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In the debt-advice business, the most confounding question always seems to be: Which debts should consumers pay off first, ones with the highest interest rates or the smallest amounts?

It evokes debate on the order of Hatfields and McCoys, blue states versus red states, toilet tissue rolling from over or under. Each side is convinced their way is clearly superior and the key to helping people get out of debt.

People with mathematical minds advise one way – paying off debts with the highest interest rates first. You retire expensive debt first, paying less interest overall as you become debt-free. It, indisputably, will cost you less.

But others advocate eliminating small debts to get them out of the way. Nobody has popularized that method more than get-out-of-debt guru Dave Ramsey on his radio and television shows and in best-selling books.

“The math does need to work, but sometimes motivation is more important than math,” Ramsey said. “Let’s face it: If you were so fabulous with math, you wouldn’t have debt.

“By paying off debts smallest to largest balance, not interest rate, you have some quick wins, which gets you fired up and keeps you motivated to stick to the plan.”

It’s math versus motivation.

Here’s what you need to know.

• Pay minimums: Implied in the great debt debate is one important caveat: make minimum payments on all debts so you don’t incur late fees and don’t default on debts, which will harm your credit scores. The question is really, “After paying minimums on all debts, which accounts do I apply extra money toward?”

Also, this assumes you are paying debts yourself and not consolidating them into a single payment with a debt-relief company, and that you’re ultimately able to pay the debts and bankruptcy is not inevitable. It also only addresses consumer debt.

• Highest interest rates first: Mathematically, the debate is a no-contest. You will pay less money in interest charges if you retire expensive debt first. But the debate doesn’t end there, because sometimes your highest-interest debt is also your largest. With this method, you could be paying on that debt for a long time, not getting the satisfaction of completely paying off anything in the near term.

• Smallest debts first: This method is often called a debt snowball because you pay off accounts with small balances first and apply money you would have been paying on those to the next-larger account, always putting more money toward the next-larger debt – a snowball effect.

The benefit is psychological, stemming from the satisfaction of fully paying off accounts. Like losing a few pounds quickly on a diet, it’s encouragement to continue with the discipline. “The debt snowball works because it’s about modifying behavior more than correct mathematics,” Ramsey said.

• Snowballing is instinctual: A 2011 academic study showed that’s what people do naturally. They instinctively want to retire more accounts, rather than those with highest interest rates, researchers reported in “Winning the Battle but Losing the War: The Psychology of Debt Management,” published in the Journal of Marketing Research. Authors called the finding “debt account aversion.”

“People will pay off the smallest loan first to reduce the total number of outstanding loans and achieve a sense of tangible progress toward debt repayment,” the authors wrote.

That leads to “non-optimal behavior” and could keep them in debt longer, the authors contend.

While snowballing is not always a mistake, “our work reveals that debt account aversion can systematically lead consumers astray when larger debts have larger interest rates. Ultimately, debt account aversion might enable consumers to win the battle but lose the war against debt.”

• Snowballing support: But in a study published last year in the same journal, Northwestern University researchers Blakeley McShane and David Gal went a different way. Instead of conducting experiments, they examined about 6,000 real debt-holders who entered a debt-relief program with mostly credit card debt. Many were probably on the verge of personal bankruptcy.

No matter how they looked at the data, they came to the same conclusion: People were more likely to become debt-free if they worked on reducing the number of debt accounts they had, not retiring debt with high interest rates, according to the study, “Can Small Victories Help Win the War?”

“There is this sort of tension: Do we snowball or do we pay off large interest rates first?” McShane said. “This idea that you need quick wins, and you get pumped up by retiring debt accounts, we find there is some truth to that.”

McShane said the snowball method might not hold for extreme examples – a $1,000 debt at 200 percent interest and a $100 debt at 1 percent interest. The high-interest account is so punitive, it makes paying it off the only good choice.

“We’re not trying to say there’s no trade-off between interest rates and size of accounts. At some point, there has to be,” he said.

In general, however, paying off debt accounts is predictive of debt elimination, regardless of balances, he said.

The findings have implications beyond debt, McShane said.

Splitting a goal – even a personal to-do list – into smaller subgoals, and completing some easily achievable ones first, might encourage you to continue toward achieving the big goal. “With any sort of goal you can ask, ‘How can I get a quick win on this?’ ” he said.

• Personality-based: If you’re the type of person who knows for a fact you will pay off all your debts, then paying the highest interest rates first makes the most sense, McShane said. It’s cheaper, and you apparently don’t need the motivation.

But if you’re like most people and uncertain about ever becoming debt-free, using the snowball method to get encouragement seems best, he said.

Gail Cunningham, spokeswoman for the National Foundation for Credit Counseling, said either system can work.

“The question becomes which one will help you stay the course until you reach the finish line,” she said. “If you’re a numbers person, you’ll likely respond well to paying down the largest debt with the highest interest rate first, because you know it’s doing you the most financial damage.

“However, for a person who needs constant motivation, that system won’t work. It simply takes too long to see significant progress. If you need that good feeling of accomplishment more often, then you’ll be better off power-paying the smallest debt and working your way up the ladder. That way you’ll stay motivated by knowing that you’ve knocked off one debt, and there’s one less statement in your mailbox.”

• Hybrid: Is compromise possible among the two warring get-out-of-debt camps? What about paying accounts first that have grossly high interest rates and then snowballing the rest?

Actually, that sounds like a good approach, McShane said, cautioning that he doesn’t have data to back the effectiveness of a hybrid approach.

• Closing accounts: Paying off debt accounts doesn’t mean you necessarily need to close them. Length of credit history is a component of credit scores, so closing old accounts could hurt your scores by making your overall credit portfolio look younger.

But if open accounts are a temptation to run up new balances and incur finance charges, it can be wiser to close them anyway.