Now the log normal distribution you saw
had a long right tail. So this gives rise
to you know, the notion of some
distributions are not symmetric about the,
the mean value. So, when we talk about
shape characteristics of a probability
distribution, the shape characteristic
that measures the symmetry of a
distribution is called Skewness. Now,
skewness is based upon taking a function
of a random variable, where the function
is the random variable minus the mean
divided by the standard deviation, all of
that cubed. All right? And then, the
skewness is the expectation of g of x.
It's the expected cubed deviation from the
mean standardized by the standard
deviation. Right? So, why is this
measuring skewness? Now, if x is way above
the mean, then x minus the mean is a
positive number. And when you cube that,
you're gonna get, a big positive number.
Right? So, if most of the values of x in,
in some sense are above the mean, then,
when you take the expectation, when you
take the probability-weighted average of
this, then you're going to have mostly
positive numbers, right? So you would have
a positive skewness if there is a tilt in
the distribution to large values, okay? So
that would be the case where if you had a
distribution that kinda looked like this,
whereas the, the mean of the distribution
might be here. But, we see that when you
have values of x below the mean, right?
They're not too large. But when you have
values of x above the mean, you can get
some really big ones. So, when you're
evaluating skewness, which is the
probability weighted average of x minus
the mean cubed multiplied by probability,
or, in a continuous random variable we
would take the interval weighed by the
density. In a distribution like this,
these big numbers would dominate this
calculation and it would become, we'd get
a positive number. So, this is a situation
where the skewness of x is greater than
zero. And then we can think of a reverse
situation. So, if we had a distribution
that kinda looke d like this. So now, say
the mean of x is h ere, then in this
distribution, we have a long left tail.
So, the values of x above the mean,
they're not too far above the mean. So
when you calculate x minus the mean cubed,
you don't have such big numbers. But now
when you look at values of x below the
mean, there are some really big ones below
the mean. So, x minus mu of x is a big
negative number, and then you cube it and
you make it even a bigger negative number.
So, when you have a distribution that
looks like this, when you perform this
calculation, these, big negative numbers
below the mean dominate and you'll get a
skewness that's negative. So, this gives
you the skewness of x that's less than
zero, okay? So, right skewed data, this is
called the long right tail, left skewed
data, long left tail. Then, in a symmetric
distribution, in a symmetric distribution
like the normal. If things are symmetric
then, the values above the mean and below
the mean cancel out. And so, when you
calculate the skewness, you get something
that's equal to zero. So, in this case,
you have the skew of x equal zero for
symmetric distribution. So, the example
discrete data set was this one where we
had probability distribution for rate of
return, and there were, you know, five
states of the economy, and then we had
probabilities associated with the states,
and then return values for the states. And
we have the discrete distribution looked
like this. So, let's compute the skewness
for this distribution. So, what do you
suppose the skewness is, just by looking
at the distribution? Yeah, it's zero. This
is a symmetric distribution. So, when you
calculate skewness, if there's any justice
in the world, then you should get zero.
So, I like this phrase, if there's any
justice in the world. A staff professor
actually it was I TA'd for staff professor
when I was in grad school since Burton
Singer and he used this phrase a lot when
he was like deriving something and it
looked like this real ly horrible
calculation. He would say, if there's any
justice in the world it would simplify
into something nice. And most of the time,
that happened. So, when we do this
calculation here, so what is the skewness?
And so, we take the return value minus the
mean cubed waived by the probability,
return minus the mean cubed times
probability. You sum them all up, then you
divide by the standard deviation cubed and
lo and behold, it is zero. Now, for a
normal distribution, so if x is a, has a
normal distribution with mean u of x and
sigma x squared, then when you evaluate
the skewness, you have to compute this
integral. And, you know? And if you're
good with calculus, then you can show
that, in fact, this integral is, is zero.
Now on the other hand, if we have a log
normal distribution, so we saw previously
that the log normal distribution had a bit
of a long right tail. So, we would expect
the skewness for a log normal to be
positive. And, in fact, if you do the
calculus on the log normal distribution,
you can show that the skewness of a log
normal random variable actually has this
analytic expression here and, and this is
positive. So the log normal distribution
exhibits a long right tail. Now, another
shape characteristic that is of particular
interest for us in finance is a shape
characteristic that tells us about the
likelihood of extreme values, right? So,
if we're looking at you know, daily stock
returns on the s and p500, we know from
time to time there's a market crash. And
so, when the market crash happens, we get
a return that's in the far left tail of
the distribution. And we can also get an
extreme positive event. There could be a
time for whatever reason the market goes
up by a huge amount in a given day. So,
you know, that's an extreme value of the
right tail of the distribution, okay? Now,
when we start looking at asset return
data, particularly if we look at daily
data, the normal distribution fails in a
big way, and one of the ways that the
normal distribution fails is
characterizing the likelihood of extreme
events. The normal distribution doesn't
imply that, I'm sorry, the normal
distribution implies that extreme events
don't happen very often at all. But we see
extreme events happening actually quite
often, and, and so, we would like to have
distributions that imply a higher
likelihood of extreme events than the
normal distribution. And, so we would be
looking at some of those kinds of
distributions in, in this course. Now, how
do we measure, the likelihood of extreme
events for a distribution? So, one type of
measure is known as Kustosis. And the
kustosis, is based upon a function of the
random variable. And the function of the
random variable is, the random variable
minus the mean divided by the standard
deviation raised to the power of four,
okay? Now, and then the kurtosis is then
the expectation of this function. So, it's
the probability weighted average of the
deviation of x from its mean, standardized
by the standard deviation raised to the
power four. So, let's draw a picture,
here. And, let's say we have a
distribution, you know, that looks like
this. And here is the mean. Okay? So,
first thing to note about kustosis is
that, we're raising the deviation of x
from the mean divided by the standard
deviation to the fourth power, right? So,
this deviation is always a positive
number, right? Now, if x is way out in the
right tail, x minus the mean is a big
number. Right? And then when you
standardize by the standard deviation,
it's still a big number. You raised it to
power of four, you're going to get
something that's huge, right? And the same
thing is true on the other end. If x is
way below the mean, then, x minus the mean
is a big negative number divided by the
standard deviation, still a big negative
number. Raised to the power of four you'll
get a big positive number. So, the
kustosis accentuates the extreme values.
