Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/85

Table of Amounts, Present Values, &c., at 5 per cent. Interest.
$i=\cdot 05.$

(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 different 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, 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. The formulas we have found above are not directly used for the calculation of the greater part of the tabular results ; but these are in practice deduced the one from the other by continuous processes, the values found by the formulas being used at intervals for the purpose of verification. Smart gives, on page 47 of his work, a description of the method he has employed, and the subject has been fully dealt with by Gray in Lis Tables and Formula, chap. 2. Since the publication of that work, the Arithmometer of M. Thomas (of Colmar) has come into extensive use for the formation of tables of this kind. For a descrip tion of the instrument, and some of its uses, the reader is refem-d to the papers in the Assurance Magazine by Major-General Hannyngton, vol. xvi. p. 244 ; Mr W. J. Hancock, xvi. 265 ; and by Gray, xvii. 249 ; xviii. 20 and 123.

Hitherto we have considered the annuity payments to be all made annually ; and the case where the payments are made more frequently now requires attention, First, sup pose that the annuity is payable by half-yearly instalments ; then, in order to find the present value of the annuity, we have first to answer the question, What is the value of a sum payable ia six months time 1 and, in order to find the amount of the annuity in n years, we must first deter mine what is the amount of a sum at the end of six months. The annual rate of interest being i, it may be supposed at first sight that the amount of 1 at the end of six months will be 1 + - ; but if this were the case, the 2i amount at the end of a second period of six months would (t\ 2 1 + - ) , or 1 + i + - . But this is contrary to our original assump tion that the annual interest is i, and the amount at the end of a year therefore 1 + i, In fact, if we suppose the interest on 1 for half a year to be -, the interest on it for . In order that the amount a year will not be i, but at the end of a year may be 1 + i, the amount at the end of six months must be such a quantity as, improved at the same rate for another six months, will be exactly 1 + i ; hence the amount at the end of six months must be ^fl + i, or (1 + 1)*. Reasoning in the same way, it is easy to see that, the true annual rate of interest being i, the amount of 1 in any number of years, n, whether integral or fractional, will always be (1 + t)". Hence, by similar reasoning to that pursued above, the present value of 1 payable at the end of any number of years, n, whether integral or frac tional, will always be (1 + i)" or v*. It is now easily seen we omit the demonstrations for the sake of brevity that the present value of an annuity payable half-yearly for n years (?4 being integral) is ^.i-l+J . 1 -(!+*)". and ttat tlie amount O f a similar 2 ^ annuity at the end of n years is %SI:f - f 1 + *)"-*. 2 i

It is to be observed, however, that when we are dealing with half-yearly payments in practice, the interest is never calculated in the way we have here supposed. On the contrary, the nominal rate of interest being $i$, the rate paid half-yearly is ${\frac {i}{2}}$, so that the true annual rate in practice is $i+{\frac {i^{2}}{2}}$; for instance, if interest on a loan is payable half-yearly, at the rate of 5 per cent. per annum, the true rate of interest is ·050625, or £5, 1s. 3d. per. per £100. Under these circumstances interest is said to be convertible into principal twice a year. Assuming that interest is thus convertible $m$ times a year, the rate of interest for the $m$^th part of a year will be ${\frac {i}{m}}$, and the amount of 1 at the end of $n$ years, that is, at the end of $mn$ intervals of conversion, will be $\left(1+{\frac {i}{m}}\right)^{mn}$. Assuming the number $m$ now to in-