ANNUITY (from Lat. annus, a year), a periodical payment, made annually, or at more frequent intervals, either for a fixed term of years, or during the continuance of a given life, or a combination of lives. In technical language an annuity is said to be payable for an assigned status, this being a general word chosen in preference to such words as " time," " term " or " period," because it may include more readily either a term of years certain, or a life or combination of lives. The magnitude of the annuity is the sum to be paid (and received) in the course of each year. Thus, if £roo is to be received each year by a person, he is said to have " an annuity of flood' If the payments are made halfyearly, it is sometimes said that he has " a halfyearly annuity of Itoo "; but to avoid ambiguity, it is more commonly said he has an annuity of ;loo, payable by halfyearly instalments. The former expression, if clearly understood, is preferable on account of its brevity. So we may have quarterly, monthly, weekly, daily annuities, when the annuity is payable by quarterly, monthly, weekly or daily instalments. An annuity
ANNUITY 75
is considered as accruing during each instant of the status for which it is enjoyed, although it is only payable at fixed intervals. If the enjoyment of an annuity is postponed until after the lapse of a certain number of years, the annuity is said to be deferred. If an annuity, instead of being payable at the end of each year, halfyear, &c., is payable in advance, it is called an annuitydue.
If an annuity is payable for a term of years independent of any contingency, it is called an annuity certain; if it is to continue for ever, it is called a perpetuity; and if in the latter case it is not to commence until after a term of years, it is called a deferred perpetuity. An annuity depending on the continuance of an assigned life or lives, is sometimes called a life annuity; but more commonly the simple term " annuity " is understood to mean a life annuity, unless the. contrary is stated. A life annuity, to cease in any event after a certain term of years, is called a temporary annuity. The holder of an annuity is called an annuitant, and the person on whose life the annuity depends is called the nominee.
If not otherwise stated, it is always understood that an annuity is payable yearly, and that the annual payment (or rent, as it is sometimes called) is £r. It is, however, customary to consider the annual payment to be, not £r, but simply r, the reader supplying whatever monetary unit he pleases, whether pound, dollar, franc, Thaler, &c.
The annuity is the totality of the payments to be made (and received), and is so understood by all writers on the subject; but some have also used the word to denote an individual payment (or rent), speaking, for instance, of the first or second year's annuity,—a practice which is calculated to introduce confusion and should therefore be carefully avoided.
Instances of perpetuities are the dividends upon the public stocks in England, France and some other countries. Thus, although it is usual to speak of £loo consols, the reality is the yearly dividend which the government pays by quarterly instalments. The practice of the French in this, as in many other matters, is more logical. In speaking of their public funds (rentes) they do not mention the ideal capital sum, but speak of the annuity or annual payment that is received by the public creditor. Other instances of perpetuities are the incomes derived from the debenture stocks of railway companies, also the feuduties commonly payable on house property in Scotland. The number of years' purchase which the perpetual annuities granted by a government or a railway company realize in the open market, forms a very simple test of the credit of the various governments or railways.
Terminable Annuities are employed in the system of British public finance as a means of reducing the National Debt (q.v.). This result is attained by substituting for a perpetual annual charge (or one lasting until the capital which it represents can be paid off en bloc), an annual charge of a larger amount, but lasting for a short term. The latter is so calculated as to pay off, during its existence, the capital which it replaces, with interest at an assumed or agreed rate, and under specified conditions. The practical effect of the substitution of a terminable annuity for an obligation of longer currency is to bind the present generation of citizens to increase its own obligations in the present and near future in order to diminish those of its successors. This end might be attained in other ways; for instance,.by setting aside out of revenue a fixed annual sum for the purchase and cancellation of debt (Pitt's method, in intention), or by fixing the annual debt charge at a figure sufficient to provide a margin for reduction of the principal of the debt beyond the amount required for interest (Sir Stafford. Northcote's method), or by providing an annual surplus of revenue over expenditure (the " Old Sinking Fund "), available for the same purpose. All these methods have been tried in the course of British financial history, and the second and third of them are still employed; but on the whole the method of terminable annuities has been the one preferred by chancellors of. the exchequer and by parliament.
