The amount of interest charged on a loan is determined by the principal (the amount borrowed) and by the method used to calculate the interest. With simple interest, the borrower pays ; fixed amount of interest for each period of the loan. The interest rate is usually quoted as an annual rate.
For a principal and an annual rate of interest , the interest owed after years is
and the total amount due on the loan is
EXAMPLE 1 Simple Interest on a Federal Direct Student Loan
The U.S. Department of Education offers direct loans to students to use for tuition, fees, housing, and textbooks (such as this one), with repayment deferred until after graduation. Any eligible student can get such a loan, regardless of financial need or credit history.
For 2014–2015, the interest rate was 4.29% for undergraduates. Interest is charged from when you receive the loan until repayment is scheduled to start, usually six months after you leave school. This interest is on the original principal only (including an origination fee of 1.073%), not on accrued (accumulated) interest, hence it is simple interest on the principal.
The interest is actually calculated according to a method called the Simplified Daily Interest Formula: The interest rate is divided by the number of days in a year, taken to be 365.25, and that daily rate (called the interest rate factor) is multiplied by the number of days in the billing period. For example, for the interest rate of 4.29%, the interest rate factor is , or 0.0117454% per day. However, for simplicity, we calculate as if the year were evenly divisible into 12 months and into 4 quarters. We also regard the origination fee as included in the original loan principal.
One option for repayment is to pay the interest as it comes due, billed on a quarterly basis, but defer starting to pay back the principal until six months after leaving school (the “grace period”).
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Suppose things played out as follows:
How much is the monthly interest, how much total interest will you have paid over the 51 months, and how much will you owe when you start to pay back the loan?
We have and , and for one quarter, we have years. So the interest for one quarter is . Over the 51 months (17 quarters), you will have paid . At the time when repayment begins, you would still owe (just) the original principal of $5500.
If instead you do not pay the interest as you go, you would owe the principal of $5500 plus the accumulated simple interest of $1002.83, for a total of $6502.83.
Suppose that you borrow $3000 at the start of your junior year. What is your quarterly interest payment?
Caution: The misleading phrase “simple interest loan” is now often used in the loan trade to refer to a loan in which interest is charged each day based on the balance owed that day, as opposed to some other kinds of loans, such as add-on loans (which we consider later). Such a “simple interest loan” is usually a loan at compound interest with a compounding period of one day. We consider compound interest loans next.