# Arithmetic Mean and Geometric Mean in Relation to Profit

## Video Transcript

My name's Robert Bonavito, New Jersey forensic accountant. This video is part of a series of videos where I discuss forensic accounting topics for educational purposes only. If this was a litigated matter, I would take a different approach, have different conclusions, based on different facts and circumstances.

Today's topic is titled "Don't Believe What You Read in the Papers". It's really about arithmetic mean and geometric mean. And it may sound a little bit bland, but this is extremely important topic if you use any type of financial data to understand the difference between these two. And the reason I call it "Don't Believe What You Read in the Papers," because a lot of times you'll see things in the papers that this company has X profit or that profit, and it's totally fabricated. It's not that they're lying, or trying to mislead you. They just don't understand what they're telling you.

And, you know, and as far as the papers, I mean when I was very young, I learned right away that you can't always believe what you read in the papers. And I just, you know, divert. When we were being interviewed for a newspaper article, and it was me, and one of my friends standing next to me. And there was a big game the next day. And my friend was a pretty cocky guy, and he was saying he was gonna do this to the other team, he's gonna do that to the other team, and the coach is that, and the cheerleaders aren't any good. He went on, and on. And I didn't say anything. So I'm like, "Boy, man he's gonna be sorry he said all that, 'cause, you know, the other team is gonna read this." The next day, I pick up the paper and it says, "Bob Bonavito predicts this." And everything he said was attributed to me. So I was pretty nervous for that game. But anyway, I learned very young that you can't always believe what you read in the papers, especially when it comes to the financial news.

And let me give you a little example here. This is example of profits in what I call New Corp. It's years one through four. Now, if you read the paper, you turn on, you know, one of these news channels or business channels they would say that the profit, you know, if you look for year one, they lost 27%, year two was a 162% increase, year three was a 101% percent increase, and year four was 84% loss. And what they would put in the paper was that the compounded annual return for New Corp was 38%, okay.

Looking at those numbers, do you think it was 38%? Do you think that's true? It is true, okay, if you use the arithmetic mean. But if you use what I use when I do my analysis for court or for analyses, the geometric mean is actually a 10% loss. And you're saying, "Bob, how can that be? How can these four numbers be calculated at a 38% compounded profit, or a 10% loss?" Trust me, it happens all the time. And when you see one of these shows, they don't say, "Oh, this was the geometric mean, this is the arithmetic mean." But, you know, I always tend to use the geometric mean, because it's more conservative.

And, you know, when I testify in court, if I was to say that the profit for this company was 38%, if the attorney doesn't know the next question, you know, well, the next question should be, "Mr. Bonavito, is it the geometric mean or the arithmetic mean?" They don't ask that question, because they don't understand this concept. But the next question should be, "Well, if I used the geometric mean, it would be a 10% loss."

So let me just do some of the math here, so you get a better feeling. You'll feel comfortable with this, and like I said, you can try this out for yourself. I've basically taken the same information, and I assumed a \$100 investment. So you'll see in year one it was a \$27 loss, okay. And my investment would've went down to, in the fourth column here, \$73. In the second year, a 162% gain. That means my investment went up \$118. So now my investment is \$191. To a \$100 investment, pretty good, probably should sell it, right? No, maybe not.

The next year, the stock goes up 101%. That's \$193, so now I have \$384, which you should be feeling pretty good about, since you've only invested 100. And unfortunately in year four, as happens more likely than not, it goes down 83%, which means a \$319 loss. So you have \$65 now. So you actually turned \$100 into \$65. And like I said before, the arithmetic profit is 38%, and the geometric's 10%. Who do you think's right, right? You have less money than you invested, so it's obviously not an increase.

And to take this a little bit further, I'm gonna give you the cumulative return calculation here. The cumulative return at 38% will be 362%, times 100 would be \$362. So if what they were telling you was correct, on the news assume that they said there was a 38% annual profit in this company, you should have \$362 in the bank. But you only have \$65. And see, the calculation works out, because a 10% annual loss would equate to a 65% cumulative return, which when you multiply that by 100, you come to \$65.

So when you see profits, or you hear profits, and there's a screen there, you may want to just take a step back, and see if it makes sense. And if you look at the first screen, it doesn't make sense that if you look at the profits and the losses for four years, it doesn't make sense that it went up 38%, right? So you have to disregard, and what you'd say is, "Hey, listen. They're probably using the wrong mean," okay. Mean is the same as average, right? So be careful with this. This is a very basic mistake a lot of people use in a lot of analysis, by using the wrong method.

My name is Robert A. Bonavito, New Jersey forensic accountant. If you have any questions about this, feel free to email me or give me a call. Thank you.