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21 1 +k 1 I1 (4.39) This expression can be evaluated by direct calculation. On occasion, either yi or may equal 1 h ; for some j with tabular interest fiinctions, in which case the interest tables can be used, li = i, formula (4.39) is undefined. However, then the present value is just nv, which is obvious from the original series. The present value of a perpetuity will exist if 0 < ilLA < 1, in which 1 t / case the sum of the geometric progression exists. If > 1 then the 1 + i geometric progression diverges and the present value of the perpetuity does not exist. Other payment patterns In practical applications the value of varying annuities in which other payment patterns exist are generally best handled from first principles. The and substituting we obtain 2. The payments represented by (Pd) can be described by n.G,(H2W„+2) and substituting we obtain . +2 I (2 nai Example 4.12 Find the present value of a perpetuityimmediate whose successive payments are 1, 2, 3, 4, . . ., at an effective rate of interest of 5%. An appropriate symbol would be (1 ) . Substimting P = 1 and Q = 1 into formula (4.35), we obtain Alternatively, we have ( ) = ; = .0025 1 + i = 420. 1.05 .0025 = 420. annuity value can be computed by finding the present value or accumulated value of each payment separately and summing the results. Appendix VI contains a further analysis of varying annuities whose payments follow a polynomial of degree n > 1. These results are required in an application contained in Section 9.9. The reader should also be careful to distinguish the term "varying annuity" from the term "variable annuity." A variable annuity is a type of life annuity in which payments vary according to the investment experience of an underlying investment account, usually one invested in common stocks. Variable annuities are beyond the scope of this book and are not considered further. Example 4.11 Use the technique involving the functions , G„, and H„ to derive: (1) formula (4.31), and (2) formula (4.33). 1. The payments represented by (la) can be described by The lefthand side represents the present value of an investment of 1 at the beginning of each period for n periods. The two terms on the righthand side can be interpreted, respectively, as the present value of the interest to be earned and the present value of the return of the principal invested. Payments varying in geometric progression Annuities with payments varying in geometric progression can be readily handled by directly expressing the annuity value as a series with each payment multiplied by its associated present or accumulated value. Since the payments and the present or accumulated values are both geometric progressions, the terms in the series for the annuity value constitute a new geometric progression. By way of example, consider an annuityimmediate with a term of n periods in which the first payment is 1 and successive payments increase in geometric progression with common ratio I + k. The present value of this annuity is V + v\\ + k) + • • + v"(l I k)"\ However, this is a geometric progession whose sum is
+ 1  nv" + nv" v" v"ajm (1  v") + (1  v") Alternatively, we can express the present value of the payments as (Hi  H„i)  (H„i  H2„+i) = ,  2 „.., + H2n+i 2v"+ + v2"+ 1  2v" + V (1  v"f id =  . As an exercise, the reader will be asked to give a verbal inte retatio to this answer. Example 4.14 Find the present value of an annuityimmediate such that payments start at 1, each payment thereafter increases by 1 until reaching 10, and then remain at that level until 25 payments in total are made. A symbol for an nperiod increasing annuityimmediate in which increases are limited to the first m periods only, 0 < m < n, is given by Thus, the present value of the annuity in this example can be written as (1 ). The reader should verify tiiat die following are all valid expressions for die present value of this annuity: . (1 ) + 10v">ai5i • lOfljsi  ( ) 1000 .07  .04 = $14,459 to the nearest dollar. 