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Now the log normal distribution you saw
had a long right tail. So this gives rise

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to you know, the notion of some
distributions are not symmetric about the,

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the mean value. So, when we talk about
shape characteristics of a probability

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distribution, the shape characteristic
that measures the symmetry of a

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distribution is called Skewness. Now,
skewness is based upon taking a function

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of a random variable, where the function
is the random variable minus the mean

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divided by the standard deviation, all of
that cubed. All right? And then, the

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skewness is the expectation of g of x.
It's the expected cubed deviation from the

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mean standardized by the standard
deviation. Right? So, why is this

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measuring skewness? Now, if x is way above
the mean, then x minus the mean is a

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positive number. And when you cube that,
you're gonna get, a big positive number.

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Right? So, if most of the values of x in,
in some sense are above the mean, then,

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when you take the expectation, when you
take the probability-weighted average of

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this, then you're going to have mostly
positive numbers, right? So you would have

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a positive skewness if there is a tilt in
the distribution to large values, okay? So

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that would be the case where if you had a
distribution that kinda looked like this,

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whereas the, the mean of the distribution
might be here. But, we see that when you

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have values of x below the mean, right?
They're not too large. But when you have

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values of x above the mean, you can get
some really big ones. So, when you're

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evaluating skewness, which is the
probability weighted average of x minus

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the mean cubed multiplied by probability,
or, in a continuous random variable we

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would take the interval weighed by the
density. In a distribution like this,

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these big numbers would dominate this
calculation and it would become, we'd get

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a positive number. So, this is a situation
where the skewness of x is greater than

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zero. And then we can think of a reverse
situation. So, if we had a distribution

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that kinda looke d like this. So now, say
the mean of x is h ere, then in this

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distribution, we have a long left tail.
So, the values of x above the mean,

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they're not too far above the mean. So
when you calculate x minus the mean cubed,

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you don't have such big numbers. But now
when you look at values of x below the

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mean, there are some really big ones below
the mean. So, x minus mu of x is a big

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negative number, and then you cube it and
you make it even a bigger negative number.

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So, when you have a distribution that
looks like this, when you perform this

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calculation, these, big negative numbers
below the mean dominate and you'll get a

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skewness that's negative. So, this gives
you the skewness of x that's less than

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zero, okay? So, right skewed data, this is
called the long right tail, left skewed

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data, long left tail. Then, in a symmetric
distribution, in a symmetric distribution

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like the normal. If things are symmetric
then, the values above the mean and below

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the mean cancel out. And so, when you
calculate the skewness, you get something

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that's equal to zero. So, in this case,
you have the skew of x equal zero for

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symmetric distribution. So, the example
discrete data set was this one where we

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had probability distribution for rate of
return, and there were, you know, five

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states of the economy, and then we had
probabilities associated with the states,

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and then return values for the states. And
we have the discrete distribution looked

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like this. So, let's compute the skewness
for this distribution. So, what do you

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suppose the skewness is, just by looking
at the distribution? Yeah, it's zero. This

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is a symmetric distribution. So, when you
calculate skewness, if there's any justice

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in the world, then you should get zero.
So, I like this phrase, if there's any

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justice in the world. A staff professor
actually it was I TA'd for staff professor

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when I was in grad school since Burton
Singer and he used this phrase a lot when

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he was like deriving something and it
looked like this real ly horrible

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calculation. He would say, if there's any
justice in the world it would simplify

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into something nice. And most of the time,
that happened. So, when we do this

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calculation here, so what is the skewness?
And so, we take the return value minus the

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mean cubed waived by the probability,
return minus the mean cubed times

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probability. You sum them all up, then you
divide by the standard deviation cubed and

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lo and behold, it is zero. Now, for a
normal distribution, so if x is a, has a

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normal distribution with mean u of x and
sigma x squared, then when you evaluate

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the skewness, you have to compute this
integral. And, you know? And if you're

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good with calculus, then you can show
that, in fact, this integral is, is zero.

