So, we'll look at data now finally.
And before we actually start looking at
data, remind yourself a covariance
stationary.
So, we're going to look at data.
And, when we start looking at asset return
data, you want to think yourself that the
asset returns that we observe are
realization of a covariance stationary
time series, okay?
And so, you think of Gaussian White Noise
as the benchmark and, and then you want to
think with a critical eye of may, of, what
a moving average process looks like, so
time dependence of, you know, one period,
or an autoregressive process where you
have decaying time dependence.
And you want to ask yourself, you know,
you know, what properties do you actually
see in the data?
And if the process were covariance
stationary process, what kind of process
would it most likely be?
Okay?
So, that's the kind of game that we want
to play.
So, all of the returns have the same mean,
the same variance.
They have a correlation that depends upon
how far apart they are, but on time, okay?
There should be no trends in the data,
and, and so on.
So, that's the process.
So, now let's look at some data.
So, for this next set of lectures, I'm
going to be looking well, two time series.
I have monthly continuously compounded
returns that I downloaded from Yahoo and
one series is going to be the monthly
returns on Microsoft, which is just an
individual asset.
And the other series we're going to look
at is the, the S and P 500 index, the
Standard and Poor's 500.
The S and P 500 think of this as a
portfolio of 500 assets, okay?
So, it's a, it has, it's a portfolio of,
of 500 stocks, and the weight of each
asset in the portfolio is based on its
market capitalization.
So, for example, if Microsoft were in the
S and P 500, the weight of Microsoft is
equal to the, the market value of
Microsoft which is the price of Microsoft
times the number of shares outstanding,
that's its market capitalization.
So, if you had to go and, if you wanted to
buy Microsoft as a company, what price
would you pay?
It would be price times number of shares
outstanding, which for Microsoft is about
$35 billion.
And then, so, you take the market
capitalization of Microsoft and then you
divide it by the sum of the market
capitalizations of all of the other 500
stocks, right?
So, the weights of, of all the assets in
the S and P 500 add to one, and assets
that have high market capitalizations have
higher weight in the S and P 500.
It's most of the, the 500 stocks that are
in the S and P 500 tend to be very large
capitalization stocks, so they're big
well-established companies.
And, and, and the weights, you know, of
these assets, it's, I don't want to say
it's equally weighted but it's fairly
equally weighted.
So, there isn't a, a huge concentration on
one or two stocks.
It's, it's a fairly diversified spread-out
portfolio.
So, the key thing to keep in mind is, this
is an individual stock.
This is a highly diversified portfolio of
500 stocks.
And what we want to do is we wanna say,
what are the, the stylized facts about the
returns on an individual security and a
highly diversified portfolio.
And we're going to see that they're very
distinct differences in the properties of
the returns of these two series, okay?
Now, here are the plots of the
continuously compounded return, and the
data goes from 1998 to 2008.
So, we don't have the financial crisis in
this period and so we'll look, we'll do a
homework assignment you know, you'll be
looking at more updated data.
And so, when you put the financial crisis
in here, you'll see a big negative
returns, okay?
So, that's you know, that's something to
be considered.
Now, when we look at these two graphs, you
want to ask yourself, what properties do
you see, and what similarities do you see
between the two series?
Okay?
So, first of all, do they look like
Gaussian White Noise?
Notice in, in the, from 1998 to 2002, the
fluctuations of Microsoft were pretty big,
so we call that the volatility.
But, after about 2002, notice the
magnitude of the fluctuations decrease,
right?
So, is that a property of a covariance
stationary time series?
Yes or no?
No.
It's because the variance appears to
depend upon time.
During this period the variance was big.
During the, this, this period the variance
was small.
So, that's an indication, perhaps, that
the volatility is, is changing over time,
okay?
So, that's one, one future to see.
What about the mean in the series?
There doesn't appear to be any obvious
trend, right?
So, the, the data appears to fluctuate
pretty close to zero in, in this plot.
And so, so if we think about covariance
stationary in terms of the mean being
constant, we don't see like the mean is up
here, and then the mean is down below, and
then the mean is up here, and, and things
like that.
So, now one of the things about, and, and
so we, I, I want to follow up on this,
this volatility comment that was made
before.
Microsoft has high volatility here than
low volatility, we also that in the S and
P 500 index.
