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

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

     1em 1em 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-