MATH 166 Spring 2016
F.3
c
Wen
Liu
F.3 Annuities and Sinking Funds
Definition:
• An annuity is a sequence of equal payments made at equal time periods.
• An ordinary annuity is one in which the payments are made at the end of the time periods of
compounding.
• The term of an annuity is the time from the beginning of the first period to the end of the last
period.
• The future value of the annuity is the total amount in the aacount, including interest, at the
end of the term of an annuity.
Future Value of an Ordinary Annuity: The future value F V of an ordinary annuity of n = mt
payments of P M T dollars paid at the end of each period into an account that earns interest at the
annual rate of r/m per period is
F V = P M T (1 + r/m)mt−1 + P M T (1 + r/m)mt−2 + · · · + P M T (1 + r/m) + P M T
r mt
1+
−1
m
= P MT
r
m
Definition: Any account that is established for accumulating funds to meet a future need is called
a sinking fund.
Sinking Fund Payment: The periodic payment P M T that is required to accumulate the sum F V
over n periods of time with interest at the rate of r/m per period is
r
m
P MT = F V r mt
−1
1+
m
Technology Center: You may use “TVM Solver” to get “FV” or “PMT”.
Examples:
1. An individual earns an extra $1000 each year and places this money at the end of each year into
an Individual Retirement Account (IRA) in which both the original earnings and the interest
in the account are not subject to taxation. If the account has an annual interest rate of 7%
compounded annually, how much is in the account at the end of 35 years?
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MATH 166 Spring 2016
c
Wen
Liu
F.3
2. Andrea, a self-employed individual, wishes to accumulate a retirement fund of $450, 000. How
much should she deposit each month into her retirement account, which pays interest at a rate
of 6.5%/year compounded monthly, to reach her goal upon retirement 40 years from now?
To use your calculator:
N
I%
PV
P MT
FV
P/Y
C/Y
P MT
Problem 1
35
7
0
1000
1
1
highlight END
Problem 2
12*40
6.5
0
450000
12
12
highlight END
3. Lauren plans to deposit $4000 into a bank account at the beginning of next month and
$225/month into the same account at the end of that month and at the end of each subsequent month for the next 3 years. If her bank pays interest at a rate of 5%/year compounded
monthly, how much will Lauren have in her account at the end of 3 years? (Assume she makes
no withdrawals during the 7-year period. Round your answer to the nearest cent.)
Interest per Period in a Sinking Fund: The interest earned during the mth period of a sinking
fund with payments of P M T and earning interest at a rate of r/m per period is
F V |m − F V |m−1 − 1
Example: A corporation creates a sinking fund in order to have $610000 to replace some machinery
in 12 years.
(a) How much should be placed in this account at the end of each month if the annual interest rate
is 5.9% compounded monthly?
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MATH 166 Spring 2016
F.3
c
Wen
Liu
(b) How much interest would they earn over the life of the account?
(c) Determine the value of the fund after 2 years.
(d) How much interest was earned during the fourth month of the 4th year?
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