Earnings per share seems delightfully simple: Divide income for common shareholders by the average outstanding shares. It's also dreadfully simplistic because, as everyone should surely know, the numerator and denominator are often manipulated. However, how many know that managers also finagle the quotient?

To answer this question, we did some research that's described in "A Penny for Your Thoughts: Sizing Up Manipulative EPS Rounding," in the July 2012 issue of Strategic Finance. (We wrote this article with Dr. Greg Martin of Indiana University-Purdue University Indianapolis.)

The findings are profound because they show that many managers and accountants cheat, which means they cannot be trusted, which in turn leads to all sorts of problems.



We began by extracting from the Compustat database the EPS numerator and denominator values from nearly 330,000 positive quarterly income results issued by about 8,200 public companies in 1995-2009. We recalculated EPS to three decimal places so we could see whether the reported two-decimal numbers had been rounded up or not. If the results were unbiased, the proportions would be 50/50, with just as many rounded up as not.

Alas, 53 percent were rounded up. That deviation may not seem significant, but the probability that it happened by chance is one out of 10 followed by 260 zeroes! It's significant because it reveals that 6 percent of the managers in the half of the sample who should not have rounded up found a way to do so. In other words, they lied.



We rightly suspected that the fudging frequency would be higher in situations where an extra penny of earning per share is more consequential. When EPS was $0.20 or less, 9 percent of the reports were rounded up when they shouldn't have been; the proportion rose to 20 percent when EPS fell to a penny.



These high rates prompted us to figure out whether we could identify specifically who was cheating, because statement users surely need to know who they are. After all, if users can't trust managers to publish honest EPS results, how can they trust anything else they report or do?

The perplexing obstacle we faced is that management can nudge the third digit from, say, four to five, with tiny boosts to the earnings that are undetectable by users or even auditors.

We solved this puzzle by hypothesizing that a management who misrounds once is likely to do it again, and then again, until it gets to be a habit. This supposition led us to compile the full stream of reported EPS for each of the 8,200 companies, which we then tested to see whether their individual rounding results were biased with more being rounded up than not. Once we had that information, we applied the binomial distribution to ascertain the probability that each company's result could have happened by chance. For example, suppose a company provided 40 quarterly results. If EPS was rounded up 23 times, the probability is 21.5 percent that it was a random occurrence; if it was rounded up 28 times, the probability is less than 1 percent.

This technique provided the mathematical microscope we needed to identify who was misrounding and how often. To begin, we set an initial threshold probability of 0.1 percent, which implied only eight of the 8,200 companies in our sample would have an abnormally high uprounding frequency. In fact, we found 320 with rounding histories that beat these long odds. Not only did we discover that manipulation through rounding is widespread, we know exactly who has done it.

Our paper provides more details, but the most astounding discovery was that one company's management rounded EPS up in all 42 of its 42 quarterly reports. The statistical probability that this result could occur by chance is only one out of 10,000,000,000,000,000 (10 quadrillion). Clearly, these people cannot be trusted, and perhaps the same is true for their auditors.



To understand more about the auditors' role, we first determined that 68 percent of all 8,200 companies in our sample engaged one of the five largest firms. We then focused on the habitual uprounders to see which auditors they used.

We found that these managers were less likely to engage a large firm as their misrounding frequency grew more outrageous. In the worst cases, where the frequency would be expected only once in a trillion observations, only 17 percent of those companies were clients of large firms.

A key question remains unanswered: We don't know whether these companies' auditors were fooled by their clients, simply looked the other way, or, what's worst, actively suggested or collaborated in the scheme to mislead. Of course, all three explanations involve reprehensible behavior.



We now make a proposal to our friends at the Financial Accounting Standards Board and the Securities and Exchange Commission on how to fix this problem. The following policy recommendation would produce no controversy, no implementation costs, and no negative impact on trustworthy managers and auditors:

Any EPS result less than one dollar should be reported with three decimal places.

