1
Annuities and Perpetuities:
The goal of these notes is to derive all perpetuity and annuity formulas. It turns
out that we only need to know one calculus formula:
T
X
j
=1
x
j
1
=
1
x
T
1
x
(1)
The proof of this calculus formula is at the end of this document (for those
should be really excited now! A growing annuity has the following cash ±ows:
t
= 0
t
= 1
t
= 2
t
= 3
...
T
0
1
1 +
g
(1 +
g
)
2
...
(1 +
g
)
T
1
(2)
Then, we can conveniently write this as: (The
P
sign just indicates summation)
PV
=
T
X
t
=1
(1 +
g
)
t
1
(1 +
r
)
t
(3)
First trick: Multiply and divide by
(1 +
r
) :
PV
=
T
X
t
=1
(1 +
g
)
t
1
(1 +
r
)
t
1 +
r
1 +
r
=
1
1 +
r
T
X
t
=1
1 +
g
1 +
r
±
t
1
Second trick: De²ne
x
±
1+
g
1+
r
so that:
PV
=
1
1 +
r
T
X
t
=1
x
t
1
(4)
We can now apply the formula in equation 1, i.e:
P
T
t
=1
x
t
1
=
1
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 Fall '07
 Berk
 Time Value Of Money, Annuity, Perpetuity, Expression, Englishlanguage films

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