0:59

Our data set is the one that we had actually explored before, and the,

Â the definition of the arithmetic mean return.

Â Basic, basically this is the average that you know from high school.

Â And an average, you know how you compute it.

Â You basically add up all the observations.

Â And you divide by the number of observations.

Â So you do that for the US, you add up those ten annual returns and

Â you divide by ten or you do the same thing for Spain or for Egypt or

Â for the world market, those are the numbers that you're going to get.

Â 9.5 for the US, 11.9 for Spain, 37.2 for Egypt, and 10.1 for the world market.

Â And of course the interesting question is, what is the meaning of these numbers?

Â Well, the meaning of these numbers, and here's one first important idea.

Â For you to keep in mind, is that this arithmetic mean return is something that

Â we do not use a whole lot in finance.

Â Most of the time when we talked about,

Â the mean return of an asset we mean something else.

Â And that something else that we mean,

Â we're going to explore a couple of minutes from now.

Â But this arithmetic mean return more often then not, is actually not what we mean.

Â I'm not saying that this is a useless number.

Â I'm saying that when we describe the performance of an asset,

Â this is not the relevant number that we tend to use.

Â What does that 9.4 percent mean?

Â Well 9.4% first of all it has the same interpretation as any average.

Â There have been high returns, low returns, positive returns,

Â negative returns, when you average all those you get 9.4.

Â So basically, you know, that's a number that maybe it's usefulness.

Â Is to compare it to the number for Spain, to compare it to the number of Egypt,

Â to compare to the number of the world market, or any other asset that you

Â may want to compare it to and see which one is higher, which one is lower.

Â Of course it will be a very incomplete, comparison,

Â because we need to bring risk into the equation, but one possibility, you know,

Â in the same way we could actually be measuring the age, of the, all the people

Â taking these smoke, and compare it to the age of the people taking some other mock.

Â And make a distinction, you know, where are the younger people or

Â where are the older people, or where are the taller people or

Â where are the skinnier people.

Â We could make all those comparisons, and, you know, the, the, usefulness of those,

Â that comparison will depend on how interesting is the question.

Â But that you actually post, but that arithmetic mean return is nothing but

Â the average of everything that happened.

Â High returns, low return, positive returns, negative returns,

Â average all that, and you get, in the case of the US for example 9.4.

Â percent.

Â Now 9.4% some people were also referred to it as the return in the typical period.

Â And again this goes back to the same idea.

Â There have been returns that have been high or low, positive or

Â negative in the typical average period you get a 9.4.

Â Percent of return, and

Â that is as far as this measure, of mean return goes there's not much more to say.

Â So if we go back to the two definitions, just to actually have them there.

Â It's again we look back, we take the average of the returns and

Â that's simply what it is, and number two.

Â Given that the returns have been high, low, positive, or

Â negative in the typical period they've been at that particular number which we

Â refer to, as the arithmetic mean return.

Â Okay, second definition of mean return.

Â Second definition is what we call the geometric mean return.

Â This looks and sounds a little bit more difficult and

Â the calculation is in fact a little bit more difficult.

Â Not, nothing to speak of.

Â Nothing very difficult.

Â But again, it's a little bit more difficult than the arithmetic mean return.

Â And let's start with the numbers.

Â And if we go back to our data, and

Â we actually calculate ex, using the formula that you're going to see.

Â And the expression that you're going to see in the technical note.

Â Then you'll get those numbers.

Â 7.6% for the US, 7.9% for Spain, 21.4 for Egypt, and 7.7%.

Â For the, world market.

Â And the first thing you see there when you compare, country by country,

Â the arithmetic and

Â the geometric mean return, is that these two numbers are different.

Â As a matter of fact, not only they are different, but

Â you should be able to see one more thing.

Â And that is that the arithmetic being returned in all cases is

Â higher than the geometric being returned.

Â Now, let, let's think a little bit and

Â try to interpret that geometric being returned.

Â And this part is really important.

Â It is important because whenever in finance we talk about the mean annual

Â return of an asset, then basically this is the number that we're referring to.

Â So let's think a little bit about the world market.

Â So that's the market that you have in the last column.

Â And we're repeating all those numbers here as well as

Â the geometric mean return of 7.7%.

Â All right, now think, let's think about, this in the following way.

Â Let's suppose that you take $100 out of your

Â pocket at the very beginning of the year 2004.

Â And as you see in that table, if during the year 2004,

Â the return that you got was 15.8%.

Â So that basically means that if you actually take $100 out of your pocket

Â at the beginning of 2004, or what is exactly the same, at the very end of 2003,

Â and you get a 15.8% return, then at the end of the year 2004.

Â You ended up with $115 and, $8 in your, in your pocket.

Â Now, if you actually keep that investment, and you go through the year 2005,

Â invested in the same world market and you get an 11.4% return.

Â That means that you started with 115.8, you obtained an 11.4%,

Â beginning from that amount and at the end of the period you've got 128.9.

Â If you keep doing that over and over again, when you get to the beginning of

Â the year 2013, you have in your pocket 170.5 dollars.

Â And in the year 2013, you obtain a 23.4% return,

Â which means that at the end of these ten years being invested, you get,

Â the number that you see on the screen, which is $210.4, $210.40.

Â That means that if you had taken a hundred dollars our of your pocket at

Â the very end of 2003, and had remained invested in the world market.

Â For the subsequent ten years, at the end of those ten years,

Â you would have in your pocket $210.40.

Â And here comes the interesting part.

Â And that is, let's think about it in a slightly different way.

Â Let's suppose you take $100 out of your pocket.

Â And you actually get a 7.7 mean annual return, over and over and over again.

Â So, in other words, imagine,

Â just imagine that you get 7.7 on top of 7.7 on top of 7.7, ten times in a row.

Â And that you let the capital accumulate over time.

Â Guess what you would get, at the end of that?

Â You would get exactly, the same $210.40.

Â So if you compare those two numbers,

Â then you get what the interpretation, of the geometric mean return is.

Â If you have been invested in the world market.

Â Between the years 2004 and 2013, starting with $100,

Â you would have ended with $210.40.

Â If you had obtained a mean annual compound return of 7.7% over those ten years,

Â you would have ended up.

Â With exactly the same amount of money.

Â So that gives you the interpretation of the geometric mean is the mean

Â annual compound return that you actually obtain by investing in

Â the world market at the end of 2003, and de-investing, or

Â getting out of the world market at the end of, 2013.

Â And that is a, a very important way, and

Â a very important definition of talking about, mean.

Â Mean returns.

Â So, two things to keep in mind.

Â First, that the geometric mean

Â return is the average rate at which an invested capital evolved over time.

Â Of course you were getting different returns over time, year after year you got

Â positive returns, negative returns, high returns and low returns.

Â But if you actually think in terms of the mean annual evolution of your capital,

Â in the case of the world market, it evolved at 7.7%,

Â so it's the average rate, in our case annual because we're using annual data.

Â Is the average annual rate at which a capital invested, evolved over time.

Â The other is that [INAUDIBLE] and it's a very complimentary definition.

Â It basically gives you, what you gain or lose, in any given period, compounded.

Â And that keyword compounded, is very important because it basically tells you.

Â At the, now it's the idea of saying over and over and over again.

Â That 7.7% that we looked at before.

Â If you had gotten that over and over and over and

Â over, over ten years, then you would have gotten to exactly the same amount of

Â money as being exposed to all those returns that we've seen.

Â In the ten years that we explore.

Â So, when, whenever you hear someone talking about the mean annual compound

Â return of an asset, they're basically referring to that geometric mean return.

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