For example, when EPS results are reported to the nearest penny, management can, say, double its reported number from $0.01 to $0.02 by nudging its unrounded EPS from $0.0149 to $0.0150. Under our proposal, there would be no point in trying because the result would still be $0.015 per share.

History shows that no new regulation is ever easily accomplished, but we find no credible objections to implementing this for both public and nonpublic companies.



Another practical contribution of our research is our "microscope." First, financial analysts now know they can look at a series of EPS numbers for a target company and determine whether its managers are playing this rounding game. If the analysts learn that bias exists, they will know that management cannot be trusted.

Second, ethical auditors should include this test in their analytical review to detect whether their clients' uprounding percentages over time indicate manipulation. If so, they will know they need to focus on that issue or even fire their clients.



Beyond these points, our findings unveil fundamental flaws in today's financial reporting practices.

Specifically, it is preposterous for managers and some capital market participants (including the media) to hyperfocus on earnings per share and attribute so much precision to the reported results. We wonder how anyone with worthwhile knowledge can believe there is enough precision to compute EPS to even the nearest dollar in light of the discretion that managers have in estimating amounts associated with product costing, depreciation, amortization, income taxes, bad debts, contingencies, warranties, pensions, etc., as well as the omission of several important comprehensive income items from earnings.

Thus, reporting EPS to the nearest cent is a real stretch. (Note: We're aware of the irony that we recommend reporting EPS to the nearest tenth of a cent, but our goal is curtailing trust-robbing misbehavior, not increased precision.)

Our point is that the EPS numerator, reported income, is greatly flawed by its imprecision. Unfortunately, this defect is another elephant in the room that no one else wants to acknowledge, much less fix.



We're not surprised to find this incontrovertible evidence of ethical corruption among managers, their internal accountants, and presumably some of their auditors, but we are nonetheless deeply disappointed.

In particular, we were dispirited when one of the manuscript's pre-publication reviewers said, in effect: "So what? Everybody knows managers put their best foot forward in their statements." We invite this person to stop and reconsider that rationalization for unethical behavior. We hope others aren't so willing to excuse fraud as unimportant.

Beyond that, if the markets know that many managers and their cronies, including accountants and auditors, are playing games to look good, then they also know that none of their numbers can be trusted. Regrettably, this situation has grave negative consequences.

Consider these "Quality Financial Reporting" axioms that we have explained in one form or another for more than 15 years:

• Untrustworthy financial reports create uncertainty for investors;

• Uncertainty produces risk for investors;

• Risk for investors makes them demand a higher rate of return; and,

• A higher rate of return produces a higher cost of capital for the company and lower prices for its shares and bonds.

Therefore, the destruction of trust in financial reports has immense implications for individual companies and their stakeholders.

Such a simple thing as manipulating earnings per share to eke out one more penny sends a clear signal that management is unreliable. Instead of smugly thinking they've put one over on the markets, their shareholders, and auditors, these people should realize they've done great damage to themselves and others.



Here's our real punch line. We invite readers to ask as frequently as you can: "In making this decision, have I acted in a trustworthy manner?" If the answer is "No," or even anything short of "Absolutely yes," then it's time to change your behavior or get out of our profession.

We realize this high standard may be unrealistic, but we cling to this unarguable concept: Society created accountants to support efficient decisions by providing trustworthy information. If an information source is suspected to be fouled and untrustworthy, it doesn't matter whether its output is actually reliable. If accountants and auditors cannot be trusted, then they are not creating value with their reports. If they're not creating value, they're wasting their time and considerable talent. And if that's so, they're under-compensated and their professional lives are less fulfilling.

So, what's wrong with a one cent boost to EPS? Virtually everything, and it costs a very pretty penny.

Paul B. W. Miller is a professor at the University of Colorado at Colorado Springs and Paul R. Bahnson is a professor at Boise State University. The authors' views are not necessarily those of their institutions. Reach them at paulandpaul@qfr.biz.