How to use the growing annuity calculator
- Enter the first payment, the growth per period and the discount rate per period. Use the same period for all three, such as years.
- Choose whether the payments last a fixed number of periods or go on forever, and enter the number of payments if they end.
- Choose whether the first payment comes at the end of period 1 or right away.
- Read the present value and the future value, then check the table, which discounts the first payments one at a time.
Formula
PV = C / (r - g) x (1 - ((1 + g) / (1 + r))^n)
If r = g: PV = n x C / (1 + r)
Forever (r > g): PV = C / (r - g)
Start of period payments: multiply PV by (1 + r)
C is the first payment, g the growth per period, r the discount rate and n the number of payments. The future value is PV x (1 + r)^n, measured at the end of the last period.
Twenty payments that grow 3% a year
The first payment is $1,000, each payment is 3% larger than the one before, the discount rate is 6% and there are 20 payments, the first at the end of year 1. The present value is $14,561.53 and the future value after year 20 is $46,700.81. The last payment is $1,753.51, and the payments add up to $26,870.37 before discounting.
The check line on the page adds the 20 payments, each divided by 1.06 to the number of years it waits, and lands on the same $14,561.53. If growth equals the discount rate at 5%, ten payments of $1,000 are worth 10 x $1,000 / 1.05 = $9,523.81. A stream that never ends, starting at $50 and growing 2% at a 7% rate, is worth $50 / 0.05 = $1,000.00.
What makes a growing annuity different
An ordinary annuity pays the same amount each period. A growing annuity pays an amount that rises or falls by a fixed percent, which fits things like a lease with yearly increases, a pension with cost of living steps or a dividend expected to grow. Using a flat annuity for these understates the value when payments rise.
The formula needs the discount rate to differ from the growth rate. When they are equal, every payment has the same present value because growth exactly offsets discounting, and the total is the number of payments times that one value.
Endless streams and their risk
A stream that never ends has a value only when the discount rate is above the growth rate. The value then depends strongly on the gap between the two. A gap of 1 point is worth five times as much as a gap of 5 points, so a small change in either assumption changes the value a lot. Treat results near that edge with care. For one-off lump sums, use the Present Value Calculator, and for solving any one unknown in a payment plan see the Time Value of Money Solver.
Assumptions and limits
- Growth and the discount rate are constant, and both apply once per period.
- Payments are made at a single point in each period: the end by default, or the start if you choose Right away.
- The discount rate you enter is an example you can change. Taxes, fees and inflation are not modeled.
- Growth can be negative to model payments that shrink, down to a decline of 99% per period.
Frequently asked questions
What does it mean when growth is higher than the discount rate?
For a fixed number of payments it is allowed, and the present value is still found. Each payment is worth more than the one before in today money, so the later payments weigh the most. An endless stream cannot be valued that way and the page refuses it.
Should I use a yearly or a monthly rate?
Use the same period for the growth, the discount rate and the number of payments. If payments are monthly, enter a monthly growth rate and a monthly discount rate, and count the payments in months.
What is the difference between ordinary and start of period payments?
With payments at the start of each period, every payment is received one period earlier, so its present value is higher by a factor of one plus the discount rate. The page applies that factor to the whole result.
Why does the page show a check line?
It discounts each payment on its own and adds them up, which does not use the shortcut formula. If the two numbers match to the cent, the formula result can be trusted for the numbers you entered.