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19 « 1.035 = 1021.035 = 20.35. By inspection of die interest tables, we have 36 < n < 37. Thus, 18 regular payments and a smaller final payment can be made. Let die smaller additional payment at die time of the final regular payment be denoted by R. Then an equation of value at the end of 18 years is R + 100 361.035 1000(1.035) 21.035 , 70.0076 R = 1000(3.45027)  100 The total final payment wotiid dius be $110.09. = $10.09. 4.4 FURTHER ANALYSIS OF ANNUITIES PAYABLE MORE FREQUENTLY THAN INTEREST IS CONVERTIBLE In this section annuities payable more frequently than interest is convertible are fiirther analyzed algebraically. In practice annuities payable more frequently than interest is convertible are more common than annuities payable less frequently than interest is convertible. This section will be subdivided into the following areas: (1) annuityimmediate, (2) annuitydue, and (3) other considerations. Annuityimmediate Let m be the number of payment periods in one interest conversion period, let n be the term of the annuity measured in interest conversion periods, and let i be the interest rate per interest conversion period. We will assume that each interest conversion period contains an integral number of payment periods; thus m and n are both positive integers. The number of annuity payments made is mn, which is also a positive integer. The present value of an annuity which pays 1/m at the end of each mth of an interest conversion period for a total of n interest conversion periods is denoted by a\ We have "1 m 1 1 1 n.l v"V 1v" 1 v" (1 + /)"  1 1v" (4.7) The accumulated value of this annuity immediately after the last payment is made is denoted by and we have (1 + 0"  1 (4.8) Formulas (4.7) and (4.8) should be compared with formulas (3.2) and (3.4), respectively. They are identical except that the denominators of (4.7) and (4.8) are i" instead of i. Since i"" is a measure of interest paid at the end of mths of an interest conversion period, the points at which interest is paid under this measure are consistent with the points at which payments are made. This property, relating the manner in which payments are made and the measure of interest in the denominator, was originally mentioned in Section 3.3. It is possible to write a and s in terms of and with an adjustment factor. The following are immediate consequences of formulas (4.7) and (4.8): « 1 (4.9) (4.10) Formulas (4.9) and (4.10) are commonly applied in obtaining numerical results when interest tables are used. Values of i/i" appear in the interest tables in Appendix I. The term i/i" is often written as s\ which is consistent with formula (4.8), setting n = 1. Example 4.6 An investment of $1000 is used to make payments of $100 at the end of each year for as long as possible with a smaller final payment to be made at the time of the last regular payment. If interest is 7% convertible semiannually, find the number ofpayments and the amount of the total final payment. The equation of value is 21.035
104 The theory ofirUerest Annuitydue The present value of an annuity which pays \lm at the beginning of each mth of ;.n interestonversion period for a total of n interest conversion periods is denoted by We have 4"" = . (4.11) The derivation of formula (4.11) is similar to formula (4.7) and is left as an exercise. The accumulated value of this annuity one mth of an interest conversion period after the last payment is made is denoted by s , and we have (1 + 0"  1 (4.12) Here, again, the relationship between the manner in which payments are made and the measure of interest in the denominator should be noted. It is an immediate consequence of formulas (4.11) and (4.12) that ..("I) (4.13) (4.14) Formulas (4.13) and (4.14) can be used in obtaining numerical results if values of d") = are available in the interest tables. Such values do appear in Appendix I, but in many interest tables they do not. However, it is possible to develop alternative formulas which can be applied. Since each payment under is made one /nth of an interest conversion period earlier than a, , we have 1 + i (4.15) and similarly. .. (rn) + ± (4.16) Formulas (4.15) and (4.16) are convenient in numerical calculations using interest tables when values of W" are available and values of i/d" are not. A number of other interesting identities involving annuities payable mthly can be developed. In particular, identities analogous to formulas (3.6) and (3.12) through (3.16) exist. These six identities and their derivations are left as exercises. Other considerations On occasion a 1 !1 payable more frequently than interest is convertible is encountered. The following formulas are analogous to formulas (3.20) and (3.21): (4.17) (4.18) A second special case, very rarely encountered in practice, is a situation in which each interest conversion period does not contain an integral number of payment periods (i.e. m > 1, but m is not integral). In this case, 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 appears in the exercises. One consideration which is important in practice involves the proper coefficients for annuities payable mthly. Each payment made is of amount 1/m, while the coefficient of the symbol is 1. In general, the proper coefficient is the amount paid during one interest conversion period, and not the amount of each actual payment. The amount paid during one interest conversion period is often called the annual rent of the annuity. This is an appropriate term if the interest conversion period is one year, as is often the case; but it is confusing if the interest conversion period is other than one year. Sometimes the term periodic rent is used instead of "annual rent" to avoid this confusion.
2. The answer is 4800 (1 + if = 4800 ,(12) .(12) Example 4.8 Rework Example 4.2 using the approach developed in Section 4.4. The rate of interest is 5% per halfyear, the term of the loan is ten interest conversion periods, and each interest conversion period contains two payment periods. Thus, the equation of value is .05 = 3000 so diat 1500 To] .05 1500 ,(2) 101 .05 1500 = $191.89. (1.012348X7.7217) The answer agrees widt diat obtained in Example 4.2. Example 4.9 At what annual effective rate of interest is the present value of a series of payments of$l every six months forever, with the first payment made immediately, equal to $10? The equation of value is 10 = 1 + + V + yi + Thus, I V ,•5 = .9 (1 + 0 = 1 which gives  1 = .2346 or 23.46%. 4.5 CONTINUOUS ANNUITIES A special case of annuities payable more frequently than interest is convertible is one in which the frequency of payment becomes infinite, i.e. payments are made continuously. Although difficult to visualize in practice, a continuous annuity is of considerable theoretical and analytical significance. Also, it is usefiil as an approximation to annuities payable with great frequency, such as daily. We will denote the present value of an annuity payable continuously for n interest conversion periods, such that the total amount paid during each interest conversion period is 1, by the symbol . An expression for is (4.19) since the differential expression vdt is the present value of the payment dt made at exact moment t. A simplified expression can be obtained by performing the integration 0=1 = logv 1 v" (4.20) Formula (4.20) is analogous to formula (3.2). Again, there is consistency between the manner in which payments are made and the denominator of the expression. Also formula (4.20) could have been obtained as follows: (m) ,. 1  v" 1  v" a=i = lim a, = lim  =  = lim a = lim (1  v") 1  v" </(«) 5 Thus, the continuous annuity is seen to be the limiting case of annuities payable mthly. It is possible to write in terms of with an adjustment factor «H1 = (4.21) Finally, it is possible to generalize the approach to finding annuity values on any date, as discussed in Section 3.4, to annuities payable more frequently than interest is convertible. Example 4.7 illustrates this approach. Example 4.7 Payments of $400 per month are made over a tenyear period. Find expressions for: (1) the present value of these payments two years prior to the first payment, and (2) the accumulated value three years after the final payment. Use symbols based on an effective rate of interest. 1. The answer is
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