Let us now talk about the Applications of the Normal
Distribution. There are two major items; one is Mean
Variance Analysis, and the main point here is that
when you evaluate a given stock or you are considering
the investment decision, you need to think about
both the mean, or the expected return as well as the
variance, which essentially talks about the risk
of that particular investment. So when we say Mean
Variance Analysis, it simply means that we look
at the mean return as well as the variance of the returns
before making a decision. The other item we will look at is the Safety First Ratio,
and that would be covered on the next slide. So the
main item here is the Safety First Rule named after
the person who first defined it called Roy so this
is often called Roy's Safety First Rule or Roy's
Safety First Criterion, which states that the optimal
portfolio minimizes the probability that the return
of the portfolio falls below some minimum acceptable
level. This minimum acceptable level is called the
"threshold level". So how do we select a optimal portfolio?
What we need to do is come up with the Safety First
Ratio using this formula. If you have a given portfolio
and the expected return on the portfolio is equal
to 10%, and let's say that you require a return of at
least 2%. Maybe the inflation is 2% and you won't be
doing more than 2% one way of putting this is that
you want to minimize the probability of the return
falling below 2% so if this is one portfolio and you
have another portfolio where the expected return
on portfolio 2 is 7% and your threshold level is
still 2%, but let's say that these two portfolios
have a different standard deviation. Here the standard
deviation is 5% and here the standard deviation
is only 1% so between these two portfolios, which
one is safer? And that is what the Safety First Rule
goes after. The safer portfolio is the one where the
probability of falling below threshold level is
minimal. So to find a portfolio we need to first come
up with the Safety First Ratio. Let us compute the
safety first ratio for this portfolio. That is simply
going to be the expected return on the portfolio which
here is 10% minus the threshold level, which is
2% divided by the standard deviation of the portfolio, which
is 5%. So here we will have 8 divided by 5, which
is equal to 1.6. So the safety first ratio for this
portfolio, let's call it portfolio A, is 1.6. The
safety first ratio for portfolio be is going to be
divide by the standard deviation which is one
so here the safety first level is 5. In order to
pick the safest portfolio we need to go with the
one that has the highest safety first ratio so between
A and B, B has the higher safety first ratio, and that is
safer. Just to get the intuition behind this, our threshold
level is 2% so clearly the portfolio with the high
expected return is better because it is giving us
a larger buffer relative to the threshold level.
However, simply looking at the expected return
is not enough, we also need to consider the standard
deviation. So even though portfolio A gives us
a large cushion relative to our threshold level, but
the variance of a portfolio A, is quite high compared
to the variance of portfolio B. Here, I'm actually
pointing at the standard deviations, but standard
deviation and variance are interchangeable. So
we need to consider both the cushion above the threshold
level and the variance. If the variance or standard
deviation is high then that pulls down the safety level
because a high standard deviation implies that
the probability of falling below a threshold level
becomes more so that's why we consider both this cushion
as well as the variation. We compute the safety first
ratio and pick the one with the higher ratio. The
one with the higher ratio has a lower probability of
falling below the threshold level. Let's see if
you can solve this problem now. For a given investor
the minimum acceptable return is 2%. What is the Safety
First Ratio for a portfolio with an expected return
on 7% and standard deviation of 10%? So let us just solve this
first. You have a situation where the expected return
is 7% so if we just regurgitate the formula first,
its return on portfolio minus the threshold level divided by the
standard deviation so for the first portfolio the
return is 7% minus the threshold level, which is 2%
divided by a standard deviation of 10% so this will give us 5/10
which is 0.5. For the second portfolio, we have a expected return of 5%, so here
we do 5 - 2 which is a threshold level divided by standard deviation
of 6, so we have 5 - 2 which is 3, divided by 6, which is
also 0.5 so it turns out that in this particular example the safety first is the same so we can actually pick either one.
In both cases, the probability of falling below the
threshold level is the same. See if you can solve
this practice problem now. The solution is given
on the next slide. Here is what you should have come
up with. The answer is C and this is the safety first
ratio which is higher compared to the other portfolios.
The Lognormal Distribution. The Lognormal Distribution
is generated by the function e^x where x is normally
distributed. If you did this normal distribution,
where x is the random variable and you transform
the normal distribution by taking e^x so x is a normal
distribution, what you end up with is a lognormal
distribution, and the way you can think of this is
if this is 0 and this is -∞ so -
a big number. When you do e to the power of a minus number,
especially if this minus number is big then you will essentially
have something that is close to zero. You can never
have e to the power of something that is a negative
number. So this log normal distribution is bounded
by 0 and just as an experiment once in your lifetime.
You can actually try to do this, take different numbers
of x starting from minus some big number like -100
and then go all the way to zero, and then take some larger
numbers like 10, 12, and so on, and you will get a distribution
that looks something like this so this is your log-
normal distribution. From a exam standpoint,
there are some basic points that you need to remember.
