INTEREST 61 series will be convergent. If, for example, the nominal rate is .05 convertible quarterly, the effective rate is ..(i^y-i ==(1.0125)4-! =4x.0125+6x(.0125)2 +4x(.0125)8+(.0125)4 =.0509+. The result, true to the fourth decimal place, is obtained by neg- lecting the last two terms in the binomial expansion. If the effective rate were .05, and we wished to find the nomi- nal rate when interest is convertible quarterly, we would have j=4:{(l+.0^-l} = 4 {l + J x .05 + ^ a ^1) (.05)2
=.0491-.
^r1^"2^05^--1}
Here again the result obtained by taking four terms of the bino- mial expansion is accurate to the fourth decimal place. 24. Instantaneous compound interest. Force of interest. From the funda- mental formula / ,\CT i=(l+J-) -1 (1) \ ml ' ' it follows that the nominal rate is identical with the effective rate when the interest is converted into principal once a year. Moreover, the effective rate increases as the number of conversion intervals increases, since the / /V" positive part of i, viz. il+—l , increases. However, it does not increase indefinitely as the number of conversion intervals in a year increases, but approaches a definite and well-known limit. This limit is found by taking the limit of both sides of equation (1). We have, then,
limi = lim[(l + J-}'"- l| insooLV W J =lim('l+2-')'"_i. »n=«>\ m/ 62 MATHEMATICAL THEORY OF INVESTMENT But, by § 10, lim (\ + 2.')" = e, m=m\ ml where e= 2.71828 +. Denoting limi by k for the moment, we have 1c=e}-l. (2) The expression e->—l, which we have denoted by k, is the effective rate corresponding to the nominal rate j convertible instantaneously. On the other hand, if the effective rate is fixed, and the number of in- tervals is increased, the corresponding nominal rate as given by the formula i j = m {(1 +;)"-1} is decreased. It is not diminished indefinitely, however, but approaches a limit which is again well known. To find this limit by rough methods, we expand (1 + t)"1 by means of the binomial theorem. This method is legiti- mate, because the resulting series is convergent (see § 9). We have, then, i_ j=m{(l+{)'--!}
r y1-!) ^i-iV1-^ I -_^ -i.1 .TOVra / ..a, m^m /\m '.. ,^ -m\l+mt+————l+——r-i-i——^+--iJ (i-i) ^-iV1-^ .. \"»___/ .„ , VTO /W / .„ ^—r'^ 1.2.3 i+-- Assuming that we may take the limit of the right member by taking the sum of the limits of the separate terms, we find - (i-i) p-iY1-^ i lim/=lim , ^__/,, \m !\m /,, m=". 1. 2 ' 1.2.3 T J . i2 , {s i1 , "-^S-?-1--- The series on the right is exactly the logarithmic series which was dis- cussed in § 11, so that ^. ^ ^ ^ ^ ^ ^ It is customary to denote the limit otj by the Greek letter 8 (delta), so that 8= log, (l+i). (4) The number 8 or its equivalent is called the force of intercut. The force of interest is the limit of the nominal rate as the number of conversion intervals in a year is indefinitely increased, corresponding to the fixed effective rate i. INTEREST 63 To express the effective rate in terms of the force of interest, it is only necessary to -write equation (4) in the exponential form. The result is l+i=el>, (5) or t=e8-l. - (6) Formula (6) is identical in form with (2), as indeed it should be, for S is precisely the limit toward which j approaches as the number of conversion intervals in a year is indefinitely increased, and f is the corresponding effective rate. The value of i in terms of S is easily found by means of equation (2) of § 10, which gives g y 33 ^-^r^i-^--1' . 8, 82 ,_83, ,-. ^=i+^T2+^-2-3+ <7) For example, if 8 = .05, 05 (.05^ (.05)' , 1 1.2 1.2.8 =.05127+. To find 8 when i is given, the logarithmic series ;2 ;S ;4 log.(l+0=«-g+3-^+... may be used. For example, if i = .05, 8= log. (1.05) _n. (-05)2 , (-05)3 (05)4, " 23 4 ' = .048790 +. It is easy to find the value of 8 by common logarithms, for, by (5) of § 16, 8= log. (1.05)=^. lo^(1.05), where - 2.302585 +. Consequently, 8 = 2.302585 x .021189 + = .048790 + ., which agrees with the result already obtained. Instantaneous compound interest and force of interest have no real ex- i