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And I want to introduce this location
scale model.

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And, this location scale model is, is very
useful, and we're going to actually, going

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use it, quite a bit.
And, this location scale model is

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motivated by the fact, by the following
result, that we deduced last time.

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So, we looked at a normal random variable.
Say, x is normal, with mean mu x and

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variance sigma x squared.
And then, we did a standardization

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process, where we, where we defined a
standardized random variable z, which was

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x minus its mean divided by its standard
deviation.

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And the standardized random variable is
normally distributed with mean zero, and

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variance one.
That is the expectation z is zero, and the

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variance of z is one.
Now, the location scale model is

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essentially the reverse of the
standardization process.

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That is, let's start with a standardized
random variable that has mean zero and

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variance one, and let's construct a
non-standardized variable by reversing

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this transformation, okay?
So then, we can think of x as being equal

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to a location mean plus a standard
deviation, which is a scale variable

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multiplied by the standardized random
variable z, okay?

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So, we can always think of a non
standardized random variable as being, you

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know, in, if it's, say, normal with mean
mu and variance sigma squared, as being

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equal to a location, a scale times a
standardized random variable, okay?

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And we'll see that this is we're actually
going to build a lot of models off this

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very simple way of looking at, at data.
Now, one of the things that's useful about

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this location scale model is that we can
deduce what the quantiles of a general

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normal distribution looks like in terms of
the mean and the standard deviation and

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the quantiles of the standardized random
variable.

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So, let's look at this.
So, quantiles of normal distribution,

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okay?
So we're going to start with z is a normal

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0,1 and by definition the alpha quantile
of z is this number z of alpha.

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And z of alpha is the z-score such that
the probability to the left of the z-score

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is equal to alpha.
Alright?

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So, again if you think of the picture,
standard normal distribution, this is z,

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here is my z alpha.
This area is equal to alpha.

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So, z alpha is equal to the alpha quantile
of normal 0,1, okay?

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So, what we want to do now, is we want to
say, well what's the quantile of a general

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normal distribution?
Well, we can use the location scale model,

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we know that x is equal to mu x plus sigma
x times z, so, we can look at the

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probability of mu x plus sigma x times z
is less than or equal to mu x plus sigma x

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times z alpha is equal to alpha.
So, we just take this probability

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statement with respect to z and then we
turn it into this probability statement

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just by adding the mean and multiply by
the standard deviation doing it on both

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sides of the probability piece.
Now, this is a general normal random

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variable.
This is the same as the probability that x

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is less or equal to the quantile of the
distribution for x, where the quantile for

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the distribution for x is the mean plus
the standard deviation times the quantile

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for the standard normal.
So, that this piece here, is the quantile

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for the standard normal.
So, I used this formula before by saying

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the quantile to standard normal, sorry,
the quantile to general normal, you can

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compute as the mean plus the standard
deviation times the quantile to standard

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normal.
So, this, you know, justifies that, that

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result.
And it comes from the use of this simple

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location of the scale model.
So now, we can review value at risk.

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Okay, so, value of risk says with some
probability, how much can you lose on an

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investment over a given period of time?
So, when we calculate value at risk you

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might say their simple return.
And, and we, say, make some assumption

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that the simple return has a normal
distribution.

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And, so, based on this normal distribution
for R, we can look at the quantile, say,

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at 0.05.
And this quantile is equal to the mean

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plus the standard deviation times the five
percent quantile of a standard normal.

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And z.05, is equal to minus 1.645, that's
the quantile of a standard normal.

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So, you can get this from the PNORM
function, sorry, the QNORM function in R,

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or the norm inverse, norm s inverse
function in Excel.

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So, once we have this five percent
quantile, so, this says, with five percent

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probability, the return could be this
value or smaller, right?

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So, the five percent quantile is telling
you the probability of loss in the left

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tail of the distribution.
So here, the example, we want to compute

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the five percent value at risk.
So, this return quantile is a rate of

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return.
But valiant risk is typically reported as

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a dollar loss.
So, in order to get the dollar loss, we

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take the return quantile and multiply it
by our initial investment.

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So, if you have an initial investment of,
say, a $100,000, then our dollar value at

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risk is the return quantile multiplied by
$100,000.

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So, if W NOT is the initial dollar
investment.

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Then, the value at risk at five percent
probability is equal to the five percent

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quantile of the return distribution times
W NOT.

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Now, this, in general, is a negative
number, right?

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Because the return, the left tail return
quantile is negative.

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But, by convention, value at risk is
recorded as a positive number.

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So, it's, when you say it's a loss, I lose
a $1,000.

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So, typically, when you report value at
risk, you take this, this value, and you

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take the absolute value of it.
And so, the absolute value is to, to give

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you the a positive dollar loss.
So, this is a dollar loss value, alright?

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So, if we think of distribution for our
losses, then, so, if this Q times, or

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sorry, if it's the return times W, so our
value at risk here is, is just the five

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percent quantile of the distribution for
the return times our initial investment,

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okay?
So, that's that again, that's value at

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risk.
And then, we have the case if so, the

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previous example was if we define the
return distribution was normal, and we had

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a simple return.
But we know that the normal distribution

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for simple returns could be problematic.
So, we often use a normal distribution for

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the continuously compounded return.
So then, we can if the continuously

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compounded return is normal, so RT, say R
is the continuously compounded return, and

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that's equal to the logarithm of one plus
the simple return.

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And that means, the simple return is E to
the continuously compounded return minus

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one.
And so here, the, this return is normal.

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Then, one + r is log normal, okay?
So, the distribution for the continuously

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compounded return is symmetric, but the
distribution for one plus the simple

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return is slightly asymmetric.
This has a long right tail.

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So, we can now say, compute the value at
risk based on this assumption.

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So, again, the steps are we compute the,
say, the five percent quantile for the

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continuously compounded return.
Sorry, this would be and then we convert

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this to the quantile for the simple
return, so that's E to the Q, little r

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minus one.
And so, that would be E^Qr plus sigma r

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times z.05 minus one.
So here, we have the quantile for

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continuously compounded return.
We turn that into a quantile for the

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simple return and then we compute the
value at risk based on the simple return.

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So, the next step is our value risk is
then our five percent quantile for the

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simple return multiplied by our initial
investment and we take the absolute value.

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Now, this number is going to be different
than the number we would get if we assume

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the simple return was normally distributed
because, in this case, the simple return

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has essentially one plus a log normal
distribution, and that, so, we have a, a

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different distribution and we'll get a
slightly different number.

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If this is normal, then one + r is e to,
to a normal and so this is a log normal.

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So, one plus the rate of return is a log
normal.

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Yeah, okay.
So, this is what we covered last time.

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And so, we'll, we'll be doing lots of r
calculations with various models as we go

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through, and again, I just wanted to
reiterate the, the computation of r, and

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particularly using the quantile that's
computed from this location scale type