Calculate an annuity's future value from regular payments, the payout a lump sum buys over N years, or how long it lasts at a set withdrawal.
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240 payments of $500.00 grow to $205,516.83 at 5% a year: $120,000.00 paid in and $85,516.83 of interest.
| Year | Start balance | Paid in | Interest | End balance |
|---|---|---|---|---|
| 1 | $0.00 | $6,000.00 | $139.43 | $6,139.43 |
| 2 | $6,139.43 | $6,000.00 | $453.53 | $12,592.96 |
| 3 | $12,592.96 | $6,000.00 | $783.71 | $19,376.67 |
| 4 | $19,376.67 | $6,000.00 | $1,130.77 | $26,507.44 |
| 5 | $26,507.44 | $6,000.00 | $1,495.60 | $34,003.04 |
| 6 | $34,003.04 | $6,000.00 | $1,879.09 | $41,882.13 |
| 7 | $41,882.13 | $6,000.00 | $2,282.20 | $50,164.33 |
| 8 | $50,164.33 | $6,000.00 | $2,705.93 | $58,870.26 |
| 9 | $58,870.26 | $6,000.00 | $3,151.34 | $68,021.60 |
| 10 | $68,021.60 | $6,000.00 | $3,619.54 | $77,641.14 |
An annuity is a series of equal payments at a fixed interest rate. Saving into one, the future value is the payment times ((1 + i)^n − 1) ÷ i, where i is the rate per period and n the number of payments; paying out of one, the payment a lump sum supports is the lump sum times i ÷ (1 − (1 + i)^−n). Payments at the start of each period (an annuity due) earn one extra period of interest. A third mode solves for n, how long a lump sum lasts.
With the defaults, 500 at the end of each month for 20 years at 5% grows to 205,516.83 from 120,000 of payments. Paid out instead, 500,000 at 5% supports 3,299.78 a month for 20 years, and 3,000 a month lasts 23.76 years: 285 full withdrawals and a final 427.00.
The rate is a nominal yearly rate divided by the number of payments a year. Insurance annuity fees, surrender charges, taxes and inflation are not included.
Saving into an annuity, the future value is payment × ((1 + i)^n − 1) ÷ i, where i is the rate per period and n the number of payments. 500 at the end of every month for 20 years at 5% (i = 0.05 ÷ 12) grows to 205,516.83, from 120,000 of payments.
Paying out of one, a lump sum supports lump sum × i ÷ (1 − (1 + i)^−n) per period. 500,000 at 5% paid out monthly over 20 years gives 3,299.78 a month.
An ordinary annuity pays at the end of each period, like loan repayments. An annuity due pays at the start, like rent, so every payment earns one more period of interest and the future value is multiplied by (1 + i).
These rows save 1,000 a year for 10 years at 5%, with nothing to start:
| Scenario | Future value |
|---|---|
| Annual savings | 12577.89 |
| Annual due | 13206.79 |
12,577.89 × 1.05 = 13,206.79. Spreadsheets mark the difference with the type argument, 0 for end and 1 for start (Microsoft's PV documentation).
Payouts work the other way. A payment at the start of the period leaves less money earning interest, so the same 500,000 supports 3,286.09 a month instead of 3,299.78.
The 500,000 payout, worked by hand:
The 240 payments total 791,946.89, so 291,946.89 of what you receive is interest. As a check, Microsoft's PMT example, 10,000 over 10 months at 8%, gives 1,037.03 a month.
Solve for the number of withdrawals: n = −ln(1 − lump sum × i ÷ withdrawal) ÷ ln(1 + i). Taking 3,000 at the end of each month from 500,000 at 5% gives n = 285.14, or 23.76 years: 285 full withdrawals and a final one of 427.00.
100,000 at 6% with 1,000 taken monthly lasts 138.98 months, about 11.58 years, ending with a payment of 975.78.
If each withdrawal is no larger than one period's interest, the balance never falls. On 500,000 at 5% that limit is 2,083.33 a month at the end of each month, and about 2,074.69 when withdrawals come at the start.
That holds only if the rate is earned every period. Variable or negative returns and fees can still exhaust the balance, and inflation shrinks what each fixed withdrawal buys.
The rate is a nominal yearly rate divided by the number of payments a year: 52 weekly, 26 every two weeks, 12 monthly, 4 quarterly, 2 half-yearly or 1. At 5% paid monthly, 0.41667% a month compounds to an effective 5.1162% a year.
If your rate is an effective annual yield instead, convert it to a nominal rate first: for monthly payments, 12 × ((1 + yield)^(1/12) − 1), so a 5% yield is a 4.8889% nominal rate.
The term also has to come out to a whole number of payments, so 2.5 years works with monthly payments but not yearly ones.
A quote for an insurance annuity also reflects lifetime guarantees, mortality pooling, fees, surrender charges and taxes. None of those are in these formulas, which value a fixed series of payments at one constant rate.
For a single lump sum, solving for its future value, rate or time, use the time value of money calculator. The retirement calculator plans a full drawdown, and the RMD calculator gives the minimum withdrawals IRS rules require from traditional accounts.
Checked against: Future value annuity factor table, 5% and 10 periods: 12.5779; exact decimal arithmetic 12,577.8925355…
Checked against: 12,577.89 × 1.05 (annuity-due factor); exact decimal arithmetic 13,206.7871623…
Checked against: Microsoft FV documentation: =FV(0.06/12, 10, -200, -500, 1) returns $2,581.40
Checked against: Microsoft PV documentation: =PV(0.08/12, 12*20, 500, , 0) returns ($59,777.15)
Multiply the payment by ((1 + i)^n − 1) ÷ i, where i is the rate per period and n the number of payments. 1,000 at the end of each year for 10 years at 5% grows to 1,000 × 12.5779 = 12,577.89; paid at the start of each year (an annuity due) it grows to 13,206.79, 5% more, because every payment earns one more year.
Divide the lump sum by the present-value factor (1 − (1 + i)^−n) ÷ i. 500,000 at 5% a year (0.4167% a month) paid over 20 years gives 3,299.78 a month, 791,946.89 in total. Microsoft's PV documentation gives the reverse: 500 a month for 20 years at 8% costs 59,777.15 today.
Solve n = −ln(1 − P·i ÷ W) ÷ ln(1 + i), with P the lump sum, i the monthly rate and W the withdrawal. 100,000 at 6% with 1,000 taken at the end of each month lasts 138.98 months: 138 full payments and a final 975.78, about 11.6 years. If W is no more than P·i (500 here), the balance never runs out.
An ordinary annuity pays at the end of each period, like loan repayments; an annuity due pays at the start, like rent. An annuity due is worth (1 + i) times the ordinary one, both in future value and in present value. At 5% a year, a 10-year annuity due of 1,000 accumulates 13,206.79 against 12,577.89.
For planning only. Lenders, tax authorities and markets apply their own rounding, fees and rules; confirm figures with them before you commit.
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