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Now on the other hand, if we have a log
normal distribution, so we saw previously

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that the log normal distribution had a bit
of a long right tail. So, we would expect

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the skewness for a log normal to be
positive. And, in fact, if you do the

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calculus on the log normal distribution,
you can show that the skewness of a log

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normal random variable actually has this
analytic expression here and, and this is

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positive. So the log normal distribution
exhibits a long right tail. Now, another

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shape characteristic that is of particular
interest for us in finance is a shape

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characteristic that tells us about the
likelihood of extreme values, right? So,

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if we're looking at you know, daily stock
returns on the s and p500, we know from

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time to time there's a market crash. And
so, when the market crash happens, we get

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a return that's in the far left tail of
the distribution. And we can also get an

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extreme positive event. There could be a
time for whatever reason the market goes

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up by a huge amount in a given day. So,
you know, that's an extreme value of the

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right tail of the distribution, okay? Now,
when we start looking at asset return

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data, particularly if we look at daily
data, the normal distribution fails in a

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big way, and one of the ways that the
normal distribution fails is

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characterizing the likelihood of extreme
events. The normal distribution doesn't

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imply that, I'm sorry, the normal
distribution implies that extreme events

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don't happen very often at all. But we see
extreme events happening actually quite

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often, and, and so, we would like to have
distributions that imply a higher

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likelihood of extreme events than the
normal distribution. And, so we would be

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looking at some of those kinds of
distributions in, in this course. Now, how

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do we measure, the likelihood of extreme
events for a distribution? So, one type of

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measure is known as Kustosis. And the
kustosis, is based upon a function of the

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random variable. And the function of the
random variable is, the random variable

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minus the mean divided by the standard
deviation raised to the power of four,

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okay? Now, and then the kurtosis is then
the expectation of this function. So, it's

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the probability weighted average of the
deviation of x from its mean, standardized

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by the standard deviation raised to the
power four. So, let's draw a picture,

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here. And, let's say we have a
distribution, you know, that looks like

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this. And here is the mean. Okay? So,
first thing to note about kustosis is

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that, we're raising the deviation of x
from the mean divided by the standard

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deviation to the fourth power, right? So,
this deviation is always a positive

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number, right? Now, if x is way out in the
right tail, x minus the mean is a big

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number. Right? And then when you
standardize by the standard deviation,

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it's still a big number. You raised it to
power of four, you're going to get

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something that's huge, right? And the same
thing is true on the other end. If x is

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way below the mean, then, x minus the mean
is a big negative number divided by the

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standard deviation, still a big negative
number. Raised to the power of four you'll

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get a big positive number. So, the
kustosis accentuates the extreme values.

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So, any distribution that has values that
are way below the mean or way above the

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mea n, is going to produce large value of
the kurtosis. Okay? So, extreme events,

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both positive and negative, lead to large
kurtosis. Now, we compute the kurtosis by

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taking the probability weight ed average
of this deviation here, weighted by the

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probabilities if we have a discrete random
variable, or we have to evaluate this

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integral if it's a continuous random
variable, right? So again, these

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calculations look a little bit horrible.
If we do this calculation for our example

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data. So for a discrete distribution, how
would we calculate kurtosis? We would say,

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the return minus the mean raised to the
power of four, multiplied by the

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probability, add, return, minus the mean
raised to the power of four, multiplied by

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probability, sum them all up and then
divide by the standard deviation raised to

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the power of four, we'd get the number
6.5. Okay? Now, when we calculate

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kurtosis, how would you know if the number
6.5 is big or small, right? Hm? Compared

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to normal, exactly. So, what's useful in
kurtosis is we need to have a benchmark

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for what is normal. You know? What is
yeah, we need to have some kind of

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benchmark. So, when we evaluate kurtosis,
the benchmark that's most often used is

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the kurtosis of a normal distribution. So,
let's look at the kurtosis of a normal

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distribution. So, if x is a general normal
distribution and you go and you do this,

