Symmetry and Skewness in Return Distributions.
Skewness or skew refers to the extent to which a distribution
is not symmetric. Now, when you have a normal distribution,
it has a skewness of 0 or we can say that a normal distribution
is symmetric. What that means is we have equal distribution
on each side of the mean. This could be your mean.
This is also the median, this is also the mode, and the
distribution on each side is the same. This is symmetric
so the skewness is equal to zero. A positively-skewed
distribution is characterized by many outliers
in the upper region or the right tail, this is called,
a right tail. Notice that we have several outliers
over here. This is called a right skew. When you have
a right skew or a positive skew these outliers on the
right pull the to the right so the mean is going to be greater
than the median which has over here, which is going
to be greater than the mode. Mode is the most frequently
occurring item. So that is over here. This is the
mode and we essentially have this relationship
where the mean is greater than the median which is
greater than the mode for a positively-skewed distribution.
For a negatively skewed distribution, we have
a large number of outliers on the left. This is the
left tail What this does is pull the mean to the left,
therefore in the situation, we have a mode which
is greater than the median which is greater than
the mean. Let us look at the practice problem. Which
of the following is most likely characterized by
by frequent small losses and a few extreme gains.
We have just seen this, the correct answer is C, positively-skewed,
where we have a few large losses and many
extreme gains which represents the outliers on
the right. Which of the following is most likely
to be true for a negatively skewed distribution?
And we just saw this, in a negative skew you have a relationship
like this where the mean is pulled out to the left,
the mode is over here, and the median is in the middle.
So you'll have a situation where the mean is less
than the median which is less than the mode. Now try
this one. The correct answer is A. In the previous
section, we've talked about skewness, which is
a measure of symmetry. What we are looking at here
is a normal distribution and here this is perfectly
symmetrical therefore, we can say that the skewness
is zero. Now another very important measure is kurtosis.
Kurtosis is a measure of the combined weight of tails
relative to the rest of the distribution. We can
have three types of kurtosis; a curve can be Mesokurtic,
Leptokurtic, or Platykurtic. The distribution we are
looking at here is a normal distribution and a normal
distribution has a certain amount of weightage in the tails.
If this way weightage in the tails is the same as a normal
distribution then we say that the curve is Mesokurtic.
So let's say we have this distribution here which
is shown in red. Notice that this distribution has
fatter tails, so there is more in the tails over here
compared to the normal distribution. Such a distribution
will we have more weightage in the tails is called leptokurtic.
Also notice that a leptokurtic distribution is more
peaked than normal. So that means that the probability
of being a little bit on either side of the mean is
more for a leptokurtic distribution relative to
a normal distribution. The 3rd category is Platykurtic. Platykurtic
distribution have thinner tails relative to a normal
distribution. So the key points to remember; Mesokurtic
means that the tails have the same amount
of weatage as a normal distribution. Leptokurtic means
that the tails are fatter than normal, and Platykurtic
means that the tails are thinner than normal.
You also need to understand this term Excess Kurtosis.
So as I said, this is a measure so we need numbers. A
mesokurtic distribution has a kurtosis equal to 3. So we
say that a normal distribution has a kurtosis equal
to 3. A Leptokurtic distribution will have a kurtosis
which is more than 3. So the fatter the tails,
the higher the kurtosis, and this leads to a measure
called Excess kurtosis. Excess kurtosis is equal
to kurtosis -3 if we say that a leptokurtic distribution
has a kurtosis equal to 4, then the excess kurtosis
will be 4 - 3 which is equal to 1, therefore on the exam
you need to be careful as to whether you are given
the kurtosis or Excess kurtosis. If you are given
excess kurtosis is equal to 1, you need to recognize
that the kurtosis is more than normal. Also if they
excess kurtosis is 1 or more than that implies that
the kurtosis is unusually high which means that the
weightage in the tails is unusually high. Now, there
is one final point I want to emphasize. For these
two distributions that we are looking at here the
mean is the same, the standard deviation is the same,
the skewness is the same. The only difference is
the kurtosis. So, if I were to ask you between these
two distributions, which is more risky, the answer
would be the one with excess kurtosis is more risky,
because the probability of being here in the negative
region is more for the distribution that has excess
kurtosis, in other words, the leptokurtic distribution.
