ANNUITIES
, a term for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c; payable from time to time; either annually, or at other intervals of time.
Annuities may be divided into such as are certain, and such as depend on some contingency, as the continuance of a life, &c.
Annuities are also divided into annuities in possession, and annuities in reversion; the former meaning such as have commenced; and the latter such as will not commence till some particular event has happened, or till some given period of time has elapsed.
Annuities may be farther considered as payable either yearly, or half yearly, or quarterly, &c.
The present value of an annuity, is that sum, which, being improved at interest, will be sufficient to pay the annuity.
The present value of an annuity certain, payable yearly, is calculated in the following manner.—Let the annuity be 1, and let r denote the amount of 1l. for a year, or 1l. increased by its interest for one year. Then, 1 being the present value of the sum r, and having to find the present value of the sum 1, it will be, by proportion | thus, r : 1 :: 1 : 1/r the present value of 1l. due a year hence. In like manner 1/r^{2} will be the present value of 1l. due 2 years hence; for r : 1 :: 1/r : 1/r^{2}. In like manner 1/r^{3}, 1/r^{4}, 1/r^{5}, &c, will be the present value of 1l. due at the end of 3, 4, 5, &c, years respectively; and in general, 1/r^{n} will be the value of 1l. to be received after the expiration of n years. Consequently the sum of all these, or (1/r)+(1/r^{2})+(1/r^{3})+(1/r^{4})+ &c, contined to n terms, will be the present value of all the n years annuities. And the value of the perpetuity, is the sum of the series continued ad infinitum.
But this series, it is evident, is a geometrical progression, whose first term and common ratio are each 1/r, and the number of its terms n; and therefore the sum s of all the terms, or the present value of all the annual payments, will be .
When the annuity is a perpetuity, it is plain that the last term 1/r^{n} vanishes, and therefore (1/(r-1))X(1/r^{n}) also vanishes; and consequently the expression becomes barely s=1/(r-1); that is, any annuity divided by its interest for one year, is the value of the perpetuity. So, if the rate of interest be 5 per cent; then (100/5)=20 is the value of the perpetuity at 5 per cent. Also 100/4 =25 is the value of the perpetuity at 4 per cent. And 100/3 = 33 1/3 is the value of the perpetuity at 3 per cent. interest. And so on.
If the annuity is not to be entered on immediately, but after a certain number of years, as m years; then the present value of the reversion is equal to the difference between two present values, the one for the first term of m years, and the other for the end of the last term n: that is, equal to the difference between .
Annuities certain differ in value, as they are made payable yearly, half-yearly, or quarterly. And by proceeding as above, using the interest or amount of a half year, or a quarter, as those for the whole year were used, the following set of theorems will arise; where <*> denotes, as before, the amount of 1l. and its interest for a year, and n the number of years, during which, any annuity is to be paid; also P denotes the perpetnity 1/(r-1), Y denotes (1/(r-1))-(1/(r-1))X(1/r^{n}) the value of the annuity supposed payable yearly, H the value of the same when it is payable half-yearly, and Q the value when payable quarterly; or universally, M the value when it is payable every m part of a year.
Theor. 1. .
Theor. 2. .
Theor. 3. .
Theor. 4.
Let the rate of interest be 4. per cent, and the term 5 years; and consequently r = 1.04, n = 5, P = 25; also let m = 12, or the interest payable monthly in theorem 4: then the present value of such annuity of 1l. a year, for 5 years, according as it is supposed payable 1l. yearly, or (1/2)l. every half year, or (1/4)l. every quarter, or (1/12)l. every month or (1/12)th part of a year, will be as follows:
Example 2. Supposing the annuity to continue 25 years, the rate of interest and every thing else being as before; then the values of the annuities for 25 years will be
Example 3. And if the term be 50 years, the values will be
Example 4. Also if the term be 100 years, the values will be
Hence the difference in the value by making periods of payments smaller, for any given term of years, is the more as the intervals are smaller, or the periods more frequent. The same difference is also variable, both as the rate of interest varies, and also as the whole term of years n varies; and, for any given rate of interest, it | is evident that the difference, for any periods m of payments, first increases from nothing as the term n increases, when n is 0, to some certain finite term or value of n, when the difference D is the greatest or a maximum; and that afterwards, as n increases more, that difference will continually decrease to nothing again, and vanish when n is infinite: also the term or value of n, for the maximum of the difference, will be different according to the periods of payment, or value of m. And the general value of n, when the difference is a maximum between the yearly payments and the payments of m times in a year, is expressed by this formula, viz, , where l. denotes the logarithm of the quantity following it. Hence, taking the different values of m, viz, 2 for half years, 4 for quarters, 12 for monthly payments, &c, and substituting in the general formula, the term or value of n for each case, when the difference in the present worths of the annuities, will be as follows, reckoning interest at 4 per cent, viz, for half-yearly payments, for quarterly payments, for monthly payments.