4.7 MORE GENERAL VARYING ANNUITIES The varying annuities considered in Section 4.6 assumed that the payment period and the interest conversion period are equal and coincide. In Section 4.7 this restriction is removed. In practice, varying annuities with payments made more or less frequently than interest is convertible occur infrequently. We will consider generalizations of the increasing annuity, (/a), in which interest is convertible more or less frequently than payments are made. Other annuities in which payments vary in arithmetic progression can be handled analogously. Consider first the case in which payments are made less frequently than interest is convertible. Let be the number of interest conversion periods in one payment period, let n be the term of the annuity measured in interest conversion periods, and let / be the rate of interest per interest conversion period. The number of payments is nik, which is integral. Let A be the present value of the generalized increasing annuity. We have (1 1 iVA = 1 t 2v* t «  1 1 ,nk + Hv" .n2k Now subtracting the first equation from the second . a[(1 + 0*  l] = 1 + V* v* h . . . g which can be expressed as + v"*  Hv" (4.40) Example 4.15 An annuity provides for 20 annual payments, the first payment a year hence being $1000. The payments increase in such a way that each payment is 4% greater than the preceding payment. Find the present value of this annuity at an annual effective rate of interest of 7%. Using formula (4.39) we have Example 4.13 Find the present value of an annuityimmediate such that payments start at 1, increase by annual amounts of 1 to a payment of n, and then decrease by annual amounts of 1 to a final payment of 1. The present value is  nv" „ (  1)  flrryi (/a)Hi + v«(Da) = i + v"
(4.41) Formula (4.41) gives the present value of nperiod annuityimmediate, payable mthly, in which each payment during the first period is 1/m, each payment during the second period is 2/m, and so forth, until each payment during the nth period is n/m. Consider next the situation in which the rate of payment changes with each payment period. Suppose that an increasing annuity is payable at the rate of 1/m per interest conversion period at the end of the first mth of an interest conversion period, 2/m per interest conversion period at the end of the second mth of an interest conversion period, and so forth. Then the first payment will be 1/m , the second will be 2/m , and so forth. Denoting the present value of such an annuity by {1" ) \ we have J m2 L + nmv nm m  nv (4.42) The derivation of formula (4.42) is left as an exercise. Annuities in which the payments vary in geometric progression and in which the payment period and the interest conversion period differ are occasionally encountered. However, such annuities present no new difficulties. They can be readily handled by expressing the annuity value as a summation of the present value or accumulated value of each payment. This summation is a geometric progression which can be directly evaluated. This technique will be illustrated in Example 4.17. (1  v) Example 4.17 Find the accumulated value at the end of ten years of an annuity in which payments are made at the beginning of each halfyear for five years. The first payment is $2000, and each payment is 98% of the prior payment. Interest is credited at 10% convertible quarterly. We count periods in quarters of a year. The accumulated value is 2000[(1.025) + (.98)(1.025)38 + (.98)(1.025) + • • • + (.98)(1.025)] = 7 > (1025)°  (.98)°(l.025)2° 1  (.98)(1.025)2 = $40,052 to the nearest dollar. 4.8 CONTINUOUS VARYING ANNUITIES The last type of varying annuity we will consider is one in which payments are being made continuously at a varying rate. Such annuities are primarily of theoretical interest. Consider an increasing annuity for n interest conversion periods in which payments are being made continuously at the rate of t per period at exact moment t. The present value of this annuity is denoted by (1 ) , and an expression for it would be tvdt (4.43) since the differential expression tvdt is the present value of the payment tdt made at exact moment t. Formula (4.40) is a generalized version of formula (4.31) and the reader should note the similarity. Consider next the case in which payments are made more frequently than interest is convertible. Two different results arise depending on whether the rate of payment is constant or varies during each interest conversion period. Consider first the situation in which the rate of payment is constant during each interest conversion period with increases occurring only once per interest conversion period. We can utilize the relationship between the manner in which payments are made and the measure of interest in the denominator to obtain the following generalized version of formula (4.31) i Example 4.16 Find the present value of a perpetuity which pays 1 at the end of the third year, 2 at the end of the sixth year, 3 at the end of the ninth year, . . . Denote the present value of this perpetuity by A. Then = v + 2v + 3v + • • • vA = + 2v + • • • Now subtracting the second equation from the first (1  v) = v + + v + • • • = 
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