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18 100(20.8170 + 9.1716) = $2999 to the nearest dollar. Example 4.2 A loan of $3000 is to be repaid with quarterly installments at the end of each quarter for five years. If the rate of interest charged on the loan is 10% convertible semiannually, find the amount of each quarterly payment. We are given an interest rate of 5% per halfyear. Let j be die equivalent rate of interest per quarter, which is the payment period. We have ; = (1.05)"2  1 = .024695. Let die quarterly payment be denoted by R. Then die equation of value is 5 . = 3000 so diat 3000 3000 15.6342 = $191.89. Example 4.3 At what annual effective rate of interest will payments of $100 at the end of every quarter accumulate to $2500 at the end of five years? Let J = !*V4 be the interest rate per quarter which accomplishes the above. Then the equation of value at the end of five years is 100 S J = 2500 = 25. We use formula (3.33) to obtain a starting value for die iteration 25 20 = .0225. We next use the NewtonRaphson method of iteration. Using formula (3.30) we generate the following successive values: (, = .022855 (2 = .022854 (3 = .022854 The annual effective rate of interest i is given by I = (1.022854)"  1 = .0946, or 9.46%. 4.3 FURTHER ANALYSIS OF ANNUITIES PAYABLE LESS FREQUENTLY THAN INTEREST IS CONVERTIBLE In this section annuities payable less frequently than interest is convertible are further analyzed algebraically. This section will be subdivided into the following areas: (1) annuityimmediate, (2) annuitydue, and (3) other considerations. Annuityimmediate Let be the number of interest conversion periods in one payment period, let be the term of the annuity measured in interest conversion periods, and let I be the rate of interest per interest conversion period. We will assume that each payment period contains an integral number of interest conversion periods; The second approach involves an algebraic analysis of such annuities. The intent is to develop algebraic expressions for such annuities in terms of annuity symbols already defined in Chapter 3, with adjustment factors sometimes being required. The reader might expect that the algebraic development for annuities payable less frequently than interest is convertible would be quite similar to that for annuities payable more frequently than interest is convertible. Surprisingly, however, the two cases have traditionally been developed in the literature with definitions and formulas which look quite different from each other. The algebraic analysis of annuities payable less frequentiy than interest is convertible is given m Section 4.3, while Section 4.4 analyzes annuities payable more frequentiy than interest is convertible. Although these algebraic approaches are not required if the sole purpose is to find numerical values for such annuities, they do provide valuable analytical insight into aimuities in general. Also, they provide an important foundation for the analysis of contingent aimuities (e.g. a montiily annuity payable for life from a pension plan valued at an effective rate of interest). Fmally, tiiey offer an alternative computational approach in which interest tables can be utilized. Example 4.1 Find the accumulated value at the end of four years of an investment fund in which $100 is deposited at the beginning of each quarter for the first two years and $200 is deposited at the beginning of each quarter for the second two years, if the fund earns 12% convertible quarterly. We are given an interest rate of 1 % per mondi. It j be die equivalent rate of interest per quarter, which is the payment period. We have j = (1.01)3 1 .030301. The value of the annuity in symbols is 100 (i" + sjij) which can be evaluted as
V + V + 1 V* 1 v" (I + 0*  I (4.1) Thus, we have an expression for the present value of this annuity in terms of annuity symbols already defined. The accumulated value of this annuity immediately after the last payment (1+0" = . (4.2) It is possible to derive formulas (4.1) and (4.2) by an alternate argument. There is a value of R such that the series of payments of 1 at the end of each interest conversion periods for n interest conversion periods can be replaced by a series of payments of R at the end of each interest conversion period so that the present values are equal. The present value of this series is Ra . Now consider any one payment period which contains k interest conversion periods. At the end of the payment period the accumulated value of payments of R at the end of each interest conversion period must equal the payment of 1 made at that point. Thus, and