Annuities may also be considered as in arrears, or as forborn, for any number of years; in which case each payment is to be considered as a sum put out to interest for the remainder of the term after the time it becomes due. And as 1l. due at the end of 1 year, amounts to r at the end of another year, and to r^{2} at the end of the 3d year, and to r^{3} at the end of the 4th year, and so on; therefore by adding always the last year's annuity, or 1, to the amounts of all the former years, the sum of all the annuities and their interests, will be the sum of the following geometrical series, 1 + r + r^{2} + r^{3} + r^{4} to r^{n-x}, continued till the last term be r^{n-x}, or till the number of terms be n, the number of years the annuity is forborn. But the sum of this geometrical progression is (r^{n}-1)/(r-1), which therefore is the amount of 1l. annuity forborn for n years. And this quantity being multiplied by any other annuity a, instead of 1, will produce the amount for that other annuity.
But the amounts of annuities, or their present values, are easiest found by the two following tables of numbers for the annuity of 1l. ready computed from the foregoing principles.
Table I.The Amount of an Annuity of 1l. at Comp. Interest.Yrs. | at 3 per cent. | 3 1/2 per cent. | 4 per cent. | 4 1/2 per cent. | 5 per cent. | 6 per cent. |
1 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 |
2 | 2.03000 | 2.03500 | 2.04000 | 2.04500 | 2.05000 | 2.06000 |
3 | 3.09090 | 3.10623 | 3.12160 | 3.13703 | 3.15250 | 3.18360 |
4 | 4.18363 | 4.21494 | 4.24646 | 4.27819 | 4.31013 | 4.37462 |
5 | 5.30914 | 5.36247 | 5.41632 | 5.47071 | 5.52563 | 5.63709 |
6 | 6.46841 | 6.55015 | 6.63298 | 6.71689 | 6.80191 | 6.97532 |
7 | 7.66246 | 7.77941 | 7.89829 | 8.01915 | 8.14201 | 8.39384 |
8 | 8.89234 | 9.05169 | 9.21423 | 9.38001 | 9.54911 | 9.89747 |
9 | 10.15911 | 10.36850 | 10.58280 | 10.80211 | 11.02656 | 11.49132 |
10 | 11.46388 | 11.73139 | 12.00611 | 12.28821 | 12.57789 | 13.18079 |
11 | 12.80780 | 13.14199 | 13.48635 | 13.84118 | 14.20679 | 14.97164 |
12 | 14.19203 | 14.60196 | 15.02581 | 15.46403 | 15.91713 | 16.86994 |
13 | 15.61779 | 16.11303 | 16.62684 | 17.15991 | 17.71298 | 18.88214 |
14 | 17.08632 | 17.67699 | 18.29191 | 18.93211 | 19.59863 | 21.01507 |
15 | 18.59891 | 19.29568 | 20.32359 | 20.78405 | 21.57856 | 23.27597 |
16 | 20.15688 | 20.97103 | 21.82453 | 22.71934 | 23.65749 | 25.67253 |
17 | 21.76159 | 22.70502 | 23.69751 | 24.74171 | 25.84037 | 28.21288 |
18 | 23.41444 | 24.49969 | 25.64541 | 26.85508 | 28.13238 | 30.90565 |
19 | 25.11687 | 26.35718 | 27.67123 | 29.06356 | 30.53900 | 33.75999 |
20 | 26.87037 | 28.27968 | 29.77808 | 31.37142 | 33.06595 | 36.78559 |
21 | 28.67649 | 30.26947 | 31.96920 | 33.78314 | 35.71925 | 39.99273 |
22 | 30.53678 | 32.32890 | 34.24797 | 36.30338 | 38.50521 | 43.39229 |
23 | 32.45288 | 34.46041 | 36.61789 | 38.93703 | 41.43048 | 46.99583 |
24 | 34.42647 | 36.66653 | 39.08260 | 41.68920 | 44.50200 | 50.81558 |
