Compound sum $3,121.60
3. The last payment is not compounded at all, the second payment is compounded for 1 year, and the first is compounded for 2 years. When the future values of each of the payments are added, their total is the sum of the annuity. In the example, this total is $3,121.60.
Expressed algebraically, with SN defined as the future value, R as the periodic receipt, n as the length of the annuity, and FVIFAin as the future value interest factor for an annuity, the formula for Sn is
Sn = R(1 + i)n-1 + R(1 + i)n-2 + . . . + R(1 + i)1 + R(1 + i)0 = R[(1 + i)n-1 + (1 + i)n-2 + . . . + (1 + i)1 + (1 + i)0]
The expression in parentheses, FVIFAin, has been calculated for various combinations of i and n.4 An illustrative set of these annuity interest factors is given in Table A.4.5 To find the answer to the 3-year, $1,000 annuity problem, simply refer to Table A.4, look down the 4% column to the row of the third period, and multiply the factor 3.1216 by $1,000. The answer is the same as the one derived by the long method illustrated in Figure A.3:
Notice that for all positive interest rates, the FVIFAin for the sum of an annuity is always equal to or greater than the number of periods the annuity runs.6
4 The third equation is simply a shorthand expression in which sigma (X) signifies sum up or add the values of n n factors. The symbol X simply says, "Go through the following process: Let t = 1 and find the first factor. Then t=1
let t = 2 and find the second factor. Continue until each individual factor has been found, and then add these individual factors to find the value of the annuity."
5 The equation given in Table A.4 recognizes that the FVIFA factor is the sum of a geometric progression. The proof of this equation is given in most algebra texts. Notice that it is easy to use the equation to develop annuity factors. This is especially useful if you need the FVIFA for some interest rate not given in the tables (for example, 6.5%).
6 It is worth noting that the entry for each period t in Table A.4 equals the sum of the entries in Table A.2 up to the period n - 1. For example, the entry for Period 3 under the 4% column in Table A.4 is equal to 1.000 + 1.0400 + 1.0816 = 3.1216.
Also, had the annuity been an annuity due, with payments received at the beginning rather than at the end of each period, the three payments would have occurred at t = 0, t = 1, and t = 2. To find the future value of an annuity due, look up the FVIFA, for n + 1 years, then subtract 1.0 from the amount to get the FVIFA, for the annuity due. In the example, the annuity due FVIFA, is 4.2465 - 1.0 = 3.2465, versus 3.1216 for a regular annuity. Because payments on an annuity due come earlier, it is a little more valuable than a regular annuity.
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