Annuities
Section 5.3
Introduction
• Let’s say you want to save money to go on a vacation, or
you want to save money now for your baby’s college
education.
• A strategy for saving a little bit of money in the present
and having a big payoff in the future is called an
annuity.
• An annuity is an account in which equal regular
payments are made.
• There are two basic questions with annuities:
– Determine how much money will accumulate over time given
that equal payments are made.
– Determine what periodic payments will be necessary to obtain a
specific amount in a given time period.
Calculating short-term annuities
• Claire wants to take a nice vacation trip, so she
begins setting aside $250 per month. If she
deposits this money on the first of each month in
a savings account that pays 6% interest
compounded monthly, how much will she have
at the end of 10 months?
• Claire’s first payment will earn 10 months
interest. So F = 250(1 + .06/12)12(10/12). Note that
the time t is 10/12. Therefore F = 250(1.005)10 =
$262.79.
• Claire’s second payment will earn 9 months
interest. Thus F = 250(1.005)9 = $261.48.
Table of future values
Payment
Future Value
1st
250(1.005)10
2nd
250(1.005)9 = $261.48
3rd
250(1.005)8 = $260.18
4th
250(1.005)7 = $258.88
5th
250(1.005)6 = $257.59
6th
250(1.005)5 = $256.31
7th
250(1.005)4 = $255.04
8th
250(1.005)3 = $253.77
9th
250(1.005)2 = $252.51
10th
250(1.005)1 = $251.25
= $262.79
Totaling up the future value
column, we see that Claire has
$2569.80 to use for her
vacation. She earned $69.80 in
interest.
Ordinary Annuity and Annuity Due
• There are two types of annuity formulas.
• One formula is based on the payments
being made at the end of the payment
period. This called ordinary annuity.
• The annuity due is when payments are
made at the beginning of the payment
period.
• We will derive the ordinary annuity formula
first.
Calculating Long Term Annuities
• The previous example reflects what actually
happens to an annuity.
• The problem is what if the annuity is for 30 years.
• Future Value of the 1st payment for an ordinary
annuity is
• F1 = PMT(1+r/n)m-1
• The future value of the next to last payment is
Fm-1 = PMT(1+r/n)
• The future value of the last payment is Fm = PMT.
• The total future value
F = F1 + F2 + F3 + … + Fm-1 + Fm
Continuing the calculation of a long
term annuity
• The future value is
• Eq1 F  pmt  pmt1  nr   pmt1  nr 2   pmt1  nr m1
• Now multiply the equation above by
(1+r/n)
• Eq2 F 1  nr   pmt1  nr   pmt1  nr 2  pmt1  nr 3  pmt1  nr m
• Take Eq2 – Eq1 F 1    F  pmt1    pmt
r m
n
r
n
• Note that m = nt. Simplifying gives the ordinary annuity
future value formula
F
nt

1  nr   1
pmt
r
n
Formulas
nt

1  nr   1
F  pmt
• ORDINARY ANNUITY
r
n
• ANNUITY DUE – receives one more
period of compounding than the ordinary
annuity so the formula is

1 
F  pmt
r nt
n
r
n
1
1  nr 
Example
• Find the future value of an ordinary
annuity with a term of 25 years, payment
period is monthly with payment size of
$50. Annual interest is 6%.
F  50
1  
.06 (12)( 25)
12
1
.06
12
• F = $34,649.70
• Note: We only put in $15,000. This means
that interest earned was $19,649.70!
Sinking Funds
• A sinking fund is when we know the future
value of the annuity and we wish to compute the
monthly payment.
• For an ordinary unity this formula is
pmt  F
r
n
r nt
n
1  
1
• For an annuity due the formula is
pmt  F
r
1   1  
r
n
n
r nt
n

1
Sinking Fund Example
• Suppose you decide to use a sinking fund
to save $10,000 for a car. If you plan to
make 60 monthly payments (5 years) and
you receive 12% annual interest, what is
the required payment for an ordinary
annuity?
pmt  10000
.12
12
.12 (12)( 5)
12
1  
1
 $122.45
Real – Life Example
• In 18 years you would like to have $50,000 saved for
your child’s college education. At 6% annual interest,
compounded monthly, what monthly deposit must be
made to accomplish this goal?
• The question does not specify when the payments will
be made so we use both formulas for comparison.
• For the ordinary annuity
pmt  50000
• For the annuity due
pmt  50000
.06
12
.06 (12)(18)
12
1  
.06
12
1  .1206 1  
.06 (12)(18)
12
1

1
 $129.08
 $128.44