ANNUITIES

What is an annuity?
        An annuity is a sequence of equal periodic payments. An ordinary annuity, the kind we'll talk about here, is one in which the payments are made at the end of each period. When you're dealing with annuities you often want to figure out the future value, or the amount of total payment plus interest.

Determining future value
      An example will help in developing the formula for finding future value. Let's say you deposit $100 once a year in an account that pays 6% yearly interest. After the first deposit, you have just $100. After the next deposit, the original $100 is now (1.06) * $100, and your total amount is $100+(1.06)*$100. After the third deposit, your original money, with interest, is now 1.06²*$100, your second deposit is now 1.06 * $100, and your total is:

Let's say we want a fast way to add this sum up.
By multiplying everything on both sides of the equation by 1.06, we get another equation:

Notice that some of the terms in the first equation and the second equation are the same, and if we subtract one equation from the other, these will get subtracted out. Let's try that! We find that:

We can factor out a 100 from the right hand side, and then divide by 0.06 to get the sum, what we were looking for in the first place! The result is:

Does this suggest a formula to you? Remember that $100 was the amount of a payment (let's call it PMT), and 0.06 was the interest rate (call it r). Also, the number of payments (n) was 3. Then the final value, say FV, is given by:

If you're not familiar with finding sums of geometric sequences, you might want to look back over the algebra above or work through it yourself, just to be sure you understand it. When you're ready, go on to the next section.