Allen Cring Productions in association with Emergent Light Studio presents...
The Illinois State Collegiate Compendium, academic lectures in Business and Economics.
This is Business Finance FIL-194, spring semester 2024.
Today, risk and return.
Now, the complicated part of this is not that complicated,
but it does call you to do some things,
and I would certainly expect you to do this in Excel.
I don't even care to do it myself by hand
because it can be tedious sometimes.
But it is, it's not terribly complicated in Excel,
but there is a caution that I have to give you
when I do it in Excel.
But before we do that, let me do a quick look at the numbers
here.
Yahoo.
And as you can see, that is about as lackluster
as a market can get.
It's just like sitting there looking stupid.
The Dow is flat right now.
And then the S&P 500 is pretty much the same.
And then the NASDAQ, a little bit of bias to the downside,
tiny, tiny bit.
But they all came, started out,
kind of dropped real fast right on the opening bell.
And well, this one didn't.
But then it's been struggling to come back up,
but there is a lot of volatility right now.
The volatility is quite noticeable, at least visually.
Now, however, very quickly, let me look at the VIX
to see what the vol is on the VIX real quick.
That's not much.
Yeah, see the volatility early this morning really spiked.
And it's dropped off really a lot
in just the last couple of minutes.
And I don't know what that's all about.
So we'll have to see how that goes from here.
But anyway, let me get back to the main show here.
And crude oil still in its band now.
About 81, I'd say, up to 88, is where I'm seeing it right now.
And it's staying in that band.
And as I said, that's going to lead to higher gas prices.
And you're seeing them at the gas pump right now, 379 locally,
379, 369.
But where it's going to go, you notice that it has been tailing
off through the day.
But that drop isn't all that noticeable.
It did fall well.
I shouldn't say that.
Because since last night, now remember,
oil commodities markets don't trade like stock markets do.
They have different time frames.
And it looks like it was up there rather noticeably
last night.
And it's been tailing downward ever since.
And that is a lot to do with the,
we know there's a lot of supply.
The supply chain is full at the sources, on the high seas,
at the refineries.
So the cause of that higher price that's
been there for the last week or so
is mostly expectations of something
that the market thought was going to happen.
And now that expectation seems to be draining out
a little bit.
But again, you have to always remember that in our markets,
expectations are everything.
Gold, it's just all over the place.
It's just really volatile right now.
It's spiked up.
But it still seems to, right now, it doesn't really
want to find out much about $2,200 an ounce.
It's keeping down a little bit more.
Silver, quieter, of course.
Now the 10-year bond, we do have at least a modest,
a tiny 4 tenths of a basis point easing off.
But it's really, for the most part,
that's just really quiet, 0.09%.
So the bond market is just sort of sitting there pretty much
flat, like the equities are.
So the market overall is just sitting back
and waiting to see what's going to happen next.
Running over here just very quickly, the Nikkei,
it started down.
And then it climbed up in the last half of the trading
last night over in Tokyo.
But it did manage to get kind of a respectable 2 thirds
of a percent out of its day.
And when we came over to London, they started out down
and then groveled back.
They're still trading.
And now they're back in positive territory too.
And if we look back over here at the equity side of it,
well, there you go.
We're just sitting there looking,
not really knowing what to do next.
It's a really quiet day.
I don't see anything exciting coming along.
Yeah, but it's just going to be one of those days
where there's nothing going on.
Very quickly, let me show you.
And I'm just going to, well, that was brilliant.
I'm just going to pull up a stock out of thin air
for a little bit here.
I don't know.
General, let me look at General Mills, something quiet.
And I have a specific purpose here.
General Mills, General Motors, I'd rather look at something
a little bit calmer.
What in the heck?
Come on.
Why am I having difficulties here?
Oh, I put it in the wrong place.
General Mills.
GM.
General Mills.
I didn't think that was as simple, but I guess there it is.
OK.
Really came off of, today, it really came out good.
Strong, strong opening.
And it's been sliding off since then.
There's some profit taking going on since the opening bell.
But man, it's really been a decent day on there.
And I've gone through the numbers before,
and I'm hopeful that you know how to look at those numbers
now.
