What is the difference between these two problems?

Problem A Kanye wants to save $10,000 in 5 years by making monthly payments into an ordinary annuity for a down payment on a condominium at the shore. If the annuity pays 0.80% annual interest compounded monthly, what will his monthly payment be?

VS.

Problem B Kanye wants to save $10,000 in 5 years by making monthly payments into an ordinary annuity for a down payment on a condominium at the shore. If the annuity pays 0.80% monthly interest compounded monthly, what will his monthly payment be?

The key is to notice that the interest rate can be given as an annual rate or a monthly rate. How do we apply the annuity formula,

In this formula,

A: future value

R: payment

m: number of times interest is compounded in a year

r: annual interest rate

n: number of compounding periods

To help us see the difference, think of ^{r}/_{m} as the interest rate per period. In Problem A, the annual interest rate is 0.80% and interest is compounded 12 times a year. To solve this problem, we set r = 0.008 and m = 12 and put the other numbers in:

The fraction on the right is equal to 61.19535446.

Solving for R in this equation gives R ≈ 163.41.

In Problem B, the interest rate given to you is the interest rate per period. In other words, ^{r}/_{m} = 0.008 This means that in this problem, you need to solve for R in

In this case, the fraction is equal to 76.62386683:

Dividing this number into 10,000 gives a payment of R ≈ 130.51.

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

ris 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,

Google Sheets has several built in functions for working with annuities. To use these functions, we’ll start with a standard Sheets worksheet.

This worksheet contains the variables used throughout Chapter 5. Values given in a problem will be entered in column B. Values calculated by Sheets will be entered in column C. We will also assume that amounts paid out are negative and amounts received are positive.

We’ll modify the worksheet shown above. This will allow us to use Sheets to calculate the different amounts in the annuity formula,

This is done using two functions in Sheets, the FV (future value) function and the PV (present value) function. Annuities have a regular payment into or out of the account. If the payment is made at the end of the compounding period, the annuity is called an ordinary annuity. Payments are made at the beginning of the compounding period for an annuity due. In Sheets, amounts that you pay out are considered negative numbers and amount you receive are positive amounts.

Find the Future Value of the Annuity

An investor deposits $500 in a simple annuity at the end of each six-month payment period. This annuity earns 10% per year, compounded semiannually.

a. Find the future value if payments are made for three years.

Solution Since the investor is paying $500 into the annuity, the payment must be entered as a negative number in Sheets.

1. Start from the basic worksheet and enter the values shown below.

2. Click in cell C6. Type =FV( in the cell. As soon as you type the parentheses, Sheets recognizes what you are trying to do.

3. Notice that the word rate is highlighted. This indicates that you need to either type in the rate or the cell reference where the rate is located. The rate for the command is actually the interest rate per period. The annual interest rate is in cell B3 and the number of periods per year is in cell B7. We need to get the interest rate per period by typing B3/B7. You can also click in cell B3, type a /, and then click the cursor in cell B7. Now type a comma.

4. The number of periods is in cell B2. Type B2 or click the cursor in that cell. To continue to the next input, type a comma.

5. The payment is located in cell B4. The value is negative since the money is paid out. Type B4 or choose cell B4 followed by a comma.

6. Since we deposit nothing into the account initially, the present value is zero. Enter B5 or select cell B5 followed by a comma and a parentheses. Press enter on the keyboard to calculate the future value.

The future value of the annuity is $3400.96. There is an optional argument for the FV function. This argument indicates whether the annuity is an ordinary annuity (use 0 for the argument) or an annuity due (use 1 for the argument).

If the last argument is not supplied, the annuity is assumed to be an ordinary annuity.

b. Find the future value if payments are made for 30 years.

Solution Thirty years of semiannual payments corresponds to n = 60. We need to make this change in the FV function, and solve for future value again. Change the value in cell B2 to 60.