Cumulative Present Value
This page presents essential tables for the Present Value Interest Factor of an Ordinary Annuity (PVIFA). These tables are fundamental computational aids in the realm of financial mathematics, built upon the core economic principle known as the time value of money. This concept highlights that, due to factors like potential earning power through interest and inflation, a sum of money available today is inherently worth more than the identical sum received at some point in the future.
Before delving into the PVIFA itself, it's important to define an annuity. An annuity is a series of financial payments of an equal amount, made or received at regular, fixed intervals over a specified period. Examples include mortgage payments, lease payments, or regular withdrawals from a retirement fund. The term "ordinary annuity" specifically refers to a series where payments occur at the *end* of each period.
The PVIFA factor represents the combined present value of receiving *one unit* of currency (such as $\textsf{₹}1$ or $1) at the end of *each* period for a specified number of periods, given a particular discount rate (interest rate) per period. In essence, the PVIFA is a single factor that encapsulates the current worth of an entire stream of identical future payments of one currency unit. These tables greatly simplify what would otherwise be repetitive and complex calculations required to determine the present value of such a stream of cash flows.
PVIFA tables are typically structured for user-friendliness, featuring rows that correspond to the number of periods ('n') and columns that correspond to different interest rates ('r') per period. To calculate the present value of an actual annuity payment, you first locate the intersection of the correct number of periods and the applicable interest rate in the table to find the relevant PVIFA factor. You then simply multiply this factor by the constant amount of the periodic payment. The result is the total present value of the entire series of future payments.
Mathematically, the PVIFA is derived by summing up the individual Present Value Interest Factors (PVIFs) for each payment in the annuity. The PVIF for a single future payment is calculated as $\frac{1}{(1+r)^t}$, where 'r' is the rate and 't' is the specific period the payment is received. The PVIFA for an 'n'-period annuity at rate 'r' is the sum of the PVIFs for periods 1 through 'n': $$ \text{PVIFA}(r, n) = \sum_{t=1}^{n} \frac{1}{(1+r)^t} $$ The tables provide the pre-calculated results of this summation for common values of 'r' and 'n'.
These tables are an indispensable tool across numerous financial applications because they allow for the efficient valuation of future payment streams. Common uses include:
- Calculating the principal amount of a loan based on the desired periodic payment (e.g., mortgage).
- Valuing bonds that pay regular coupon interest.
- Determining the present value of a planned series of retirement income withdrawals.
- Evaluating investments expected to generate consistent returns over time.
- Comparing different investment or financing options with varying payment structures.
By providing direct access to these calculated factors, this resource empowers users to quickly and accurately perform the necessary discounting calculations required for sound financial analysis and decision-making regarding annuities.
This table shows the annuity factor for an amount at the end of each year for n years at r%.
| Periods | Interest Rates (r) | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| (n) | 1% | 2% | 3% | 4% | 5% | 6% | 7% | 8% | 9% | 10% |
