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20 (1 + 0 log,(l + 0 (1 + 0"  1 ,. (m) ,. ..(m) = nm 51 = lim S, (4.23) (4.24) Additional insight into continuous annuities can be obtained by differentiating formula (4.22) with respect to its upper limit n, and then replacing n with t, which gives (4.25) = l+5i upon substituting formula (4.23). Formula (4.25) has an interesting verbal inte retation. Consider an investment fund in which money is continuously being deposited at the rate of 1 per interest conversion period. The fund balance at time t is equal to 1 . The fund balance is changing instantaneously for two reasons. First, new deposits are occurring at the constant rate of 1 per interest conversion period, and second, interest is being earned at force 5 on the fund balance 1 . Similarly, by differentiating formula (4.19) we have dIt = 1  5 a (4.26) Formula (4.26) also has a verbal 1 1 11 , which is deferred until some additional material is presented in Chapter 6. We can express the value of continuous annuities strictly in terms of the force of interest &. When this is done, formula (4.20) becomes 1 e (4.27) e"\ = 3. 5 5 which factors into However, 1 =0 implies that 5 = 0, which clearly is an extraneous root. Thus, we have , 2 so diat ,2 5 = = .0693, or 6.93%. 10 4.6 BASIC VARYING ANNUITIES Thus far all the annuities considered have had level payments. We now remove this restriction and consider annuities with varying payments. In Section 4.6 it will be assumed that the payment period and interest conversion period are equal and coincide. As in Chapter 3, we will just use the term "period" for both. Naturally, any type of varying annuity can be evaluated by taking the present value or the accumulated value of each payment separately and summing the results. On occasion, this may be the only feasible approach. However, there are several types of varying annuities for which relatively simple expressions can be developed, and we will consider these. The following types of varying annuities will be discussed in this section: (1) payments varying in arithmetic progression, (2) payments varying in geometric progression, and (3) other payment patterns. Payments varying in arithmetic progression Consider a general annuityimmediate with a term of n periods in which payments begin at P and increase by Q per period thereafter. The interest rate is / per period. Figure 4.4. is a time diagram for this annuity. It should be and formula (4.23) becomes Values of /75 = 1 can be readily calculated directly and also appear in the interest tables in Appendix I. The accumulated value of a continuous annuity at the end of the term of the annuity is denoted by 7 . The following relationships hold: \\ + i)dt (4.22) = . (4.28) In a sense, this is a case of an annuity in which the payment period and interest conversion period are equal, but which was not cons dered in Chapter 3. Example 4.10 Find the force of interest at which "s = 1 . Using formula (4.28) we have
,1 v" . nv nv The accumulated value is given by since it must be the present value accumulated for n periods. (4.29) (4.30) P+Q P+2Q  P+(n2)Q P+(nl)Q Figure 4.4 Time diagram for formulas (4.29) and (4.30) Formulas (4.29) and (4.30) can be used in solving any problem in which payments vary in arithmetic progression. However, there are two special cases which often appear and have special notation. The first of these is the increasing annuity in which P = 1 and Q = I. Figure 4.5 is a time diagram for this annuity. The present value of this annuity, denoted by (1 ) , can be obtained from formula (4.29)  nv lv" + a  nv in+l)v" (4.31) The accumulated value of this annuity, denoted by (/s), is ( ) = (1 + 0" n+T] (« + 1) (4.32) Formula (4.31) can be derived by an alternative approach which considers an increasing annuity to be the summation of a series of level deferred annuities. Applying formula (3.17), we have («)1 =E va /=0 nl E v (4.31) («)„n Figure 4.5 Time diagram for an increasing annuity noted that P must be positive but that Q can be either positive or negative as long as P + (n  1)2 > 0 to avoid negative payments. Let A be the present value of the annuity. Then we have A = Pv + iPV Q)v + {P + 2Q)v + • + [P + (n 2)Q]v" + [P+(n l)Q]v". This series is a combination of an arithmetic and a geometric progression. Such a series can be solved algebraically by multiplying by the common ratio in the geometric progression to give (1 + i)A = P + iP + Q)v + iP + 2G)v2 + {P + 3Q)v + • • + [P + {n l)Q]v"K Now subtracting the first equation from the second iA =P+Q{v + v + v + • • • + v")  Pv"  (n  l)Gv = 1  v") + j2(v + + + • • • + v" + v")  Qnv" . Thus, A = P/ +Q
) = 0=1 nv n  nv"  + nv" n  a (4.33) The accumulated value of this annuity, denoted by (Ds) , is (Ds) = ( ) (1 + i)" n(l + 0"  (4.34) Formula (4.33) can also be derived by the alternative approach which considers a decreasing annuity to be the summation of a series of level annuities. Following this approach, we have (4.33) since lim =  and lim nv" = 0. 7!» 00 Note that P and Q must both be positive to avoid negative payments. An alternative approach to finding expressions for varying annuities is to make use of the following three quantities: F = (4.36) The present value of a payment of 1 at the end of n periods. G„ = (4.37) The present value of a level perpetuity of 1 per period, first payment at the end of n periods. Again, the alternative derivation is quite efficient and provides valuable insight into the nature of a decreasing annuity. n1 n2 „ = (Oa)„ (o)«n Figure 4.6 Time diagram for a decreasing annuity (4.38) = The present value of an increasing perpetuity of 1, 2, 3, . . . ,first payment at the end of n periods. These symbols are very useful in setting up expressions for varying annuities. By appropriately describing the pattern of payments, expressions for the annuity can be immediately written down. This will be illustrated in the examples. The formulas developed in this subsection have interesting verbal interpretations. For example, consider formula (4.31) written as 0 =:(/a) +nv". Not only is this derivation quite efficient, but it also provides valuable insight into the nature of an increasing annuity. The second of these is the decreasing annuity in which P = n and Q = I. Figure 4.6 is a time diagram for this annuity. The present value of this annuity, denoted by iDa), can be obtained from formula (4.29) All the above formulas are for annuitiesimm.ediate. However, we may use the previously mentioned relationship between the manner in which payments are made and the denominator of the expression for the annuity value to find formulas for annuitiesdue. Changing / in the denominator of any of the above formulas to d will produce values for annuitiesdue. Also, it is possible to have varying perpetuities. We can find the general form for a perpetuity by taking the limit of formula (4.29) as n approaches infinity, obtaining (4.35)
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