The one I want to focus on right here, right now,
is this guy right here, Beta.
And as I've said before, in finance,
this is our go-to measure of risk.
Now, we do have other measures of risk.
The standard deviation of returns is a measure.
And everything is on returns.
These are price metrics.
These are returns metrics.
So you're taking this Beta is a measure
of the volatility of the returns.
Now, we can talk about, pretend it's the volatility of the price,
because the price is related to the returns.
But technically speaking, where we have to be now in the course
is volatility of returns.
And this is our go-to.
It is not just the standard deviation.
And I'll get into that in just a few minutes here,
what I mean by that.
But for the time being, just understand
that this is different.
Now, we've got to understand that.
This is different.
Now, we do have some other ones.
The standard deviation is a measure of risk.
The Beta is a measure of risk, too.
And there's another one here, too, the Sharpe ratio.
And there are different books have different letters for that.
I'll just write it as SR, the Sharpe ratio.
It is a measure.
And it's a very popular measure in mutual funds.
I mean, I'm not sure where that came from.
It's mostly historical, because mutual funds have a Beta.
And I look at the Beta.
But I swear, whenever you're looking at mutual funds,
as you notice with GM here, Sharpe ratio is nowhere.
But if you're looking at a mutual fund,
all the Sharpe ratio is going to show.
And I hope you'll see why it's not a really great measure.
But it is popular in that niche market of mutual funds.
Whatever.
Now, there's one other one that I would bring up here, too,
the coefficient of variation, the CV.
Now, this is where I never can remember.
Did you take the statistics course yet,
the management course?
I can't remember.
I think it's 150 or something like that.
Have you had that course yet?
OK.
It's OK, because I'm going to teach it a lot better
than they would anyway, because I'm
going to focus on what these things mean to us in our work.
And I do use Excel like a boss.
I do not use the formulas, because the formulas
can get to be a lot like work.
And it's not that they're hard, complex.
But man, you just have to keep calculating over,
adding and taking differences, then squaring them,
then weighting it by the probability,
and then going on to the next one,
and then adding all of those up, then taking
the square root of it.
That's way too much like accounting work to me.
So we take Excel very seriously.
And all these we can calculate with Excel.
One caution, and then I'll get off this subject,
because it's so important that it's not something
that I want to talk a whole lot about.
A lot of times, we will calculate
these with historical data.
And that's not what we should be doing.
We should be calculating it with future data,
or future expected data.
And that gets into the kind of obvious question,
how the hell do you do that?
If we haven't had the future yet, how do you get it?
The typical way that we do that, we
do an adjustment from history to the future.
How far do I want to take this?
OK.
We do it actually the way we taught artificial intelligence
systems to do a lot of things that they
do in modeling and data analysis.
What we'll do is we will take, let's say, a year of data.
And we will get the expected, we'll get, say,
the average of that.
And then what we'll do is we'll go up, say, two months.
And we'll take data that, let me do it this way.
We've got a vector of data, a historical vector of data.
We'll take this, OK?
And we'll get the average of this data.
This is just one of the metrics.
Let's say the average of the returns.
And we put an R bar.
That's for an average.
And then we will say, OK, let's take this last data.
This is the simulated history.
And then we will pretend that this was the future data,
R capital F, I better put, because it
looks like a risk-free rate.
And then we'll look at how much the historical data deviated
from this quote unquote future data.
It was all historical, of course,
but we're going to pretend that this was future data.
And we'll see how much different this future
was than what we thought it would
be from the historical data.
And then that will be our adjustment
for all of the historical data.
Ouch.
That difference tells us how much different the history
is from what happens next.
And that's a simple way of doing it.
We have much more complex ways, but honestly, they all
kind of boil down to this.
And that way, we can get an estimate
of how much another data set.
So in other words, we would now take a whole set of data
and get its historical.
And we say that up here, it would be R bar
plus this adjustment figure.
That's how we would get a future expected value.
This is done all the time.
So when I talk about these metrics over there,
we are doing historical.
We're not going to do that adjustment.
But that's how you would do an adjustment
so that you're looking at the expected future standard
deviation.
