# Future Value of a Growing Annuity

A growing annuity is a payout plan in which the payments increase by a fixed percentage each year. The future value of a growing annuity (FVGA) is the effective monetary value of those funds when the payments stop. For instance, suppose it is January 1, 1999 and you will receive a payment on January 1 for the next four years. With each passing year the payment amount increase by 5%. Your payments are thus

Jan 1, 2000: $4000

Jan 1, 2001: $4200

Jan 1, 2002: $4410

Jan 1, 2003: $4630.50

If you could invest each payment into an account that earns 6% interest per year, then at the end of four years you would have

$4000(1.06^{3}) + $4200(1.06^{2}) + $4410(1.06) + 4630.50

= $18788.28.

This is the future value of the annuity. Computing the FVGA gives you the true value of an annuity so that you can accurately compare it to other investments and payment plans.

## FVGA 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 equationFV = P[(1+r)

^{N}- (1+g)

^{N}]/(r-g)

In case r = g, the formula is

FV = PN(1+r)

^{N-1}

## Example

Suppose you are presented with two options for a growing annuity. Each annuity begins on the same day and lasts for 10 payments, with an annual interest rate of 4%.In Option 1, the initial payment is $750 and the growth rate is 6%. In Option 2, the initial payment is $1000 and the growth rate is 1%. Which option has the larger future value on the day of the last payment?

The future value of Option 1 is

FV = 750[(1.04)

^{10}- (1.06)

^{10}]/(0.04-0.06)

= 7500[1.480244-1.790848]/(-0.02)

= 11647.65

The future value of Option 2 is

FV = 1000[(1.04)

^{10}- (1.01)

^{10}]/(0.04-0.01)

= 7500[1.480244-1.104622]/(0.03)

= 12520.73

This means that Option 1 is the better choice.

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