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13 1111 1 1 0 12 3 6 7 h 9 10 11 12 T T T2 Figure 3.4 Time diagram for illustration in Section 3.4 Present values more than one period before the first payment date In this case, the present value of the annuity at the beginning of the 1st period is seen to be the present value at the end of the 2nd period discounted for two periods, i.e. It is possible to develop an alternate expression for this present value strictly in terms of annuity values. Temporarily assume that imaginary payments of 1 are made at the end of the 1st and 2nd periods. Then the present value of all nine payments at time t = 0 is . However, we must remove the present value of the imaginary payments, which is . Thus, an alternate expression for the present value is  . This expression is convenient for calculation, if interest tables are being used. This type of annuity is often called a deferred annuity, since payments commence only after a deferred period. In general, the present value of an annuityimmediate deferred for m periods with a term of n periods after the deferred period is (3.17) V = a m+n\ nil It should be noted that since payments are made at the ends of periods, the first payment under such a deferred annuity is m I 1 periods after the evaluation date, not m periods. We have used annuitiesimmediate instead of annuitiesdue, since numerical values are directly available from the interest tables for annuitiesimmediate. However, it is possible to work with a deferred annuitydue. The reader should .verify that the answer to this case, expressed as annuitydue, is 71 = Accumulated values more than one period after the last payment date In this case, the accumulated value of the annuity at the end of the 12th period is seen to be the accumulated value at the end of the 9th period, accumulated for three periods, i.e. h 7l(l+0 Here it is also possible to develop an alternate expression strictly in terms of annuity values. Temporarily, assume that imaginary payments of 1 are made at the end of the 10th, 11th, and 12th periods. Then the accumulated value of all 10 payments is s . However, we must remove the accumulated value of the imaginary payments, which is Sj. Thus, an alternate expression for the accumulated value is In general, the accumulated value of an period annuity, m periods after the last payment date, is (1+•)" = (3.18) It is also possible to work with annuitiesdue instead of annuitiesimmediate. The reader should verify that the answer in this case expressed as an annuitydue is We will see that it is possible to develop values for all three cases in terms of annuity symbols already defined. The above three cases can best be illustrated by example. Consider an annuity under which seven payments of 1 are made at the end of the 3rd through the 9th periods inclusive. Figure 3.4 is a time diagram for this annuity. The values at the end of the 2nd, 3rd, 9th, and 10th periods are given directly, either by annuitiesimmediate or by annuitiesdue, as labeled on the time diagram. The present value at the beginning of the 1st period is an example of case 1, the accumulated value at the end of the 12th period is an example of case 2, and the current value at the end of the 6th period is an example of case 3. These three cases are denoted by arrows 1, 2, and 3, respectively, on the time diagram.
Here it is again possible to develop an alternate expression strictly in terms of aimuity values. Separate the seven payments into the first four payments and the last three payments. The accumulated value of the first four payments is and the present value of the last three payments is a. Thus, an alternate expression for the current value is 41 +31 In general, the current value of an nperiod annuity immediately after the mth payment has been made (m < n) is (1+0" = 1  + n=m\ (3.19) It is also possible to work with annuitiesdue instead of annuitiesimmediate. The reader should verify that the answer in this case expressed as an annuitydue is (1 + 0 = 4 a Summary In general, it is possible to express the value of an annuity on any date which is an integral number of periods from each payment date as the sum or difference of annuitiesimmediate. Other equivalent expressions do exist, and the reader should practice translating one form of an answer into alternate forms. The reader should not try to work problems by memorizing formulas (3.17), (3.18), and (3.19). Any problem of this type can be best handled from first principles, as illustrated in this section. The reader should also be careful to observe that the labels on the time diagram are merely an aid in visualizing the payments involved, but they do not affect the answer. For example, we could label the time diagram in Figure 3.4 from 8 to 20 instead of 1 to 12 and the answers to the various examples would not change. The elements on the time diagram that do affect the answer are the have = V + + 4 1  V (3.20) provided v < 1, which will be the case if / > 0. Alternatively, we have 1 v" n»oo since lim v n»oo = 0. Formula (3.20) can be interpreted verbally. If principal of l/i is invested at rate /, then interest of / • 1 = 1 can be paid at the end of every period forever, leaving the original principal intact. By an analogous argument, for a perpetuitydue, we have ii (3.21) Current values between the first and last payment dates In this case, the current value of the annuity at the end of the 6th period is seen to be the present value at the end of the 2nd period accumulated for four periods or the accumulated value at the end of the 9th period discounted for three periods, i.e. number of payments and the location of the evaluation date or comparison date in relation to the payment dates. If it is necessary to find the value of an annuity on a date which is not an integral number of periods from each payment date, the value should be found on a date which is an integral number of periods from each payment date and then the value on this date can be accumulated or discounted for the fractional period to the actual evaluation date. This situation will be illustrated in the exercises. 3.5 PERPETUITIES A perpetuity is an annuity whose payments continue forever, i.e. the term of the aimuity is not finite. Although it seems unrealistic to have an annuity with payments continuing forever, examples do exist in practice. The dividends on preferred stock with no redemption provision and the British consols, which are nonredeemable obligations of the British government, are examples of perpetuities. Preferred stock will be defined and discussed further in Chapter 7. The present value of a perpetuityimmediate is denoted by , and we
It should be noted that accumulated values for pe etuities do not exist, since payments continue forever. It is instructive to use the concept of perpetuities to give a verbal interpretation to formula (3.2) 1  v" (3.2) Consider two perpetuities. The first pays 1 at the end of each period and has present value l/i. The second is deferred n periods and, thus, has present value v" . The difference is the present value of payments of 1 at the end of each period during the deferred period, which is a Some readers feel that pe etuities are of limited practical significance. However, a broader range of transactions than payments continuing forever can be looked upon as equivalent to pe etuities. For example, consider the investment of $1000 at a 10% annual effective rate of interest, payable in installments each year, for any particular period of time, at the end of which the principal of $10( is returned. Such a transaction is equivalent to determining the present value of a pe etuity paying $100 at the end of each year at an effective rate of interest of 10%. Example 3.5 A leaves an estate of $100,000. Interest on the estate is paid to beneficiary for the first 10 years, to beneficiary for the second 10 years, and to charity D thereafter. Find the relative shares ofB, C, and D in the estate, if it assumed the estate will earn a 7% annual effective rate of interest. The value of Bs share is 1 = 7000(7.0236) = $49,165 to the nearest dollar. The value of Cs share is 7000(20]  ) = 7000(10.5940  7.0236) = $24,993 to the nearest dollar. The value of Ds share is 7000(a «201 ) = 7000 " 105940 = $25,842 to die nearest dollar. Note that die sum of die shares of B, C, and D is equal to $100,000 as expected. Also note diat die present value of die estate at die end of 20 years is 100,000(1.07)= $25,842, to die nearest dollar, which is equal to Ds share. ITus confirms die fact diat charity D continuing to receive die interest into perpetuity or jecciving the estate value in a lumpsum at the end of 20 years are equivalent in value. 3.6 NONSTANDARD TERMS AND INTEREST RATES Thus far we have assumed that n is a positive integer and that > 0 in any of our annuity symbols. This section considers the implications if these conditions are not satisfied. Most of the results in this section are of greater mathematical than practical interest. However, the results in this section have appeared in the literature and thus need to be addressed. Also, the analysis is of instrucfive value. Consider first what the symbol a , where n is a positive integer and 0 < Jt < 1, might represent. Formula (3.1) cannot be applied, since it requires that be a positive integer. However, it is possible to derive a result which is consistent with formula (3.2) as follows: 1  V 1 v"b v"V I V (1 + 01 (3.22) Thus, an inte retation for the symbol aj consistent with the formula (3.2) is that it is the present value of an nperiod annuityimmediate of 1 per period, (1 4 i)k I plus a final payment at rime n I A: of f. The amount of this final irregular payment seems rather unusual. A payment that might be more "comfortable" to some readers is k, i.e. the payment would be proportional to the fracrional rime involved. Fortunately, k is reasonably close to (1 I 0* 1 As an exercise, the reader will be asked to find the error involved in this approximarion. However, it is possible to give other reasonable inte etations to the symbol  : . For example, how about letting the final payment be k, but having it paid at time n I 1? Other possibiliues also suggest themselves. Thus, is not really a welldefined symbol without further description of the payments actually involved. Formula (3.22) is one possible inte retation, but other inte etations are also reasonable. Unfortunately, the need to inte ret annuity symbols for nonintegral terms does arise in practice. For example, many courts use an annuitycertain for a
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