The expected future beta.
It would be the historical with that adjustment factor
added in or subtracted from.
OK.
Now let me get down to some brass tacks here on this.
I'm going to pull up.
And I'm actually just going to pull up some live data for us
to use to actually do these calculations here.
But before I do that, a little bit of notation here
and terminology.
The book is using a term that has come up
as the way to describe it.
But it hasn't been around for all that many years.
But it seems to be pretty popular.
Distinguishing between standalone risk and portfolio
risk.
And there's even another one too.
But unfortunately, this edition of the book takes that out.
But before we talk too much about these,
we have to talk about standalone with an E.
We have to talk about risk.
Risk is the variability of possible outcomes.
If there is only one possible outcome, there's no risk.
If there is more than one possible outcome,
then there is risk.
Now it's not just how many outcomes.
It's how much those outcomes can be different from one another.
That's the next step.
So in other words, there is the how many outcomes.
That's the big one first.
Oh, that didn't work.
The next one would be how different
are the possible outcomes.
That's the next one.
Then the next one is how, for lack of a better word,
how clustered are the possible outcomes.
So in other words, you have 100 different possible outcomes.
Well, that's noticeable right there.
That's important.
But what if all of them are really close to each other,
the outcomes are?
Then that's not a very risky situation compared to one
where the outcomes are quite different.
And then one last one that I will somewhat explore,
and I will show you something.
You haven't had a management course,
so I don't want to get into it too much.
But I will show you in Excel how you can calculate it.
There are a couple of other ones.
One that's kind of important to me
is how biased is the clustering.
And by that I mean you might have 100 possible outcomes.
95 of them are very close.
But then you have a tail that is on one side,
a tail of potential outcomes.
That would be important too, because there
is this typical assumption we make.
And that is that the outcome is not
as good as the outcome of the other.
There is this typical assumption we make.
And it's actually not as good.
It's used all of the time that outcomes
are distributed normally.
They are the bell curve, the famous bell curve.
Now there is a law of mathematics
that says that if you make a bell curve with a single data,
it will begin to behave like this bell curve,
what we call the normal curve.
That's great.
However, the reality is that there
is some data that just you cannot make that assumption.
Is we always, well, it's a bell curve.
Your average is 75%.
And it's a bell curve and all that.
Because there are all these mathematical simplifications
we can make, very cool things you can do with a normal curve.
So we hope that it is normal.
And even if it's not, we pretend it's normal.
But I can tell you right now, there's a problem with it.
Well, in education, there's a great problem with it.
You see, if the average of an exam is 75%,
that's our ideal, perfect 75%.
Of course, you screwed that up.
You got an 80%.
But the problem is that, OK, the highest score possible
would be 100%.
The lowest score possible would be a 0.
So do you see that there would be a tail down here
of scores that would not be matched
by a tail of the same distance from the average
on the outside?
In other words, I could have an outcome on the outside no more
than 25 points above the average.
But I could have a test score that was 75% 75 points
below the average.
So we could get a 0 or a 5 or something like that.
So in other words, the assumption of a bell curve
is invalid.
The same, unfortunately, is true of stock returns.
And it's actually, OK.
In other words, stock returns, we assume
that there is an expected return to a stock.
And we would assume that it could possibly deviate
from that on either side.
So it would be a normal distribution.
And we use all of these cool little metric tricks
for finding the mean, the standard deviation,
all that kind of cool stuff.
Unfortunately, we know for a fact
that it is not a normal curve.
In fact, it has a very, at first glance,
stock returns do behave, look like a normal.
But if you look a little more carefully
over a long period of data, historical data,
it looks actually like this.
The first thing that it is biased upward.
We know that stock markets, stocks in general,
their returns are biased upward.
In other words, over a long period of time,
returns are positive.
The second thing is these two little bastards right here.
A normal distribution has very thin tails.
After the clumping, in a normal distribution,
67% of the returns are between one standard deviation above
and one standard deviation below.
95% are here.
99% are here.
That's normal.
Unfortunately, we have what are called black swans
and white swans.
In certain periods of time, you can have too many black swans.
In other times, we can have too many white swans.