Here are the sorts of questions you might see on
the exam. Which of the following is most likely to
be a measure of when a distribution is more or less peaked
compared to a normal distribution. So all you really
need to do here is remember the term. The correct
answer as you have just seen is C. A distribution identical
to a normal distribution is...and here again, you just
have to remember the fact that a mesokurtic distribution
has a kurtosis of 3 and this is your normal distribution.
The final section in this very long reading now. Using
Geometric and Arithmetic Means. A geometric mean
is appropriate for making investment statements
about past performance. Now, let us look at the straightforward
example. Say you had $100, two years ago, and you had
a return of 100% in year 1, - 50% in year 2. What was your
mean return? If you simply take the arithmetic mean,
then you will have a 100 and then -50 so that is
50 divided by 2 which is equal to 25%. So the arithmatic mean is 25% but look
at what's actually happening when you have $100,
in the first year you get 100% return. So
you end up with $200 at the end of the first year, but
then in the second year you lose 50% which means that
you are back down to 100. So in reality you have not made
any money. You started with 100 you end up with
100. So your mean return really is 0%. Arithmetic mean
however is telling you that the return is
25%. So clearly the arithmetic mean is not the right
measure when you are looking at returns over a period of
time. A better measure is the geometric mean. So let's
look at how we would calculate the geometric mean.
If you recall with the geometric mean you do one plus the
return in the first year, a 100 percent return
means you just do one plus one. Remember a 50% return
would admit that we do 1+1. Remember, a 50% return would have meant that we do 1+0.5 so a 100% return
means that we (1+1) and then you multiply by the return
for the 2nd year 1 + the return for the 2nd,
year which is actually - 0.5, and then you raised to the
power of 1/2 we have 2 multiplied by 0.5 raised to the power of 1/2.
So what you have is 2 * 0.5 is 1. 1^1/2 is still 1 and
then you minus the 1 to get a rate and you end up with
0. So your geometric mean is 0, which means
that on average you are getting a 0% return and that is
a much more accurate depiction of what is happening.
Arithmetic mean is appropriate for making investment statements in a forward-looking context.
If you have $100 now and the two possible returns
for the upcoming year are 100% gain or a -50% loss.
What is the expected value? /here it makes sense to
look at the arithmetic mean of 100 and
-50 and to say that our expected return is 100 -50
over 2 which is equal to 25%. So saying that our expected
return is 25% is appropriate. However, in the first
scenario where we have time series data, there the appropriate
measure is the geometric mean. So that is it, we
are done with this reading. Just a few major summary
points: we talked about measures of Central Tendency
and here you need to understand the various kinds of means.
We talked about arithmetic mean, we talked about
geometric mean, harmonic mean, the weighted-average,
and so on. You need to understand all these different
kinds of means and how to calculate them. Then we
talked about Measures of Dispersion. The most important
measure is standard deviation, this tells us how
spread out our data is. You need to know how to use the
formula, but it is more important that you understand
how to use the calculator to determine standard
deviation. Recognize also that standard deviation
is different for a population versus a sample. You
also need to understand Chebyshev's inequality,
which is 1 - (1/k squared), that k stands for the number
of standard deviations and the strength of Chebyshev's
inequality is that it allows us to see how
much data will be within a certain range assuming
without making any assumptions about the nature
of the underlying data. And finally you need to understand
the basics of skewness and kurtosis. In terms of
the work that you now need to do go, over the learning
objectives outlined in the curriculum, go over the
examples and practice problems, especially the
shorter problems or the multiple-choice oriented
problems in the curriculum. The other questions
might look a little bit difficult and that's okay,
at least read the solutions and try to figure out
what the curriculum is trying to say. I would say,
it is more important that you do practice questions
from other sources like Schwesers and try to do as many
questions as possible to make sure that you are on
top of this material. That is it.