# Two Ways To Work Out How Much An Annuity Has Accumulated

Let’s take a look at two ways you can calculate the amount of money accumulated in an ordinary annuity. Recall that an annuity is a series of payments made into account that accrues interest over time. Below we will look at an example in which we calculate the interest on each payment and add the amounts to figure out how much has accumulated in the account. This is how we initially introduced the annuity. After this calculation, we’ll use a formula to get the same accumulated amount. This is how you will do it for typical problems.

Suppose a payment of \$1000 is made semiannually to the annuity over a term of three years. If the annuity earns 4% per year compounded semiannually, the payment made at the end of the first six-month period will accumulate

This means \$1000 is multiplied by 1.02 five times, once for each of the remaining six-month periods.

The next payment also earns interest, but over 4 six-month periods. This payment has a future value of

This process continues until we have the future value for each payment.

The last payment occurs at the end of the last period and earns no interest. Now add the amounts on the right to obtain the accumulated amount,

A ≈ 1104.08+1082.43+1061.21+1040.40+1020+1000 ≈ 6308.12

In this sum, each term has been rounded to the nearest penny resulting in a potential inaccuracy. This sum  is the sum of the five payments of \$1000 plus any accumulated interest.

Now let’s look at how you would get the same number in practice…with the annuity formula.

For a few payments over a few periods, creating this sum and adding the terms on your calculator is not too intimidating. However, if there are many payments over many years the task is exhausting. Luckily, there is a formula for calculating the sum of these terms,

In this formula,

R is the amount of the payment

r is the annual interest rate

m is the number or payments per year

n is the number of payments over the life of the annuity.

Let’s use this formula to calculate the accumulated amount A in the annuity described above:

With this formula, we have only rounded once…at the very end of the problem. If you add the terms individually (rounding each of them), you can potentially end up with a slightly different amount.

When completing problems on the homework or quizzes, make sure you use the appropriate strategy. If they indicate that you should work out how much each payment accumulates to and add the results, use the strategy at the top. However, if they indicate that you should use the annuity formula, then use the formula above. This may be why some of you are ending up with incorrect answers.

Another potential issue is using the calculator to compute the formula. The best strategy is to think of the formula like this,

On a TI graphing calculator, this looks like