Terminable annuities, as employed by the British government, fall under two heads:—(a) Those issued to, or held by private
persons; (b) those held by government departments or by funds under government control. The important difference between these two classes is that an annuity under (a), once created, cannot be modified except with the holder's consent, i.e. is practically unalterable without a breach of public faith; whereas an annuity under (b) can, if necessary, be altered by interdepartmental arrangement under the authority of parliament. Thus annuities of class (a) fulfil most perfectly the object of the system as explained above; while those of class (b) have the advantage that in times of emergency their operation can be suspended without any inconvenience or breach of faith, with the result that the resources of government can on such occasions be materially increased, apart from any additional taxation. For this purpose it is only necessary to retain as a charge on the income of the year a sum equal to the (smaller) perpetual charge which was originally, replaced by the (larger) terminable charge, whereupon the difference between the two amounts is temporarily released, while ultimately the increased charge is extended for a period equal to that for which it is suspended. Annuities of class (a) were first instituted in 18o8, but are at present mainly regulated by an act of 1829. They may be granted either for a specified life, or two lives, or for an arbitrary term of years; and the consideration for them may take the form either of cash or of government stock, the latter being cancelled when the annuity is set up. Annuities (b) held by government departments date from 1863. They have been created in exchange for permanent debt surrendered for cancellation, the principal operations having been effected in 1863, 1867, 187o, 1874, 1883 and 1899. Annuities of this class do not affect the public at all, except of course in their effect on the market for government securities. They are merely financial operations between the government, in its capacity as the banker of savings banks and other funds, and itself, in the capacity of custodian of the national finances. Savings bank depositors are not concerned with the manner in which government invests their money, their rights being confined to the receipt of interest and the repayment of deposits upon specified conditions. The case is, however, different as regards forty millions of consols (included in the above figures), belonging to suitors in chancery, which were cancelled and replaced by a terminable annuity in 1883. As the liability to the suitors in that case was for a specified amount of stock, special arrangements were made to ensure the ultimate replacement of the precise amount of stock cancelled.
Annuity Calculations.The mathematical theory of life annuities is based upon a knowledge of the rate of mortality among mankind in general, or among the particular class of persons on whose lives the annuities depend. It involves a mathematical treatment too complicated to be dealt with fully in this place, and in practice it has been reduced to the form of tables, which vary in different places, but which are easily accessible. The history of the subject may, however, be sketched. Abraham Demoivre, in his Annuities on Lives, propounded a very simple law of mortality which is to the effect that, out of 86 children born alive, r will die every year until the last dies between the ages of 85 and 86. This law agreed sufficiently well at the middle ages of life with the mortality deduced from the best observations of his time; but, as observations became more exact, the approximation was found to be not sufficiently close. This was particularly the case when it was desired to obtain the value of joint life, contingent or other complicated benefits. Therefore Demoivre's law is entirely devoid of practical utility. No simple formula has yet been discovered that will represent the rate of mortality with sufficient accuracy.
The rate of mortality at each age is, therefore, in practice usually determined by a series of figures deduced from observation; and the value of an annuity at any age is found from these numbers by means of a series of arithmetical calculations. The mortality table here given is an example of modern use.
The first writer who is known to have attempted to obtain, on correct mathematical principles, the value of a life annuity, was Jan De Witt, grand pensionary of Holland and West Friesland. Our knowledge of his writings on the subject is derived from twopapers contributed by Frederick Hendriks to the Assurance Magazine, vol. ii. p. 222, and vol. iii. p. 93. The former of these contains a translation of De Witt's report upon the value of life annuities, which was prepared in consequence of the resolution passed by the statesgeneral, on the 25th of April 1671, to negotiate funds by life annuities, and which was distributed to the members on the 3oth of July 1671. . The latter contains the translation of a number of letters addressed by De Witt to Burgomaster Johan Hudde, bearing dates from September 167o to October 1671. The existence of De Witt's report was well known among his contemporaries, and Hendriks collected a number of extracts from various authors referring to it; but the
Number Living and Dying at each Age, out of ro,000
entering at Age 10.
Age. Living. Dying. Age. Living. Dying.
