Annuity value and payout
An annuity is a stream of equal payments, and its value depends on when each one lands. This prices the stream today and at the end of the term, handles the one-period difference between an ordinary annuity and one paid in advance, and inverts the same formula to show what a lump sum pays out.
| Period | Payment | Balance |
|---|---|---|
| 6 | $1,500 | $9,104 |
| 12 | $1,500 | $18,461 |
| 18 | $1,500 | $28,078 |
| 24 | $1,500 | $37,963 |
| 30 | $1,500 | $48,123 |
| 36 | $1,500 | $58,565 |
| 42 | $1,500 | $69,298 |
| 48 | $1,500 | $80,329 |
| 54 | $1,500 | $91,668 |
| 60 | $1,500 | $103,321 |
| 66 | $1,500 | $115,299 |
| 72 | $1,500 | $127,610 |
| 78 | $1,500 | $140,264 |
| 84 | $1,500 | $153,269 |
| 90 | $1,500 | $166,636 |
| 96 | $1,500 | $180,375 |
| 102 | $1,500 | $194,497 |
| 108 | $1,500 | $209,011 |
| 114 | $1,500 | $223,929 |
| 120 | $1,500 | $239,261 |
| 126 | $1,500 | $255,021 |
| 132 | $1,500 | $271,218 |
| 138 | $1,500 | $287,867 |
| 144 | $1,500 | $304,978 |
| 150 | $1,500 | $322,565 |
| 156 | $1,500 | $340,642 |
| 162 | $1,500 | $359,221 |
| 168 | $1,500 | $378,317 |
| 174 | $1,500 | $397,945 |
| 180 | $1,500 | $418,118 |
| 186 | $1,500 | $438,853 |
| 192 | $1,500 | $460,164 |
| 198 | $1,500 | $482,068 |
| 204 | $1,500 | $504,582 |
| 210 | $1,500 | $527,722 |
| 216 | $1,500 | $551,505 |
| 222 | $1,500 | $575,950 |
| 228 | $1,500 | $601,075 |
| 234 | $1,500 | $626,899 |
| 240 | $1,500 | $653,441 |
The balance column is what the account holds after that payment. Total paid in is $360,000; the difference is $293,441 of compound interest.
PV = PMT · (1 − (1+i)^−n) / i · Multiply by (1+i) for an annuity due.
FV = PMT · ((1+i)^n − 1) / i · The mirror image of the present value factor.
PMT = PV · i / (1 − (1+i)^−n) · The same identity solved for the payment.
PV = PMT/(i − g) · (1 − ((1+g)/(1+i))^n) · Valid while g ≠ i; the equal case reduces to n/(1+i).
A clear path from inputs to results.
An annuity is a stream of equal payments, and what it is worth depends on when each one lands. This prices the stream today and at the end of the term, handles the one-period difference between an ordinary annuity and one paid in advance, and inverts the same identity to show what a lump sum pays out.
How to use
- Enter the payment you make or receive in each period.
- Set the annual rate, the term in years, and how often payments occur.
- Choose whether payments land at the end of the period or the start.
- Add a growth rate if the payment rises with inflation.
- Read the present value, the future value, and the payout a lump sum would buy.
The formula
Reference: Broverman, Mathematics of Investment and Credit, ch. 2; the same identity underlies the Reg Z amortisation convention.
Worked example
$1,500 a month for twenty years at 5.5%
- Payment $1,500 per month.
- Annual rate 5.5%, paid monthly for 20 years — 240 periods.
- Ordinary timing, no growth.
- 1
The periodic rate is 5.5% / 12 = 0.4583%.
- 2
The present value factor is (1 − 1.0045833⁻²⁴⁰) / 0.0045833 = 145.37.
- 3
$1,500 × 145.37 = $218,059 — the lump sum equivalent today.
- 4
The future value factor is (1.0045833²⁴⁰ − 1) / 0.0045833 = 435.63.
- 5
$1,500 × 435.63 = $653,441 at the end, of which $360,000 was paid in.
- 6
The remaining $293,441 is compound interest.
$218,059 today, $653,441 in twenty years, $293,441 of it interest.
Interpreting the result
Present value and future value are the same relationship read in two directions, and the gap between the future value and the total paid in is the whole argument for starting early. The timing choice is worth exactly (1 + i): about half a percent a month at these rates, which compounds into a meaningful difference over twenty years.
Assumptions & limitations
- The rate is nominal and compounds at the payment frequency, so a 5.5% annual rate paid monthly is an effective 5.64%.
- Payments are equal, or grow at a constant rate; irregular payments need a cash-flow model instead.
- An ordinary annuity pays at the end of each period, a due annuity at the start.
- There are no fees, taxes or mortality charges — an insurance annuity priced with these would pay less.
- The growth rate must differ from the interest rate; the equal case is handled as its limit.
Frequently asked questions
What is the difference between an ordinary annuity and an annuity due?
One period of interest. An annuity due is worth exactly (1 + i) times an ordinary annuity, because every payment arrives one period earlier and earns for one period longer. At 6% monthly that is 0.5% on the whole present value.
How does a growing annuity work?
The payment rises at a constant rate, so the discounting has to account for both the interest rate and the growth. The formula reduces to the level case when the two rates are equal, which is handled as a limit rather than by dividing by zero.
What payout does $500,000 buy?
At 5.5% over twenty years, $3,439 a month. Raise the rate and the payout rises; lengthen the term and it falls. The exhaustion check confirms the balance reaches zero exactly at the end of the term.
Why is the future value so much larger than the present value?
Because they are measured at different dates. The future value includes all the interest the payments earn over the term; the present value strips it out. Neither is more correct — they answer different questions.
Can this price an insurance annuity?
It prices the mathematical stream, not the product. A commercial annuity also prices mortality, expenses and a profit margin, so its quoted payout is lower — often by 10–20% for a lifetime income option.
Methodology last reviewed 2026-10-21. Results are informational and not financial or engineering advice — see our disclaimer.