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 +* )" ; and the value of the deferred annuity to continue for n years is therefore
[ math ]
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 -f-t )" 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 +* )" 1 ; and by proportion the amount of an annuity of 1 similarly improved will be r*.
One of the principal applications of the theory of annui ties 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 j 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 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. 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 desir able for various purposes, and in particular for the ascer tainment 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 instal ment of the principal. The problem of determining the separate amounts of these has been considered by Turn- bull, Tables, p. 128 ; and by Gray, Ass. 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 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.) The amount of 1 in any number of years, n ; or
(2.) The present value of 1 due in any number of years, n; or (1 +i) n .
(3.) The amount of an annuity of 1 in any number of (l+i)-l years, n; or s ?
(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 num ber 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; i or j- 7 - .
, 1
The following table, on p. 75, in which the rate of in terest 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 (6) were altered so as to relate to annuities payable in advance (or annuities-due). In that case (3) would give the amount at com pound 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 ; that is to say, the values of the functions - - and 7equalsign . +1 _[_> 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
