# Present Value of a Growing Annuity

 Initial Payment P = \$ Number of Payments N = Annual Interest Rate (percent) r = % Annual Growth Rate (percent) g = %

Growing annuities are payment plans in which the payouts increase by a fixed percent each period. The present value of a growing annuity (PVGA) is the current monetary value of the annuity. For instance, suppose it is January 1, 1999 and you will receive a payment on January 1 for the next five years. Each passing year the payments increase by 2%. Your payments are thus:

Jan 1, 2000: \$1250
Jan 1, 2001: \$1275
Jan 1, 2002: \$1300.50
Jan 1, 2003: \$1326.51
Jan 1, 2004: \$1353.04

Imagine you could invest each payment into an account that earns 4% interest per year, then on Jan 1, 2004 you would have

\$1250(1.044) + \$1275(1.043) + \$1300.50(1.042) + 1326.51(1.04) + 1353.04
= \$7035.76.

But if you were given \$5782.88 right now, and you invested it in an account that earned 4% annually, then on Jan 1, 2004 you would also have

\$5782.88(1.045) = \$7035.76.

This means that \$5782.88 is the present value of the annuity. Rather than work the arithmetic out by hand with large sums, you can use a more compact formula, or use the calculator above.

## PVGA Formula

Let P be the amount of the initial payment, N be the number of payments, g be the growth rate (decimal), and r be the annual percentage rate (decimal). Then the future value is computed with the equation

PV = P[1 - (1+g)N(1+r)-N]/(r-g)

In case r = g, the formula is

PV = PN/(1+r)

## Example

You will be given annual payments for the next ten years stating a year from now. The first payment will be \$1000, and each passing year it will increase by 3.4%. Assuming you could invest the money in an account that bears 3.4% annual interest, what is the present value of this growing annuity?

Here we have P = 1000, N = 10, r = 0.034, and g = 0.034. Since r = g in this example, we must use the second formula. Therefore the present value of this growing annuity is

1000(10)/(1.034) = \$9671.18