Derivation of Annuity FormulasFollowing are derivations for annuity formulas. It is actually easier to start withthe 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 geomet-ric progression:

(4A-1)

At first blush, it might seem that X must equal infinity, since the sum goes toinfinity. But notice that as t gets large, the term At gets very small, because A is lessthan 1. For example, suppose that A is equal to 1–2. Then the sum is

Notice that as we continue to add terms, X approaches but never exceeds 1.Perhaps from an algebra or calculus course you recall that the sum of a geometricprogression actually has a solution:

(4A-2)

For example, in the case of A � 1–2 , the sum is

(4A-3)

Now consider a perpetuity with a constant payment of PMT and an interestrate of I. The present value of this perpetuity is

(4A-4)PV � aq

t�1

PMT(1 � I 2 t

X � aq

t�1(1>2)t �

1>21 � (1>2)

� 1

X � aq

t�1At �

A1 � A

X � 1>2 � 1>4 � 1>8 � 1>16 � p

X � aq

t�1At

web extension 4A

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4A-2 Web Extension 4A Derivation of Annuity Formulas

This can be written as a geometric progression:

(4A-5)

Because 1 � I is positive and greater than 1 for reasonable values of I, the summa-tion in Equation 4A-5 is a geometric progression with A � 1/(1 � I). Therefore,using Equation 4A-2, we can write the summation in Equation 4A-5 as

(4A-6)

Substituting this result into Equation 4A-4 gives us the present value of a perpetuity:

(4A-7)

Now consider the time lines for a perpetuity that starts at time 1 and a perpe-tuity that starts at time N � 1:

PV �PMT

I

aq

t�1a 1

1 � Ib t

�

a 11 � I

ba1 � a 1

1 � Ib b

�1I

PV � PMTaq

t�1

111 � I 2 t � PMTa

q

t�1a 1

1 � Ib t

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10

PMT

2

PMT

N

PMT

N�1

PMT

N�2

PMT

N�3

PMT

10 2 N N�1

PMT

N�2

PMT

N�3

PMT

Notice that if we subtract the second time line from the first, we get the time linefor an ordinary annuity with N payments:

10 2 N

PMT PMT PMT

Therefore, the present value of an ordinary annuity is equal to the present valueof the first time line minus the present value of the second time line. The presentvalue of the first time line, which is a perpetuity, is given by Equation 4A-7:

(4A-8)

If we apply Equation 4A-7 to the second time line, it gives the value of thepayments discounted back to time N (because if we just look at the time line from

PV of first time line �PMT

I

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Derivation of Annuity Formulas 4A-3

N on, it is an ordinary annuity that starts at time N � 1). To find the present valueof the second time line, we just discount this perpetuity value back to time 0:

(4A-9)

Subtracting Equation 4A-9 from 4A-8 gives the present value of an ordinary annuity, PVA:

(4A-10)

This can be rewritten as

(4A-11)

The future value of an ordinary annuity is equal to the present value com-pounded out to N periods:

(4A-12)

This can be rewritten as

(4A-13)FVA � PMT c a 11 � I 2 NI

b � a 1Ib d

FVA � PVA 11 � I 2 N � PMT c a 1Ib � a 1

I11 � I 2 N b d 11 � I 2 N

PVA � PMT c a 1Ib � a 1

I11 � I 2 N b d

PVA � a PMTIb � a PMT

Ib

111 � I 2 N

PV of second time line � a PMTIb

111 � I 2 N

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