**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 £100 is to be received each year by a person, he is said to have “an annuity of £100.” If the payments are made half-yearly, it is sometimes said that he has “a half-yearly annuity of £100”; but to avoid ambiguity, it is more commonly said he has an annuity of £100, payable by half-yearly 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 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, half-year, &c., is payable in advance, it is called an annuity-due.

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 £1. It is, however, customary to consider the annual payment to be, not £1, but simply 1, 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 £100 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 feu-duties 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 1808, 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, 1870, 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, 1 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 two papers 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 states-general, on the 25th of April 1671, to negotiate funds by life annuities, and which was distributed to the members on the 30th 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 1670 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 report is not contained in any collection of his works extant, and had been entirely lost for 180 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.

Age. | Living. | Dying. | Age. | Living. | Dying. |
---|---|---|---|---|---|

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 |
10,000 9,921 9,921 9,881 9,846 9,806 9,784 9,784 9,743 9,684 9,616 9,560 9,493 9,434 9,361 9,297 9,249 9,185 9,125 9,054 8,987 8,913 8,848 8,774 8,701 8,625 8,554 8,479 8,398 8,311 8,223 8,142 8,057 7,970 7,886 7,793 7,696 7,600 7,493 7,387 7,274 7,154 7,030 6,910 |
79 0 40 35 40 22 0 41 59 68 56 67 59 73 64 48 64 60 71 67 74 65 74 73 76 71 75 81 87 88 81 85 87 84 93 97 96 107 106 113 120 124 120 119 |
54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 |
6791 6662 6509 6359 6207 6051 5898 5714 5528 5337 5137 4931 4716 4496 4276 4039 3793 3580 3358 3090 2847 2547 2306 2061 1837 1611 1392 1196 1005 832 660 541 424 332 260 186 150 116 80 44 15 15 10 |
129 153 150 152 156 153 184 186 191 200 206 215 220 220 237 246 213 222 268 243 300 241 245 224 226 219 196 191 173 172 119 117 92 72 74 36 34 36 36 29 0 5 10 |

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 6¼ 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 half-year 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 80, it is greater in the ratio of 3 to 1; 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 80, when all will be dead.

De Witt calculates the value of an annuity in the following way. Assume that annuities on 10,000 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 11, 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 10 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 10,000. 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 80 young lives (who, from the context, may be taken as of the age 6) about 1 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 upon mere 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 three-fourths of his property in legacies, so that at least one-fourth 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 1 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 60, 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’ Purchase. |
Age. | Years’ Purchase. |
---|---|---|---|

Birth to 20 20 ” 25 25 ” 30 30 ” 35 35 ” 40 40 ” 41 41 ” 42 42 ” 43 43 ” 44 44 ” 45 |
30 28 25 22 20 19 18 17 16 15 |
45 to 46 46 ” 47 47 ” 48 48 ” 49 49 ” 50 50 ” 55 55 ” 60 60 and upwards |
14 13 12 11 10 9 7 5 |

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 Insurance.

*Commutation tables*, aptly so named in 1840 by Augustus De Morgan (see his paper “On the Calculation of Single Life Contingencies,” *Assurance Magazine*, xii. 328), show the proportion in which a benefit due at one age ought to be changed, so as to retain the same value and be due at another age. The earliest known specimen of a commutation table is contained in William Dale’s *Introduction to the Study of the Doctrine of Annuities*, published in 1772. A full account of this work is given by F. Hendriks in the second number of the *Assurance Magazine*, pp. 15-17. William Morgan’s *Treatise on Assurances*, 1779, also contains a commutation table. Morgan gives the table as furnishing a convenient means of checking the correctness of the values of annuities found by the ordinary process. It may be assumed that he was aware that the table might be used for the direct calculation of annuities; but he appears to have been ignorant of its other uses.

The first author who fully developed the powers of the table was John Nicholas Tetens, a native of Schleswig, who in 1785, while professor of philosophy and mathematics at Kiel, published in the German language an *Introduction to the Calculation of Life Annuities and Assurances*. This work appears to have been quite unknown in England until F. Hendriks gave, in the first number of the *Assurance Magazine*, pp. 1-20 (Sept. 1850), an account of it, with a translation of the passages describing the construction and use of the commutation table, and a sketch of the author’s life and writings, to which we refer the reader who desires fuller information. It may be mentioned here that Tetens also gave only a specimen table, apparently not imagining that persons using his work would find it extremely useful to have a series of commutation tables, calculated and printed ready for use.

The use of the commutation table was independently developed in England—apparently between the years 1788 and 1811—by George Barrett, of Petworth, Sussex, who was the son of a yeoman farmer, and was himself a village schoolmaster, and afterwards farm steward or bailiff. It has been usual to consider Barrett as the originator in England of the method of calculating the values of annuities by means of a commutation table, and this method is accordingly sometimes called Barrett’s method. (It is also called the commutation method and the columnar method.) Barrett’s method of calculating annuities was explained by him to Francis Baily in the year 1811, and was first made known to the world in a paper written by the latter and read before the Royal Society in 1812.