| 1 | .990 | .980 | .971 | .962 | .952 | .943 | .935 | .926 | .917 | .909 |
| 2 | 1.970 | 1.942 | 1.913 | 1.886 | 1.859 | 1.833 | 1.808 | 1.783 | 1.759 | 1.736 |
| 3 | 2.941 | 2.884 | 2.829 | 2.775 | 2.723 | 2.673 | 2.624 | 2.577 | 2.531 | 2.487 |
| 4 | 3.907 | 3.808 | 3.717 | 3.630 | 3.546 | 3.465 | 3.387 | 3.312 | 3.240 | 3.170 |
| 5 | 4.853 | 4.713 | 4.580 | 4.452 | 4.329 | 4.212 | 4..100 | 3.993 | 3.890 | 3.791 |
| 6 | 5.795 | 5.601 | 5.417 | 5.242 | 5.076 | 4.917 | 4.767 | 4.623 | 4.486 | 4.355 |
| 7 | 6.728 | 6.472 | 6.230 | 6.002 | 5.786 | 5.582 | .5.389 | 5.206 | 5.033 | 4.868 |
| 8 | 7.562 | 7.325 | 7.020 | 6.733 | 6.463 | 6.210 | 5.971 | 5.747 | 5.535 | 5.335 |
| 9 | 8.566 | 8.162 | 7.786 | 7.435 | 7.108 | 6.802 | 6.515 | 6.247 | 5.995 | 5.759 |
| 10 | 9.471 | 8.983 | 8.530 | 8.111 | 7.722 | 7.360 | 7.024 | 6.710 | 6.418 | 6.145 |
| 11 | 10.368 | 9.787 | 9.253 | 8.760 | 8.306 | 7.887 | 7.499 | 7.139 | 6.805 | 6.495 |
| 12 | 11.255 | 10.575 | 9.954 | 9.385 | 8.863 | 8.384 | 7.943 | 7.536 | 7.161 | 6.814 |
| 13 | 12.134 | 11.348 | 10.635 | 9.986 | 9.394 | 8.853 | 8.358 | 7.904 | 7.487 | 7.103 |
| 14 | 13.004 | 12.106 | 11.296 | 10.563 | 9.899 | 9.295 | 8.745 | 8.244 | 7.786 | 7.367 |
| 15 | 13.865 | 12.849 | 11.938 | 11.118 | 10.380 | 9.712 | 9.108 | 8.559 | 8.061 | 7.606 |
| 16 | 14.718 | 13.578 | 12.561 | 11.652 | .10.838 | 10.106 | 9.447 | 8.851 | 8.313 | 7.824 |
| 17 | 15.562 | 14.292 | 13.166 | 12.166 | 11.274 | 10.477 | 9.763 | 9.122 | 8.544 | 8.022 |
| 18 | 16.398 | 14.992 | 13.754 | 12.659 | 11.690 | 10.828 | 10.059 | 9.372 | 8.756 | 8.201 |
| 19 | 17.226 | 15.679 | 14.324 | 13.134 | 12.085 | 11.158 | 10.336 | 9.604 | 8.950 | 8.365 |
| 20 | 18.046 | 16.351 | 14.878 | 13.590 | 12.462 | 11.470 | 10.594 | 9.818 | 9.129 | 8.514 |
| Periods | Interest Rates (r) | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| (n) | 11% | 12% | 13% | 14% | 15% | 16% | 17% | 18% | 19% | 20% |
| 1 | .901 | .893 | .885 | .877 | .870 | .862 | .855 | .847 | .840 | .833 |
| 2 | 1.713 | 1.690 | 1.668 | 1.647 | 1.626 | 1.605 | 1.585 | 1.566 | 1.547 | 1.528 |
| 3 | 2.44 | 2.402 | 2.361 | 2.322 | 2.283 | 2.246 | 2.210 | 2.174 | 2.140 | 2.106 |
| 4 | 3.102 | 3.037 | 2.974 | 2.914 | 2.855 | 2.798 | 2.743 | 2.690 | 2.639 | 2.589 |
| 5 | 3.696 | 3.605 | 3.517 | 3.433 | 3.352 | 3.274 | 3.199 | 3.127 | 3.058 | 2.991 |
| 6 | 4.231 | 4.111 | 3.998 | 3.889 | 3.784 | 3.685 | 3.589 | 3.498 | 3.410 | 3.326 |
| 7 | 4.712 | 4.564 | 4.423 | 4.288 | 4.160 | 4.039 | 3.922 | 3.812 | 3.706 | 3.605 |
| 8 | 5.146 | 4.968 | 4.799 | 4.639 | 4.487 | 4.344 | 4.207 | 4.078 | 3.954 | 3.837 |
| 9 | 5.537 | 5.328 | 5.132 | 4.946 | 4.772 | 4.607 | 4.451 | 4.303 | 4.163 | 4.031 |
| 10 | 5.889 | 5.650 | 5.426 | 5.216 | 5.019 | 4.833 | 4.659 | 4.494 | 4.339 | 4.192 |
| 11 | 6.207 | 5.938 | 5.687 | 5.453 | 5.234 | 5.029 | 4.836 | 4.656 | 4.486 | 4.327 |
| 12 | 6.492 | 6.194 | 5.918 | 5.660 | 5.421 | 5.197 | 4.988 | 4.793 | 4.611 | 4.439 |
| 13 | 6.750 | 6.424 | 6.122 | 5.842 | 5.583 | 5.342 | 5.118 | 4.910 | 4.715 | 4.533 |
| 14 | 6.982 | 6.628 | 6.302 | 6.002 | 5.724 | 5.468 | 5.229 | 5.008 | 4.802 | 4.611 |
| 15 | 7.191 | 6.811 | 6.462 | 6.142 | 5.847 | 5.575 | 5.324 | 5.092 | 4.876 | 4.675 |
| 16 | 7.379 | 6.974 | 6.604 | 6.265 | 5.954 | 5.668 | 5.405 | 5.162 | 4.398 | 4.730 |
| 17 | 7.549 | 7.120 | 6.729 | 6.373 | 6.047 | 5.749 | 5.475 | 5.222 | 4.990 | 4.775 |
| 18 | 7.702 | 7.250 | 6.840 | 6.467 | 6.128 | 5.818 | 5.534 | 5.273 | 5.033 | 4.812 |
| 19 | 7.839 | 7.366 | 6.938 | 6.550 | 6.198 | 5.877 | 5.584 | 5.316 | 5.070 | 4.843 |
| 20 | 7.963 | 7.469 | 7.025 | 6.623 | 6.259 | 5.929 | 5.628 | 5.353 | 5.101 | 4.870 |