Future value of an annuity calculator

Enter the payment, the annual rate, how many years the payments run, how often they happen, and whether each one lands at the end or the start of the period. The result is the value on the last day, split into the money you paid in and the interest you did not.

Everything here assumes a level payment: the same amount every period, with interest added at the same frequency. That is what a standing order into a savings account, a payroll deduction into a pension, or a fixed premium actually looks like. If the amount changes from year to year, the formula below stops applying and each payment has to be grown on its own.

A payment of $PMT$ made at the end of every period, for $n$ periods, at a period rate of $i$, grows to:

$$FV = PMT \times \frac{(1 + i)^n - 1}{i}$$

The fraction is the annuity factor: what one unit of payment per period is worth at the end. Multiply it by the payment and the answer falls out, which is why the factor is worth reading on its own. At 0.5 percent a period for 240 periods it is 462.040895, so the ending value is 462.040895 times whatever the payment is.

The rate $i$ is the rate for ONE period, not the annual rate, and $n$ counts periods, not years. At 6 percent a year compounded monthly over 20 years, $i$ is 0.005 and $n$ is 240, never 0.06 and 20. The compound interest calculator handles the other half of the same arithmetic, which is one amount left alone to grow.

An ordinary annuity pays at the END of each period. An annuity due pays at the START. The arithmetic is identical except that every payment in an annuity due sits in the account one period longer, so the whole total is grown one more time:

$$FV_{due} = PMT \times \frac{(1 + i)^n - 1}{i} \times (1 + i)$$

One extra factor of $(1 + i)$, applied once to the total, not once per payment. The gap between the two is therefore exactly the period rate: at 0.5 percent a month the annuity due ends 0.5 percent higher, whether the payments run for two years or thirty. The money paid in is the same either way, so the whole difference lands in the interest.

Which one applies is a fact about the payment, not a preference. Payroll deductions, month-end standing orders and most savings plans pay at the end. Rent, insurance premiums and many pension contributions are taken at the start. The second worked example runs one stream of payments both ways.

$500 a month for 20 years pays in $120,000 and ends at $231,020.45, so 48 percent of the total is interest. Keep the same payment for 30 years and you pay in $180,000 and end at $502,257.52, where 64 percent is interest.

Half as much again paid in, more than twice as much out. The early payments are what does it: in the 30 year run the first payment earns interest for 359 periods and the last earns none at all, so the payments made in the first ten years carry most of the compounding.

That asymmetry is why a modest payment started early beats a larger one started late. It is also the reason to work the problem backwards when there is a number you have to hit: the savings goal calculator takes the target and the date and returns the payment that gets there.

The formula is not arbitrary. Each payment grows by one more compounding period than the next, so the payments form a geometric series and the annuity factor is that series summed: geometric series.