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. 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 £1 00 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. When an annuity is deferred for any number of years, say n, it is said, indifferently, to commence, or to be entered upon, after n years,'or to run from the end of n years; and if it is payable yearly, the first payment will be made at the end of (π+l) years; if half-yearly, the first half-yearly payment will be made at the end of (ra + ∣) years; if quarterly, the first quarterly payment will be made at the end of (n + ∣) years; and so on. 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. Of late years, however, it has become 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. It is much to be desired that this course should be followed in any tables that may be published in future.

The annuity, it will be observed, 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.

The theory of annuities certain is a simple application of algebra to the fundamental idea of compound interest. According to this idea, any sum of money invested, or put out at interest, is increased at the end of a year ^by the addition to it of interest at a certain rate; and at the end of a second year, the interest of the first year, as well as the original sum, is increased in the same proportion, and so on to the end of the last year,—the interest being, in technical language, converted into principal yearly. Thus, if the rate of interest is 5 per cent., £1 improved at interest will amount at the end of a year to £1’05, or, as we shall in future say, in conformity with a previous remark, 1 will at the end of a year amount to 1Ό5. At the end of a second year this will be increased in the same ratio, and then amount to (Γ05) 2 . In the same way, at the end of a third year, it will amount to (1O5) 3 , and so on.

Let i denote the interest on 1 for a year; then at the end of a year the amount of 1 will be 1 + i. Reasoning as above, at the end of two years the amount will be (1 +i) 2 , at the end of three years (1 +t) 3 , and so on. In general, at the end of n years the amount will be (1 +i)“; or this is the amount of 1 at compound interest in n years. The present value of a sum, say 1, payable at the end of n years, is such a sum as, being improved at compound interest for n years, will exactly amount to 1. We have seen that 1 will in n years amount to (1 +t)", and by proportion we easily see that the sum which in n years will amount to 1, must be > or (1 +i)~∙. It is usual to put V for ~j~pρ so t ħat v is the value of 1 to be received at the end of a year, and tΓ the value of 1 to be received at the end of n years.

Since 1 placed out at interest produces i each year, we see that a perpetuity of i is equal in value to 1; hence, by proportion, a perpetuity of 1 is equal in value to -.. At ⅛

5 per cent., t=O5, and -=»20; or a perpetuity is worth 20 years’ purchase: at 4 per cent., it is worth 25 years’ purchase, = 25^: at 3 per cent., it is worth 33⅜ years’ purchase, (,⅛ = 33⅜) .

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, this £100 is a mere conception or ideal sum; and the reality is the £3 a year which the Government pays by half-yearly instalments. The practice of the French in this, as in many other matters, is more logical In speaking of their public funds, 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 now issued so largely by various 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 realise in the open market, forms a very simple te3t of the credit of the various governments or railways. Thus at the present time (May 1874) the British perpetual annuity of £3, derived from the 3 per cent, consols, is worth £93, or 31 years’ purchase; and a purchaser thus obtains 3’226 per cent, interest on his investment. Other examples are given in the subjoined tables, the figures in which are deduced from the Stock Exchange quotations of the irredeemable stocks issued by the various govern-mp∏t,R∙- » . . [table]

The following are a few other examples of perpetui« ties ·— τ . . [table]

We may mention in passing that the more usual practice of foreign governments when borrowing is not to grant the lender a perpetual annuity, but to issue to him bonds of say £100 each, bearing an agreed rate of interest, these bonds being usually issued at a discount, and redeemed at par by annual drawings during a specified term of years.

We have seen that the present value of any sum payable at the end of n years is found by multiplying it by (1+i) - "; hence the value of a perpetuity of 1 deferred n years is ^ 1+ 9~ · Nowan annuity for n years is clearly the difference between the value of a perpetuity to commence at once and a perpetuity deferred n years; its value is therefore τ - -—≠- = —; —> 0r P utt i n g « for the value of the annuity, we have— a = 1 -< 1 + i >-.

By means of this equation, having any two of the three quantities, a, i, n, we can determine the third either exactly or approximately. Thus for n we have—

log (l-tα), log (1+0 '

There is no means of determining the value of t exactly, but it may be found to any degree of accuracy required by methods of approximation which our limits will not allow us to describe.