25 | 36.45926 | 38.94986 | 41.64591 | 44.56521 | 47.72710 | 54.86451 |
26 | 38.55304 | 41.31310 | 44.31174 | 47.57064 | 51.11345 | 59.15638 |
27 | 40.70963 | 43.75906 | 47.08421 | 50.71132 | 54.66913 | 63.70577 |
28 | 42.93092 | 46.29063 | 49.96758 | 53.99333 | 58.40258 | 68.52811 |
29 | 45.21885 | 48.91080 | 52.96629 | 57.42303 | 62.32271 | 73.63980 |
30 | 47.57542 | 51.62268 | 56.08494 | 61.00707 | 66.43885 | 79.05819 |
31 | 50.00268 | 54.42947 | 59.32834 | 64.75239 | 70.76079 | 84.80168 |
32 | 52.50276 | 57.33450 | 62.70147 | 68.66625 | 75.29883 | 90.88978 |
33 | 55.07784 | 60.34121 | 66.20953 | 72.75623 | 80.06377 | 97.34316 |
34 | 57.73018 | 63.45315 | 69.85791 | 77.03026 | 85.06696 | 104.18375 |
35 | 60.46208 | 66.67401 | 73.65222 | 81.49662 | 90.32031 | 111.43478 |
36 | 63.27594 | 70.00760 | 77.59831 | 86.16397 | 95.83632 | 119.12087 |
37 | 66.17422 | 73.45787 | 81.70225 | 91.04134 | 101.62814 | 127.26812 |
38 | 69.15945 | 77.02889 | 85.97034 | 96.13820 | 107.70955 | 135.90421 |
39 | 72.23423 | 80.72491 | 90.40915 | 101.46442 | 114.09502 | 145.05846 |
40 | 75.40126 | 84.55028 | 95.02552 | 107.03032 | 120.79977 | 154.76197 |
41 | 78.66330 | 88.50954 | 99.82654 | 112.84669 | 127.83976 | 165.04768 |
42 | 82.02320 | 92.60737 | 104.81960 | 118.92479 | 135.23175 | 175.95054 |
43 | 85.48389 | 96.84863 | 110.01238 | 125.27640 | 142.99334 | 187.50758 |
44 | 89.04841 | 101.23833 | 115.41288 | 131.91384 | 151.14301 | 199.75803 |
45 | 92.71986 | 105.78167 | 121.02939 | 138.84997 | 159.70016 | 212.74351 |
46 | 96.50146 | 110.48403 | 126.87057 | 146.09821 | 168.68516 | 226.50812 |
47 | 100.39650 | 115.35097 | 132.94539 | 153.67263 | 178.11942 | 241.09861 |
48 | 104.40840 | 120.38826 | 139.26321 | 161.58790 | 188.02539 | 256.56453 |
49 | 108.54065 | 125.60185 | 145.83373 | 169.85936 | 198.42666 | 272.95840 |
50 | 112.79687 | 130.99791 | 152.66708 | 178.50303 | 209.34800 | 290.33590 |
51 | 117.18077 | 136.58284 | 159.77377 | 187.53566 | 220.81540 | 308.75606 |
52 | 121.69620 | 142.36324 | 167.16472 | 196.97477 | 232.85617 | 328.28142 |
53 | 126.34708 | 148.34595 | 174.85131 | 206.83863 | 245.49897 | 348.97831 |
54 | 131.13750 | 154.53806 | 182.84536 | 217.14637 | 258.77392 | 370.91701 |
Yrs. | at 3 per cent. | 3 1/2 per cent. | 4 per cent. | 4 1/2 per cent. | 5 per cent. | 6 per cent. |
1 | 0.97087 | 0.96618 | 0.96154 | 0.95694 | 0.95238 | 0.94340 |
2 | 1.91347 | 1.89969 | 1.88610 | 1.87267 | 1.85941 | 1.83339 |
3 | 2.82861 | 2.80164 | 2.77509 | 2.74896 | 2.72325 | 2.67301 |
4 | 3.71710 | 3.67308 | 3.62990 | 3.58753 | 3.54595 | 3.46511 |
5 | 4.57971 | 4.51505 | 4.45182 | 4.38998 | 4.32948 | 4.21236 |
6 | 5.41719 | 5.32855 | 5.24214 | 5.15787 | 5.07569 | 4.91732 |
7 | 6.23028 | 6.11454 | 6.00205 | 5.89270 | 5.78637 | 5.58238 |
8 | 7.01969 | 6.87396 | 6.73274 | 6.59589 | 6.46321 | 6.20979 |
9 | 7.78611 | 7.60769 | 7.43533 | 7.26879 | 7.10782 | 6.80169 |
10 | 8.53020 | 8.31661 | 8.11090 | 7.91272 | 7.72173 | 7.36009 |
