Rounding Numbers; Let Me Count The Ways

Last week, Dave Michaels at The Wall Street Journal reported that the Securities and Exchange Commission is looking into whether companies were improperly rounding up their earnings per share to the next highest cent.  The SEC's inquiry was reportedly incited by a paper by Nadya Malenko and Joseph GrundfestQuadrophobia: Strategic Rounding of EPS Data.  However, Professors Malenko and Grundfest are not the only academics to claim that managers are using rounding to manipulate earnings per share.  See Miller, Martin & Bahnson, A Penny For Your Thoughts: Sizing Up Manipulative EPS RoundingStrategic Finance (July 2012).  Oddly, Professors Malenko and Grundfest fail to mention this paper in the current version of their paper (last revised in 2014).

Rounding may seem very basic.  After all, we were taught in school, and most of us remember, how to round numbers.  But as with a great many other things in life, rounding is not as uncomplicated as it might appear.  Although companies report earnings per share to the nearest cent, assume, for ease of discussion, that we are asked to round 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8 and 7.9 to the nearest integer.  

We were generally taught a single rounding technique - rounding half up (aka arithmetic rounding).  Using this method, 7.1 - 7.4 would be rounded to the nearest integer, 7 and 7.6-7.9 would be rounded to the nearest integer, 8; leaving 7.5 to be rounded up to 8.  Of course, it is important to know which way is "up".  If we are rounding negative values, -7.5 would be rounded up to -7 and not down to -8.  It is not written in stone, however, that one must round half up.  One could just easily follow a rounding half down rule.  In this case, 7.5 would be rounded down to 7, rather than up to 8.  Similarly, -7.5 would be rounded down to -8 and not up to -7.

Both rounding half up and rounding half down present a potentially problem.  If we always round in one direction, there is an obvious bias in that direction.  As we round more and more numbers, this bias grows.  A number of rounding techniques are employed to minimize this error.   These include round half even or rounding half odd that involve rounding the half (7.5 in our example) to the nearest even (8) or odd number (7), as the case may be.  It turns out that there are even more techniques for dealing with rounding.  If you are using Excel, you should pay attention to the "round" function that you select so that you understand how your values will be rounded.  I don't know whether the SEC has ever adopted a formal rule for rounding.

In tomorrow's post, I plan to discuss the somewhat related technique of truncation.