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this integral, it turns out, this thing is
three, okay? So, that's kinda nice. So,

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the benchmark value of the kustosis that
is, the, the kurtosis value for normal

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distribution is three. So, if random
variable has a kurtosis that's bigger than

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three, then you know that the distribution
for that random variable has fatter tails

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than the normal distribution. That is, it
implies more extreme values than the

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normal distribution does, okay? Now, on
the opposite side, what happens if the

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kurtosis is less than three? Kurtisis less
than three, then you have thinner tails

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than the normal distribution. So, you can
think of two paces to cons ider. So if

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kurtosis say, this is my benchmark normal,
here. And then, distribution with fatter

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tails than the normal might look something
like this, okay? So here, the kurtosis for

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x would be greater than three for the red
distribution beca use we have more

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probability in the tails than the normal
distribution in that case. One of the

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greatest stories. Okay? So, and then the
opposite case to consider. So again, this

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is my benchmark normal and then in this
case, what if the distribution looks like

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00:13:09,013 --> 00:13:15,867
this? So in this case, the distribution
for x has thinner tails than the normal,

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so this red distribution will have a
kurtosis that's less than three So, now

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because the normal is the benchmark, then
very often when kurtosis is reported, it,

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what's being reported is actually
something called Excess Kurtosis. An

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00:13:33,552 --> 00:13:39,259
excess kurtosis is the kurtosis of the
random variable minus the kurtosis of a

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00:13:39,259 --> 00:13:44,714
normal, which is three. So, if excess
kurtosis is zero, then, you have the same

138
00:13:44,714 --> 00:13:50,144
kurtosis as a normal. If excess kurtosis
is positive, then you're in this case. And

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00:13:50,144 --> 00:13:55,734
if excess kurtosis is negative, then
you're in this case. So, we start looking

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00:13:55,734 --> 00:14:01,040
at data, we'll actually do some
probability calculations of extreme events

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00:14:01,040 --> 00:14:06,328
based upon a normal distribution and then
we'll actually look at the likelihood of

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00:14:06,328 --> 00:14:10,986
extreme events based upon the actual data.
And that, that's another way that we'll

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00:14:10,986 --> 00:14:16,267
see that the normal distribution is not a
very good distribution for certain kinds

144
00:14:16,267 --> 00:14:20,475
of returns because it, it doesn't imply
that extreme values happen very often. So,

145
00:14:20,475 --> 00:14:26,031
for example, you look at, like, the crash
in 1980, October, 1987. So, if you assume

146
00:14:26,031 --> 00:14:32,025
returns are normally distributed, the, the
crash is an event that would happen you

147
00:14:32,025 --> 00:14:38,027
know, once every six trillion years. And
so essentially, the likelihood of that

148
00:14:38,027 --> 00:14:44,037
happening is zero. And but you know, it
did happen. And whereas if we have other

149
00:14:44,037 --> 00:14:49,792
kinds of distributions that have thicker
tails, then we would actually have you

150
00:14:49,792 --> 00:14:55,282
know, a reasonable prob ability of, of
such an event like October 1987 happening.

151
00:14:55,282 --> 00:14:59,569
So, for the point of view of risk
modelling, and this is a thing that I want

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00:14:59,569 --> 00:15:04,566
to emphasize as we go through the course,
the distribution that you use for returns

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00:15:04,566 --> 00:15:10,698
is a, is, is incredibly important because
the distribution for returns tells you the

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00:15:10,698 --> 00:15:16,356
likeliho od of, of good or bad events.
And, and if you're, you know, trying to

155
00:15:16,356 --> 00:15:22,198
characterize the risks associated with
investments, then you know, your

156
00:15:22,198 --> 00:15:29,215
distribution is, is paramount to giving
you the information for how to make

157
00:15:29,215 --> 00:15:34,538
decisions. So, getting the right
distribution is, is an important thing to

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00:15:34,538 --> 00:15:41,169
do. And, and we'll see that that's not
always an easy thing to do in, in