The normal distribution would say it's a very unusual event.
To give you an idea, once in 100 years,
you would have a black swan if returns are normally
distributed.
We have so far had seven black swans in the last century.
We've had three white swans.
There should be one of each on each side
if it's normally distributed.
So unfortunately, it's not normally distributed.
In fact, we even know what that one is.
The mathematics of probability theory is very old
and it's very complete.
This one down there, it was discovered over 200 years ago.
It's called a Cauchy distribution.
You don't need to know that.
But Cauchy distributions are awful.
I dealt with them in probability theory classes
in graduate school.
They don't have standard deviations, oddly enough.
That's something that nice, quiet, pleasant distributions
have.
Cauchy is a monster.
We pretend that stock returns are normally distributed,
even though we know that by using those for forecasting,
we're going to get our butts kicked every now and then
by a black swan or happily by a white swan.
I'm taking that just as in context here.
Get that son of a bitch off the board.
Fortunately, computers can do this now,
work with Cauchy distributions.
You have to work with them numerically.
But they still carry an underlying little nastiness.
OK, so let me get this off the board here for you
for just a minute.
OK, first thing first.
That's the average.
This is our measure of central tendency, the point where
things want to be.
They're up here, down here.
The x bar is simply, and I'm going
to write this in fancy mathematical notation,
with a will.
Well, actually, I should put this as an r bar.
These are returns.
Our expected return is the sum of each
of the individual returns divided by n.
Add them up, divide by the number of data points.
However, we do have to appreciate that sometimes we
don't have.
This would be true if we had a lot of data.
Each one would have 1 over n probability of occurrence.
In general, if you are running into data
that's a little more complicated,
it would be the sum from i equals 1
to n of each return times the probability
of occurrence of that term.
And let me do it in Excel so you see what I mean by this.
Watch.
Where the hell is my Excel?
I'm just going to whip up a sheet here.
Excel.
Open.
And you can pull up your own Excel for this.
So I'm going to write this as a
and you can pull up your own Excel for this.
I would encourage you to.
I'm sticking with returns.
So return probability.
And I'll go through a couple here.
Let's say that a return of negative 12% on a stock
has a probability of 0.10.
A return of negative 3% has a probability of 0.15.
A return of 5% has a probability of 0.30.
A return of 10% has a probability of 0.25, let's say.
And a return of 14% has a probability 0.2.
So now, the expected return.
Let me separate these.
You would take this times this plus this times this plus this
times this plus this times this.
But we don't have to do it that way.
We can do it by some product.
Are you familiar with some product?
Some product.
This vector, what Excel calls array,
by this vector or array.
And that would be a percent.
So the expected return to the portfolio would be 5.15%.
And then the spread, the standard deviation,
I would need to take each of these returns.
Oops, I did that wrong.
Not an equal sign.
Equals.
The next step is, and I'll put this one up on the web for you
if you're getting behind a little bit.
I would take each of the individual returns
minus the expected return.
And I would make that an absolute
so I could just run it down the line.
And then I would square the result times the probability.
I don't even have to do that, but I will.
And then I would add them up.
I'd get that for every one of them.
And then the standard deviation would be the square root
of the sum of those.
And that is a percent.
It's a measure of the spread.
And the last one that I would, another one.
You see, the problem with the standard deviation
is that it would depend upon the size of the data, what
the numbers were in the original data.
If I wanted to look at different stocks against each other
to measure their standalone risk,
I would need to scale those numbers.
So the coefficient of variation, I have to expand this one more
time, I would take the standard deviation divided
by the expected return.
This would make it so that I could look at different data
sets where the returns had different possibilities
and different probabilities.
And they would all be in the same units.
My ass.
Really?
Good grief.
Did I do that right?
Equals the standard deviation divided by the expected return.
That should not be that, oh, it's doing it as a percent,
and it's not a percent.
It's just a number.
Come figure out.
So in other words, this is just a pure number.
It's not a percent.
It's just a factor.
And that makes it so that I could
see another set of data that had much different looking return,
spreads, and probabilities.
And I could compare the two because they would be unitless.
So I can say, well, I had this other returns.