10 ro,000 79 54 6791 129
11 9,921 0 55 6662 153
12 9,921 40 56 65o9 150
13 9,881 35 57 6359 152
14 9,846 40 58 6207 156
15 9,806 22 59 605 1 153
16 9,784 0 6o 5898 184
i
17 9,784 41 61 5714 186
r8 9,743 59 62 5528 191
19 9,684 68 63 5337 200
20 9,616 56 64 5137 206
21 9,560 67 65 4931 215
22 9,493 59 66 4716 220
23 9,434 73 67 4496 220
24 9,361 64 68 4276 237
25 9,297 48 69 4039 246
26 9,249 64 70 3793 213
27 9,185 6o 71 358o 222
28 9,125 71 72 3358 268
29 9,054 67 73 3090 243
30 8,987 74 74 2847 300
31 8,913 65 75 2547 241
32 8,848 74 76 2306 245
33 8,774 73 77 2061 224
34 8,701 . 76 78 1837 226
35 8,625 71 79 1611 219
36 8,554 75 8o 1392 196
37 8,479 81 81 1196 191
38 8,398 87 82 1005 173
39 8,311 88 83 832 172
40 8,223 81 84 66o 119
41 8,142 85 85 541 117
42 8,057 87 86 424 92
43 7,970 84 87 332 72
44 7,886 93 88 260 74
45 7,793 97 89 186 36
46 7,696 96 90 15o 34
47 7,600 107 91 116 36
48 7,493 Io6 92 8o 36
49 7,387 113 93 44 29
50 7,274 120 94 15 0
51 7,154 124 95 15 5
52 7,030 120 96 10 10
53 6,910 119
report is not contained in any collection of his works extant, and had been entirely lost for 18o years, until Hendriks discovered it among the state archives of Holland in company with the letters to Hudde. It is a document of extreme interest, and (notwithstanding some inaccuracies in the reasoning) of very great merit, more especially considering that it was the very first document on the subject that was ever written.
It appears that it had long been the practice in Holland for life annuities to be granted to nominees of any age, in the constant proportion of double the rate of interest allowed on stock; that is to say, if the towns were borrowing money at 6 %, they would be willing to grant a life annuity at 12 %, and so on. De Witt states that " annuities have been sold, even in the present century, first at six years' purchase, then at seven and eight; and that the majority of all life annuities now current at the country's expense were obtained at nine years' purchase "; but that the price had been increased in the course of a few years from eleven years' purchase to twelve, and from twelve to
fourteen. He also states that the rate of interest had been successively reduced from 6; to 5 %, and then to 4 %. The principal object of his report is to prove that, taking interest at 4 %, a life annuity was worth at least sixteen years' purchase; and, in fact, that an annuitant purchasing an annuity for the life of a young and healthy nominee at sixteen years' purchase, made an excellent bargain. It may be mentioned that he argues that it is more to the advantage, both of the country and of the private investor, that the public loans should be raised by way of grant of life annuities rather than perpetual annuities. It appears conclusively from De Witt's correspondence with Hudde, that the rate of mortality assumed as the basis of his calculations was deduced from careful examination of the mortality that had actually prevailed among the nominees on whose lives annuities had been granted in former years. De Witt appears to have come to the conclusion that the probability of death is the same in any halfyear from the age of 3 to 53 inclusive; that in the next ten years, from 53 to 63, the probability is greater in the ratio of 3 to 2; that in the next ten years, from 63 to 73, it is greater in. the ratio of 2 to 1; and in the next seven years, from 73 to 8o, it is greater in the ratio of 3 to I; and he places the limit of human life at 80. If a mortality table of the usual form is deduced from these suppositions, out of 212 persons alive at the age of 3, 2 will die every year up to 53, 3 in each of the ten years from 53 to 63, 4 in each of the next ten years from 63 to 73, and 6 in each of the next seven years from 73 to 8o, when all will be dead.
