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So, in the last class we were discussing the normal distribution and I had
mentioned that the normal distribution is used a lot in finance to characterize the
probability distribution of rates of return and once we start working with
data, so since a week after next, we will see why normal distribution is used a lot
because it matches, what we observe in the data, you know, pretty well.
But, I went through an example to, highlight some problems with, you know,
taking the normal distribution, on faith as a good distribution to characterize a
rate of return. And so I wanna go through this example
again. And to, you know, reiterate the points
that, that I made last time. So we're gonna discuss why the normal
distribution may not be appropriate for simple returns.
So again, we're looking at a simple return, which is just the percentage
change in price over some period of time. So let's consider, say we have an annual
investment. So we're gonna buy Microsoft today, and
we're gonna sell it one year from now, and RT is gonna represent the annual rate of
return on that investment. So let's just assume that.
The annual rate of return follows a normal distribution.
And the mean of this distribution's five%. And the standard deviation is 50%, okay?
And so if we think of, any time you have a problem where you write down a
distribution, it's worthwhile to, to draw it.
So I'm, I'm telling you I have a normal distribution.
You have a normal distribution it has a bell shaped curve.
And here. My rate of return is normal. With mean
five%. And, standard deviation, 50%.
Okay. So we know the center of this distribution
is at point zero five. And the standard deviation, you know,
gives us a spread about the average. And we go out + or -one standard
deviation, we get 67 percent of the area. We go out + and -two standard deviations,
we get 95 percent of the area. You go out + and -three standard
deviations, you get 99 percent of the area.
Okay? So, If we think of this rate of return,
so. The mean.
What's the standard deviation? So if this is Mu of R, we can go Mu R plus
Sigma R, and then we can go down, Mu R minus Sigma R.
And this would be 0.05 plus 0.50 and this would be 0.05 minus 0.50.
So if you go mean plus the standard deviation your out at 55%.
You go to mean minus the standard deviation your down to minus 45%.
So understand in this, this, in this example.
You know, going out one standard deviation below the mean, you're losing 45 percent
of your money. So that's almost half your money you've
lost. Now, if you go out two standard
deviations, you're down five percent minus 100%, and so you're down 95 percent of
your money. If you go out three standard deviations,
you're down more than 100%, right? So you've lost more than your initial
investment. Well that can't happen right?
So. The problem is, is that the distribution
is defined for all real values. It's defined for returns between minus
infinity and infinity and we know that a rate of return the smallest value it can
be is 100 percent. So.
So we know the return has to be bigger than minus 100 percent and based on this
distribution what is the likelihood that we can actually lose more than our initial
investment? And so what's the probability that the
return is less than minus one? Okay.
So you're in the far left tail of the distribution.
And if you use the norm dysfunction in Excel, for example.
We find that there's a 1.8 percent chance that, with this normal distribution, you
can lose more than 100 percent of your money.
Okay? And so that doesn't make any sense at all.
And another way of viewing this is that this distribution implies there is a 1.8
percent chance that the price in the future is less than zero.
Okay. So again, that doesn't make any sense
either. So this is.
An example to illustrate. Why a particular.
A normal distribution may not be appropriate for simple rates of return.
Because it could. It, cause it.
Could imply. Negative prices.
Now. What can we do about this?
Well, it turns out that, if we consider continuously compounded rates of return.
And again, the continuously compounded rate of return is the logorithm of 1plus
the simple rate of return, right? And given the continuously compounded
return, we can solve back for the simple return by taking E to the continuously
compounded return minus one. Okay.
Now, instead of assuming a normal distribution for the simple rate of
return, we can assume a normal didtribution for the continuously
compounded rate of return. Okay?
So let's take this same rate of distribution, the continuously compounded
return has mean five percent and standard deviation 50%.
Now, the thing that's a bit counter-intuitive about continuously
compounded returns is that it's perfectly okay for the continuously compounded
return to be smaller than minus one. For example, suppose the continuously
compounded return is -two. So we're saying the continuously
compounded return is -200%. Does that make sense?
Well, what is the simple return that's implied by this, continuously compounded
return. It's either the -two, -one.
Well that's minus 86.5%. So if the continuously compounded return
is minus 200%. You've actually only lost 86 percent of
your money. So and in fact, the continuously
compounded return is defined for any value between minus infinity and infinity,
right? Suppose the continuously compounded return
is minus infinity, right? What is the implied simple return in that
case? Minus one, right?
Cuz you're gonna get e to the minus infinity, which is zero, minus one, so
it's minus one. So if the continuously compounded return
is minus infinity, you've lost all of your money, okay?
So, the continuously compound return is perfectly sensible from minus infinity to
infinity, and so, the normal distribution is perfectly okay to describe the
distribution of the continuously compounded return.
