Definition of Annuity

Chapter 3
An annuity is any sequence of equal periodic payments.
Mathematics of Finance An ordinary annuity is one in which payments are made at
the end of each time interval. If for example, $100 is deposited
into an account every quarter (3 months) at an interest rate of
8% per year, the following sequence illustrates the growth of
money in the account after one year:
Section 3
0.08
Future Value of an Annuity;
y; Sinkingg Funds 100 + 100 1 + + 100 (1.02 ) (1.02) + 100(1.02)(1.02)(1.02)
4
100 + 100(1.02) + 100(1.02) 2 + 100(1.02)3
3rd qtr 2nd quarter 1st quarter

This amount was just put in at the end of the 4th quarter,
so it has earned no interest.
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General Formula for

Example
Future Value of an Annuity

(1 + i) 1
n
Suppose a $1000 payment is made at the end of each
FV = PMT quarter and the money in the account is compounded
i quarterly at 6.5% interest for 15 years. How much is in the
where account after the 15 year period?
FV = future value (amount)
PMT = periodic payment
i = rate per period
n = number of payments (periods)

Note: Payments are made at the end of each period.

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 Example Amount of Interest Earned

Suppose a $1000 payment is made at the end of each How much interest was earned over the 15 year period?
quarter and the money in the account is compounded
quarterly at 6.5% interest for 15 years. How much is in the
account after the 15 year period?
Solution: (1 + i ) n 1
FV = PMT
i
0.065 4(15)
1 + 1
FV = 1000
4 = 100,336.68
0.065
4

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Amount of Interest Earned

Graphical Display
Solution

How much interest was earned over the 15 year period?

Solution:
Each periodic payment was $1000. Over 15 years,
15(4)=60 payments were made for a total of $60,000.
Total amount in account after 15 years is $100,336.68.
Therefore, amount of accrued interest is $100,336.68 -
$60 000 = $40,336.68.
$60,000 $40 336 68

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 Balance in the Account Sinking Fund
at the End of Each Period

Definition: Any account that is established for

accumulating funds to meet future obligations or debts is
called a sinking fund.
The sinking fund payment is defined to be the amount
that must be deposited into an account periodically to have
a given future amount.

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Sinking Fund
Sinking Fund Payment Formula
Sample Problem

To derive the sinking fund payment formula,

formula we use How much must Harry save each month in order to buy a new
algebraic techniques to rewrite the formula for the future car for $12,000 in three years if the interest rate is 6%
value of an annuity and solve for the variable PMT: compounded monthly?

(1+ i)n 1
FV = PMT
i
i
FV = PMT
(1+ i) n
1

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 Sinking Fund Approximating Interest Rates
Sample Problem Solution Example

How much must Harry save each month in order to buy a new Mr.
Mr Ray has deposited $150 per month into an ordinary
car for $12,000 in three years if the interest rate is 6% annuity. After 14 years, the annuity is worth $85,000. What
compounded monthly? annual rate compounded monthly has this annuity earned
Solution: during the 14 year period?
i
FV = PMT
(1 + i ) n
1
0 06
0.06

12000 12 = pmt = 305.06
0.06 36

1 + 1
12

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Approximating Interest Rates Solution

Solution (continued)

((1 + i ) n 1 Graph
p each side of the
Mr. Ray has deposited $150 per month into an ordinary
Mr FV = PM T last equation separately
annuity. After 14 years, the annuity is worth $85,000. What i
annual rate compounded monthly has this annuity earned on a graphing calculator
during the 14 year period? (1 + i)14(12) 1 and find the point of
85,000 = 150 intersection.
Solution: Use the FV formula: Here FV = $85,000, PMT = i
$150 and n, the number of payments is 14(12) = 168.
85,000 (1 + i)168 1
Substitute these values into the formula. Solution is =
approximated graphically. 150 i
(1 + x)168 1 85, 000
y= = = 566.67
x 150

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 Solution
(continued)

Graph
p of y =
566.67

Graph of
y = (1 + x) 1
168

The monthly interest rate is about 0.01253 or 1.253%.

The annual interest rate is about 15%.

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