If the annuity for n years is not to be enjoyed at once, but only after the lapse of t years, its value will be reduced in the proportion of 1 to the value of 1 payable in t years, or 1: (1 + i)~ t ', and the value of the deferred annuity to continue for n years is therefore—

It Remains to find the amount at compound interest at the end of n years of an annuity payable for that term. The amount of 1 in n years being (1 +i) n , its increase in that time is (1 + i) n - 1; but this increase arises entirely from the simple interest, i, of 1 being laid up at the end of each year and improved at compound interest during the remainder of the term. Hence it follows that the amount at compound interest of an annuity of i in n years must be (1 +⅛)" - 1; and by proportion the amount of an annuity of 1 similarly improved will be -- + 9—- ·

One of the principal applications of the theory of annuities certain is the valuation of leasehold property; another is the calculation of the terms of advances in consideration of an annuity certain for a term of years. At present a large sum of money is annually borrowed by corporations and other public bodies upon the security of local rates in the United Kingdom. It is sometimes arranged in these transactions that a fixed portion of the loan shall be paid off every year; but it is more commonly the case that, in consideration of a present advance, an annuity is granted for a term of years, usually 25 or 30, but in some instances extending to 50. 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 rentcharge upon the fee-simple for a term not exceeding 30 pears. 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 4∣ per cent.—than they could obtain upon ordinary mortgages with equally good security; the reason for this, of course, being that these loans are not so suitable as others for private lenders. In this case, as in all others, the price is determined by the laws of supply and demand; and the number of lenders being less than in the case of ordinary mortgages, the terms paid by the borrowers are higher. When a loan is arranged in this way, it is desirable for various purposes, and in particular for the ascertainment of the proper amount of income-tax, to consider each year’s payment as consisting partly of interest on the outstanding balance of the loan and partly as an instalment of the principal. The problem of determining the separate amounts of these has been considered by Turnbull, Tables, p. 128; and by Gray, Λss. Mag., xi. 172.

In making calculations for these and similar purposes, it is but seldom necessary to use the formulas given above. 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 1726 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, 3⅜, 4, 5, 6, 7, 8, 9, 10 per cent.; the five most important tables giving—

(1.) The amount of £1 in any number of years, n ; or (l+⅛)*.

(2.) The present value of £1 due in any number of years, n∙, or(l+i)~".

(3.) The amount of an annuity of £1 in any number of (l+i)∙-l years, n; or i —

(4.) The present value of an annuity of £1 for any number of years, n; or —— : —-— ·

(5.) The annuity which £1 will purchase for any number of years, n; or -— ∖ ·

The scheme would be more complete if we add, with Corbaux, whose tables will be described below— (6.) The annuity which would amount to £1 in n years; or (i + i)∙-ι∙

Smart gives the values of these functions not only for all integral values of n up to 100, but also for the half-years. In this respect, however, he has not been followed by subsequent writers. Some (as Milne) have stated that he “has given the amounts and the present values both of sums and annuities certain, wrong for every odd number of half-years.” This remark requires some explanation. The fact is that Smart’s tables are quite correct, but are not applicable according to the conventions that are usually observed in practice. Thus, for example, he gives the amount of 1 at the end of half a year at 4 per cent, interest as equal to √Γ04 = lO1980390, this being the strictly correct theoretical value at 4 per cent, interest. But in practice the invariable rule is, when interest is calculated for a shorter term than a year, to take a proportionate part of the year’s interest, thus virtually charging a higher rate for the odd time, so that in the above case the amount at the end of half a year will be 1Ό2. This circumstance renders it undesirable to tabulate the amounts and values for half-years; and this is never now done. The course adopted by Smart of giving the values and amounts of annuities for the odd half-years is open to more serious objection. His figures in these cases are merely the values of the algebraical functions above given, corresponding to the values of n, ⅜, 1∣, 2∣, &c., and do not give the amounts and values of annuities payable half-yearly. In fact, we are unable to see that any practical use could ever be made of his results in these cases.

. The following table, on p. 75, in which the rate of interest is 5 per cent., will serve to illustrate the nature of the tables in question, as reprinted by Baily, D. Jones, and others.

