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:

  • You took out such a loan for $5500 (the maximum amount for a dependent first-year student) on September 1, 2014, at the start of your freshman year.
  • You pay the interest as you go, on a quarterly basis as billed.
  • Paying back the loan is to begin on December 1, 2019, after you graduate on June 1, 2019 (so you will have had the loan for ).
  • We disregard the origination fee, which is deducted from what you get.

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.