Future Value of an Annuity Due

Builds onFuture Value of an Ordinary Annuity — if this page feels steep, start there.

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

$FV_{due}$: the subscript 'due' marks payments made at the START of each period
$PMT$: the equal payment, deposited at the beginning of every period
$\frac{(1+r)^n - 1}{r}$: the ordinary annuity factor — the value if payments came at period-END
$\times(1+r)$: one extra growth factor, because every payment arrives one period earlier and earns one more period of interest

Why it must be true

An annuity due pays at the start of each period — rent, leases, insurance premiums. Every payment therefore compounds for exactly one more period than in an ordinary annuity. That is the entire difference: take the ordinary factor and multiply once by $(1+r)$ .

Under exam pressure, don't memorize two annuity tables. Remember one principle: due = ordinary × (1+r), because each cash flow shifts one period earlier.

The derivation

Shift every ordinary-annuity payment one period earlier. Each cash flow now compounds for one extra period, which multiplies each term — and hence the whole sum — by $(1+r)$ :

FV_{due} = \underbrace{PMT \times \frac{(1+r)^n - 1}{r}}_{\text{ordinary annuity}} \times (1+r)

When to reach for it

Equal payments at the START of each period — rent collected in advance, insurance premiums, contributions on the 1st of the month.

Listen for

“at the beginning of each period”“paid in advance”“first payment made today”“rent / lease / premium schedules”

Back-of-the-envelope

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

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Compute the ordinary annuity, then add r% on top — 'due' is exactly one extra period of growth on the whole sum.
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In multiple choice, the due answer and the ordinary answer are usually BOTH options: pick the one exactly (1 + r) times the other, on the correct side.

Traps in applying it

✗Applying the (1+r) timing shift twice — each payment moves exactly one period earlier, so the whole sum scales once.
✗On the calculator: forgetting to switch BGN mode on — or worse, forgetting to switch it back off for the next problem.
✗Assuming 'due' from the product name instead of reading when payments actually occur.

Limits & criticisms

The ordinary-vs-due distinction is purely contractual timing — the formula can't tell you which applies; the instrument's payment schedule does. Everything else inherits the standard annuity caveats: level payments, constant rate, no taxes or fees.

Where it came from

The "due" distinction comes straight from commercial practice: rent, leases and insurance premiums have always been collected in advance, so actuaries needed a version of the annuity mathematics shifted one period earlier. The terminology (annuity-due) is inherited from the life-insurance offices that standardized premium schedules in the 18th and 19th centuries.

In practice the adjustment matters wherever payment timing is negotiable — lease accounting, premium pricing, and any savings plan that deposits on the first of the month rather than the last.

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.

Start-of-period accumulation

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

The ordinary annuity shifted one period earlier — every deposit compounds once more, so the whole sum scales by (1+r).

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Required start-of-period saving

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

Because start-of-period money works one period harder, the required deposit is smaller than the ordinary-annuity answer by exactly (1+r).

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On the BA II Plus

Worked example: A saver wants $54,182.17 after 14 periods, making deposits at the start of each period at 2.5% per period. What deposit is needed?

1.[2ND] [CLR TVM]always clear the worksheet first
2.[2ND] [BGN] [2ND] [SET] [2ND] [QUIT]payments at the beginning of each period (screen shows BGN)
3.14 [N]
4.2.5 [I/Y]the rate per period, as a percent
5.0 [PV]
6.54,182.17 [FV]
7.[CPT] [PMT]compute the unknown
8.[2ND] [BGN] [2ND] [SET] [2ND] [QUIT]set timing back to END when you're done

→ $3,200.00

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