It will be seen that the figures in the column numbered (2) are the reciprocals of those in (1), and the figures in column (5) the reciprocals of those in (4). Also, that the figures in (4) are the sums of the first 1, 2, 3, &c., terms of (2). Again, the figures in (3) are derived by the successive addition of those in (1) to the first term, 1 ’ 000000; and the figures in (4) are equal to the product of those in (2) and (3). We have added the column (6) from Cor-baux’s tables. These figures are the reciprocals of those in (3), and are equal to the product of those in (5) and (2), while the figures in (5) are the products of those in (1) and (6).

It would perhaps be more convenient in practice if tables (3) and (6J were altered so as to relate to annuities payable in advance (or annuities-due). In that case (3) would give the amount at compound interest in n years of an annuity-due of 1, and (6) the annuity-due which would, at compound interest, amount to 1 in n years;

(l+i) n + 1 -l-i that is to say, the values of the functions and (1 + ⅞)**+ 1 1 ' ¾ > respectively. One very common application of table (3) is to find the amount of the premiums paid upon a life policy, and these premiums are always payable in advance. If that table were arranged as here suggested, the figures contained in it would be derived from those in (1), in precisely the same way as Table of Amounts, Present Values, &c., at 5 per cent. Interest, i = ∙05. [table]

(4) from (2). It would also be an improvement, for a reason to be mentioned presently, if the heading of the tables were altered, so that, for example, instead of (1) being called a table of the amounts of “£1” at the end of any number of “years,” it were called a table of the amounts of “1” at the end of any number of “terms.” It is not to be understood that the tables are arranged in the, manner here shown. Smart gives, in his First Table of Compound Interest, the values of our (1) for the various rates of interest arranged side by side; in his Second Table he gives the values of our (2) at diiferent rates of interest similarly arranged; and so for (3), (4), and (5). This arrangement has been followed by most authors, not only by those mentioned above as having copied Smart’s tables, but also by Chisholm, who states that the compound interest tables in his work (Commutation Tables, 1858) have been specially computed for it. He gives the tables (1), (2), (3), (4), and (5), at the rates of interest 3, 3⅛, 4, 5, 6 per cent., to any number of years up to 105. Hardy s Doctrine of Simple and Compound Interest, 1839, contains tables (1), (2), (3), (4), for the rates of interest ⅛, ⅜, f, 1, 1∣, 1 i, 1J, 2, 2⅜, 2⅜, 2∣, 3, 3∣, 3⅜, 3∣, 4, 4⅜, 4⅞, 4∣, 5, 6, 7, and 8 percent., for any number of years up to 100. The arrangement in our specimen is that adopted by Rance in his Tables of Compound Interest, 1852. He gives tables (1), (2), (3), and (4), at rates of interest proceeding, by steps of ⅛ per cent., from ∣ to 10 percent., for any number of years from 1 to 100. The same arrangement is also followed in the tables appended to Francis Corbaux’s Doctrine of Compound Interest, 1825; but this author gives in one opening of the book not only the six tabular results in our specimen, but also the corresponding results when interest at the nominal rate of 5 per cent, is payable half-yearly and quarterly, thus giving in one opening 24 tabular results side by side. He gives results for every quarter of a year up to 16 years, and then for each year up to 100; and his rates of interest range from 3 to 6 per cent., at intervals of ∣ per cent. The only other tables we think it needful to mention are Turnbull’s, 1863. His work gives tables (1), (2), (3), (4), and (5), for the Tates of interest 3, 3⅜, 4, 4⅜, 5, and 6, payable yearly; for 3, 3⅛, 3⅜, 3f, 4, 4⅛, 4⅜, 4J, 5, 5⅜, payable half-yearly; and 3, 3⅜, 4, 4⅛, 5, 6, payable quarterly. The results extend in every case to 80 terms of conversion—that is, 80 years, 80 half-years, and 80 quarters respectively. The arrangement of the tables differs from that cf all the others mentioned, as each page contains only the results pertaining to one table at one rate of interest. The results are given not only in decimals, but in £ s. d. in addition, which also is unusual.

A few words may be here added as to...