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Derivation of annuity formula PV and FV
Master of economics (DKN2A)
University of Madras
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Perpetuities and Annuities: Derivation of shortcut formulas
- Perpetuity formula Outline
- The mathematical derivation of the PV formula
- Derivation of the perpetuity formula using the Law of One Price......................................................................
- Annuity formulas
- The mathematical derivation of the PV formula
- Derivation of the annuity formula using the Law of One Price
- Growing Perpetuity formula
- The mathematical derivation of the PV formula
- Derivation of the perpetuity formula using the Law of One Price....................................................................
- Growing Annuity formula
- The mathematical derivation of the PV formula
- The formula for the growing annuity encompasses all of the other formulas
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Perpetuity formula
A perpetuity is a stream of equal cash flows that occur at regular intervals and last for ever
The mathematical derivation of the PV formula
The present value of a perpetuity P with payment C and interest r is given by:
=
1 +
+
1 +
+
1 +
+ ⋯
= C ∗
1
1 +
+
1
1 +
+
1
1 +
+ ⋯
= ∗
1
1 +
∞
You may recognize this, from Calculus classes, as a geometric progression:
=
∞
Where Z is a positive constant that is less than 1, and X is the sum of the geometric progression
Recall that the sum of such a series actually has a closed-form solution:
=
∞
=
1 −
The Present Value of the perpetuity can then be written as a geometric progression, where =
:
= ∗
1
1 +
∞
= ∗
1 −
= ∗
1
1 +
1 −
1
1 +
= ∗
1
!"
!#$%#&'(&)
=
*
$
0 1 2 3
...
C C C
...
fbenabdelkader
Annuity formula
An ordinary annuity is a stream of N equal cash flows paid at regular intervals.
The mathematical derivation of the PV formula
The present value of an N-period annuity A with payment C and interest r is given by:
+=
1 +
+
1 +
+
1 +
+ ⋯+
1 +
,
+= ∗
1
1 +
,
You may recognize this, from Calculus classes, as a finite geometric series. The formula for the sum of such
a series is:
=
,
=
∗
1 −
,
1 −
The Present Value of the N-period annuity can then be written as a geometric progression, where =
:
+
= ∗
1
1 +
,
= ∗
∗
1 −
,
1 −
= ∗
1
1 +
∗
1 −
1
1 +
,
1 −
1
1 +
This equation can be simplified by multiplying it by
which is to multiply it by 1. Notice that (1+r) is
canceled out throughout the equation by doing this. The formula is now reduced to:
+= ∗
1 −
1
1 +
,
1 + − 1
!"N
NN
N-
--
-period An
period Anperiod An
period Annuity
nuitynuity
nuity=
:
$
∗
;
< −
<
< + $
=
>
0 1 2 3 N
...
C C C C
...
fbenabdelkader
Alternative derivation:
Now consider the time lines for a perpetuity that starts at time 1,
Perpetuity C, t 1
and a perpetuity that starts at time N+1:
Perpetuity C, t N+
Notice that if we subtract the second time line from the first, we get the time line for an ordinary annuity with
N payments:
The present value of an ordinary annuity is then equal to the present value of the first time line minus the
present value of the second time line.
The present value of the Perpetuity C, t 1
, is given by:
!"
!#$%#&'(&) *,&
<
=
$
<
The present value of the Perpetuity C, t N+
, is given by:
!"
!#$%#&'(&) *,&
=<
=
$
∗
<
< + $
=
@
0 1 2 3 N N+1 N+2 N+
... ...
C C C C C C C
0 1 2 3 N N+1 N+2 N+
... ...
C C C
0 1 2 3 N
...
C
C C C
...
...
... ...
fbenabdelkader
Derivation of the perpetuity formula using the Law of One Price......................................................................
To derive the shortcut, we calculate the value of an annuity by creating our own annuity.
Suppose you could invest $100 in a bank account paying 5% interest per year. Suppose also you withdraw the
interest and reinvest the $100 every year. You might decide after 20 years to close the account and withdraw
the principal. By doing this, you can create a 20-year annuity of $5 per year. In addition, you will receive an
extra $100 at the end of 20 years.
The Law of One Price:
"BS'# QTT#& =!"U'&'$# VBTW USOXT %B(P Y) &W# BTT#&
The present value of all cash flows in our example is $100 (the initial investment required to create these cash
flows).
$100 =
20_year annuity of $5 per year
)
+
(
$100€ in 20 years
)
(
20_year annuity of $5 per year
)
= $100 −
(
$100€ in 20 years
)
= $100 −
$
(1 + 5%)
d
= $62.
So the PV of the 20-year annuity of $5 per year is $62.
Let’s generalize: suppose we invest an amount P in the bank. Every period we can withdraw the interest,
C=r*P, leaving the principal P. After N periods, we close the account and we get back the original investment
P. According to law of one price, P is the present value of all future cash flows.
=
(
Annuity of C for N periods
)
+
(
P in period N
)
(Annuity of C for N periods)= −(P in period N)
= −
(
1 +
)
,
= ∗;1 −
1
(
1 +
)
,
>
Recall that the periodic payment C is the interest earned every period: = ∗
fbenabdelkader
Equivalently,
=
The present value of an ordinary Annuity
!"(=%#$(OPT QRR'(&))=
:
$
∗;< −
<
(< + $)
=
>
fbenabdelkader
The Present Value of the perpetuity can then be written as a geometric progression,
where =
l
and g < r
(i)=
(1 + r)
+
(1 + r)
∗
1
(1 + )
∞
=
(1 + r)
+
(1 + r)
∗
1 + j
1 +
1 −
1 + j
1 +
(
i
)
=
(1 + r)
+
(1 + r)
∗
(1 + j)
(
− j
)
(i)=
(1 + r)
∗ 1 +
(1 + j)
( − j)
(
i
)
=
(1 + r)
∗
(
1 +
)
( − j)
!"(m$OX(Rn !#$%#&'(&))=
$ − n
; oHMℎ n &lt;fbenabdelkader
Derivation of the annuity formula using the Law of One Price
To derive the shortcut, we calculate the value of a growing perpetuity by creating our own perpetuity.
Suppose you want to create a perpetuity growing at 2%. You could invest $100 in a bank account paying 5%
interest per year forever. At the end of the first year, suppose you withdraw only $3 from the interest ($5).
You will have $102to reinvest (2% more than the initial investment). By doing this, you can create a growing
perpetuity that starts at $3 and grows 2% per year.
The Law of One Price: the value of the growing perpetuity must be the same as the cost we incurred to
create the perpetuity.
Let’s generalize: suppose we invest an amount P in the bank. If we want to increase the amout we withdraw
each year by g, then P will have to grow by the same factor g.
Every year we should reinvest (1 + j) and withdraw =( − j)= ( − j)
The present value of the growing perpetuity is then the upfront cost: =
r
kl
.
!"(m$OX(Rn !#$%#&'(&))=
$ − n
The formula for the growing
!"(m$OX(Rn
Example:
Consider a growing perpetuity:
E = ∞ ;j q
l
q 1
<n
<$
=
→ D wx =→
!"m$OX(Rn !#$%#&
The formula for the growing annuity encompasses all of the other formulas
$OX(Rn QRR'(&)
:
$ n
∗ <
< n
< $
→∞
!#$%#&'(&)
:
$kn
∗< D
:
$kn
fbenabdelkader
mulas
n
$
=
Derivation of annuity formula PV and FV
Course: Master of economics (DKN2A)
University: University of Madras
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