And so for example, you know, the probability that the continuously
compounded return is less than minus two is the same that the probability that the
simple return is less than minus 86.5 percent and, you know, it's a very, very
small chance that, that will happen. So this is an example why.
We like continuously compounded returns because.
We can, use a normal distribution. We don't run into any of these problems.
That, you know. We have returns can be, you know, less
than minus one or prices can be negative or things like that.
So yeah that's. I wanna emphasize that.
This is another reason why throughout this course we will often do probability
modeling with a continuously compounded return instead of the simple return.
Now. The.
Previous two examples, give rise to, the following log normal distribution.
So, for example. If we start with a random variable that's
normally distributed. For example, if a continuously compounded
return is normally distributed. And we define a new random variable that's
the exponential of the first one. So remember, the, so one+ the simple rate
of return is E to the continuously compounded return.
It turns out that Y, which is the exponential of X, has what's called a log
normal distribution, okay? In the log normal distribution, as
parameters mew x and sigma x squared and these two parameters are exactly the same
as the parameters of the underlying normal distribution.
Okay, Now, the mean of the continuous of the, of Y, which is the exponential of X
is not mew of X. In fact it takes a little work but you can
show that the mean of Y is equal to the exponential of the mean of X, right
because Y is the exponential of X you might think that the mean of Y would be
the exponential mean of X. But it's the exponential of the mean of X
plus the variance of X divided by two. Okay.
So there's a, a slight adjustment. And, and this adjustment comes from the
change of variable formula in calculus and, and, and so on.
So we can, we can deduce the mean of Y as the exponential of the mean of X plus
sigma squared divided by two. And then, more tedious calculations can be
done to show that the variance of Y, or single Y squared is the exponential of two
times the mean of X plus the variance of X squared, times the exponential of the
variance of X minus one. So it's a pretty complicated formula.
So, lets look a little bit at the Relationship between the normal
distribution and the log normal distribution in the context of returns.
So, its, we will start with the assumption that the continuously compound return
follows a normal distribution with mean five percent and standard deviation 50%.
Now, we know that, the, the. One plus the simple return is equal to E
to the RT, right?. So we can think that's, in this
relationship here, X is R, and Y is one+R. So one+ the simple rate of return, which
is E to the contingency compounded return, will follow a log normal distribution.
And the parameters of this log normal distribution are the same as these things
up there. But the mean of one+ the simple rate of
return is the exponential of the mean of. The continuously compounded return plus
one-half the variance of the continuously compounded return.
This turns out to be 1.191. [inaudible].
So this is the mean of the distribution of, of one plus R.
So the mean of the distribution of R would be 1.91 minus one.
So this is saying that the simple rate of return would have a mean of nineteen%.
Where as the mean of the continuously compounded return has a mean of five%.
So notice that the log normal distribution kind of pushes out the, the mean a bit,
for the simple return. And then we can deduce the variance, of
one plus the simple rate of return, by, you know, evaluating this horrible
formula. We get something that's, 0.563.
And, and so again, that's different from the variance formula that we have.
The variance for the continuously compounded return is.5 squared.
So here I am graphing the two distributions.
So this is the distribution for. The continuously compounded return.
And so we can see that. You know, its mean is around.
You know five percent here and it has a big standard deviation.
And the blue line is the, the distribution for the yeah, this would be for one plus
the rate of return. And so, notice that this thing starts at
zero, because when R is minus one, one plus R is zero.
So, the gross return starts at zero, and, we've seen that it, it has a shape that
is, a little bit asymmetric, so the log normal distribution, because when you take
E to the. X, you always have something that's
positive that produces a bit of skewness or asymmetry, in the distribution relative
to the normal distribution. So we see this log normal distribution has
a bit of a long I guess, right tail and, and so this is referred to as positive
skewness, and, and, and we'll see that later on.
So again we this is an example where you start with a normal distribution for a
variable, you make a transformation of that.
And then we can deduce a distribution for a transformation of, of that variable.
And, and, and then we can, you know, work with that.
Now if you're working in R. So one of the nice things about R is that
it has functions for essentially any distribution you can possibly think of.
So if you want to, do computations with a log normal distribution.
There are four functions. Rl norm.
So L norm stands for log normal. And this allows you to simulate data from
the log normal. The you can compute the cumulative
distribution function with PL norm. You compute, compute quantiles with QL
norm. And you can com, compute the height of the
curve. So when I plotted the graph I used the DL
norm function. And, and so again, so an, anything you
want to do with the, the log normal there's an R function to do this.
Now in Excel there is no, there, there are some functions.
You can compute the CDF for the log normal.
But you can't simulate random variables. And you can't compute quantiles, and you
can't compute the density. So Excel only has, some functions for
dealing with the log normal, but not, a full, Array.
And again this is a reason why. You know we're using R for the statistical
stuff. Is that it has a lot more things built in.
Whereas Excel does not.