Download Future Value of an Annuity

Future Value of an
Annuity
After paying all your bills, you have $200
left each payday (at then end of each month)
that you will put into savings in order to
save up a down payment for a house. If you
invest this money at 5% interest per year,
compounded monthly, how much money
will you have saved toward the down
payment at the end of 5 years?
First, a little background:
Definitions:
1. An annuity is a succession of equal
payments made at equal periods of time.
An ordinary annuity has payments due at
the end of each time period.
2. A sequence is a list of numbers.
4. A geometric sequence is a list of
numbers, starting with a number, a,
where each number is some constant
r times the preceding number.
5. A geometric series is the sum of a
geometric sequence.
3. A series is the sum of the numbers in a
sequence.
In general, a geometric series has the form:
Example:
a + ar + ar 2 + ar 3 + . . . + ar n + . . .
1, 2, 4, 8 is a geometric sequence where
a = 1and r = 2.
The sum of the first n terms of a geometric
series can be calculated by the formula:
1 + 2 + 4 + 8 is a geometric series.
Sum =
a( r n − 1)
(r − 1)
1
So, if we invest $200 a month, using
r

A = P 1 + 
 m
m⋅t
the future
maturity value
59
 .05 
of the first
2001 +

 12 
deposit is:
58
of the second:
 .05 
2001 +

 12 
57
of the third:
 .05 
2001 +

 12 
So the value of our annuity in 5 years is:
  .05  59 
200 1 +
− 1
  12 

a (r n − 1)

=
=
(r − 1)
 .05 
1 +
 −1
 12 
$13,601.22
(rounded to the nearest penny).
The sum of the future value of all the
deposits is:
59
58
 .05 
 .05 
2001 +
 + 2001 +
 + L + 200
12


 12 
which is a geometric series, where


.05 
a = 200, r = 1 + 12 

, and n = 59.
Future Value of an Ordinary Annuity:
Let r be the annual interest rate, m be the
number of times per year that interest is
compounded, and let i =
r
. Also, R is the
m
payment, and n the number of payments,
made at the end of each interest period,
n = m . t. Then the future value of an annuity,
S, is:
 (1 + i )n − 1
S = R

i


In the formula
 (1 + i )n − 1
S = R

i


n
 (1 + i ) − 1

 is sometimes denoted sn¬ i ,
i


read “s angle n at i”.
Example: After 40 years of work, it’s
time to retire. Your company invested
$500 at the end of each month in an
ordinary annuity with annual interest rate
of 8% compounded monthly. If you take
a lump-sum retirement payment, how
much should it be?
2
 (1 + i )n − 1
S = R

i


where R = 500, i =
.08
, the number of
12
payments, n, is (12 per year)(40 years) = 480,
so your lump sum =
  .08  480 
 − 1
 1 +
12 


 = $1,745,503.92
500
.08




12


 (1 + i )n − 1
S = R

i


.08
where i = 12 , n = 480, and now S =
2,000,000, and we need to solve for R where
  .08  480 
 − 1
 1 +
12 

2000000 = R  
.08




12


Classwork:
Find the future value of an ordinary
annuity with payments of $150
made semiannually for 20 years in
an account that pays 6% interest
compounded semiannually.
Let’s say you work for the same
company, and you want to have
$2,000,000 when you retire after
40 years. The company will allow
you to make additional monthly
payments. How much more do
you have to pay each month to
retire after 40 years with 2
million?
Simplifying, you have:
2000000 = R(3491.007831)
Divide to find R = 572.90 (to the nearest
penny).
Keep as many decimal places as possible
for greatest accuracy.
Definition: A sinking fund is an annuity
set up to reach a specified value at a
specified time. Typically, sinking funds
are set up by an organization to retire a
debt.
3
Example: A department of
transportation wishes to build a bridge at
a cost of $1.2 million. They plan to
finance the bridge with the sale of bonds
that will come due in 5 years and have a
face value at that time of $2 million.
At the end of each fiscal year, the
department can invest part of their annual
budgeted funds in a mutual fund that
returns 10% per year compounded
quarterly. How much do they need to
budget each quarter in order to be able to
repay the $2 million debt in 5 years?
We can use the formula for the future
value of an annuity.
 (1 + i )n − 1
S = R

i


  0.1 

 − 1
 1 +
4 
 = R( 25.54465761)
2000000 = R  
0. 1




4


2000000 = R (25.54465761)
20
R = $78,294.26
2000000
= $78308.54 ≠ $78,294.26
25.54
Hold on to as many decimal places as
possible in intermediate calculations.
When units are dollars, it is reasonable to
round you final answer to 2 decimal
places.
Since we’re rounding the final
answer two decimal places (the
nearest penny), why hold on to 8
decimal places in the intermediate
answer (25.54465761)? Could we
round it to two decimal places and
get the same answer?
Annuity due:
An annuity due differs from an ordinary
in that payments to an annuity due are
made at the beginning of each period,
not at the end of the period.
4
The future value, S, of an annuity due
is:
 (1 + i )n+1 − 1
S = R
−R
i


where R is the payment, i is interest
per period, and n is number of
payments (same as for an ordinary
annuity.
5