So, any distribution that has values that
are way below the mean or way above the
mea n, is going to produce large value of
the kurtosis. Okay? So, extreme events,
both positive and negative, lead to large
kurtosis. Now, we compute the kurtosis by
taking the probability weight ed average
of this deviation here, weighted by the
probabilities if we have a discrete random
variable, or we have to evaluate this
integral if it's a continuous random
variable, right? So again, these
calculations look a little bit horrible.
If we do this calculation for our example
data. So for a discrete distribution, how
would we calculate kurtosis? We would say,
the return minus the mean raised to the
power of four, multiplied by the
probability, add, return, minus the mean
raised to the power of four, multiplied by
probability, sum them all up and then
divide by the standard deviation raised to
the power of four, we'd get the number
6.5. Okay? Now, when we calculate
kurtosis, how would you know if the number
6.5 is big or small, right? Hm? Compared
to normal, exactly. So, what's useful in
kurtosis is we need to have a benchmark
for what is normal. You know? What is
yeah, we need to have some kind of
benchmark. So, when we evaluate kurtosis,
the benchmark that's most often used is
the kurtosis of a normal distribution. So,
let's look at the kurtosis of a normal
distribution. So, if x is a general normal
distribution and you go and you do this,
this integral, it turns out, this thing is
three, okay? So, that's kinda nice. So,
the benchmark value of the kustosis that
is, the, the kurtosis value for normal
distribution is three. So, if random
variable has a kurtosis that's bigger than
three, then you know that the distribution
for that random variable has fatter tails
than the normal distribution. That is, it
implies more extreme values than the
normal distribution does, okay? Now, on
the opposite side, what happens if the
kurtosis is less than three? Kurtisis less
than three, then you have thinner tails
than the normal distribution. So, you can
think of two paces to cons ider. So if
kurtosis say, this is my benchmark normal,
here. And then, distribution with fatter
tails than the normal might look something
like this, okay? So here, the kurtosis for
x would be greater than three for the red
distribution beca use we have more
probability in the tails than the normal
distribution in that case. One of the
greatest stories. Okay? So, and then the
opposite case to consider. So again, this
is my benchmark normal and then in this
case, what if the distribution looks like
this? So in this case, the distribution
for x has thinner tails than the normal,
so this red distribution will have a
kurtosis that's less than three So, now
because the normal is the benchmark, then
very often when kurtosis is reported, it,
what's being reported is actually
something called Excess Kurtosis. An
excess kurtosis is the kurtosis of the
random variable minus the kurtosis of a
normal, which is three. So, if excess
kurtosis is zero, then, you have the same
kurtosis as a normal. If excess kurtosis
is positive, then you're in this case. And
if excess kurtosis is negative, then
you're in this case. So, we start looking
at data, we'll actually do some
probability calculations of extreme events
based upon a normal distribution and then
we'll actually look at the likelihood of
extreme events based upon the actual data.
And that, that's another way that we'll
see that the normal distribution is not a
very good distribution for certain kinds
of returns because it, it doesn't imply
that extreme values happen very often. So,
for example, you look at, like, the crash
in 1980, October, 1987. So, if you assume
returns are normally distributed, the, the
crash is an event that would happen you
know, once every six trillion years. And
so essentially, the likelihood of that
happening is zero. And but you know, it
did happen. And whereas if we have other
kinds of distributions that have thicker
tails, then we would actually have you
know, a reasonable prob ability of, of
such an event like October 1987 happening.
So, for the point of view of risk
modelling, and this is a thing that I want
to emphasize as we go through the course,
the distribution that you use for returns
is a, is, is incredibly important because
the distribution for returns tells you the
likeliho od of, of good or bad events.
And, and if you're, you know, trying to
characterize the risks associated with
investments, then you know, your
distribution is, is paramount to giving
you the information for how to make
decisions. So, getting the right
distribution is, is an important thing to
do. And, and we'll see that that's not
always an easy thing to do in, in