The first one is that the lognormal distribution
is bounded from below by 0. So it is useful for
modelling asset prices which never take negative
values. So if you are modelling an asset price, obviously
the lowest an asset can go to is 0 therefore,
it makes more sense to use a lognormal distribution
versus a normal distribution, which can take on
negative values. When you are modeling returns, returns
can be negative. So it is okay to use a normal distribution
to model returns. Clearly, the lognormal distribution
is skewed to the right. So you see a fatter tail over
here. And remember the phrase, 'the log is normal.'
So what that means is given that this is e ^ x, the log-
normal distribution. If you take the natural log
of e ^ x you simply get x, so that is basic log mathematics.
So the log of the lognormal is X which we already defined
as a normal distribution. So the log of the lognormal
distribution is normal. That's why we say that the
log is normal. And finally, we will talk about the
Monte Carlo simulation. What you need to do is simply read what is given on the slide.
I will give you a very basic explanation of how this
works, but make sure you read this first. Let us say
that you are trying to price a complex security and
you have a certain model for pricing that security
given different factors. So let's say the first
factor is interest rates, you can expect different
possible interest rates in the future, that's one factor.
Similarly, there might be other factors. Let's call
them factor 2, factor 3 and so on. Then there is
a certain model that you apply and you come up with
the price of the security based on the model. What
Monte Carlo simulation allows us to do is predict
or project different possible values for the interest
rate. So you tell your computer system, which is where
the simulation is going to happen, that for interest
rates you are expecting a certain value, of say 4%,
and there is a certain distribution. So you feed
this distribution into the computer. Similarly,
you feed the distributions for factors 2 and factors
3, so whatever you feel the appropriate distribution
might be, you into all this into the computer. What
the computer than does is takes a particular value
for the first factor, a particular value for the
second factor, and the third factor assuming that
we just had three factors over here. And obviously
we could have more and then based on these factors
and the model that is specified in the computer,
it comes up with a certain price. Then it runs a second
iteration where it will take another possible value
for the factor based on the distribution, another
value for factor 2, another value for factor 3,
and it will come up with another possible price, and
it will do this a million times and based on running
this, say a million times, there will be a certain distribution
for the price and essentially by running millions
or thousands of iterations, we will get a certain
distribution for the price of our complex security.
Now, as long as you understand the general concept you
are fine for the exam because clearly you will not
be required to do a Monte Carlo simulation on the
exam. In the real world though, this a technique that is used
very often because in the real world you will have
access to the technology that allows you to do such
simulations. From a exam standpoint, you need to understand
the limitations of the Monte Carlo simulation as
well as the limitations of a Historical simulation.
The Monte Carlo simulation is what we've just talked
about and you simply need to read what is on this slide
to understand the basic limitations. The sorts of
models that we just talked about. So just understand
what we mean by historical simulation and the limitations
are quite self-evident, you can read what is given on the
slide over here. So we are now at the end of this reading.
Just a brief summary. The most important points that
you need to understand, the basics related to the
distribution of discrete random variable and continuous
random variables. With a discrete random variable
you can either have a uniform distribution, and here
you can think of the example where we said, let's
roll a dice and there are 6 possible outcomes, each with the
probability of 1/6 so uniform because each outcome
has the same probability. With a binomial, remember
the fact that we have a combination of Bernoulli's.
The binomial distribution essentially deals with
the probability of success. So to quickly summarize
this point, let's say you flip a coin 10 times, the
probability of being successful seven times would
be an example. So 7 is your binomial random distribution.
The formula that you need to remember is nCx, where
n is the total number of times you perform the experiment,
x is the number of successes, and then p, which is the
probability of success to the power of x times (1 - p) ^ (n - x). This is the
probability of having 7 successes. Then you also
need to remember the expected value is n times the
probability and the variance is n times the probability
of success times (1 - p). Then we moved on to continuous random
variables. So here we are simply saying that between
a bound of a and b, a continuous random variable can
take on a value and the probability of being within
any small range is the same as long as this distance
is the same, and then we spent a fair amount of time
on the normal distribution. You need to memorize
the critical formula, which is z=x -
mu over the standard deviation. Plus you need to
also remember the area under the curve for one standard
deviation from the mean, two standard deviations
from the mean and so on, and you need to learn this transformation
and practice several problems. Just to conclude,
as I keep saying you need to go over the learning objectives
presented in the curriculum. Go over the examples,
but these examples in the curriculum can be fairly
complex. So if you'll find them difficult, don't
worry too much just read them once or twice. Also,
the practice problems in the curriculum, some of
them can be complex. Read the solutions, but for this
particular topic, I would say that you practice
problems from Schweser or Finn quiz or other sources
to get a better sense of what you will eventually
see on the exam. That is it for this reading.