The formula for the future value of an ordinary annuity
/What is an Ordinary Annuity?
An ordinary annuity is a series of payments made at the end of each period in a series of payments. It has the following characteristics:
All payments are in the same amount (such as a series of payments of $1,000).
All payments are made at the same intervals of time (such as once a month or quarter, over a period of a year).
All payments are made at the end of each period (such as payments being made only on the last day of the month).
Usually, payments made under the ordinary annuity concept are made at the end of each month, quarter, or year, though other payment intervals are possible (such as weekly or even daily).
The Future Value of an Ordinary Annuity
A common financial planning concept is to calculate the amount of money that will be paid back to an investor on a future date if the investor makes a series of payments prior to that date, assuming that the funds are invested at a certain interest rate. Future value is the value of a sum of cash to be paid on a specific date in the future. Therefore, the formula for the future value of an ordinary annuity refers to the value on a specific future date of a series of periodic payments, where each payment is made at the end of a period.
The formula for calculating the future value of an ordinary annuity (where a series of equal payments are made at the end of each of multiple periods) is:
P = PMT [((1 + r)n - 1) / r]
Where:
P = The future value of the annuity stream to be paid in the future
PMT = The amount of each annuity payment
r = The interest rate
n = The number of periods over which payments are made
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This value is the amount that a stream of future payments will grow to, assuming that a certain amount of compounded interest earnings gradually accrue over the measurement period. Usually, the key variable in the equation is the interest rate assumption, which could be severely misstated from the interest rate that is actually experienced in future periods.
Example of the Future Value of an Ordinary Annuity
The treasurer of ABC International expects to invest $100,000 of the firm's funds in a long-term investment vehicle at the end of each year for the next five years. He expects that the company will earn 7% interest that will compound annually. The value that these payments should have at the end of the five-year period is calculated as:
P = $100,000 [((1 + .07)5 - 1) / .07]
P = $575,074
As another example, what if the interest on the investment compounded monthly instead of annually, and the amount invested were $8,000 at the end of month? The calculation is:
P = $8,000 [((1 + .005833)60 - 1) / .005833]
P = $572,737
The .005833 interest rate used in the last example is 1/12th of the full 7% annual interest rate.
Rate Table for the Future Value of an Ordinary Annuity of 1
| n | 1% | 2% | 3% | 4% | 5% | 6% | 8% | 10% | 12% |
| 1 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 2 | 2.0100 | 2.0200 | 2.0300 | 2.0400 | 2.0500 | 2.0600 | 2.0800 | 2.1000 | 2.1200 |
| 3 | 3.0301 | 3.0604 | 3.0909 | 3.1216 | 3.1525 | 3.1836 | 3.2464 | 3.3100 | 3.3744 |
| 4 | 4.0604 | 4.1216 | 4.1836 | 4.2465 | 4.3101 | 4.3746 | 4.5061 | 4.6410 | 4.7793 |
| 5 | 5.1010 | 5.2040 | 5.3091 | 5.4163 | 5.5256 | 5.6371 | 5.8666 | 6.1051 | 6.3529 |
| 6 | 6.1520 | 6.3081 | 6.4684 | 6.6330 | 6.8019 | 6.9753 | 7.3359 | 7.7156 | 8.1152 |
| 7 | 7.2135 | 7.4343 | 7.6625 | 7.8983 | 8.1420 | 8.3938 | 8.9228 | 9.4872 | 10.0890 |
| 8 | 8.2857 | 8.5830 | 8.8923 | 9.2142 | 9.5491 | 9.8975 | 10.6366 | 11.4359 | 12.2997 |
| 9 | 9.3685 | 9.7546 | 10.1591 | 10.5828 | 11.0266 | 11.4913 | 12.4876 | 13.5795 | 14.7757 |
| 10 | 10.4622 | 10.9497 | 11.4639 | 12.0061 | 12.5779 | 13.1808 | 14.4866 | 15.9374 | 17.5487 |
| 11 | 11.5668 | 12.1687 | 12.8078 | 13.4864 | 14.2068 | 14.9716 | 16.6455 | 18.5312 | 20.6546 |
| 12 | 12.6825 | 13.4121 | 14.1920 | 15.0258 | 15.9171 | 16.8699 | 18.9771 | 21.3843 | 24.1331 |
| 13 | 13.8093 | 14.6803 | 15.6178 | 16.6268 | 17.7130 | 18.8821 | 21.4953 | 24.5227 | 28.0291 |
| 14 | 14.9474 | 15.9739 | 17.0863 | 18.2919 | 19.5986 | 21.0151 | 24.2149 | 27.9750 | 32.3926 |
| 15 | 16.0969 | 17.2934 | 18.5989 | 20.0236 | 21.5786 | 23.2760 | 27.1521 | 31.7725 | 37.2797 |
| 16 | 17.2579 | 18.6393 | 20.1569 | 21.8245 | 23.6575 | 25.6725 | 30.3243 | 35.9497 | 42.7533 |
| 17 | 18.4304 | 20.0121 | 21.7616 | 23.6975 | 25.8404 | 28.2129 | 33.7502 | 40.5447 | 48.8837 |
| 18 | 19.6148 | 21.4123 | 23.4144 | 25.6454 | 28.1324 | 30.9057 | 37.4502 | 45.5992 | 55.7497 |
| 19 | 20.8109 | 22.8406 | 25.1169 | 27.6712 | 30.5390 | 33.7600 | 41.4463 | 51.1591 | 63.4397 |
| 20 | 22.0190 | 24.2974 | 26.8704 | 29.7781 | 33.0660 | 36.7856 | 45.7620 | 57.2750 | 72.0524 |
| 21 | 23.2392 | 25.7833 | 28.6765 | 31.9692 | 35.7193 | 39.9927 | 50.4229 | 64.0025 | 81.6987 |
| 22 | 24.4716 | 27.2990 | 30.5368 | 34.2480 | 38.5052 | 43.3923 | 55.4568 | 71.4028 | 92.5026 |
| 23 | 25.7163 | 28.8450 | 32.4529 | 36.6179 | 41.4305 | 46.9958 | 60.8933 | 79.5430 | 104.6029 |
| 24 | 26.9735 | 30.4219 | 34.4265 | 39.0826 | 44.5020 | 50.8156 | 66.7648 | 88.4973 | 118.1552 |
| 25 | 28.2432 | 32.0303 | 36.4593 | 41.6459 | 47.7271 | 54.8645 | 73.1059 | 98.3471 | 133.3339 |
| 26 | 29.5256 | 33.6709 | 38.5530 | 44.3117 | 51.1135 | 59.1564 | 79.9544 | 109.1818 | 150.3339 |
| 27 | 30.8209 | 35.3443 | 40.7096 | 47.0842 | 54.6691 | 63.7058 | 87.3508 | 121.0999 | 169.3740 |
| 28 | 32.1291 | 37.0512 | 42.9309 | 49.9676 | 58.4026 | 68.5281 | 95.3388 | 134.2099 | 190.6989 |
| 29 | 33.4504 | 38.7922 | 45.2189 | 52.9663 | 62.3227 | 73.6398 | 103.9659 | 148.6309 | 214.5828 |
| 30 | 34.7849 | 40.5681 | 47.5754 | 56.0849 | 66.4389 | 79.0582 | 113.2832 | 164.4940 | 241.3327 |