11 | 9.25262 | 9.00155 | 8.76048 | 8.52892 | 8.30541 | 7.88687 |
12 | 9.95400 | 9.66333 | 9.38507 | 9.11858 | 8.86325 | 8.38384 |
13 | 10.63496 | 10.30274 | 9.98565 | 9.68285 | 9.39357 | 8.85268 |
14 | 11.29607 | 10.92052 | 10.56312 | 10.22283 | 9.89864 | 9.29498 |
15 | 11.93794 | 11.51741 | 11.11839 | 10.73955 | 10.37966 | 9.71225 |
16 | 12.56110 | 12.09412 | 11.65230 | 11.23402 | 10.83777 | 10.10590 |
17 | 13.16612 | 12.65132 | 12.16567 | 11.70719 | 11.27407 | 10.47726 |
18 | 13.75351 | 13.18968 | 12.65930 | 12.15999 | 11.68959 | 10.82760 |
19 | 14.32380 | 13.70984 | 13.13394 | 12.59329 | 12.08532 | 11.15812 |
20 | 14.87747 | 14.21240 | 13.59033 | 13.00794 | 12.46221 | 11.46992 |
21 | 15.41502 | 14.69797 | 14.02916 | 13.40472 | 12.82115 | 11.76408 |
22 | 15.93692 | 15.16712 | 14.45112 | 13.78442 | 13.16300 | 12.04158 |
23 | 16.44361 | 15.62041 | 14.85684 | 14.14777 | 13.48857 | 12.30338 |
24 | 16.93554 | 16.05837 | 15.24696 | 14.49548 | 13.79864 | 12.55036 |
25 | 17.41315 | 16.48151 | 15.62208 | 14.82821 | 14.09394 | 12.78336 |
26 | 17.87684 | 16.89035 | 15.98277 | 15.14661 | 14.37519 | 13.00317 |
27 | 18.32703 | 17.28536 | 16.32959 | 15.45130 | 14.64303 | 13.21053 |
28 | 18.76411 | 17.66702 | 16.66306 | 15.74287 | 14.89813 | 13.40616 |
29 | 19.18845 | 18.03577 | 16.98371 | 16.02189 | 15.14107 | 13.59072 |
30 | 19.60044 | 18.39205 | 17.29203 | 16.28889 | 15.37245 | 13.76483 |
31 | 20.00043 | 18.73628 | 17.58849 | 16.54439 | 15.59281 | 13.92909 |
32 | 20.38877 | 19.06887 | 17.87355 | 16.78889 | 15.80268 | 14.08404 |
33 | 20.76579 | 19.39021 | 18.14765 | 17.02286 | 16.00255 | 14.23023 |
34 | 21.13184 | 19.70068 | 18.41120 | 17.24676 | 16.19290 | 14.36814 |
35 | 21.48722 | 20.00066 | 18.66461 | 17.46101 | 16.37419 | 14.49825 |
36 | 21.83225 | 20.29049 | 18.90828 | 17.66604 | 16.54685 | 14.62099 |
37 | 22.16724 | 20.<*>7053 | 19.14258 | 17.86224 | 16.71129 | 14.73678 |
38 | 22.49246 | 20.84109 | 19.36786 | 18.04999 | 16.86789 | 14.84602 |
39 | 22.80822 | 21.10250 | 19.58448 | 18.22966 | 17.01704 | 14.94907 |
40 | 23.11477 | 21.35507 | 19.79277 | 18.40158 | 17.15909 | 15.04630 |
41 | 23.41240 | 21.59910 | 19.99305 | 18.56611 | 17.29437 | 15.13802 |
42 | 23.70136 | 21.83488 | 20.18563 | 18.72355 | 17.42321 | 15.22454 |
43 | 23.98190 | 22.06269 | 20.37079 | 18.87421 | 17.54591 | 15.30617 |
44 | 24.25427 | 22.28279 | 20.54884 | 19.01838 | 17.66277 | 15.38318 |
45 | 24.51871 | 22.49545 | 20.72004 | 19.15635 | 17.77407 | 15.45583 |
46 | 24.77545 | 22.70092 | 20.88465 | 19.28837 | 17.88007 | 15.52437 |
47 | 25.02471 | 22.89944 | 21.04294 | 19.41471 | 17.98102 | 15.58903 |
48 | 25.26671 | 23.09124 | 21.19513 | 19.53561 | 18.07716 | 15.65003 |
49 | 25.50166 | 23.27656 | 21.34147 | 19.65130 | 18.16872 | 15.70757 |
50 | 25.72976 | 23.45562 | 21.48218 | 19.76201 | 18.25593 | 15.76186 |
51 | 25.95123 | 23.62862 | 21.61749 | 19.86795 | 18.33898 | 15.81308 |
