Math Problems

By Brad Thomason, CPA

 

This one has the potential to be kind of nerdy, so you may want to just skip it.  You’ve been warned.

We have a love/hate relationship with math.  We don’t really like to do it, but we like the fact that it gives us definitive answers.  The numbers never lie, right?

Well, maybe.  But whether the numbers are lying is often a much less pressing question than whether or not the person receiving the answer understands what those numbers are saying.

If someone who doesn’t understand the underlying mathematics relies on their own interpretation of what the numbers are saying as a basis for certainty, doesn’t that open the door for some really scary mistakes and bad decisions?  Let me share with you 3 places that I often see people go off the tracks when placing their reliance on the numbers.

1.        The use of statistics.  In school statistics and probability are taught together.  Probability deals with what might happen in some future event; statistics deals with what happened in the past.  We all start learning the principles via the coin toss.  The probability of heads versus tails is 50% or 1:1.  We expect it to happen half the time.  And if we actually flip a coin a bunch of times and write down the answers, lo and behold, it ends up producing a more or less even distribution.  This plants a seed in our minds.  It establishes the notion that statistics and probability exist along a 2 way street.  If you know the probability you can project what the statistics will show.  And if you have the stats, you can reverse engineer what the probability was.  This relationship is solid and enduring and provable.  When you’re dealing with coins that is.  Because you see, coins don’t change.  They always have 2 sides.  The immutable nature of the coin itself is what makes the relationship enduring.  But we use this fundamental understanding of how coins work as our basic building block for statistical analysis of all sorts of things that are far more complex.  Like financial markets and entire economies.  That is, things which do not remain static.  To the extent that the conditions of the past do not repeat themselves in the future, then the presumed 2-way relationship between probability and statistics is irrevocably broken.  Past results are not a guarantee of future performance.  You’ve read that many time before but have you ever stopped to think of why that would be the case?  Because the conditions that were present when the statistics were being made are not likely to be the conditions that exist when the future unfolds.  Yet, how often does the idea of “due diligence” center on an analysis of the past as a means of determining whether or not a proposed action is prudent to take up in the future?  Our failure to understand this aspect of the probability/statistics construct causes us to reject things that would probably be fine; and accept with confidence and certainty things for which we really have no reasonable basis for either.

2.       Letting Excel do the math.  We use spreadsheets for a simple reason: Excel doesn’t make math errors, and people sometimes do.  But there’s a hidden problem here when we start using advanced formulas, and even more so with pre-programmed functions.  Because even though Excel is not going to screw up the calculation, that doesn’t mean that the programmer used the right function for the job.  I’ll give you a recent example.  A presentation of an investment opportunity concluded that if all noted assumptions came to pass, the internal rate of return over a multi-year period of time would be 18%.  I asked for and received the cash flow data, which I plugged into Excel and applied the IRR function.  It did indeed return 18% on my spreadsheet just as the promoters said it would.  Except that the actual rate of return was closer to 12%.  You see, the IRR concept was developed as a means to relate long-term operating projects back to a common denominator from the investment world, such as annuity payments or compound earnings from bond interest.  IRR works best when there is one cash inflow, and one cash outflow many years into the future.  But the cash flow data for this particular investment was not a one-in/one-out; it was multiple cash flows over a period of years.  Which is what lead to the problem.  You see, the IRR calculation in Excel makes the assumption that if there are intermittent cash flows during the life of the project they are reinvested at the same rate as the project itself.  But that’s certainly not a foregone conclusion in the real world, and it wasn’t in this case either.  There was no mechanism for reinvesting and compounding this project’s cashflows.  To make things worse, this assumption creates a feedback loop that causes the disparity between what the spreadsheet says and reality to grow the further into the future you go (in other words, instead of compounding earnings, you compound the error).  So what was probably an honest attempt to present some data failed because the people who applied the Excel function apparently didn’t understand all that the use of the function implied.  (A more involved use of multiple NPV calculations would have produced a more accurate answer in this particular case, by the way.)

3.       Math is a language (i.e. not perfect).  I think Gallileo was the first one to formally advance the idea of math as a language, and in the years since the notion that “math is the language of science” has become ubiquitous.  But there is an often unappreciated implication of this idea, a trade-off, that makes a difference (I’m reminded of that line in the Disney movie Aladdin where they talk about the trade-offs of being a genie: Phenomenal Cosmic Power!…itty-bitty living space).  If math is a language then it is subject to the same weaknesses as all other languages.  That is to say there will be times when math simply is not up to the task of fully accurate expression.  I’ll bet everyone reading this knows what lemonade tastes like.  But I’ll also bet that none of us could describe the taste of lemonade, using words, in a way that would lead to a listener having the same sensory experience that would come from actually drinking lemonade.   We may know the taste and know it well, but the fact that taste buds don’t respond to the spoken word means that language can’t do the same job as actual experience.  That’s the inherent weakness of all languages.  In math, we have similar problems.  In the real world I can take a piece of rope and cut it definitely and absolutely into three, equal, separate pieces.  But I can’t do the same thing with math (because that .33333 goes on forever to infinity, even though the real rope has a finite stopping point).  Same problem with pi; and with the old saw about the arrow never reaching its target because it has to cover half the distance before it can cover the whole distance, and those halves go on forever.  Maybe so.  But the rope goes to three pieces, and I’m not standing in front of the guy with the bow.  Do these examples mean that math doesn’t work?  Of course not.  It just means that math, as a language, has the same types of weaknesses as any other language.  Doesn’t mean we shouldn’t use it.  But it probably does mean that we ought not to expect perfection from it either, because that’s asking more than it can deliver.  Even math has its limitations.

So perhaps the numbers don’t lie.  But the users don’t always understand what they are saying, nor are the numbers always going to be up to the task of telling us everything we need to know.  Something to keep in mind the next time you attempt a math problem.  Please calculate with care.

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