substituting R =1/5 into Roj , formula (4.1) is obtained. Formula (4.2) can be derived by a similar argument. Figure 4.1. is a time diagram clarifying the above argument. [R R  R R][R R  R R] [R R R R] Figure 4.1 Time diagram for an annuityimmediate payable less frequently than interest is convertible Annuitydue The present value of an annuity which pays 1 at the beginning of each interest conversion periods for a total of n interest conversion periods is 1 + V* + v* + 1v* (4.3) The accumulated value of this annuity interest conversion periods after the last payment is /7, .Vm (4.4) It is possible to derive formulas (4.3) and (4.4) by an alternate argument analogous to the argument used above for the annuityimmediate. The reader should fill in the details for this argument. Figure 4.2 is a time diagram for this case. [R R R K\[R R R R\ [R R R ] Figure 4.2 Time diagram for an annuitydue payable less frequently than interest is convertible Other considerations On occasion a pe etuity payable less frequently than interest is convertible is encountered. The present value of such a pe etuityimmediate is thus k and n are both positive integers. The number of aimuity payments made is n/k, which is also a positive integer. The present value of an annuity which pays 1 at the end of each interest conversion periods for a total of n interest conversion periods is
1 v (1 + if 1 (4.5) which is also the limit of formula (4.1) as n approaches infinity. Similarly, the present value of a pe etuitydue is (4.6) A second special case occasionally encountered is to find the value of a series of payments at a given force of interest 5. Although coming under the category of annuities payable less frequently than interest is convertible, this situation is not adequately handled by the methods discussed above, since n and k are both infinite. This situation can best be handled by writing an expression for the value of the annuity as the sum of present values or accumulated values of each separate payment, replacing v* with and (1 + /)* with e**. This expression can be summed as a geometric progression. An illustration of this type appears in the exercises. A third special case, very rarely encountered in practice, is a situation in which each payment period does not contain an integral number of interest conversion periods (i.e. A:>1, but A: is not integral). Here again, the best approach would be to resort to basic principles, i.e. to write an expression as the sum of present values or accumulated values of each separate payment, and then to sum this expression as a geometric progression. An illustration of this type also appears in the exercises. It should be observed that no onesymbol expressions for the annuity values given by formulas (4.1) through (4.6) have been defined. The only expressions are those in terms of ordinary aimuity symbols given by formulas (4.1) through (4.6). Finally, it is possible to generalize the approach to finding annuity values on any date, as discussed in Section 3.4, to annuities payable less frequently than interest is convertible. Example 4.4 illustrates this approach. Example 4.4 Find an expression for the present value of an annuity in which there are a total of r payments ofl, the first to be made at the end of seven years, and the remaining payments at threeyear intervals, at an annual effective rate i, expressed as: (1) an annuityimmediate, and (2) an annuitydue. Figure 4.3 is a time diagram for this example. 0 1 2 3 4 5 6 7 t 9 10 11 12 13 7+3(/l) 7+3/ Figure 4.3 Time diagram for Example 4.4 The present value of tiiis annuity is given by 3r+4 1. Summing die geomettic progression, we have  + 4 + (1 ,3r+4) (i ,4) a  Note diat die annuityimmediate form is characterized by die s in die denominator. 2. Summing die geometric progression, we have 1 v Note diat die annuitydue form is characterized by die in die denominator. Widi practice die reader should be able to direcdy write down die answer to problems of fliis type in eidier die annuityimmediate form or die annuitydue form widiout acUially summing the geometric progression. Example 4.5 Rework Example 4.1 using the approach developed in Section 4.3. The rate of interest is 1% per mondi, die term of die annuity is 48 interest conversion periods, and each payment period contains tiuee interest conversion periods. Since diis is an annuitydue, die accumulated value is 10Q48].oi + 24].oi = iffl 61.2226b 26.9735 2999 «31.01 2.9410 using die interest tables and rounding to die nearest dollar. The answer agrees widi diat obtained in Example 4.1.
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