# 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/r2 will be the present value of 1l. due 2 years hence; for r : 1 :: 1/r : 1/r2. In like manner 1/r3, 1/r4, 1/r5, &c, will be the present value of 1l. due at the end of 3, 4, 5, &c, years respectively; and in general, 1/rn 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/r2)+(1/r3)+(1/r4)+ &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/rn vanishes, and therefore (1/(r-1))X(1/rn) 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/rn) 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.

Example 1.

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 r2 at the end of the 3d year, and to r3 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 + r2 + r3 + r4 to rn-x, continued till the last term be rn-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 (rn-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
| Table II.The present Value of an Annuity of 1l.
 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
The Use of Table I.

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.

The Use of Table II.

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.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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