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All right. So two things to keep in mind.

Â And, and these are, are important things that for you to keep in mind.

Â And you'll see in a minute why.

Â At the first.

Â A, and remember if you actually keep in mind the data that we've seen.

Â We highlighted already not only.

Â That the arithmetic and the geometric mean return were different for

Â each of the countries that we're looking at.

Â But also we looked at the pattern.

Â And the pattern was that the arithmetic mean return,

Â was higher than the geometric mean return.

Â Now strictly speaking if you want to be mathematically correct,

Â what we can say for sure, is that the arithmetic mean return is higher than or

Â equal to, the geometric mean return.

Â Now, I'm saying that with that only to be mathematically correct,

Â because strictly speaking, that is the case.

Â Now, if you really think about it, there's only one circumstance in which

Â the arithmetic and the geometric mean are going to be the same number.

Â And that is when you get the same return over and over and over again.

Â So for example, you buy an asset, and you get 10%, 10%, 10%, 10%,

Â 10% for all the periods that you're looking at.

Â Then when you calculate the arithmetic and the geometric mean return.

Â They're going to be the same.

Â And I'm saying that well that's not very interesting because none of

Â the assets that we work with in finance actually have that characteristic.

Â They typically fluctuate over time and

Â whenever you have a fluctuation in the value of an asset, however little,

Â that implies a difference between the arithmetic and the geometric mean return.

Â Now, characteristic number two.

Â The difference between the arithmetic, and the geometric mean return, which,

Â as we said before, is always a positive different,

Â is increasing in the variability of the asset.

Â In fact, it's increasing in the volatility of the asset.

Â But since we haven't yet

Â defined volatility, I'm going to, I'm not going to try to use that word just yet.

Â So, think about that depending on how much assets fluctuate over time,

Â the higher that fluctuation, the larger it's going to

Â be the difference between the arithmetic mean and the geometric mean.

Â And let me give you an example that would actually highlight,

Â 2:12

why that is actually important.

Â So this is actually a, a, very, an asset with very little risk.

Â And that asset with very little risk as you're seeing in there.

Â These are one US Treasury Bills and basically they have no risk.

Â They will not give you a whole lot of returns, but

Â they will not actually scare you along the way.

Â So as you see in those numbers that are in front of you between 2004 and

Â 2013 all the numbers have been positive, some, sometimes a little higher,

Â sometimes a little lower.

Â But, you know, you haven't gotten huge returns,

Â you haven't gotten disappointment either.

Â Now- .

Â If you add up all those returns and divide by ten which is the number of returns that

Â we have there, then you're going to get an arithmetic mean return of 1.95%.

Â If you calculated the geometric mean return instead-

Â . Then what you're going to get is 1.93%.

Â A difference of two basis points.

Â Remember if you have never heard about the concept of basis points,

Â 100 basis points is equal to one percent.

Â So that basically means that two basis points is .02%.

Â And if all the differences between.

Â Arithmetic and geometric mean of return were of that size,

Â then we wouldn't worry too much, the diff, the difference between the two.

Â But, we do need to worry and here's why.

Â Let's consider now the Russian market.

Â Now, I should clarify that this is a Russian equity market, and

Â as you see there, I don't need to tell you much about the risk of this market, in

Â some periods you actually more than double your capital, in some periods your loss.

Â About 80% of your capital,

Â in some other periods you lost about one-third of your capital.

Â A market with huge variability, with huge volatility,

Â with huge fluctuations in returns from very positive to very negative.

Â Here comes the interesting thing.

Â Let's look first the whole period that we have there in terms of

Â returns between 1995 and 2004.

Â If we were to calculate the arithmetic mean return we would get

Â a huge number 52.5%.

Â Now, let's suppose that the following scenario.

Â I'm someone who wants you to buy Russian equity.

Â So here's a story that I tell you.

Â Look, you should be investing in Russian equities, and the reason is this.

Â Between the years 1995 and 2004,

Â the mean annual return of the Russian equity market was over 52%.

Â I haven't lied to you.

Â I really haven't lied to you.