I had eight returns and eight associated probabilities.
And their standard deviation was like 12.7.
Well, because the numbers were bigger, probably.
And this, the coefficient of variation,
scales the numbers so that they all are unitless.
They're no longer percents.
However, fortunately, we don't have to do this too much.
Because if your data set is large enough,
these probabilities don't mean anything.
All of the probabilities will just be 1, in this case,
1 fifth, 0.20.
If you have enough data, the probabilities kind of wash out
over a large period of time.
And I'm going to show you that in just a minute here
with the much larger data sets where
we don't have to worry about this nonsense.
So in Excel, in Excel, the x bar is,
the way you write that is average of the data set,
average of the data set.
And the standard deviation, STDEV,
is STDEV, open the parentheses.
Now, you've got to be careful here.
Because the formula for a sample is not the same
as the formula for a population.
You will almost never have a population.
Let me show you what happens.
And we'll get one here.
I'll pull up some data, and we'll
play with it here for a little bit.
Is there anything else I want to do here?
I could do, technically, I could do sharp.
But it doesn't really have any meaning in this one
because I didn't have a risk-free rate involved.
But let me take some data here.
OK.
Now, in order to do this, I'm going to go to a place called,
it's an old site.
And everything, all the data sets,
data is everything to us in finance.
Data, data, data.
Everything is data-driven.
And unfortunately, we don't have a data set.
Unfortunately, the data is almost always costly.
You have to pay a subscription fee.
And sometimes that subscription fee can be insane.
And that's the kind of stuff that artificial intelligence
needs to do its modeling.
It has to have millions of data points
for it to have reliability.
So that means you have to have the ability
to subscribe the AI into these expensive data sets.
And they're all over the place.
Now, the Federal Reserve has data.
But it actually isn't that dense.
In order for you to go and get real actual data that
is like millions and millions of data points,
you have to subscribe to some of these services.
I mean, I just ran into one a couple of weeks ago
called PIG.
And PIG is just vast.
But you subscribe to it.
S&P has databases, too.
They are vast.
But you have to subscribe to those.
And there are dozens of other services.
This site that I'm going to try to find here for you,
it's been around forever.
Its data isn't that vast.
But it's free.
And it's very reliable.
We know that the data doesn't have any mistakes in it.
And every year, they update it.
Let me see.
Stock returns historical data.
And the place that I'm going to look for is one stock one.
One stock one.
One stock one dot com.
There it is.
Now I want you to look at something.
What's cool about this data set is that they have these indexes.
The Dow, the return to the Dow since 1975, the return to S&P.
But it has ETFs.
Any stock you could think of that is publicly traded, you can find here.
You can also find data for other countries, for their markets in here as well.
So if you're not trying to solve the problems of the world,
you just want to get data and do some modeling or testing
or just try out your mad Excel skills, you can't beat this.
So watch, what I'll do here, there are a couple of ways in Excel
that you can capture data from a website.
And I'm almost reticent to do this here, but what the hell.
I'll try it and see.
First things first, in order to use Excel to capture a data table,
I'll go to this.
Oh, will you stop it with that?
Okay, here is the Dow Jones.
Now what I'm going to need to do to accomplish this is I'm going to actually
copy, control C, the URL, the web address.
Now I'm going to go over here to Excel and I'm going to say data,
get a data set.
Oh, come on.
There's one for get it from the web.
Oh, yeah, this new one actually can get a picture, the data from a picture.
No, is that maybe what I want?
Okay, close.
If I can find, this is, oh, quit it, cancel.
I want to get data from the web.
And I'm not familiar with this 365.
I've got the professional version.
From, yeah, it's got some of the major ones, from a file, no,
from a power platform, online, from other sources, from the web.
There we go.
There's the one I want from the web.
Apologies for that.
And it will say, all right, where is this data?
Control V, that's when I'll paste in the URL.
And I'll say, okay.
Now it's going to run over there and it's going to say, okay, I see it right there.
I think.
Now, connect.
And it's going to say, I see a bunch of tables there.
Now, one of the problems with websites, and this is, this wasn't how we did it in the old days,
but that top thing you see in websites, the menus, and the clickdowns and all that,
that's actually set up as a table.
