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.
11.000001.000001.000001.000001.000001.00000
22.030002.035002.040002.045002.050002.06000
33.090903.106233.121603.137033.152503.18360
44.183634.214944.246464.278194.310134.37462
55.309145.362475.416325.470715.525635.63709
66.468416.550156.632986.716896.801916.97532
77.662467.779417.898298.019158.142018.39384
88.892349.051699.214239.380019.549119.89747
910.1591110.3685010.5828010.8021111.0265611.49132
1011.4638811.7313912.0061112.2882112.5778913.18079
1112.8078013.1419913.4863513.8411814.2067914.97164
1214.1920314.6019615.0258115.4640315.9171316.86994
1315.6177916.1130316.6268417.1599117.7129818.88214
1417.0863217.6769918.2919118.9321119.5986321.01507
1518.5989119.2956820.3235920.7840521.5785623.27597
1620.1568820.9710321.8245322.7193423.6574925.67253
1721.7615922.7050223.6975124.7417125.8403728.21288
1823.4144424.4996925.6454126.8550828.1323830.90565
1925.1168726.3571827.6712329.0635630.5390033.75999
2026.8703728.2796829.7780831.3714233.0659536.78559
2128.6764930.2694731.9692033.7831435.7192539.99273
2230.5367832.3289034.2479736.3033838.5052143.39229
2332.4528834.4604136.6178938.9370341.4304846.99583
2434.4264736.6665339.0826041.6892044.5020050.81558
2536.4592638.9498641.6459144.5652147.7271054.86451
2638.5530441.3131044.3117447.5706451.1134559.15638
2740.7096343.7590647.0842150.7113254.6691363.70577
2842.9309246.2906349.9675853.9933358.4025868.52811
2945.2188548.9108052.9662957.4230362.3227173.63980
3047.5754251.6226856.0849461.0070766.4388579.05819
3150.0026854.4294759.3283464.7523970.7607984.80168
3252.5027657.3345062.7014768.6662575.2988390.88978
3355.0778460.3412166.2095372.7562380.0637797.34316
3457.7301863.4531569.8579177.0302685.06696104.18375
3560.4620866.6740173.6522281.4966290.32031111.43478
3663.2759470.0076077.5983186.1639795.83632119.12087
3766.1742273.4578781.7022591.04134101.62814127.26812
3869.1594577.0288985.9703496.13820107.70955135.90421
3972.2342380.7249190.40915101.46442114.09502145.05846
4075.4012684.5502895.02552107.03032120.79977154.76197
4178.6633088.5095499.82654112.84669127.83976165.04768
4282.0232092.60737104.81960118.92479135.23175175.95054
4385.4838996.84863110.01238125.27640142.99334187.50758
4489.04841101.23833115.41288131.91384151.14301199.75803
4592.71986105.78167121.02939138.84997159.70016212.74351
4696.50146110.48403126.87057146.09821168.68516226.50812
47100.39650115.35097132.94539153.67263178.11942241.09861
48104.40840120.38826139.26321161.58790188.02539256.56453
49108.54065125.60185145.83373169.85936198.42666272.95840
50112.79687130.99791152.66708178.50303209.34800290.33590
51117.18077136.58284159.77377187.53566220.81540308.75606
52121.69620142.36324167.16472196.97477232.85617328.28142
53126.34708148.34595174.85131206.83863245.49897348.97831
54131.13750154.53806182.84536217.14637258.77392370.91701
| 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.
10.970870.966180.961540.956940.952380.94340
21.913471.899691.886101.872671.859411.83339
32.828612.801642.775092.748962.723252.67301
43.717103.673083.629903.587533.545953.46511
54.579714.515054.451824.389984.329484.21236
65.417195.328555.242145.157875.075694.91732
76.230286.114546.002055.892705.786375.58238
87.019696.873966.732746.595896.463216.20979
97.786117.607697.435337.268797.107826.80169
108.530208.316618.110907.912727.721737.36009
119.252629.001558.760488.528928.305417.88687
129.954009.663339.385079.118588.863258.38384
1310.6349610.302749.985659.682859.393578.85268
1411.2960710.9205210.5631210.222839.898649.29498
1511.9379411.5174111.1183910.7395510.379669.71225
1612.5611012.0941211.6523011.2340210.8377710.10590
1713.1661212.6513212.1656711.7071911.2740710.47726
1813.7535113.1896812.6593012.1599911.6895910.82760
1914.3238013.7098413.1339412.5932912.0853211.15812
2014.8774714.2124013.5903313.0079412.4622111.46992
2115.4150214.6979714.0291613.4047212.8211511.76408
2215.9369215.1671214.4511213.7844213.1630012.04158
2316.4436115.6204114.8568414.1477713.4885712.30338
2416.9355416.0583715.2469614.4954813.7986412.55036
2517.4131516.4815115.6220814.8282114.0939412.78336
2617.8768416.8903515.9827715.1466114.3751913.00317
2718.3270317.2853616.3295915.4513014.6430313.21053
2818.7641117.6670216.6630615.7428714.8981313.40616
2919.1884518.0357716.9837116.0218915.1410713.59072
3019.6004418.3920517.2920316.2888915.3724513.76483
3120.0004318.7362817.5884916.5443915.5928113.92909
3220.3887719.0688717.8735516.7888915.8026814.08404
3320.7657919.3902118.1476517.0228616.0025514.23023
3421.1318419.7006818.4112017.2467616.1929014.36814
3521.4872220.0006618.6646117.4610116.3741914.49825
3621.8322520.2904918.9082817.6660416.5468514.62099
3722.1672420.<*>705319.1425817.8622416.7112914.73678
3822.4924620.8410919.3678618.0499916.8678914.84602
3922.8082221.1025019.5844818.2296617.0170414.94907
4023.1147721.3550719.7927718.4015817.1590915.04630
4123.4124021.5991019.9930518.5661117.2943715.13802
4223.7013621.8348820.1856318.7235517.4232115.22454
4323.9819022.0626920.3707918.8742117.5459115.30617
4424.2542722.2827920.5488419.0183817.6627715.38318
4524.5187122.4954520.7200419.1563517.7740715.45583
4624.7754522.7009220.8846519.2883717.8800715.52437
4725.0247122.8994421.0429419.4147117.9810215.58903
4825.2667123.0912421.1951319.5356118.0771615.65003
4925.5016623.2765621.3414719.6513018.1687215.70757
5025.7297623.4556221.4821819.7620118.2559315.76186
5125.9512323.6286221.6174919.8679518.3389815.81308
5226.1662423.7957621.7475819.9693318.4180715.86139
5326.3749923.9572621.8726720.0663418.4934015.90697
5426.5776624.1133021.9929620.1591818.5651515.94998
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 years21.48218
tabular value for 10 years8.11090
the difference13.37128
mult. by20
gives267.4256
or2671. 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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ANEMOMETER
ANEMOSCOPE
ANGLE
ANGULAR
ANNUAL
* ANNUITIES
ANNULETS
ANNULUS
ANOMALOUS
ANOMALY
ANSER