Â The problem is that I gave you the incentive to run

Â the following calculation.

Â That is, I gave you the incentive to think well,

Â if I had started with $100 at the beginning of 1995.

Â My money had compounded at 52.5%.

Â Over ten years, I would've ended with over $6800.

Â So it started with $100, I ended up with $6800.

Â I multiplied my capital by 68 times in only ten years.

Â That's fantastic.

Â I do want to invest in the Russian market now.

Â What's the problem with that?

Â Well remember, the arithmetic mean return number,

Â doesn't tell you at which rate your money evolved over time.

Â What tells you that is the geometric mean return.

Â And guess what, when we calculate the geometric mean return, it's 18.4.

Â Now 18.4 is a great number, I mean we would like to

Â get many assets in our portfolios in which we get 18.4% per year.

Â Over. Ten years.

Â And we probably will not be able to find all those many.

Â But, the thing is that 18.4 is far, far lower than 52.5% per year.

Â And as a matter of fact, when you compound 18.4 over ten years.

Â Had you started with $100 at the beginning of 1995,

Â at the end of 2004, you would have $542.

Â Now, $542 is still a great return, but of course, it's far, far lower.

Â Than 68, hundred dollars.

Â So, that means that what really happened to your money is that

Â it evolved at 18.4% per year.

Â Over ten years and your capital went from $100 to $542.

Â Again, that may be a very good rate of return for those ten years,.

Â But it's far, far lower than the $6,800 that I led you to believe.

Â Now, this is why the difference between the arithmetic mean and

Â the geometric mean is important.

Â If I don't tell you, if I'm a little wishy-washy, if I'm not very specific.

Â About what I mean by mean return then I may be actually lying to

Â you without lying to you.

Â Because I haven't lied when I say that the mean annual return was 52.5%.

Â I was just a little wishy-washy so to give you the incentive.

Â To run a calculation that is not the correct one.

Â Now it actually gets worse than that.

Â And the reason it gets worse than that is the following.

Â Let's focus now on that period,.

Â that shorter period between 1995 and 1998.

Â Now let's look at that period between 1995 and 1998.

Â What we see there is that if we calculate the arithmetic mean return of

Â those four numbers, is 38.7%, just under 39%.

Â And again, let's suppose and let's go back to a hypothetical story that for

Â whatever reason, I want you to invest in Russian equities.

Â And I tell you, look, between 95 and

Â 98, the mean annual return of this market was almost 39%.

Â And I'm not lying to you,

Â the numbers would back up that the mean on your return is 38.7 percent.

Â But, at the same time that I'm not lying to you, I'm not being very specific, and

Â I give you the incentive to run that calculation.

Â That had you started with $100 at the beginning of 1995,

Â and obtained those four returns between 95 and 98,

Â at the end of that period, you would have ended with $371 in your pocket.

Â What's the problem with that?

Â Well, that if you calculate the geometric mean return,

Â that number was actually minus 9.7%.

Â That means that you almost lost 10% per year, on a compounded basis.

Â And I'm not lying to you there either.

Â You can actually calculate those two numbers.

Â And remember.

Â The relationship between the arithmetic and

Â the geometric mean is such that the first is higher than the second.

Â But, being higher than the second does not prevent the situation in

Â which the first is positive and the second is negative.

Â As it is the case at the, here with the Russian market between '95 and '98.

Â So we have a very large and positive arithmetic mean return.

Â And then awful and negative geometric mean return.

Â So. Your money.

Â Actually lost at the mean annual rate, of almost 10% per year.

Â Which means that you start at the year 1995,

Â with $100 and you end the year 1998, with $67 in your pocket.

Â And that happen.

Â With an arithmetic mean return of 38.7%.

Â So, I sort of rest my case in terms of trying to impress

Â upon you the importance of the difference between these two types of return.

Â They're very different because, they answer different questions,

Â they're numerically different and

Â one can tell you that you're actually making money over time.

Â But, the other many show you that you're losing money over time.

Â Or you're making a lot less money than you thought you were making.

Â To begin with.

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