Thomas Bond Sprague

“ An annuity de.- pending 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. ”

“ Thus, for example, if a country gentleman of the age of 40 wished to borrow 10,000 upon a landed estate, the law forbade him to pay, or the lender to receive, more than 5 per cent, interest, say 500 a year ; but the law did not forbid his granting an annuity of 1000 for his life, secured upon the estate. Speaking roughly, an annual payment of 300 would be required to insure 10,000 upon the borrower s life, and the annuity would therefore return the lender about 7 per cent, interest, in addition to the premium on the insurance necessary to return his capital. ”

“ All the preceding methods require a considerable amount of calculation in order to obtain the value of an annuity on a life of any particular age. We will now explain some methods of approximation, by means of which we can calculate with much less labour the value of an annuity at a single age, when we do not require a complete table of annuities. ”

“ But it is un certain that the life will exist to the end of a year, and the value of the annuity must therefore be reduced in the proportion of this uncertainty, or be multiplied by the probability that the given life will survive a year. ”

“ BY an annuity is meant 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, as will be more fully explained further on. ”

“ The fact that Gompertz s law does not correctly represent the mortality throughout the whole of life, proves that the above-described practical method of finding the value of an annuity on three joint lives is accurate only in certain cases. ”

“ 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 arith metical calculations. ”

“ Landed proprietors also, who possess only a life interest in their property, have been authorised by various Acts of Parliament to borrow money for the purpose of improving their estates, and can grant a rent- charge upon the fee-simple for a term not exceeding 30 years. These are very favourite investments with the life insurance companies of the country, as they are thus enabled to obtain a somewhat higher interest from 4 to 4f per cent. ”

“ We learn from De Wit s above-mentioned report, that the Governments of Holland and West Fries- land had granted annuities systematically for one hundred and fifty years before any correct estimate was formed of the value of a life annuity. The British Government has at various times granted life annuities, more especially on the Tontine principle, for the purpose of raising money when it was difficult to obtain the sums required for the public service by the ordinary methods. ”

“ The theory of continuous annuities has been fully developed by Woolhouse (Ass. Mag., xv. 95) . Assuming the number living in the mortality table at any age x to be represented by l x , the value of a continuous annuity on a nominee 1 /"*oo -I y-oo *- ., of the age x is j- I l x ifdx eq j I l x e Sx dx, putting m^/ x IxJ x I 8 eq log e (l+t) . From the nature of the case, l x must be a function that is never negative for positive values of x ; and as x becomes larger, l x must continually diminish, and must vanish when x becomes infinite. ”

“ As the reader will have observed, neither Gompertz s nor Makeham s formula represents correctly the rate of mortality for very young ages. Various formulas have been given which are capable of representing with sufficient accuracy the number living at any age from birth to extreme old age, but they are all so complicated that they are of little "more than theoretical interest. ”

“ 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. ”

“ The computer usually has recourse to one of the tables which have been published, containing values and amounts calculated for various rates of interest. An extensive set of tables of this kind was published in 172G by John Smart ; and many subsequent writers, as Dr Price, Baily, Milne, Davies, D. Jones, J. Jones, have reprinted or abridged portions of these tables. They show the amount and the present value both of a payment and of an annuity of 1 for every term of years not exceeding 100, at the several rates of interest, 2, 2, 3, 3
(1.) ”

“ This may be defined as the pro portion of the persons of that age who would die in the course of a year, if the intensity of the mortality remained constant for a year, and the number of persons under obser vation also remained constant, the places of those who die being constantly replaced by fresh lives. ”

“ But we have seen above that the value of ^- (1*03) - 10 = 664779, and that a w = 16*4744; and multiplying these together, we get the value of the deferred annuity, 10 9518, as before.
We have, in conformity with popular usage, called our auxiliary table a " D and N table." It is also called a " commutation table," a name proposed by De Morgan in his paper " On the Calculation of Single Life Contingencies," which appeared in the Companion to the Almanac for the year 1840, and which is reprinted in the Assurance Magazine, xii. ”

