web extension 28b - derivation of annuity formulas
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Web Extension 28B
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 pro-gression, where A is a positive constant that is less than 1 and X is the sum of the geometric progression:
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ikad
a/iS
tock
pho
to.c
om
(28B–1)X At
t 1
∞
55
∑
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 progres-sion actually has a solution:
For example, in the case of A = 1/2, the sum is
(28B–2)X AAA
t
t
5 5251 1
∞
∑
(28B–3)X t
t
5 52
55
( / )/( / )
1 21 2
1 1 21
1
∞
∑
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28WB-2 Web Extension 28B Derivation of Annuity Formulas
Now consider a perpetuity with a constant payment of PMT and an interest rate of I. The present value of this perpetuity is
(28B–4) PVt
515
PMTI t( )11
∞
∑
This can be written as a geometric progression:
(28B–5) PV PMTI
PMTIt
t t
t
51
515 5
11
111 1( )
∞ ∞
∑ ∑
Because 1 + I is positive and greater than 1 for reasonable values of I, the sum-mation in Equation 28B-5 is a geometric progression with A = 1/(1 + I). Therefore, using Equation 28B-2, we can write the summation in Equation 28B-5 as
(28B–6)1
1
11
1 11
1 15
1
21
5 II
It
∞
∑t
51I
Substituting this result into Equation 28B-4 gives us the present value of a perpetuity:
(28B–7) PVPMTI
5
(28B–8)
Now consider the time lines for a perpetuity that starts at time 1 and a perpetu-ity that starts at time N + 1:
PV of first time line =PMTI
0 1 2 N N+1 N+2 N+3├───┼───┼───∙∙∙───┼────┼───┼───┼─∙∙∙ PMT PMT PMT PMT PMT PMT
0 1 2 N 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 2 N├───┼───┼───∙∙∙───┤ PMT PMT 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 28B-7 as follows:
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28WB-3Web Extension 28B Derivation of Annuity Formulas
If we apply Equation 28B-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:
Subtracting Equation 28B-9 from 28B-8 gives the present value of an ordinary an-nuity, PVA:
(28B–9)PV of second time line = PMTI I N
11( )1
(28B–10)PVA PMTI
PMTI I N5 2
1
11( )
(28B–11)PVA PMTI I I N
5 21
1 11
( )
(28B–12)FVA = PVA (1+1)N 5 21
1PMTI I I
IN
N1 11
1
( )( )
(28B–13)FVAN
5 12PMT I
I I( )1 1
This can be rewritten as
The future value of an ordinary annuity is equal to the present value com-pounded out to N periods:
This can be rewritten as
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