Math 141 Lecture Notes
Section 5.2 Annuities
Future Value of an Annuity
An annuity is a sequence of payments made at regular time intervals. The time period in
which these payments are made is called the term of the annuity.
An annuity in which the payments are made at the end of each payment period is called
an ordinary annuity, whereas an annuity in which the payments are made at the
beginning of each period is called an annuity due.
An annuity in which the payment period coincides wit the interest compounding period is
called a simple annuity, whereas an annuity in which the payment period differs from
the interest compounding period is called a complex annuity.
Annuities considered here have terms given by fixed time intervals, periodic payments
equal in size, payments made at the end of the payment periods, and payment periods
coincide with the interest compounding period.
The future value S of an annuity of n payments of R dollars each, paid at the end of
each investment period into an account that earns interest at the rate of i per period, is
 (1  i) n  1
S  R
.
i


Example 1: Find the amount of an ordinary annuity of 12 monthly payments of $100
that earn interest at 10% per year compounded monthly.
Present Value of an Annuity
The present value P of an annuity of n payments of R dollars each, paid at the end of
each investment period into an account that earns interest at the rate of i per period, is
1  (1  i)  n 
P  R
.
i


Example 2: Find the present value of an ordinary annuity of 24 payments of $100 each
made monthly and earning interest at 8.5% per year compounded monthly.
Example 3: As a savings program toward Alice’s college education, her parents decide
to deposit $100 at the end of every month into a bank account paying interest at the rate
of 5% per year compounded monthly. If the savings program began when Alice was 5
years old, how much money would have accumulated by the time she turns 18?
Example 4: After making a down payment of $3500 for an automobile, Enrico paid $250
per month for 36 months with interest charged at 11% per year compounded monthly on
the unpaid balance. What was the original cost of the car? What portion of Murphy’s
total car payments went toward interest charges?
Example 5: Cynthia is planning to make a contribution of $2000 on January 31 of each
year into an IRA earning interest at an effective rate of 8.6% per year. After she makes
her 28th payment on January 31 of the year following her retirement, how much will she
have in the IRA?
Example 6: Both Tom and Tony are salaried individuals, 45 years of age, who are saving
for their retirement 22 years from now. Both Tom and Tony are also in the 28% marginal
tax bracket. Tom makes a $1200 contribution annually on December 31 into a savings
account earning an effective rate of 7.5% per year. At the same time, Tony makes a
$1200 annual payment to an insurance company for an after-tax-deferred annuity. The
annuity also earns interest at an effective rate of 7.5% per year. (Assume that both men
remain in the same tax bracket throughout this period and disregard state income taxes.)
a. Calculate how much each man will have in his investment account at the end of 20
years.
b. Compute the interest earned on each account.