Colorado Mathematicians Reveal Best Way to Repay Student Loans

As the government grapples with helping U.S. students pay down their debt – now estimated at a staggering $1.7 trillion owed by nearly 45 million borrowers – a team of Colorado mathematicians has come up with an eye-opening finding: income-based repayment doesn’t work in every graduate’s favor.

In an article published in the SIAM Journal on Financial Mathematics, math experts show how they’ve come up with a formula to calculate the optimal strategy for repaying student debt and the results are clear – surprisingly not by using complex equations, but rather, simple high school mathematics.

“The math shows there’s a simple way to repay student loans in order to minimize the overall cost to the student, yet nobody appears to be promoting it,” said Yu-Jui Huang, an Assistant Professor in Applied Mathematics at University of Colorado, who collaborated on the study with Paolo Guasoni, Head of Mathematical Sciences at Dublin City University in Ireland, with support from Saeed Khalili, Research Assistant in the Mathematics Department at University of Colorado.

What the mathematicians discovered is that, contrary to guidelines pushed by popular online student loan calculators, which typically present two repayment options – either maximize payments to pay debt down quickly or enroll in an income-based repayment program to keep payments as low as possible until the balance qualifies for loan forgiveness – there’s a third option that is actually better for many students, especially those with graduate degrees.

“According to our research, the optimal strategy for some borrowers is to pay down a lot at the beginning of the loan term and defer enrolling in an income-based repayment plan until a later date, or what we call the critical horizon,” explained Guasoni, who says it’s time for people to pay as much attention to their student loans as they pay to their mortgages.

“It’s a simple change in strategy, yet just like renewing a mortgage to take advantage of a lower interest rate, it can make a huge difference, resulting in tens of thousands of dollars in savings over time,” he said.

The option proposed by the researchers is most beneficial to students with large loans, such as graduates of advanced degree programs like dental, medical, or law school, who tend to carry debt in excess of $100,000. Their approach sheds light on the fact that, although student loan forgiveness sounds beneficial, by the time loans are forgiven by the government, some 20 or 25 years after graduation, the balance owing can swell to more than $1 million due to compound interest over time and will be subject to income tax that can reach 40 percent or more.

“So, the year you have your student loan forgiven, you actually have to pay taxes as if you received the forgiven amount as income that year,” Guasoni explained. “If you let your student loan increase over time, the amount you’re going to owe in taxes is so large, you would have been better paying off the loan more quickly to begin with. For large student loans, such taxes can reach hundreds of thousands of dollars.

If a student loan balance is small enough, the researchers argue that one should pay it off as quickly as possible. Otherwise, the basic algebraic equation they propose uses the loan term, income tax rate, interest rate of the student loan, and interest rate of the borrower’s next most expensive loan to come up with a number. If the result is negative, they say a borrower’s optimal strategy is to enroll in an income-based repayment plan right away; if the result is positive, it equals the optimal number of years to wait before enrolling in a plan – and the borrower should pay down as much as possible in the meantime.

“Anyone who has studied high school math can do our calculation,” said Guasoni, noting that the formula is publicly available on the SIAM website. If you’re saddled by the burden of a student loan, the numbers will tell you if there’s a better option out there for you.”

Using the scenario of a dental school graduate carrying $300,000 in debt at 7.08 percent interest, they showed the possible savings. For example, keeping up maximum payments to repay the loan as quickly as possible yields an overall loan cost of $512,000; enrolling in an income-based repayment plan from the start to keep payments down yields a total loan cost of $524,000 when taxes on the forgiven amount are accounted for; and using the new formula proposed by the researchers yields the lowest total loan cost of $490,000, representing a savings of up to $34,000 – more than 10 percent of the original loan.

Each student loan is different, and the researchers caution that income-based repayments aren’t optimal for everyone. “If your student loan is $50,000 or less, for example, in most cases it’s best to pay it off as quickly as you can,” Huang said.

“The untold story of debt forgiveness is that the interest rate continues to increase your balance,” Huang explained. “It sounds very attractive to enroll in a repayment plan and pay very little on your student loan today, but if you’re not paying attention, it will catch up with you tomorrow.”

Read more in SIAM News, a publication of Society for Industrial and Applied Mathematics.