π Annuity Calculator
Grow regular payments into a lump sum, or see what monthly income a lump sum buys.
Future Value of Payments
Estimates for planning only, not financial advice.
How to Use This Calculator
Pick a direction. On the Grow Payments tab, enter what you will pay each period to see the future lump sum. On the Buy Income tab, enter a lump sum to see the payment it supports. Both tabs share the same annuity math, just solved for different unknowns.
Choose the tab that matches your question: growing savings into a lump sum, or converting a lump sum into income.
Enter the payment or lump sum, the payment frequency, the annual rate, and the number of years.
Read the headline result plus the breakdown: total paid in, interest earned or paid, and the effective rate per period.
Compare tabs: grow $500 a month for 20 years, then flip to Buy Income with the resulting lump sum to see the income it funds.
The Annuity Formulas
Both formulas describe the same stream of payments seen from opposite ends of time. The future value formula compounds each payment forward; the payment formula discounts the lump sum backward across the payout period. If you grow payments on one tab and feed the result into the other tab at the same rate and term, you get your payment back, which is a good way to check the math.
Worked Example
You save $500 at the end of every month for 20 years at 6 percent nominal, so r = 0.005 per month and n = 240. Future value = 500 x ((1.005^240 - 1) / 0.005) = $231,020.45. You put in $120,000 and interest added $111,020.45. Now flip it: a $250,000 lump sum at 5 percent nominal over 20 years (r = 0.0041667, n = 240) buys a monthly payment of 250000 x 0.0041667 / (1 - 1.0041667^-240) = $1,649.89. Total payouts would be $395,973.60.
Pure Math vs Real Products
These formulas assume a fixed rate and ignore every cost in the real world: annuity contracts add mortality and expense charges, surrender periods, and rider fees that can total 2 to 4 percent a year. Taxes matter too: annuity payouts are part return of principal and part taxable gain. Use the calculator for the benchmark number, then subtract the real costs before you decide.
Estimates for planning only, not financial advice.
Frequently Asked Questions
What is an annuity in math terms?
An annuity is a series of equal payments at regular intervals. The future value of an annuity tells you what regular deposits grow into; the present value tells you what lump sum a stream of future payments is worth today. Mortgages, car loans, and retirement payouts are all annuities wearing different clothes.
How much will my monthly deposits grow to?
That is the future value of an annuity: payment x (((1+r)^n - 1) / r), with r the rate per period and n the number of payments. At $500 a month and 6 percent nominal for 20 years, the answer is $231,020.45. Small changes in the rate move the result a lot over long periods, so test a conservative rate too.
How much monthly income does a lump sum buy?
Flip the formula: payment = lump sum x r / (1 - (1+r)^-n). A $250,000 lump sum at 5 percent over 20 years buys about $1,649.89 a month. This is the math behind immediate annuity quotes, before the insurer adds its fees and profit margin.
What is the difference between an ordinary annuity and an annuity due?
An ordinary annuity pays at the end of each period, like a typical loan payment; an annuity due pays at the beginning, like rent. Annuity-due values are higher by exactly one period of interest. This calculator uses the ordinary annuity convention, which matches most loan and savings illustrations.
Why do insurer annuity quotes pay less than this formula?
Because the formula is pure math with no costs, while an insurer prices in administrative fees, commissions, profit, and the risk that you outlive the average. A quoted payout 10 to 20 percent below the pure-math number is normal. Use this calculator as the benchmark, then judge quotes against it.
Does inflation matter for annuity planning?
A great deal. A fixed $1,650 monthly payment buys noticeably less in 20 years: at 3 percent inflation its purchasing power falls to about $913. If you need income that keeps up with prices, model a lower real return or look for inflation-adjusted payout options, and save more than the nominal math suggests.