ANNUITIES 79 31. The amount of an annuity. PROBLEM. To find the amount of a, forborne annuity of 1 per annum, payable annually for n years. We will first find the required formula in terms of the effective rate i. The first payment made one year from the beginning of the term of the annuity will be accumulated for n—1 years and will amount to (l+iy~1. Likewise, the second will amount to (14- {y-\ the third to (1 +1)"'3, and so on, while the last term will be cash. The series of amounts will be (l+O-1, (l+O-^-.-.Cl+O, 1. Denoting the total amount by s^, according to the notation established in § 30, and writing the terms in reverse order, we have ^=l+(l+0+(l+^)2+...+(l+0»-2+(l-^0n-l. The expression on the right is a geometrical progression of n terms with ratio 1 + i, and the sum is found by the formula g^rL^. (See formula (2)'), § 2.) r — 1 Substituting for r, ;, and a their values 1+z, (l+i)"~\ and 1, we find, as the required formula,
(1+t)'1-!
^^——T
(1)
If interest is convertible m times a year at nominal rate j, we may replace i and (l+i)" by their values as obtained in the fundamental relations (1) and (2) of § 23. The result will be („ \mn i+u -i in/ .ON ^=7——T^i——' <2) (l+^) -1 \ m/ Formula (2) gives the amount of the annuity in terms of the nominal rate./. 80 MATHEMATICAL THEOKY OF INVESTMENT PROBLEM. To find the amount of an annuity of 1 per annum, payable p times a year for n years. Suppose the rate of interest is the effective rate i. The time to elapse between the making of the first payment and the end of the whole transaction will be n — — years; and, similarly, the times for the second and succeeding payments will be "-„ years, n-- years, and so on. The last payment will be cash. The amounts of the various payments, beginning with the first, will be ^1+0^, ^14-0"-;, ..., ^^ ^ Writing these amounts in reverse order and denoting their sum by ^), according to § 30, we have ^=^+^1+^+^1+^++^(1+^)B4 This is again a geometrical progression with first term 1-, ratio 1. 1 _T_ p (l+z'y, and last term -(1+i)" ". By formula (TV) of § 2 the sum will be JL ,(^1(1+^-1. "1 p i_ \.°) (1+y-l If we desire the formula for the amount in terms of a nominal ratey, convertible m times a year, we have only to replace 1 + i by its value in terms of./, viz. by fl+^T, and we obtain the formula \ m' i^T-1 <„, 1\ m ^ T,—~~' ^ : (i-^y-i \ m) ANNUITIES 81 Finally, if the interval between the successive payments of the annuity coincides with the conversion interval, and, con- sequently, m=p, formula (4) reduces to (,)'\"P i 1+L} -1 l ^) = 1 ———PL——— = -!- s at rate J (5) ir1 p y_ p np} p P The special case for which the solution is given by formula (5) is important in later work. If the annual rent of an annuity is -R, the amount K will be ^^d±^l, (0 f ^^>=^_0±0'^1_, (6') ^[(i+0"-i] as the case may be. The corresponding formulas when the rate is nominal, payable m times a year, will be found by replacing / i \" l+.by(l-^). EXAMPLES 1. A man puts $100 into a savings bank at the end of every year for 5 years. If the bank pays 4%, payable annually, what will his savings amount to at the end of the 5 years ? Solution. The amount will be found by substituting the values n = 5 and i = 4 in formula (1), and then multiplying by 100. The result is $100. (li04)5^1^ ^641.63. .04 2. Suppose $50 is deposited semiannually and the savings bank pays in- terest twice a year at nominal rate .04; to what would the savings amount in the previous example? In this case./ = 4, »i = 2, p = 2, and formula (4) or C5) becomes ,„, , /-i o2t10 — 1 $100 s^ = $100 i- i——L——- = $547.49. 3. Find the amount of an annuity of $500 a year for 10 years accumu- lated at 4%. 4. A man saves $400 a year for 10 years; what will bis savings amount to in 10 years if they are accumulated at 4% ?