De Witt calculates the value of an annuity in the following way. Assume that annuities on Io,000 lives each ten years of age, which satisfy the Hm mortality table, have been purchased. Of these nominees 79 will die before attaining the age of r1, and no annuity payment will be made in respect of them; none will die between the ages of 11 and 12, so that annuities will be paid for one year on 9921 lives; 40 attain the age of 12 and die before 13, so that two payments will be made with respect to these lives. Reasoning in this way we see that the annuities on 35 of the nominees will be payable for three years; on 40 for four years, and so on. Proceeding thus to the end of the table, 15 nominees attain the age of 95, 5 of whom die before the age of 96, so that 85 payments will be paid in respect of these 5 lives. Of the survivors all die before attaining the age of 97, so that the annuities on these lives will be payable for 86 years. Having previously calculated a table of the values of annuities certain for every number of years up to 86, the value of all the annuities on the 10,000 nominees will be found by taking 40 times the value of an annuity for 2 years, 35 times the value of an annuity for 3 years, and so on—the last term being the value of to annuities for 86 years—and adding them together; and the value of an annuity on one of the nominees will then be found by dividing by ro,000. Before leaving the subject of De Witt, we may mention that we find in the correspondence a distinct suggestion of the law of mortality that bears the name of Demoivre. In De Witt's letter, dated the 27th of October 1671 (Ass. Mag. vol. iii. p. 107), he speaks of a " provisional hypothesis " suggested by Hudde, that out of 8o young lives (who, from the context, may be taken as of the age 6) about r dies annually. In strictness, therefore, the law in question might be more correctly termed Hudde's than Demoivre's.
De Witt's report being thus of the nature of an unpublished state paper, although it contributed to its author's reputation, did not contribute to advance the exact knowledge of the subject; and the author to whom the credit must be given of first showing how to calculate the value of an annuity on correct principles is Edmund Halley. He gave the first approximately correct mortality table (deduced from the records of the numbers of deaths and baptisms in the city of Breslau), and showed how it might be employed to calculate the value of an annuity on the life of a nominee of any age (see Phil. Trans. 1693; Ass. Mag. vol. xviii.).
Previously to Halley's time, and apparently for many years subsequently, all dealings with life annuities were based uponmere conjectural estimates. The earliest known reference to any estimate of the value of life annuities rose out of the requirements of the Falcidian law, which (40 B.c.) was adopted in the Roman empire, and which declared that a testator should not give more than threefourths of his property in legacies, so that at least onefourth must go to his legal representatives. It is easy to see how it would occasionally become necessary, while this law was in force, to value life annuities charged upon a testator's estate. Aemilius Macer (A.D. 230) states that the method which had been in common use at that time was as follows:—From the earliest age until 30 take 30 years' purchase, and for each age after 30 deduct i year. It is obvious that no consideration of compound interest can have entered into this estimate; and it is easy to see that it is equivalent to assuming that all persons who attain the age of 30 will certainly live to the age of 6o, and then certainly die. Compared with this estimate, that which was propounded by the praetorian prefect Ulpian was a great improvement. His table is as follows:
Age. Years' Age. Years'
Purchase. Purchase.
Birth to 20 , 30 45 to 46 14
20 „ 25 28 46 ,, 47 13
25 „ 30 25 47 ,, 48 12
30 ,, 35 22 48 ,, 49 II
35 ,, 40 20 49 ,, 50 to
40 „ 41 19 50 „ 55 9
41 „ 42 18 55 „ 60 7
42 ,, 43 17 6o and c
43 ,, 44 16 upwards 1
44 „ 45 15
Here also we have no reason to suppose that the element of interest was taken into consideration; and the assumption, that between the ages of 40 and 50 each addition of a year to the nominee's age diminishes the value of the annuity by one year's purchase, is equivalent to assuming that there is no probability of the nominee dying between the ages of 40 and 50. Considered, however, simply as a table of the average duration of life, the values are fairly accurate. At all events, no more correct estimate appears to have been arrived at until the close of the 17th century.
The mathematics of annuities has been very fully treated in Demoivre's Treatise on Annuities (1725); Simpson's Doctrine of Annuities and Reversions (1742); P. Gray, Tables and Formulae; Baily's Doctrine of Life Annuities; there are also innumerable compilations of Valuation Tables and Interest Tables, by means of which the value of an annuity at any age and any rate of interest may be found. See also the article INTEREST, and especially that on
End of Article: ANNUITY (from Lat. annus, a year) 

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