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Unformatted text preview: WEB EXTENSION Derivation of Annuity Formulas
In this extension we give derivations for annuity formulas. It is actually easier to start with the formula for a perpetuity. First, consider the following geometric progression, where A is a positive constant that is less than 1 and X is the sum of the geometric progression: 4B X t1 A t (4B-1) At first blush, it might seem that X must equal infinity, since the sum goes to infinity. But notice that as t gets large, the term At gets very small, because A is less than 1. For example, suppose that A is equal to 1/2. Then the sum is X = 1/2 + 1/4 + 1/8 + 1/16 + Notice that, as we continue to add terms, X approaches but never exceeds 1. From an algebra or calculus course you may recall that the sum of a geometric progression actually has a solution: A A 1-A
t X (4B-2) t1 For example, in the case of A = 1/2, the sum is X t1 1=2 1 -1=2 1 1=2
t (4B-3) Now consider a perpetuity with a constant payment of PMT and an interest rate of I. The present value of this perpetuity is PV PMT t t1 1 I (4B-4) 1 2 Web Extension 4B: Derivation of Annuity Formulas This can be written as a geometric progression: t 1 1 PV PMT PMT t t1 1 I t1 1 I (4B-5) Because 1 + I is positive and greater than 1 for reasonable values of I, the summation in Equation 4B-5 is a geometric progression with A = 1/(1 + I). Therefore, using Equation 4B-2, we can write the summation in Equation 4B-5 as t1 1 1I t 1 1 1I 1 I 1- 1I (4B-6) Substituting this result into Equation 4B-4 gives us the present value of a perpetuity: PV PMT I (4B-7) Now consider the time lines for a perpetuity that starts at time 1 and a perpetuity that starts at time N + 1:
0 1 PMT 0 1 2 PMT 2
... ... N PMT N N+1 PMT N+2 PMT N+3 PMT ... N+1 N+2 N+3 PMT PMT PMT ... Observe that, by subtracting the second time line from the first, we get the time line for an ordinary annuity with N payments:
0 1 PMT 2 PMT
... N PMT Therefore, the present value of an ordinary annuity is equal to the present value of the first time line minus the present value of the second time line. The present value of the first time line, which is a perpetuity, is given by Equation 4B-7 as follows: PV of first time line PMT I (4B-8) If we apply Equation 4B-7 to the second time line, it gives the value of the payments discounted back to time N (because if we look at the time line just from N on, it is an ordinary annuity that starts at time N + 1). To find the present value of the second time line, we just discount this perpetuity value back to time 0: Web Extension 4B: Derivation of Annuity Formulas 3 PV of second time line PMT 1 I 1 IN (4B-9) Subtracting Equation 4B-9 from 4B-8 gives the present value of an ordinary annuity, PVA: PVA PMT PMT 1 - I I 1 IN (4B-10) This can be rewritten as " !# 1 1 PVA PMT - I I1 IN (4B-11) The future value of an ordinary annuity is equal to the present value compounded out to N periods: " !# 1 1 - 1 IN FVA PVA 1 I PMT I I1 IN
N (4B-12) This can be rewritten as " FVA PMT ! # 1 IN 1 - I I (4B-13) ...
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