Now, going back to other characteristics,
essentially what I'm, what I'm doing is
I'm mirroring with sample statistics, the
kinds of things we did in probability
theory.
Alright.
So an important thing for us is a quantile
of the distribution.
So, when we did value at risk we found,
you know, if you want to know with five
percent probability, how much money we
could loose, we compute the five percent
quantile of the return distribution.
Now, we have, we, if we have a probability
distribution, we can determine the
quantile from the inverse of the CDF.
But with data, we can also estimate or
compute a sample quantile, right?
And the sample quantile is often called
the empirical quantile or the percentile.
So, for some alpha and 01, we can get the
hundred times alpha percentile that is,
like, the fifth percentile of the data.
The fifth percentile of the data is just
the data points such that or, sorry, the
alpha percentile is the data points such
that alpha percent of the data is less
than that value, okay?
And so essentially, how do you find these
empirical quantiles in this ordered data
from largest to smallest and the five%,
the, the, the fifth percentile is just
that point in the data such that five
percent is to the left, okay?
And, and then, we also have these things
called quartiles, the, you know, the first
quartile is the twenty-fifth percentile,
the second quartile is the median, the
third quartile is the seventy-fifth
percentile and, and so on.
There's something called the interquartile
range which is a measure of the, the, you
know, the bulk of the distribution.
It's just the difference between the third
quartile and the first quartile.
Now, let's look at actual data.
So, if we're looking at the Microsoft
data, there's a function of R called the
quantile.
The quantile function computes the
empirical quantiles of the data.
And if you just use the quantile function
by itself, it gives you the minimum, so
the minimum return is -42%.
It gives you the first quartile, so it's
the twenty-fifth percentile, so -5.3%.
25 percent of the returns are smaller than
this value, okay?
The, this is the median, the, and the
median return is one%, okay?
So, and then, the on the right side of the
distribution so 75 percent of the returns
are less than six percent or 25 percent of
the returns are bigger than six%.
And then, this is the largest return, it's
34%, okay?
Now, for value at risk calculations, we're
often interested in the one percent and
the five percent empirical quantiles.
So, if we look at Microsoft, we use the
quantile function, we say, prod is equal,
you know, we combine 0.01 and 0.05.
Then, we get the one percent quantile and
the five percent quantile.
So here the one percent empirical quantile
is -26%.
So , one percent of the returns are less
than 26 percent and the five percent
empirical quantile is -fourteen percent so
five percent of the returns are smaller
than -fourteen%, okay?
Now again, we can compare, so here are
quantiles.
These are from the actual data, right?
Now, what about quantiles from a normal
distribution?
So, if I take a normal distribution that
has the same mean and standard deviation
as the data, then I get these quantiles.
So, the one percent is, is -24 percent
where as in the data it is -26%.
So, the one percent quantile in the data
is smaller than the one%t normal quantile,
okay?
And if we look at the five percent
quantile, it's kind of interesting that
the normal five percent quantile is
actually bigger than, in the empirical
data, right?
So, that's kind of like saying that data
distribution is kind of squished with fat
tails relative to the normal, okay?
Now, when we're doing value at risk we use
quantiles to do the value at risk
calculations.
So, we can do value at risk with the
empirical quantiles, or we can do value at
risk from the normal quantiles that use
the mean and standard deviation from the
actual data.
Okay, so, those are two different types of
assumptions for doing for doing VAR.
Now, if we look at the S and P 500, we can
do the same comparison.
So, the empirical quantiles for the S and
P 500 data, the one percent quantiles
minus eleven percent and the five percent
quantiles minus 7.5%.
Now, it's interesting to compare this to
Microsoft.
So, with the S and P 500 with say, one
percent probability, you could lose eleven
percent or more.
But with Microsoft, with one percent
probability, you could lose 26 percent or
more, right?
So, that's almost three times as much as
what you can lose in the S and P 500.
