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How long is a fiscal quarter? Maybe you thought it was a quarter of a year, or three months? You'd be right most of the time, but for a handful of public companies, that assumption is dead wrong. A few firms, including big names like PepsiCo, Kroger and AutoZone, use a rare reporting system that makes one quarter of each year a full month longer than the rest. For market watchers comparing companies or attempting seasonal analysis, those dramatically different quarter lengths could throw a wrench in calculations, because one quarter is 33 percent longer. And most people have no idea those long quarters even exist. "I discovered it totally by accident and said, 'I can't believe this,'" said Martin Gosman, an economics professor at Wesleyan University who has co-written a paper about the long quarters and warns students about them in his classes. "I asked people who have audited and been in public accounting, and they said they'd never heard of it." There's nothing in SEC rules that says a company can't have uneven quarters, as long as they're consistent year to year. But it is extremely unusual – only seven companies in the S&P 500 use the system, according to a CNBC analysis.

Basically, instead of breaking out four quarters into even 13-week periods, these outliers report for three 12-week periods and one 16-week period. Why do some companies use such a strange fiscal calendar? CNBC asked analysts, accountants, academics, trade groups, regulators and the companies themselves. No one knew exactly why the accounting move started, but there are a few hypotheses that could explain the phenomenon.

Wonky food quarters

One clue that sheds light on the mysterious quarters is the type of companies that use them: they tend to be in the business of feeding people. In fact, out of thousands of publicly traded companies tracked by FactSet, only 30 use the long-quarter system, and 20 of those are food-related. Even AutoZone, one of the few nonfood-related companies on the S&P list, said its current system seems to be a holdover from when it was a division of a grocery store; AutoZone was spun off from food distributor Malone and Hyde in 1986. It could be that food companies historically sought to even out fluctuations in the summer or the holiday season by extending that period, suggested restaurant consultant Malcolm Knapp. Or it could be a one-time industry standard that persists for a few companies today. "There are a lot of historical things in fiscal calendars that people just don't want to make the effort to change," Knapp said. "Even if they're dumb and have no meaning." CNBC contacted dozens of companies over several weeks, but none shared a meaningful explanation for why they chose that system over a more traditional 13-week quarter. When companies responded, it appeared that the schedule was simply a matter of convention. "Advance Auto Parts has used its current accounting and reporting schedule for many years, which includes a longer first quarter," a company rep said in a statement. "We have maintained this schedule for consistency in our financial reporting."

It is clear some companies inherited the system from corporate parents. Yum Brands, for example, was spun off from PepsiCo in 1997, which seems to have been using the system for decades (quarterly earnings forms weren't required until the late 1970s). On the other hand, it's unlikely to be purely a historical artifact because some companies on the list just had IPOs, meaning they specifically picked this reporting system rather than being stuck with it for historical reasons. And at least one company, Kroger, has changed the part of the year where the long quarter is located, while others, like Panera Bread, have given up the system entirely without giving any reason why.

You can't have it all in accounting

The best explanation for why companies would adopt such a system may have to do with how they adapt their external reporting to pre-existing internal accounting procedures — calendars are naturally uneven, but businesses need to come up with systems for comparing weeks, months and quarters, and every system has tradeoffs. Say a company is keeping track of its numbers on a monthly basis. That makes it easy to report relatively even fiscal quarters of three months each (that's why most quarters are about 90 days long). But what if a company gets most of its business on weekends? In that case, a quarter in a monthly system would end up with a different number of weekends from year to year, causing calendar-based fluctuations. That's why the National Retail Federation recommends retailers use something called the 52/53 week year, a system that keeps the number of Saturdays constant by defining a year as 52 weeks (plus an extra week every few years). Generally, companies include 13 weeks in each quarter, divided into two four-week "months" and a five-week "month." The system allows companies to handle inventory and payroll on a weekly basis, while still fitting their results into the quarterly system required by the SEC. But the months are different lengths, so month-to-month results are difficult to compare.