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All right. So two things to keep in mind.

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

And you'll see in a minute why.

At the first.

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

We highlighted already not only.

That the arithmetic and the geometric mean return were different for

each of the countries that we're looking at.

But also we looked at the pattern.

And the pattern was that the arithmetic mean return,

was higher than the geometric mean return.

Now strictly speaking if you want to be mathematically correct,

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

equal to, the geometric mean return.

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

because strictly speaking, that is the case.

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

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

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

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

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

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

They're going to be the same.

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

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

They typically fluctuate over time and

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

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

Now, characteristic number two.

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

as we said before, is always a positive different,

is increasing in the variability of the asset.

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

But since we haven't yet

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

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

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

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

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

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why that is actually important.

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

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

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

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

they will not actually scare you along the way.

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

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

sometimes a little lower.

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

you haven't gotten disappointment either.

Now- .

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

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

If you calculated the geometric mean return instead-

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

A difference of two basis points.

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

100 basis points is equal to one percent.

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

And if all the differences between.

Arithmetic and geometric mean of return were of that size,

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

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

Let's consider now the Russian market.

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

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

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

About 80% of your capital,

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

A market with huge variability, with huge volatility,

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

Here comes the interesting thing.

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

returns between 1995 and 2004.

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

a huge number 52.5%.

Now, let's suppose that the following scenario.

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

So here's a story that I tell you.

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

Between the years 1995 and 2004,

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

I haven't lied to you.

I really haven't lied to you.

The problem is that I gave you the incentive to run

the following calculation.

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

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

My money had compounded at 52.5%.

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

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

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

That's fantastic.

I do want to invest in the Russian market now.

What's the problem with that?

Well remember, the arithmetic mean return number,

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

What tells you that is the geometric mean return.

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

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

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

Over. Ten years.

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

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

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

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

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

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

Than 68, hundred dollars.

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

it evolved at 18.4% per year.

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

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

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

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

the geometric mean is important.

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

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

you without lying to you.

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

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

To run a calculation that is not the correct one.

Now it actually gets worse than that.

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

Let's focus now on that period,.

that shorter period between 1995 and 1998.

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

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

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

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

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

And I tell you, look, between 95 and

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

And I'm not lying to you,

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

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

I give you the incentive to run that calculation.

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

and obtained those four returns between 95 and 98,

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

What's the problem with that?

Well, that if you calculate the geometric mean return,

that number was actually minus 9.7%.

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

And I'm not lying to you there either.

You can actually calculate those two numbers.

And remember.

The relationship between the arithmetic and

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

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

which the first is positive and the second is negative.

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

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

And then awful and negative geometric mean return.

So. Your money.

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

Which means that you start at the year 1995,

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

And that happen.

With an arithmetic mean return of 38.7%.

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

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

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

they're numerically different and

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

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

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

To begin with.

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