CF 9-4A – Calculating Arithmetic and Geometric Returns – Corporate Finance

Let's take a look at problem nine, four, eight, we're doing arithmetic returns and geometric returns. Remember, arithmetic returns is just taking a simple average. We figure out what our return was for a given year or for several years, and we take the average just in the conventional way. Geometric returns are, we don't take the average in a conventional way, we consider compounding principles. And, you know, we are looking at the stock market, stock returns tend to be compounding returns. We've looked at compound growth, you know, or year over year returns. So here's why arithmetic returns can be problematic. I'll give you an example where arithmetic, using arithmetic returns or arithmetic average is just gives you a ludicrous answer. So let's say I start with $100 stock, that's in your time zero, a year goes by, and my stock goes down to $50, so let's make it $60. And then another year goes by and my stock goes back up to $100. So year two, I'm back up to $100. Okay. So you and I can look at this and say, over the two years, my return is zero, right? I put in $100 at the start, two years later, I've still got $100, I've had zero return. But unfortunately, if we calculate arithmetic return as the way we've learned to do it in this course, we're going to show a funny number. So we went from $100 down to $60, this is minus 40%. We went from 60 back up to 100, I actually don't know the math here. Well, actually, I do know the math. Let's see. So we went up by 40 on an investment of 60, 66% growth, right? So we grew by 40 on our $60 investment, 40 divided by 60 is plus zero or plus 66.7%. So what is our average return? Well, we total these two and divide by two. So let's see, minus, no, let's do it this way, 40 negative plus 66.7 positive equals 26.7, divide by two, it's 13.33333 is what the answer would be, 13.333%. So if I did an arithmetic average here, I would say, oh, we had 13% return. You and I know just looking at it, we had 0% return. So I'll teach you a technique here called geometric averages or geometric average returns. That gets over that if you run these same numbers at the end of the video once you know how to do geometric returns, you'll see it works out to zero, which is sort of what a normal human would think of this investment. So anyway, I just wanted to introduce this concept to you to say, arithmetic returns don't always produce reasonable results. They usually do. Usually arithmetic returns are fine. I think in most finance textbooks, most questions you're expected to calculate arithmetic returns. But if you're asked to do geometric returns, you're going to learn how to do it in this video. Let's go ahead and calculate though the arithmetic average returns. So we got year one to year two, year one to year two, and what's happened with our stock? We went from 50 to $52. So the stock price went up $2 and so change in price and then the dividend and then we'll do total return and then we'll do percentage return. I know it's a lot to sort of take in here, but let's look at this one and actually let's go through each of them. So year one to year two, there's a $2 price change. Year two to year three, it's a negative $3.5 price change, right from 52 down to 48. From 48 up to 61.75, it's a big number, 61.75 minus 48.5. It grew by 13.25, so that's year four and into year five, it grew by $3.25 more. So if all we had to do is the price change, that would be fine, but we also have dividends. That's part of our return. So our dividend was plus $1, plus $1.25, plus $1.25, plus $1.40. So that's part of our return too. So our total return we saw in year two was $3 positive. Total return in year three, we lost money, even with the dividend, $3.5 negative plus $1.25 is $2.25. Zoom in on this a bit, $13.25 plus $1.25 is $14.50, and $3.25 plus $1.40 is $4.65 positive. Okay, so those are all my returns. The percentage return, to figure that out, you just take the return, divide by the stock price in the previous year. So I got $3 on a $50 investment, three divided by 50, 6%. I got $2.25 negative, why did I put a positive there, $2.25 negative on a $52 investment, $2.25 negative divided by 52 is negative 4.33%. I got $14.50, again, this is a year four return. So on an investment that started the year at $48.50, I returned $14.5. So $14.5 positive divided by $48.50, 29.9%. So positive 29.9%. And year five, $4.65 on an investment that started the year at $61.75, $4.65 divided by $61.75, 7.53%. Okay, so there we have our returns. Now, the average return, the arithmetic average, you just add these up, divide by four. So let's do that. So 6 minus 4.33 plus 29.9 plus 7.53 is 39.1. And I sum up that list, I get 39.1% of my total divide by four, 9.775%, wonderful, okay. So that's the arithmetic return, the arithmetic average return on the stock, 9.775%. The geometric return works a little differently. You take your percentage return, and here's how I do it. I state that one as a decimal. So I'll just put decimal as my header, and I'm going to go over a couple more columns here. So 0.06, negative 0.0433, positive 0.299, and positive 0.0753. Okay, add one. We're going to take one out at the end here, but we add one. So I'll just call this my plus one column. So 1.06, 1 minus 0.04, let's see, 1 minus 0.0433 is 0.9567, 1.299, 1.0753. Then I compound the growth, compound, I always just do, I say compound a dollar. So pretend I invested a dollar in this stock. After one year, I would have 1.06, I would have a dollar six. So I go one times 1.06 times, so after one year, it's 1.06. After two years, so times 0.9567, after two years, it's 1.014. After three years, so you might be saying to yourself, okay, I invest a dollar, right, in this $50 stock. Well, you can invest a dollar, but just pretend. I invest a dollar. I get 6% return in year one, so I'm up to 1.06, including my dividends. I lose money on the stock in year two, and I'm back down to a dollar one. You may be saying to yourself, well, wait, the 50, the stock's down to 48. If I invest a dollar at 50 and I'm down to 48, shouldn't I have less than a dollar today? And the answer is no, because of the dividends, right? We had a dollar and a dollar 25 in dividends, which more than make up for our buck 50 loss. So in any event, let's continue. You just compound this number, so you multiply this number. So you just multiply one by each number in this, and you just keep going down the list. So 1.014 times 1.299 is 1.317 times 1.070753, and I get 1.4165. Okay, so there's my number. Now, this is a funny little mathematical thing. You might think, okay, I minus the one, I divide by four. We actually don't do that. We raise this to the power of a fourth. Because we're compounding the growth, we want to say, okay, well, this is four years of growth, got us to 1.41. What does one year of growth look like? You go 1.4165 to the power of one divided by the number of years of growth we're compounding here, in this case, four. So 1.4165 to the power of a fourth. So now let's go y to the x, 0.25. I get 1.09095, 1.09095. Take away the one. We put the one in arbitrarily. Now we take it away. We get 0.09095 or 9.095%. And there we have it. We've computed the geometric average. And it's just a different way of averaging that takes compounding into consideration. And of course, so much of finance is based on compound growth. So geometric average will sometimes produce a better answer than arithmetic average. Hopefully I've produced a reasonably good answer for you. And if I have, hit that button. Thanks so much for watching. See you in the next video. Bye-bye.