“ In dealing with annuities in which three lives are in volved, we are met by the difficulty that no tables exist iu which the values of such annuities are given to the extent required in practice. Such tables as those computed for the Carlisle 3 per cent, table by Herschel Filipowski are of too limited extent to be of any practical utility; for the values being given only for certain ages differing by multiples of five years, a considerable amount of labour is required to deduce the values for other ages. ”

“ Various local bodies have at different times raised money on the security of the local rates in consideration of the grant of life annuities ; and, at the present time, the Manchester Cor poration grants annuities on favourable terms for the pur pose of obtaining funds to defray the expense of the water works belonging to the city. During the existence of the usury laws, it was very common for persons borrowing money upon the very best security to grant annuities upon their lives in consideration of a present advance. ”

“ For a temporary insurance for n years,
The annual premiums payable during life for the same benefits are found by substituting N. e _ 1 for D x in the denominator; and the annual premiums payable for m years, by putting N,_, - N xini _ l in the denominator instead of D.,..
Before quitting this subject, we should mention that in practice other columns are added to the table besides the D and N columns. A column, S, is given for the purpose of calculating the values of increasing annuities; a column, M, for calculating the values of assurances ”

“ But when a very large num ber of drawings are made, we can prove that the ratio of white balls drawn to the black will differ very little from the ratio of a to b, and will exactly equal it if the number of drawings is supposed to be indefinitely large. ”

“ The student who wishes to pursue the subject more thoroughly, and to become acquainted with all the improvements in the theory of annuities that have been introduced of late years, should care fully study the various articles contributed to the Journal of the Institute of Actuaries, particularly those of Woolhouse and Make- ham. ”

“ The existence of De Wit s report was well known among his contemporaries, and Mr Hendriks has 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 Mr Hendriks conceived the happy idea of searching for it among the state archives of Holland, when it was found together with the letters to Hudde. ”

“ The employment of formulas such as those given in the last paragraph, and the application of the differential calculusfto the theory of life contingencies, have naturally led to an improvement in the theory which is probably destined to become of very great importance we refer to the introduction of the idea of " continuous " annuities and assurances. ”

“ Assuming the number now to increase indefinitely, or interest to be convertible momently, the above amount becomes e *, where e is the base of the
natural (or Napierian) logarithms.
In consequence of the above-mentioned practice as to half-yearly interest, the values given in Smart s tables for the odd half-years, though theoretically correct, are prac tically useless, and they have been superseded by the other tables above mentioned. ”

“ Age. ^Number Number Age. Number Number Age. Number Number Number | dying in Number dying in Number dying in Number dying in Number dying in Number dvingin alive at the fol alive at the fol alive at the fol alive at the fol alive at the fol alive at the fol each age. lowing each age. lowing each age. lowing each age. lowing each age. lowing each age. ”

“ Neison s Contributions to Vital Statistics, 1857. Mortality of England and Wales (males) , 3, 3, 4, 44, 5, 6, 7, 8, 9, 10 per cent., columns D and N, with logarithms ; two joint lives, males, 7 per cent., columns D and N ; also D and N columns relating to the mortality of master mariners, and to that among friendly societies, and in particular the Manchester Unity.
Jardiue Henry, Government Life Annuity Commutation Tables, 1866 and 1873, single lives male and female, 0, 1, 2, 2, 3, 3, 4, 4, 5, 5J, 6, 7, 8, 9, 10 per cent.
Institute of Actuaries Life Tables, 1872. ”

“ Other instances of perpetuities are the incomes derived from the deben ture stocks now issued so largely by various railway com panies, also the feu-duties commonly payable on house property in Scotland. ”

“ He gives the tables (1) , (2) , (3) , (4) , and (5) , at the rates of interest 3, 3i, 4, 5, 6 per cent., to any number of years up to 105. Hardy s Doctrine, of Simple ami Compound Iiitcrcst, 1839, contains tables (1) , (2) , (3) , (4) , for the rates of interest \, 4, f, 1, li, 14, IJ, 2, 2
A few words may be here added as to the practical method of constructing compound interest tables. ”