So again, if we think about risk as being
the probability of loss, then there's a
higher probability of loss associated with
Microsoft than there is with the S and P
500.
Okay, so okay, so here is the empirical
quantiles.
And R functions there are lot of functions
you can sort data, you can take the
minimum and the maximum, you can compute
range which is the difference between the
minimum and the max, compute your
quantiles, the median, you can compute
interquartile ranges.
So, you have all these, you know, nice
functions you know, that, you can work
with, the homework assignment, you know,
takes you through many things.
I want to make a comment about value at
risk using empirical quantiles.
So, there's something known in, in the
finance literature as historical value at
risk.
When you, when you hear the term
historical value at risk, you're computing
value at risk using the empirical
quantiles, not the quantiles from a normal
distribution or something like that,
you're just using the empirical quantiles
from the data.
So, what is actually being computed?
So, we have our simple returns and the
historical value at risk is your initial
investment times the empirical quantile of
your simple returns, you know, at the
alpha level, okay?
So, this is just the, the data point that
we use from the quantile function, okay?
And, and so, this, this could be an
estimate of, of our loss, okay?
If we have continuously compounded
returns, remember the value at risk then
would be e to the continuously compounded
quantile -one times your initial
investment.
So, this would be your value at risk with
continuously compounded data.
And the key thing is, again, we're using
the, just the empirical quantile, we're
not using the quantile from, say, a normal
distribution, alright?
Now so, here's the histogram that gives us
the general shape of the distribution.
Now, what about shape characteristics?
If we want to know the sensor, then we
compute the sample average.
The sample average is a, is a measure of
the center of the distribution and that's
just an equally-weighted average of the
data points, okay?
If we want to know the spread about the
average for the histogram, we compute the
sample variance.
The sample variance, we take one over the
sample size minus one, and then we sum all
of the observations, we subtract the mean
value, we square it.
So this is, this is called the sample
average.
Notice we divide by t - one and not t,
okay?
We'll come back to this a little bit later
on.
The reason for dividing by t - one is to
make the estimates slightly, a little bit
better.
We, we have what's called an unbiased
estimate of spread when you divide by t
minus one.
The sample standard deviation is the
square root of the sample variance.
Again, that's the an estimate of the
spread about the average in the same units
as the data.
One of the things I want to point out
about notation you know, the sample
average is often denoted as x bar, and
sometime as just mu hat.
So, the Greek letter represents the
probably, the expected value of a random
variable, and when you put a hat on top of
it, that means we're estimating that
expected value.
And here, we're estimating the expected
value using the sample average.
This is known as the plug in principle in
Statistics.
So, we plug in a sample statistic for a
population expectation.
And for sample variance, we often, you
often see little s ^two for sample
variance and sometimes we also call it
sigma hat squared.
So, again that's just terminology.
And then, similarly, we can compute a
sample skewness.
Sample skewness, we take the average of
the observations minus the mean cube
divided by the standard deviation cube and
again, we divide by t - one to make this
an unbiased estimate.
And then, sample kurtosis, this is
measuring the thickness in the tails.
We sum up the observations, we subtract
the mean, we raise it to the power of
four, we divide by the standard deviation
to a fourth.
Okay, that's the estimated kurtosis and
estimated axis kurtosis would be this
kurtosis -three, okay.
And again we, axis kurtosis gives us a
measure of tail thickness relative to a
normal distribution, okay?
So, we have sample statistics that give us
the shape characteristics of the
histograms.
So, in R, we have set of functions.
So, the mean function computes the sample
mean.
The bar function can be the variance, SD
compute standard deviation.
Core R actually doesn't have functions for
skewness and kurtosis.
So, I like to use the skewness and
kurtosis functions from the performance
analytics package.
This performance analytics package in
general, is a very nice collection of
functions that are useful in Finance.
They are very nice charting functions and
they have a number of utility functions
that are, that are quite useful.