And that's not what you probably want.
So I'll bet it's the second table.
And it'll give me a preview.
Ah, it is.
Well, spank me, Jesus.
Look at that.
Do you see it?
Okay, that's the one I want.
Okay, so load that mother.
Oh, that data set.
You know what I mean, for God's sake.
Look, it's right there.
Well, isn't that a very convenient thing?
Okay, so I will now call, give that a meaningful name.
I will call it George.
No, I won't.
I'll call it Dow 30.
And I've got it.
So now let me do sheet two.
Go over here.
And I'm going to go back to, back up one, the S&P 500.
And I'm going to get its URL.
And I'm going to do the same stupid pet trick, now that I remember.
Data from the web.
Give it that webpage and say Okie Dokie.
And I'll also say, let's have a look at table.
Yes, that's table one there.
So load that mother, or that thing.
See it?
And I got that one.
And I'll call that S&P 500.
And now I'm going to do the trick one more time.
I'm going to go to the site and I'm going to get the NASDAQ,
historical data, or the health.
There it is.
And I will capture its web address.
And I'll go back to Excel and I'll say data, get data from the web.
Ok.
And it's table one.
It is every time.
So load that data.
Now if you do this right, it saves you a lot of pain.
Because a long time ago, we actually had to type in each data entry on our own,
or do a copy and paste and have Excel misunderstand how it worked.
This way is a lot more powerful.
And you can just draw down masses.
If you can find a place like, well, FRED, the Federal Reserve's database,
that data is kind of, they do it by day sometimes.
You might want ten years.
Well you just do this trick I did here and it will slam that data right,
all of it, into Excel.
As a matter of fact, FRED is even better than that.
You can actually tell it to download the data as an Excel sheet.
So they're cheating these days.
Ok.
There we go.
Now let's go back here and do some stupid pet tricks.
Remember that what I want are returns.
Ok.
I want returns.
So these other ones really, the year obviously matters,
but what I really want are the returns for each of these.
So, whoops.
Oh, I should have, I forgot to label that NASDAQ.
So here we are.
We got all our returns.
So here's what we're going to do.
And this is another stupid pet trick.
I'm going to highlight all of these numbers.
All of these.
I hold down the control key and I click on the ones that I want.
Now I'm going to go down here and I'm going to put in the average.
And I'm going to put in one more here too that I haven't talked about.
The median.
The median is the halfway point in the data.
That's not the average.
And in fact, the difference between them is kind of important.
But then I'm also going to do the standard deviation.
And I'm going to do also the coefficient of variation.
Now remember, I've got all three sheets highlighted.
So what I do to one will be done to all of them.
So the first thing I'm going to do is I'm going to say equals average of all the date, of the returns.
The average of the returns.
Now I'll turn these into percents eventually here.
And then I'm going to do the median.
The central point of all the data.
Now this next one I caution. Watch out for what you do.
I'm going to write the standard deviation equals STDEV.
Now notice that it gives you a whole menu.
The second one,.S, is the one we want.
But if you don't put anything there, it will do.S.
You don't have to. It defaults to.S.
Thank heaven. I never remembered to do that.
Okay, STDEV.
Close the parentheses.
And finally, the coefficient of variation equals,
and this one you just do by, you just do the standard deviation divided by the average.
Now let me turn all these, these first three are percentages.
So I'll turn those into percentages to two decimal places.
Okay, so there's my Uncle Bob.
Now let's go back through and look at what we found.
The average.
For the Dow, the average over the last, now this data was from 1975 to 2023.
So that would make it, what, 25, 48 years.
This is 48 years of data.
The average return to the Dow was 9.79%.
The average return to the, whoops, it didn't work, equals average, I went too far.
Median.
I have to do median of this data.
I took that too far with that, all sheets.
It took the first set of data for all of the sheets.
Again, equals median, median of the second tape, that table of data.
Standard deviation of all that data.
There we go.
Now let me do this S&P 500 again, average of that data.
Equals median of that data.
Yeah, I was taking the first table for all of them.
And for that one, equals STDEV of this data.
Wow.
