INTEREST 59 EXAMPLES 1. In what time would $236.41 amount to $421.32, with interest con- vertible quarterly at 6% nominal ? By formula (8) _ log 421.32 - log 236.41
., /, .06\ 41og^l+^)
_ 0.2509463 - 0.0258640 = 9.70. 2. In what time will $2394.62 amount to $10,000 when placed at compound interest at 5%, convertible semiannually ? 23. Nominal and effective rates of interest. The two problems for which formulas (1) and (3) of the previous section are solu- tions make necessary a distinction between two kinds of rates of interest when the conversion interval is less than a year, viz. nominal rate and effective rate. The nominal rate is the rate which would be realized if the interest received at the end of each conversion interval were not productively invested until the end of the year, while the effective rate is the total return on the unit principal for one year. For example, if $100 were invested at 6%, payable semiannually, with the understanding that the rate per half year be one half the rate per year, the in- terest for the first half year would be $3, and the amount avail- able at the beginning of the second half year would be $103. The interest on $103 for the second half year would be $3.09; so that the total amount at the end of the year would be $106.09. The return on $1 is therefore .0609, and the effective rate is 6.09%. If the $3 received at the middle of the year had not been productively invested, the total amount at the end of the year would have been $106, and the.rate would have been 6%. This rate is the nominal rate. The nominal rate may also be de- nned as the product of the rate per conversion interval by the number of conversion intervals in a year. In the example the rate per conversion interval is .03, and the number of conversion intervals in a year is 2, giving .06 as the nominal rate. 60 MATHEMATICAL THEORY OF INVESTMENT PROBLEM. To express the effective rate in terms of the nominal rate, and vice versa. Let i be the effective rate and j the nominal rate convertible m times a year. By formula (3) of § 22 the compound amount of 1 for one year, when the interest is convertible m times a year at rate -- per conversion interval, is (i+^r. \ w The total return on 1 for one year, i.e. the effective rate of interest, is therefore <=(l+;;)"-l -w We obtain a very useful form of the relation between effective and nominal rates by simply transposing the 1; thus, l+l=(l+:r (2) To obtain j in terms of i we have only to solve (1) or (2) for j. Extracting the mth root of both sides of (2), transposing 1, and, finally, multiplying through by m, we find j=m{(l+i):n-l}- (3) If we wish to emphasize the fact that j depends upon m as well as i, we write j^; thus, .?<»)= m{(l+ir-l}. (3') The quantity j^ is an important factor in many computations that will occur later on. Clearly, ^=i. (4) The computation required in finding the effective rate when the nominal rate is given, or vice versa, is easily accomplished by means of the binomial expansion, since, according to § 9, the