So, very often, the volatility of the S
and P 500 index is called market
volatility because the S and P 500 is
thought to be a proxy for the overall
movements in the stock market.
And so here, market volatility was high,
and then down here market volatility was
low.
So, notice that Microsoft's volatility is
tracking the market volatility as well.
So, we see some common behavior between
the series and in this regard.
Now, these two graphs are a little bit
misleading because the scales are
different.
Notice that here, this, this is -40%, this
is twenty percent for Microsoft.
For the S and P 500, this is -fifteen%,
this is six%.
So, the volatility of the market is much,
much lower than the volatility of
Microsoft, but that's not apparent because
of the way that the graphs are drawn.
If we put these two asset, assets on the
same scale.
So, here, I have Microsoft in red, and the
S and P 500 in blue.
Then, we can see I think, more clearly,
some similarities and differences between
the series, okay?
So, very clearly, the volatility of
Microsoft is much greater than the
volatility in the S and P 500, okay?
So, this one fact about assets.
When you have a highly diversified
portfolio like the S and P 500, you, you
do, you have a smaller volatility than an
individual security, okay?
And remember, volatility is very often
used as a proxy for risk.
Volatility is telling you uncertainty
about the mean.
So, the uncertainty about the mean for the
S and P 500 is a lot smaller than the
uncertainty associated with, with
Microsoft.
So, we might think there's less risk in
the market than there is in Microsoft by
itself, okay?
Now, what's another feature that we can
see, or, or another commonality between
Microsoft and the S and P 500, that we can
see in this graph, that we couldn't see
very clearly in the other graph?
Are Microsoft and the S and P 500
correlated with each other?
So, Microsoft and the S and P 500 are kind
of moving up and down together at the same
time.
Well, that means they're positively
correlated, right?
When one, when the market goes up,
Microsoft goes up.
When the market goes down, the Microsoft
goes down.
Now, I'm not saying that there is a causal
relationship between the two.
They're, they're just moving at the same
time, okay?
Right.
So, those are some important properties
and these are the kinds of things that we
want to do when we look at data.
We want to see, what do we see?
What are commonalities?
What are features?
Is it covariance stationary?
If so, why?
If not, why not?
Those kinds of things.
Is there a relationship between the
volatility of the series and the expected
return?
So, if you invest, if you take a stock
that has a high volatility, do you tend to
get a high average return as well?
And the answer is, is the stylized fact
is, is generally yes.
When we look at stocks with high
volatility, they tend to have higher
expected returns than lower volatility
stocks.
Not always.
This is the idea that risk and return go
together.
So, and we'll be, we'll be doing some
plots later on while we're say, put
expected average return or expected return
on this axis and volatility on this axis,
and we'll see that you know, things tend
to move up in a direction like this and
it's, it's tends to be a somewhat stylized
fact but it's not always true.
Okay.
So, our observed sample is a realization
of this covariance stationary time series.
And, now, what we want to do is we want to
compute some of descriptive statistics,
okay?
Now, in, in a very simple term, you know,
a statistic is just a data summary.
So, we compute descriptive statistics,
that means we're creating data summaries
and we're trying to capture particular
features.
Now, when we studied random variables and
looked at probability distributions, I
said there were certain shape
characteristics of a probability
distribution.
There was the center of the distribution,
which was the expected value, the
variance, the spread, the skewness,
asymmetry, kurtosis, fat tails.
Well, all of these properties that we've
talked about for random variables, we can
create a descriptive statistics from the
sample.
So, if we want to know what is the
probability distribution look like?
We can compute a histogram which is a
graphical descript, description of the of
the distribution of the data.
And then, of that histogram, we can think
where's the histogram centered?
You know, what's the spread in the
histogram?
What's it's skewness, or kurtosis?
So, for all of the probability model
concepts, there's going to be a sample
analogue, a descriptive statistics that
gives us, you know, the same kind of
information, okay?
So, we're going to create data summaries
to describe certain features of the data,
to learn about the unknown probability
distribution that we think is actually
generating the data.
So, the, the thing about, you know,
probability modeling and statistics in the
real world is, we don't know what the true
model is that generated the data.
I mean, we'll never know what it is,
right?
And so, what we're trying to do is we're
trying to kinda guess at what we think is
the underlying model.
And, we're going to use these descriptive
statistics to try to help us pick good
models for describing the data versus you
know, bad models, and, and so on.
Alright.