And I want to introduce this location
scale model.
And, this location scale model is, is very
useful, and we're going to actually, going
use it, quite a bit.
And, this location scale model is
motivated by the fact, by the following
result, that we deduced last time.
So, we looked at a normal random variable.
Say, x is normal, with mean mu x and
variance sigma x squared.
And then, we did a standardization
process, where we, where we defined a
standardized random variable z, which was
x minus its mean divided by its standard
deviation.
And the standardized random variable is
normally distributed with mean zero, and
variance one.
That is the expectation z is zero, and the
variance of z is one.
Now, the location scale model is
essentially the reverse of the
standardization process.
That is, let's start with a standardized
random variable that has mean zero and
variance one, and let's construct a
non-standardized variable by reversing
this transformation, okay?
So then, we can think of x as being equal
to a location mean plus a standard
deviation, which is a scale variable
multiplied by the standardized random
variable z, okay?
So, we can always think of a non
standardized random variable as being, you
know, in, if it's, say, normal with mean
mu and variance sigma squared, as being
equal to a location, a scale times a
standardized random variable, okay?
And we'll see that this is we're actually
going to build a lot of models off this
very simple way of looking at, at data.
Now, one of the things that's useful about
this location scale model is that we can
deduce what the quantiles of a general
normal distribution looks like in terms of
the mean and the standard deviation and
the quantiles of the standardized random
variable.
So, let's look at this.
So, quantiles of normal distribution,
okay?
So we're going to start with z is a normal
0,1 and by definition the alpha quantile
of z is this number z of alpha.
And z of alpha is the z-score such that
the probability to the left of the z-score
is equal to alpha.
Alright?
So, again if you think of the picture,
standard normal distribution, this is z,
here is my z alpha.
This area is equal to alpha.
So, z alpha is equal to the alpha quantile
of normal 0,1, okay?
So, what we want to do now, is we want to
say, well what's the quantile of a general
normal distribution?
Well, we can use the location scale model,
we know that x is equal to mu x plus sigma
x times z, so, we can look at the
probability of mu x plus sigma x times z
is less than or equal to mu x plus sigma x
times z alpha is equal to alpha.
So, we just take this probability
statement with respect to z and then we
turn it into this probability statement
just by adding the mean and multiply by
the standard deviation doing it on both
sides of the probability piece.
Now, this is a general normal random
variable.
This is the same as the probability that x
is less or equal to the quantile of the
distribution for x, where the quantile for
the distribution for x is the mean plus
the standard deviation times the quantile
for the standard normal.
So, that this piece here, is the quantile
for the standard normal.
So, I used this formula before by saying
the quantile to standard normal, sorry,
the quantile to general normal, you can
compute as the mean plus the standard
deviation times the quantile to standard
normal.
So, this, you know, justifies that, that
result.
And it comes from the use of this simple
location of the scale model.
So now, we can review value at risk.
Okay, so, value of risk says with some
probability, how much can you lose on an
investment over a given period of time?
So, when we calculate value at risk you
might say their simple return.
And, and we, say, make some assumption
that the simple return has a normal
distribution.
And, so, based on this normal distribution
for R, we can look at the quantile, say,
at 0.05.
And this quantile is equal to the mean
plus the standard deviation times the five
percent quantile of a standard normal.
And z.05, is equal to minus 1.645, that's
the quantile of a standard normal.
So, you can get this from the PNORM
function, sorry, the QNORM function in R,
or the norm inverse, norm s inverse
function in Excel.
So, once we have this five percent
quantile, so, this says, with five percent
probability, the return could be this
value or smaller, right?
So, the five percent quantile is telling
you the probability of loss in the left
tail of the distribution.
So here, the example, we want to compute
the five percent value at risk.
So, this return quantile is a rate of
return.
But valiant risk is typically reported as
a dollar loss.
So, in order to get the dollar loss, we
take the return quantile and multiply it
by our initial investment.
So, if you have an initial investment of,
say, a $100,000, then our dollar value at
risk is the return quantile multiplied by
$100,000.
So, if W NOT is the initial dollar
investment.
Then, the value at risk at five percent
probability is equal to the five percent
quantile of the return distribution times
W NOT.
Now, this, in general, is a negative
number, right?
Because the return, the left tail return
quantile is negative.
But, by convention, value at risk is
recorded as a positive number.
So, it's, when you say it's a loss, I lose
a $1,000.
So, typically, when you report value at
risk, you take this, this value, and you
take the absolute value of it.
And so, the absolute value is to, to give
you the a positive dollar loss.
So, this is a dollar loss value, alright?
So, if we think of distribution for our
losses, then, so, if this Q times, or
sorry, if it's the return times W, so our
value at risk here is, is just the five
percent quantile of the distribution for
the return times our initial investment,
okay?
So, that's that again, that's value at
risk.
And then, we have the case if so, the
previous example was if we define the
return distribution was normal, and we had
a simple return.
But we know that the normal distribution
for simple returns could be problematic.
So, we often use a normal distribution for
the continuously compounded return.
So then, we can if the continuously
compounded return is normal, so RT, say R
is the continuously compounded return, and
that's equal to the logarithm of one plus
the simple return.
And that means, the simple return is E to
the continuously compounded return minus
one.
And so here, the, this return is normal.
Then, one + r is log normal, okay?
So, the distribution for the continuously
compounded return is symmetric, but the
distribution for one plus the simple
return is slightly asymmetric.
This has a long right tail.
So, we can now say, compute the value at
risk based on this assumption.
So, again, the steps are we compute the,
say, the five percent quantile for the
continuously compounded return.
Sorry, this would be and then we convert
this to the quantile for the simple
return, so that's E to the Q, little r
minus one.
And so, that would be E^Qr plus sigma r
times z.05 minus one.
So here, we have the quantile for
continuously compounded return.
We turn that into a quantile for the
simple return and then we compute the
value at risk based on the simple return.
So, the next step is our value risk is
then our five percent quantile for the
simple return multiplied by our initial
investment and we take the absolute value.
Now, this number is going to be different
than the number we would get if we assume
the simple return was normally distributed
because, in this case, the simple return
has essentially one plus a log normal
distribution, and that, so, we have a, a
different distribution and we'll get a
slightly different number.
If this is normal, then one + r is e to,
to a normal and so this is a log normal.
So, one plus the rate of return is a log
normal.
Yeah, okay.
So, this is what we covered last time.
And so, we'll, we'll be doing lots of r
calculations with various models as we go
through, and again, I just wanted to
reiterate the, the computation of r, and
particularly using the quantile that's
computed from this location scale type