By what has been universally considered an unfortunate error of judgment, this paper was not recommended by the council of the Royal Society to be printed, but it was given by Baily as an appendix to the second issue (in 1813) of his work on life annuities and assurances. Barrett had calculated extensive tables, and with Baily’s aid attempted to get them published by subscription, but without success; and the only printed tables calculated according to his manner, besides the specimen tables given by Baily, are the tables contained in Babbage’s *Comparative View of the various Institutions for the Assurance of Lives*, 1826.

In the year 1825 Griffith Davies published his *Tables of Life Contingencies*, a work which contains, among others, two tables, which are confessedly derived from Baily’s explanation of Barrett’s tables.

Those who desire to pursue the subject further can refer to the appendix to Baily’s *Life Annuities and Assurances*, De Morgan’s paper “On the Calculation of Single Life Contingencies,” *Assurance Magazine*, xii. 348-349; Gray’s *Tables and Formulae* chap. viii.; the preface to Davies’s *Treatise on Annuities*; also Hendriks’s papers in the *Assurance Magazine*, No. 1, p. 1, and No. 2, p. 12; and in particular De Morgan’s “Account of a Correspondence between Mr George Barrett and Mr Francis Baily,” in the *Assurance Magazine*, vol. iv. p. 185.

The principal commutation tables published in England are contained in the following works:—David Jones, *Value of Annuities and Reversionary Payments*, issued in parts by the Useful Knowledge Society, completed in 1843; Jenkin Jones, *New Rate of Mortality*, 1843; G. Davies, *Treatise on Annuities*, 1825 (issued 1855); David Chisholm, *Commutation Tables*, 1858; Nelson’s *Contributions to Vital Statistics*, 1857; Jardine Henry, *Government Life Annuity Commutation Tables*, 1866 and 1873; *Institute of Actuaries Life Tables*, 1872; R. P. Hardy, *Valuation Tables*, 1873; and Dr William Farr’s contributions to the sixth (1844), twelfth (1849), and twentieth (1857) *Reports* of the Registrar General in England (English Tables, 1. 2), and to the *English Life Table*, 1864.

The theory of annuities may be further studied in the discussions in the English *Journal of the Institute of Actuaries*. The institute was founded in the year 1848, the first sessional meeting being held in January 1849. Its establishment has contributed in various ways to promote the study of the theory of life contingencies. Among these may be specified the following:—Before it was formed, students of the subject worked for the most part alone, and without any concert; and when any person had made an improvement in the theory, it had little chance of becoming publicly known unless he wrote a formal treatise on the whole subject. But the formation of the institute led to much greater interchange of opinion among actuaries, and afforded them a ready means of making known to their professional associates any improvements, real or supposed, that they thought they had made. Again, the discussions which follow the reading of papers before the institute have often served, first, to bring out into bold relief differences of opinion that were previously unsuspected, and afterwards to soften down those differences,—to correct extreme opinions in every direction, and to bring about a greater agreement of opinion on many important subjects. In no way, probably, have the objects of the institute been so effectually advanced as by the publication of its *Journal*. The first number of this work, which was originally called the *Assurance Magazine*, appeared in September 1850, and it has been continued quarterly down to the present time. It was originated by the public spirit of two well-known actuaries (Mr Charles Jellicoe and Mr Samuel Brown), and was adopted as the organ of the Institute of Actuaries in the year 1852, and called the *Assurance Magazine and Journal of the Institute of Actuaries*, Mr Jellicoe continuing to be the editor,—a post he held until the year 1867, when he was succeeded by Mr T. B. Sprague (who contributed to the 9th edition of this *Encyclopaedia* an elaborate article on “Annuities,” on which the above account is based). The name was again changed in 1866, the words “Assurance Magazine” being dropped; but in the following year it was considered desirable to resume these, for the purpose of showing the continuity of the publication, and it is now called the *Journal of the Institute of Actuaries and Assurance Magazine*. This work contains not only the papers read before the institute (to which have been appended of late years short abstracts of the discussions on them), and many original papers which were unsuitable for reading, together with correspondence, but also reprints of many papers published elsewhere, which from various causes had become difficult of access to the ordinary reader, among which may be specified various papers which originally appeared in the *Philosophical Transactions*, the *Philosophical Magazine*, the *Mechanics’ Magazine*, and the *Companion to the Almanac*; also translations of various papers from the French, German, and Danish. Among the useful objects which the continuous publication of the *Journal* of the institute has served, we may specify in particular two:—that any supposed improvement in the theory was effectually submitted to the criticisms of the whole actuarial profession, and its real value speedily discovered; and that any real improvement, whether great or small, being placed on record, successive writers have been able, one after the other, to take it up and develop it, each commencing where the previous one had left off.