In our, in our labs we are going to be
using the performance analytics functions
quite a lot.
So, you should become familiar with, with
this package, okay?
Now, one of the things I wanted to say
about R and I'll come back to this here
when you calculate, when you use the mean
function, you only work on either a vector
of data or one column of data at a time.
If you have many return series, say, in a
matrix, often, very often you want to
compute the mean for each column, alright?
The mean function doesn't work column
wise.
And so, in R, when you want to apply a
function to the columns of data, there's a
function called apply.
And, and so this, the apply function
essentially implement a loop.
You loop over the columns and then you
apply the function to the column itself.
I'll illustrate how this works.
These apply functions are very, very, very
useful particularly, say, if you have a
thousand assets and you want to compute
the means for each of the thousand assets,
you can do it in one command by, by using
the apply function.
Alright.
So, let's look at some of the sample
statistics.
So, for the mean return for Microsoft is
0.006.
So, this is the monthly mean and it's
about it's a little less, a little less
than one percent per month, okay.
The variance is 0.01, now that's in
squared returns, the standard deviation is
in returns.
The typical spread about the average is
ten%, okay?
Remember, this is a monthly return.
So remember mean plus and minus the
standard deviation for bell-shaped data
gives you roughly two-thirds of the
observations.
So, two-thirds of the observations on
Microsoft are between -ten percent and
ten%, okay?
So, losing ten percent in a month is, is
quite a lot actually.
The sample skewness is negative.
So in the picture, we saw a long left tail
and when you use the skewness function for
Microsoft, you do get a negative skewness
number.
And the kurtosis function, actually gives
you axis kurtosis.
So don't, don't be fooled by this.
The fact that it's kurtosis, when you look
at the help and performance analytics, you
find that what's actually been computed as
sample kurtosis -three.
So, axis kurtosis is 1.8 so the actual
kurtosis is 4.8.
So, the Microsoft data have fatter tails
than the normal distribution.
So, we're seeing a mean return of, you
know, 0.6 percent per month, a standard
deviation of ten percent per month, a
negative skewness, and an axis kurtosis,
okay?
Now, suppose we had a matrix of data that
contains, the first column contains the
data from Microsoft and the second column
contains the date for the S and P 500.
And I want to compute the means of each of
the columns of this, this matrix.
This is where we use the apply function.
So you, say, apply your data matrix and
then this parameter two means apply to the
columns.
If this parameter was one then you would
apply the function to the rows, okay?
So, this means apply to the columns and
what are you applying?
You're applying the mean function.
And so, what do you get?
You get the mean for the first column, the
mean for the second column, okay?
So here, we can compare Microsoft and the
S and P 500.
The average return for Microsoft is 0.6%.
The average return for the S and P 500 is
0.3%.
So, on average, we get about twice as much
per month on Microsoft than we do for the
S and P 500, okay?
Now, what about the standard deviations?
So, the standard deviation for Microsoft
is ten%, actually it's, it's about
eleven%, and the standard deviation for
the S and P 500 is only four%, right?
So here, the spread about the average for
Microsoft is, is about 2-1/2 times bigger
than that for the S and P 500.
Notice that the mean for Microsoft is
higher than the mean for the S and P 500,
and the standard deviation for Microsoft
is bigger than the standard deviation for
the S and P 500.
So, we're seeing assets that have higher
volatility, has, has a higher mean, the
asset with a lower volatility has a lower
mean, okay?
If you were using daily data instead of
monthly data, data, we would get a better
analysis of the tail.
And in general, the answer is yes, because
we have more daily data than monthly data.
And so, we have more information in the
tail with daily data than we do with
monthly data.
Okay.
So and then we have the skewness for
Microsoft, S and P 500, both have negative
skewness, and the skewness for S and P 500
is quite a bit bigger, is a bigger
negative number than for Microsoft.
And the axis kurtosis is bigger for
Microsoft than it is for the S and P 500.
So, again, just some comparisons.