Section 4: Annuities
Definition 4.1: An annuity is a sequence of equal payments made at regular intervals of time.
Remark: The regular intervals are usually months.
Example 4.1: Suppose that someone deposits $100 a month into a savings account paying an annual
rate of interest of 4.25% (compounded monthly) for 20 years.
Let’s examine the mathematics of an annuity via Example 4.1. For the sake of terminology, the
monthly payment of an annuity is called the rent and we will denote the rent by R . In Example 4.1,
R 100 .
The amount in the account after 20 years of monthly payments are made is called the future
value of the annuity and we will denote this quantity by F .
We’d like to find a formula for F . For convenience, let R be the rent, r be the annual rate of
interest, and n the compound period of an annuity. Define the compound interest rate as i ,
where i .
Note, after the first period, F R because no interest is paid in the first period. After two
periods, F R R1 i , because interest is paid on the first rent payment. After three periods,
F R R1 i R1 i , etc., so that, after k rent payments,
F R R1 i R1 i R1 i 1
2 k 1
We’d like to come up with a compact form for 1 . Note first that we can factor R from every term in
1 to obtain
F R 1 1 i 1 i 1 i
2 k 1
The trick is to let x 1 i . The expression in brackets then becomes the k 1 degree polynomial
1 x x 2 x k 1 . It is well-known in algebra that
1 x x x
2 k 1
Hence, 1 1 i 1 i 1 i
2 k 1 1 i k 1 1 i k 1 , so that
1 i 1 i
1 i 1
F R .
We can now solve Example 4.1: we want to know the future value of an annuity when R 100 ,
n 12 , and k 240 . Hence,