Formula for the present value of an annuity due
/What is the Formula for the Present Value of an Annuity Due?
The present value of an annuity due is used to derive the current value of a series of cash payments that are expected to be made on predetermined future dates and in predetermined amounts. The calculation is usually made to decide if you should take a lump sum payment now, or to instead receive a series of cash payments in the future (as may be offered if you win a lottery).
The present value calculation is made with a discount rate, which roughly equates to the current rate of return on an investment. The higher the discount rate, the lower the present value of an annuity will be. Conversely, a low discount rate equates to a higher present value for an annuity.
The formula for calculating the present value of an annuity due (where payments occur at the beginning of a period) is:
P = (PMT [(1 - (1 / (1 + r)n)) / r]) x (1+r)
Where:
P = The present 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 is the same formula as for the present value of an ordinary annuity (where payments occur at the end of a period), except that the far right side of the formula adds an extra payment; this accounts for the fact that each payment essentially occurs one period sooner than under the ordinary annuity model.
The factor used for the present value of an annuity due can be derived from a standard table of present value factors that lays out the applicable factors in a matrix by time period and interest rate. For a greater level of precision, you can use the preceding formula within an electronic spreadsheet.
Example of the Present Value of an Annuity Due
ABC International is paying a third party $100,000 at the beginning of each year for the next eight years in exchange for the rights to a key patent. What would it cost ABC if it were to pay the entire amount immediately, assuming an interest rate of 5%? The calculation is:
P = ($100,000 [(1 - (1 / (1 + .05)8)) / .05]) x (1+.05)
P = $678,637
Rate Table for the Present Value of an Annuity Due 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 | 1.9901 | 1.9804 | 1.9709 | 1.9615 | 1.9524 | 1.9434 | 1.9259 | 1.9091 | 1.8929 |
| 3 | 2.9704 | 2.9416 | 2.9135 | 2.8861 | 2.8594 | 2.8334 | 2.7833 | 2.7355 | 2.6901 |
| 4 | 3.9410 | 3.8839 | 3.8286 | 3.7751 | 3.7232 | 3.6730 | 3.5771 | 3.4869 | 3.4018 |
| 5 | 4.9020 | 4.8077 | 4.7171 | 4.6299 | 4.5460 | 4.4651 | 4.3121 | 4.1699 | 4.0373 |
| 6 | 5.8534 | 5.7135 | 5.5797 | 5.4518 | 5.3295 | 5.2124 | 4.9927 | 4.7908 | 4.6048 |
| 7 | 6.7955 | 6.6014 | 6.4172 | 6.2421 | 6.0757 | 5.9173 | 5.6229 | 5.3553 | 5.1114 |
| 8 | 7.7282 | 7.4720 | 7.2303 | 7.0021 | 6.7864 | 6.5824 | 6.2064 | 5.8684 | 5.5638 |
| 9 | 8.6517 | 8.3255 | 8.0197 | 7.7327 | 7.4632 | 7.2098 | 6.7466 | 6.3349 | 5.9676 |
| 10 | 9.5660 | 9.1622 | 8.7861 | 8.4353 | 8.1078 | 7.8017 | 7.2469 | 6.7590 | 6.3282 |
| 11 | 10.4713 | 9.9826 | 9.5302 | 9.1109 | 8.7217 | 8.3601 | 7.7101 | 7.1446 | 6.6502 |
| 12 | 11.3676 | 10.7868 | 10.2526 | 9.7605 | 9.3064 | 8.8869 | 8.1390 | 7.4951 | 6.9377 |
| 13 | 12.2551 | 11.5753 | 10.9540 | 10.3851 | 9.8633 | 9.3838 | 8.5361 | 7.8137 | 7.1944 |
| 14 | 13.1337 | 12.3484 | 11.6350 | 10.9856 | 10.3936 | 9.8527 | 8.9038 | 8.1034 | 7.4235 |
| 15 | 14.0037 | 13.1062 | 12.2961 | 11.5631 | 10.8986 | 10.2950 | 9.2442 | 8.3667 | 7.6282 |
| 16 | 14.8651 | 13.8493 | 12.9379 | 12.1184 | 11.3797 | 10.7122 | 9.5595 | 8.6061 | 7.8109 |
| 17 | 15.7179 | 14.5777 | 13.5611 | 12.6523 | 11.8378 | 11.1059 | 9.8514 | 8.8237 | 7.9740 |
| 18 | 16.5623 | 15.2919 | 14.1661 | 13.1657 | 12.2741 | 11.4773 | 10.1216 | 9.0216 | 8.1196 |
| 19 | 17.3983 | 15.9920 | 14.7535 | 13.6593 | 12.6896 | 11.8276 | 10.3719 | 9.2014 | 8.2497 |
| 20 | 18.2260 | 16.6785 | 15.3238 | 14.1339 | 13.0853 | 12.1581 | 10.6036 | 9.3649 | 8.3658 |
| 21 | 19.0456 | 17.3514 | 15.8775 | 14.5903 | 13.4622 | 12.4699 | 10.8181 | 9.5136 | 8.4694 |
| 22 | 19.8570 | 18.0112 | 16.4150 | 15.0292 | 13.8212 | 12.7641 | 11.0168 | 9.6487 | 8.5620 |
| 23 | 20.6604 | 18.6580 | 16.9369 | 15.4511 | 14.1630 | 13.0416 | 11.2007 | 9.7715 | 8.6446 |
| 24 | 21.4558 | 19.2922 | 17.4436 | 15.8568 | 14.4886 | 13.3034 | 11.3711 | 9.8832 | 8.7184 |
| 25 | 22.2434 | 19.9139 | 17.9355 | 16.2470 | 14.7986 | 13.5504 | 11.5288 | 9.9847 | 8.7843 |
| 26 | 23.0232 | 20.5235 | 18.4131 | 16.6221 | 15.0939 | 13.7834 | 11.6748 | 10.0770 | 8.8431 |
| 27 | 23.7952 | 21.1210 | 18.8768 | 16.9828 | 15.3752 | 14.0032 | 11.8100 | 10.1609 | 8.8957 |
| 28 | 24.5596 | 21.7069 | 19.3270 | 17.3296 | 15.6430 | 14.2105 | 11.9352 | 10.2372 | 8.9426 |
| 29 | 25.3164 | 22.2813 | 19.7641 | 17.6631 | 15.8981 | 14.4062 | 12.0511 | 10.3066 | 8.9844 |
| 30 | 26.0658 | 22.8444 | 20.1885 | 17.9837 | 16.1411 | 14.5907 | 12.1584 | 10.3696 | 9.0218 |
The Difference Between an Annuity Due and an Ordinary Annuity
The key difference between an annuity due and an ordinary annuity is that the payments for an annuity due are scheduled to be issued at the beginning of each payment period, while the payments for an ordinary annuity are scheduled for the end of each period. This timing differential causes another difference, which is that the earlier payments for an annuity due give it a higher present value than the present value of an ordinary annuity.
Related Articles
Formula for the Future Value of an Annuity Due
Formula for the Future Value of an Ordinary Annuity