My cool pet trick didn't work there.
Okay, so now let's go back.
The Dow, for the last 48 years, the average return.
If you had just put money in and left it there, your average annual return would have been 9.79%.
For the S&P 500, a somewhat riskier portfolio, it would have been 10.32%.
For the NASDAQ, if you had just stuck the money in there and left it, the average return would have been 14.84%.
There is the risk-return relationship.
The greater the risk, the greater the return.
Now let's go back here and look at some things.
The median tells us half of the numbers are below that number and half are above that number.
When the average and the median are different, that tells us about the skewedness.
Here's what it means.
To a certain extent, it tells us about how much different from a perfect bell curve the data is.
Look at this one.
Suppose that you had this long tail of data like that.
These low numbers would actually pull down the average from what it should be.
So you have, on this case, the average would be higher than the median.
The median doesn't care the size of the numbers.
See all these big numbers up here?
They are being counterbalanced by this long tail down here.
The same would go the other way.
So I could have some high returns that would pull the average upward.
That's a problem on tests where a lot of people are not doing all that great.
But you have just a small cluster, maybe five, eight students who ace the test.
They make the average look higher than the data is really telling you it is.
So in this case, the average is being drawn down by heavy losses in a few years.
That's the black swans.
The median, half the numbers are below it, half of them are above it, is telling us a different story.
So in a case like this, I would say that the average is being affected by heavy losses that make the average,
those really low numbers, those negative numbers, are affecting the average,
but they don't do anything to the median.
Look at the S&P 500, the same problem is occurring there.
The black swans are pulling the average downward.
But look at the NASDAQ.
The black swans don't affect the distribution as much.
In other words, small cap companies are not as sensitive to black swans as the big 500 or the massive Dow are.
Now look at the coefficients of variation.
For the Dow, the CV, well, let me put this to about 2%.
For the Dow, the coefficient of variation is 1.5.
For the S&P 500, it's 1.56.
In other words, there's a little more absolute spread.
And for the NASDAQ, there's more spread, absolutely.
So that's why the CV, see if I just look at the numbers, you get that.
For the Dow, you get 14.67. For the S&P 500, you get 16.8.
But that's because the Dow's numbers are bigger.
Once I get that bigness, that difference in size of the numbers out of there,
we can see the true risk-return relationship coming through.
So look at this. 9.79 on an absolute risk metric of 1.5.
10.32, 1.56. 14.84, 1.68, the risk-return relationship.
Now I'll put this up for you to play around with, but this gives you visually what we constantly are talking about,
that risk-return relationship, the greater the risk, the greater the return.
And here, over a very large amount of data, we see it shining through.
We also see some hints, too.
We see some hints that, for example, like I said, black swans pulled down the average for the Dow
and for the S&P 500 far more than they do for the small cap stocks.
And we also see that risk-return relationship coming through again.
And you can do this with tons of different companies, tons of different indexes.
It always shows up. It always does.
The same phenomenon, lower risk portfolios, lower expected returns, more skew in the data.
And as you get higher and higher risk of the portfolio, you get less skew and you get higher spread, absolutely speaking,
which is what the CV does.
That's good to know for all of us in our business.
And we can play with this on Monday.
I'm going to, if you don't have it, I'll try to do it real quick here.
Let me show you something.
In files, options, I think it's options, go to add-ins.
See this right here, this analysis tool pack?
You click on that one and say go.
And you want the analysis tool pack, okay.
Now what that will do is it will give you the access to a pile of data analytics.
You can do all kinds of things like this and all kinds of other things with that data analysis pack.
And all you have to do is say here's the data, use the data analysis pack,
and it will make an extra spreadsheet with all of these statistics on it for you.
So again, in order to do that, you do files, options, and you go down to the add-ins,
and you choose the data analysis pack and you say go.
And it will put it in there for you.
Now for some of you, you'll have to do it every time you,
if you want to do a new data analysis, some Office 365s,
you have to do this little thing every time to get the pack loaded.
Others, it will be in there permanently.
But anyway, you have a quiz to take.
Please take that quiz, and once you're finished with it, that's all I have for you today.
Thank you.