Student Loan Arithmetic

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We see some student loan borrowers complaining that after paying for many years, they still owe a large portion of the loan. Here’s how the math works.

A meme popular on social media holds this: “My wife and I left graduate school 23 years ago with a combined total of \$70,000 debt. Since then we’ve made \$500 monthly payments for 23 years (\$120,000+). Today, we still owe \$60,000. Explain to me again why student loan debt shouldn’t be cancelled.”

Let’s look at the arithmetic and see why this couple owes so much after paying for so long. I constructed an amortization table that illustrates the situation described in the meme.

I used an interest rate of 8.37 percent. I show the start and end of the table, and after 23 years (276 months), the balance is \$60,212, very nearly the balance mentioned in the meme. (Although my calculations show that 276 payments of \$500 equals \$138,000, which is different than mentioned in the meme.)

This is what happens when a borrower seeks to make small monthly payments: Most of the payment goes to interest, and very little to pay off the loan balance. In this example, at the start of the table, only \$12 is applied to the loan balance, with \$488 going to interest. (I show values rounded to whole dollars, but the underlying calculations use more precision.)

In fact, after 23 years, only \$79 of the monthly payment goes to paying the balance, with \$421 going to paying interest on the remaining balance.

What if the borrower decided to have higher payments, say \$600? The following table illustrates:

The difference is profound. With a higher payment, the loan is paid at month 242. A nearby chart shows how higher payments reduce the loan balance faster than lower payments.

This arithmetic should inform the debate over student loan forgiveness.

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