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Chapter 5
Section 5.3
Annuities
Annuities
The compound interest investment in which all of the money that is to be invested
(i.e. the principal (P)) is put in the investment all at once. This type of investment is
called a lump sum investment.
A second type of investment is where the money is invested in a series of regular
fixed payments (pymt) at the end of each compounding period is called an ordinary
annuity. This is the most common sort of annuity.
If the money is invested in a series of regular fixed payments (pymt) at the
beginning of each compounding period is called an annuity due.
Holiday Club Investment
In order to save money for Christmas presents a person will invest $200 at the end
of September, October and November for Christmas shopping starting in December.
The account pays 3.6% compounded monthly. How much will they have saved for
Christmas shopping?
This is a type of annuity investment. They want to know the future value (FV) of the
annuity. We can look at it as 3 different lump sum investments, one for that is 2
periods (n = 2), one that is 1 period (n = 1) and 1 that is 0 periods (n = 0).
FV  2001 

.036 2
12
 2001 
  200  601.80
.036 1
12
Annuity Formulas
Annuities are often used for investment that run for many periods such as loans and
retirement savings accounts. This may involve hundreds of payments and treating
each one as a separate lump sum investment is not practical. Mathematicians have
developed an ingenious method involving algebra to generate a formula for this.
If we let pymt stand for the amount of each payment and n is the number of payments
then the total future value of all payments is given by:
FV  pymt1  i 
n 1
 pymt1  i 
n2
   pymt1  i   pymt
1
Multiply both sides of the equation above (i.e. every term on the right side) by the
value (1+i). That will cause each exponent to go up by 1.
FV 1  i   pymt1  i   pymt 1  i 
n 1
n
 pymt 1  i 
n 1
 FV 
FV 1  i   FV  pymt 1  i   pymt
   pymt1  i   pymt1  i 
2
   pymt 1  i   pymt1  i   pymt
2
1
n


FV  iFV  FV  pymt 1  i   1

n

iFV  pymt 1  i   1
n
 1  i   1 

FV  pymt

i


n
We then subtract the first expression from the
second noticing what cancels.
Some factoring, canceling and regrouping
gives the formula!
Annuity Formulas
There are 2 types of annuities we mentioned, one where the payment is made at
the end of each period call an ordinary annuity which its future value is given by
FV(ord) and the other where each payment is made at the beginning of each period
which we call and annuity due where the future value is given by FV(due).
Future Value of Ordinary Annuity
Future Value of Annuity Due
 1  i n  1 

FV (ord )  pymt

i


 1  i n  1 
1  i 
FV (due )  pymt

i


A person has $300 dollars taken out of their paycheck at the end of each biweekly
period and put in a retirement account that pays 5.2% compounded biweekly. How
much will they have in this account when they retire in 25 years?
This is asking for the future value of an
ordinary annuity.
pymt = 300
i
.052
26
n  26  25  650
 1  i n  1 

FV (ord )  pymt

i


650
 1  .052

 1 
26

 300
.052


26


 399,680.40
Example
A person wants to have 250,000 in their retirement account when they retire in 20
years. The decide to invest at the end of each week into a Tax Deferred Account
(TDA) that pays 7.8% compounded weekly.
a) How much will their weekly payments need to be?
We still use the FV(ord) formula but we
want to know the value for pymt. We
will use some algebra to solve for this
value.
.078
i

FV(ord)=250,000
52
n = 20·52 = 1040
 1  i n  1 

FV (ord )  pymt

i


1040
 1  .078

 1 
52

250000  pymt
.078


52


250000  pymt 2502.17482 
pymt 
250000
 99.91
2502.17482
b) How much money does he make in interest over the life of this investment?
He makes 1040 payments of $99.91.
His total investment is 1040·99.91 = $103,906
He will have $250,000 in the TDA when he retires so the amount he has earned
will be $250000-$103,906 = $146,094 in interest over the 20 years!
c) We he retires he switches to an account that pays 6% compounded monthly
and withdraws $1000 each month for living expenses. Fill in the table below.
Month
Beginning
Balance
Interest
Withdraw
Ending
Balance
1
250,000.00
1,250.00
1,000.00
250,250.00
2
250,250.00
1,251.25
1,000.00
250,501.25
3
250,501.25
1,252.51
1,000.00
250,753.76
1. He begins the first month with the 250,000 in the retirement account.
2. In month 1 he earns 250,000*(.06/12) = 1250 in interest.
3. After the 1000 dollar withdrawal he has a gain of $250.
4. The next month begins with what was at the end of the previous month.
5. In month 2 he earns 250,250*(.06/12) = 1,251.25 in interest.
6. After the 1000 dollar withdrawal he has a gain of $251.25
7. The next month begins with what was at the end of the previous month.
8. In month 3 he earns 250,501.25*(.06/12) = 1252.51 in interest.
9. After the 1000 dollar withdrawal he has a gain of $252.51