52 | 26.16624 | 23.79576 | 21.74758 | 19.96933 | 18.41807 | 15.86139 |
53 | 26.37499 | 23.95726 | 21.87267 | 20.06634 | 18.49340 | 15.90697 |
54 | 26.57766 | 24.11330 | 21.99296 | 20.15918 | 18.56515 | 15.94998 |
To find the Amount of an annuity forborn any number of years. Take out the amount from the 1st table, for the proposed years and rate of interest; then multiply it by the annuity in question; and the product will be its amount for the same number of years, and rate of interest.
And the converse to find the rate or time.
Exam. 1. To find how much an annuity of 50l. will amount to in 20 years at 3 1/2 per cent. compound interest.—On the line of 20 years, and in the column of 3 1/2 per cent, stands 28.27968, which is the amount of an annuity of 1l. for the 20 years; and therefore 28.27968 multiplied by 50, gives 1413.9841. or 1413l. 19s. 8d. for the answer.
Exam. 2. In what time will an annuity of 20l. amount to 1000l. at 4 per cent. compound interest?—Here the amount of 1000l. divided by 20l. the annuity, gives 50, the amount of 1l. annuity for the same time and rate. Then, the nearest tabular number in the column of 4 per cent. is 49.96758, which standing on the line of 28, shews that 28 years is the answer.
Exam. 3. If it be required to find at what rate of interest an annuity of 20l. will amount to 1000l. forborn for 28 years.—Here 1000 divided by 20 gives 50 as before. Then looking along the line of 28 years, for the nearest to this number 50, I find 49.96758 in the column of 4 per cent. which is therefore the rate of interest required.
Exam. 1. To find the present value of an annuity of 50l. which is to continue 20 years, at 3 1/2 per cent.— By the table, the present value of 1l. for the same rate and time, is 14.21240; therefore 14.2124 X 50 = 710.62l. or 710l. 128. 4d. is the present value sought.
Exam. 2. To find the present value of an annuity of 20l. to commence 10 years hence, and then to continue for 40 years, or to terminate 50 years hence, at 4 per cent. interest.—In such cases as this, it is plain we have to sind the difference between the present values of two equal annuities, for the two given times; which therefore will be effected by subtracting the tabular value of the one term from that of the other, and multiplying by the annuity. Thus,
tabular value for 50 years | 21.48218 | |
tabular value for 10 years | 8.11090 | |
the difference | 13.37128 | |
mult. by | 20 | |
gives | 267.4256 | |
or | 2671. 8s. 6d. | the answer. |
The foregoing observations, rules, and tables, contain all that is important in the doctrine of annuities certain. And for farther information, reference may be had to arithmetical writings, particularly Malcolm's Arithmetic, page 595; Simpson's Algebra, sect. 16; Dodson's Mathematical Repository, page 298, &c; Jones's Synopsis, ch. 10; Philos. Trans. vol. lxvi, page 109.
For what relates to the doctrine of annuities on lives, see Assurance, Complement, Expectation, Life Annuities, Reversions, &c.