An Annuity

Date: 12/13/95 at 17:1:35
From: Anonymous
Subject: Compound Interest w/ Monthly Installments

Question:
One of your previous 'columns' dealt with compound interest of
a predetermined amount of money that was deposited and then 
accumulated interest compounded monthly without any additional 
funds being added, similar to a savings bond.  Now, how would you 
derive the future amounts at set points in time of a bank account 
in which a set amount of money was deposited MONTHLY and in
addition accrued interest monthly?

Date: 6/18/96 at 13:57:5
From: Doctor Robert
Subject: Re: Compound Interest w/ Monthly Installments

The arrangement you speak of is called an annuity.  Suppose that the 
money you deposit earns interest  i  each month.  Suppose interest is 
calculated at the end of each month and you deposit P dollars at the 
end of each month.  Let us see what we have at the end of 6 months.  

It is easiest to start at the endpoint rather than at the beginning. 
The total amount you have is 
 
 S = P + P(1+i)+P(1+i)^2 + P(1+i)^3 + ... + P(1+i)^5

where the first term is the amount you deposited at the end of the 
sixth month, the second term is the amount that your fifth-month 
deposit is worth, the third term is the amount that your fourth-month 
deposit is worth, etc.  It works out this way because each deposit 
only earns interest from the time that it is deposited. 

Now the expression for S is nothing but a geometric series.  The first 
term is P and the common ratio is (1+i).  If we are to work this for n 
months, then

 S = P + P(1+i)+ P(1+i)^2 + ... + P(1+i)^(n-1).

The formula for the sum of a geometric series can be used to get the 
following result:

 S = P[(1+i)^n - 1]/i  

which is the formula which gives the total value of the account, where 
P is the monthly payment, i is the MONTHLY interest rate, and n is the 
number of months.  Note that this result is only good for the 
situation where the payments are made at the END of the interest 
period.  The equation is different when payments are made at the 
beginning of the period.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/