $1,067,925 x [1 - ((1 + 0.02)/(1 + 0.06))^5].

Step 7

Calculate the numbers in the remaining brackets. In this example, add 1 to 0.02 to get 1.02. Add 1 to 0.06 to get 1.06. Divide 1.02 by 1.06 to get 0.9623. Raise 0.9623 to the fifth power to get 0.8252. Subtract 0.8252 from 1 to get 0.1748. This leaves $1,067,925 x 0.1748.

Step 8

Multiply the remaining numbers to calculate the present value of the growing annuity. Concluding the example, multiply $1,067,925 by 0.1748 to get a present value of $186,673. This means the value of the growing payment stream is worth $186,673 today.

About the Author

Bryan Keythman has performed stock investment research and writing for a consulting firm since 2008. He also has prior experience sourcing and underwriting commercial real-estate investment and development opportunities for a commercial real-estate developer. Keythman holds a Bachelor of Science in finance.

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