How annuity future value works

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. 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.

On $500 a month that factor gives $231,020.45. You paid in $500 \times 240 =$ $120,000, so $111,020.45 of the total is interest, a shade under half.

The future value of an annuity calculator on this page is that identity, with a switch for payments at the start of the period. How compound interest works is the other half of the same arithmetic: one amount left alone to grow.

The annuity timing explorer is ordinary against due as two bars. Drag the years and watch the extra period of growth stay a fixed share of the ordinary total.

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:

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

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.

On this sheet, paying at the start reaches $232,175.55 against $231,020.45 at the end. Interest rises from $111,020.45 to $112,175.55. The gap is $232{,}175.55 - 231{,}020.45 = 1{,}155.10$, which is 0.5 percent of $231,020.45, one month of interest on the whole balance.

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. Ordinary against due annuity is that pair in a table.

$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: how savings goals work takes the target and the date and returns the payment that gets there.

This formula is the savings stream on its own. If you also hold an opening balance, grow that at the same rate with the compound interest calculator and add the two totals. The two parts do not interact, so adding them is exact.

This formula is the savings stream on its own. If the account already holds a lump, grow that lump at the same rate with the compound interest calculator and add the two totals. The two parts do not interact, so adding them is exact.

What is not exact is treating a stream as if it were a lump. Putting $120,000 in on day one and leaving it for 20 years at 6 percent monthly is a different object from paying $500 a month. The lump earns interest on the whole sum from month one. The stream earns interest only on what has arrived. That is why the ordinary annuity on this sheet ends at $231,020.45 rather than at the much larger figure a day-one lump of $120,000 would reach.

The inverse question, a target and a date with a payment still to find, is how savings goals work. That page rearranges this factor. The two pages are the same identity run forwards and backwards.

Mixing ordinary with due moves the answer by exactly one period of growth. At 0.5 percent a month that is always 0.5 percent of the ordinary total, $232{,}175.55 - 231{,}020.45 = 1{,}155.10$ on the 20 year sheet, whether you noticed the payment date or not. Mixing an annual rate with a monthly period count is worse. Compound 6 percent for 240 periods instead of 0.5 percent and you are modelling a 6 percent monthly return, which is not 6 percent a year.

One check catches nearly all of it: the annuity factor can never be smaller than the number of payments, because every payment is worth at least what was put in. A factor of 462 on 240 payments is plausible. A factor below 240 is a sign that $i$ or $n$ is in the wrong units.

Ordinary against due annuity is the timing in a table. Annuities explained is the insurance product that swaps a lump for an income, which is a different object: that payout hands capital back with the interest. Do not read a 6 percent savings factor as a 6.3 percent life-annuity payout.

It is not an insurance annuity, not a payout rate, and not a mortality credit. Annuities explained is the product that swaps a lump for an income. This page is a standing order into an account that compounds. Mixing the two is how a 6 percent savings factor gets read as a 6.3 percent life-annuity payout, which is a different object: that payout hands capital back with the interest.

The two mistakes that move the savings answer most are mixing ordinary with due, and mixing an annual rate with a monthly one. Compound 6 percent for 240 periods instead of 0.5 percent and you are modelling a 6 percent monthly return. One check catches nearly all of it: the annuity factor can never be smaller than the number of payments, because every payment is worth at least what was put in.

The three sheets are $500 a month for 20 years at 6 percent ordinary ($231,020.45, $111,020.45 of interest), the same stream as a due annuity ($232,175.55), and the ordinary stream for 30 years ($502,257.52, $322,257.52 of interest). This is educational material, not financial advice.