Future Value of an Ordinary Annuity

Builds onFuture Value of a Single Sum — if this page feels steep, start there.

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

$FV$: what the whole savings plan is worth at the end
$PMT$: the equal payment you deposit at the END of every period
$(1+r)^n - 1$: total compound growth over n periods, minus the starting 1 — i.e. just the accumulated interest
$\frac{(1+r)^n - 1}{r}$: the 'annuity factor': one number that adds up the growth of every single deposit
$n$: the number of payments made

Why it must be true

An ordinary annuity is just $n$ separate deposits, each made at the end of its period. The last deposit earns nothing; the first compounds for $n-1$ periods. The annuity factor $\frac{(1+r)^n - 1}{r}$ is simply the sum of all those individual growth factors — a geometric series collapsed into one fraction.

Sanity anchor: with $r \to 0$ the factor tends to $n$ (you just stack the deposits), and it grows past $n$ as the rate rises.

The derivation

Each payment of $PMT$ compounds from the end of its own period to the end of period $n$ :

FV = PMT\left[(1+r)^{n-1} + (1+r)^{n-2} + \dots + (1+r) + 1\right]

The bracket is a geometric series with ratio $(1+r)$ . Multiply it by $(1+r)$ , subtract the original, and everything cancels except two terms:

(1+r)S - S = (1+r)^n - 1 \quad\Rightarrow\quad S = \frac{(1+r)^n - 1}{r}

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

When to reach for it

Equal payments made at the END of every period, accumulating with interest — savings plans, sinking funds, retirement contributions.

Listen for

“deposits X at the end of each period”“saves … per month / per year”“equal contributions”“balance immediately after the final deposit”“sinking fund”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Bracket it: the answer is more than $n \times PMT$ (deposits alone) and less than $n \times PMT \times (1+r)^n$ . Usually the only choice inside that band is correct.
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Quick estimate: total deposits earn interest for about half the term on average, so $FV \approx n \cdot PMT \cdot (1 + r)^{n/2}$ .
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Sanity: at r = 0 the factor is exactly n. If your factor came out below n, a sign or exponent slipped.

Traps in applying it

✗Payment timing: start-of-period payments (rent, premiums) need the annuity-due version, one extra (1+r).
✗Payment frequency ≠ rate frequency — monthly deposits need a monthly rate and n in months.
✗Valuing at a date after the last payment — the factor lands exactly on the final payment date; further growth needs another lump-sum step.

Limits & criticisms

It requires perfectly level payments and a constant rate for the entire span. Real savings plans escalate contributions, skip months, and earn floating returns — so treat the answer as a planning baseline, not a projection. Taxes and fees, which compound just as relentlessly, are outside the formula.

Where it came from

Annuities are older than the formula — Rome sold annua (yearly stipends) two millennia ago, and Britain financed wars with them. The mathematics of summing a geometric series of payments was worked out by early actuaries and appears polished in Jacob Bernoulli's Ars Conjectandi (1713) era; 18th-century sinking funds — schemes to retire national debt by regular deposits — ran on exactly this factor.

Today it is the mathematics of every savings plan: 401(k) accumulation, education funds, and the sinking-fund provisions still written into bond indentures.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Accumulate a savings plan

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

The forward face: what a stream of equal deposits grows into once every payment compounds from its own date.

Drill this face →

Required periodic saving

PMT = \frac{FV \cdot r}{(1+r)^n - 1}

The planning face: given a goal, divide it by the annuity factor to find the deposit each period must contribute.

Drill this face →

On the BA II Plus

Worked example: A saver wants $52,860.65 after 14 periods of end-of-period deposits earning 2.5% per period. What deposit is required each period?

1.[2ND] [CLR TVM]always clear the worksheet first
2.14 [N]
3.2.5 [I/Y]the rate per period, as a percent
4.0 [PV]
5.52,860.65 [FV]
6.[CPT] [PMT]compute the unknown

→ $3,200.00

Where it leads

Master this and the following come almost for free:

Present Value of an Ordinary Annuity Future Value of an Annuity Due

Drill this formula Back to the library