Problem 5: Compound Annuities

Suppose you are looking to buy a new four-wheeler to remove snow from your driveway. Instead of borrowing the $7,000 you would need to pay for the four-wheeler, you want to save for the purchase. You need to ask yourself two questions:

a. How much will you need to save each month if you want to buy the four-wheeler in fifty months and you can earn 7 percent interest on your investment?

b. How much will you have to save each month if you want to buy the four-wheeler in twenty-four months and you can earn 7 percent interest on your investment?

Note: The method you use to calculate the monthly payments will depend on the type of financial calculator you have. Some calculators require you to set the number of payments to twelve (for monthly payments) and also divide the interest rate by twelve months. Other calculators only require you to set the number of payments to twelve. Determine what your calculator requires before you solve problems requiring monthly data.

Before solving for the monthly payment, follow these steps: (1) clear your calculator's memory, (2) set your number of payments to twelve so that your calculator will calculate monthly payments instead of annual payments, and (3) make sure your calculator is operating in “end mode,” since the payments are received at the end of each period.

To solve the first question, input the following information:

FV = –$7,000

N = 50

I = 7

PMT = ?

If you earn 7 percent interest on your investment, you will need to save $120.98 each month to save $7,000 in fifty months. If you do not have a financial calculator, use the following to solve this problem:

PMT = FV_n,i/[(1 + (i/12))ⁿ – 1]/(i/12)

PMT = $7,000 * [(1 + (0.07/12))⁵⁰ – 1]/(0.07/12) = $120.98

To solve the second question, input the following information:

FV = ­$7,000

N = 24

I = 7

PMT = ?

After solving for the payment, you will discover that you need to save $272.57 each month to save $7,000 in twenty-four months. If you do not have a financial calculator, use the following to solve this problem:

PMT = $7,000 * [(1 + (0.07/12))²⁴ – 1]/(0.07/12) = $272.57

As a general rule, it is better to save for a purchase than to borrow money for it because when you borrow you will have to pay interest instead of earning interest.

Present Value of Annuities

Let's try another sample problem using annuities; this time, we will be calculating the present value instead of the set payment amount.