So at this point now I wanna walk through
a, a, a fairly lengthy example to
illustrate this very important risk
measure that's known as value at risk.
And, valiant risk is, is, It's a risk
measure that is answering the question.
How much money could you lose with a
certain probability?' Okay?
And so, that's a very natural form of
risk, right?
You have an investment, you have your
money at risk, you wanna know how much
money could you lose with a certain
probability, right?
And, and so value at risk gives you
exactly that number.
So, a value at risk number might be
something like $1,000, and so you might
say with five percent probability, if I
start with a $10,000 investment, I could
lose $1,000.
Okay?
And, and that's a nice way to talk about
risk because it.
You know, it, it hits you right in the
pocket book.
It says you know, in, in dollar terms.
You know, what, what's at stake.
And this is in contrast to.
From a statistical point of view, we often
characterize risk as the standard
deviation over the rate of return.
And a lot of people just don't have
intuition about what standard deviation
is.
And so even though it turns out that
standard deviation and value at risk are
very closely related to one another, value
at risk being a dollar loss number is
something that most people, understand
much more clearly.
Alright.
So where does value of risk come from, and
how do we compute it?
So we're gonna walk through a simple
example.
We start with a $10,000 investment on
Microsoft and, or Investment Horizons in
one month.
So, we buy Microsoft today.
We put $10,000 that might buy, you know
100 shares of Microsoft stock.
We hold it for a month and we sell it.
And we wanna know, how much can we lose
with say, five percent probability, right.
So that would be known as the five percent
value of risk or the five percent bar.
Now we're gonna get to that in a sa,
slightly systematic way.
Now the random variable of interest is the
monthly rate of return, that's the
percentage change in price over the month,
and we're gonna assume that the monthly
rate of return has this following normal
distribution, okay?
So the, the mean of the monthly rate of
return is gonna be five percent and the
standard deviation is gonna be ten%, all
right?
So mu R is.05, sigma R is.10.
Okay.
And the goal is to calculate how much we
can lose with the specified probability
alpha.
And alpha is going to be some number.
It's usually five percent or one%.
Okay?
Now, we're gonna approach this issue in a
couple of intermediate steps.
And, and partially this is just to
illustrate you know, working with linear
functions and random variables and doing
probability calculations and so on.
And then another goal of this is to build
more of the intuition about where the
value at risk number that we come up with
comes from.
So, some questions that we're gonna answer
in this example are the following: so,
what's the probability distribution of end
of month wealth?
So end of month wealth is W1, that's our
initial wealth.
Times one plus the rate of return, so
that's the future value of our
investment.'Kay?
And so notice that end of month wealth is
a linear function of R, and R is a normal
random variable, so we know that end of
month wealth is also gonna be a normal
random variable, and we, we just wanna
calculate its mean and its variance.'Kay?
Once we have the distribution for end of
period wealth, then we can answer
questions like, what's the likelihood that
our end of period wealth is less than a
certain amount.
Right.
So this is if we start with $10,000 then
this question is saying what's the
likelihood that we could lose $1000 or
more.
Right.
And then we can ask, another question.
What is the rate of return that produces
our end of period wealth equal to $9,000?
Right?
In percentage terms, you know, what, in
saying, what is the, So, and what rate of
return produces this end of period wealth?
And then finally, you wanna ask, what is
the monthly value at risk on a $10,000
investment?
That is, how much could we lose, if the
rate of return is, less than its five
percent quantile?
Okay, so let's walk through this here, So
our wealth, end of period wealth is our
initial investment times one plus the rate
of return.
So that's just the future value of our
initial investment.
It's a linear function of R.
R is normally distributed, so W1 is
normally distributed.
What's the mean of W1?
Well, it's the expectation of W1.
It's the expectation of 10,000 times one
plus R, and using the rules of the
linearity of expectation, that's equal to
10,000 times one plus the expected return.
The expected return is five%.
So, our expected future wealth.
Is equal to, $10,500.
So, right.
If we.
On average if we expect to get a five%.
Rate of return.
Then our.
Future wealth is expected to be $500
higher than our initial wealth.
Okay?
What about the variance of our future
wealth?
Well the variance of our future wealth is
equal to our initial wealth square.
Times the variance of the rate of return.
And so that's equal to 10,000 square
times, .1 squared.
So that's, that's a $1,000,000.
Now notice that the variance of
inter-period wealth is in units of wealth
squared, so this is dollars squared, so
this doesn't have a good interpretation.
We take the standard deviation, we take
the square root of this, it turns out the
standard deviation of our inter-period
wealth is $1,000.
And that's in the same units as, as our
wealth.
So, wealth distribution is normal and has
mean 10500 and has a standard deviation of
1000.
Okay, again this just using a linear
properties of random variables and the
packet stuff is normal.
Alright, so now that we know that this
distribution of'NF' period wealth, we can
answer questions like what's the
probability of'NF' period wealth is less
than 9000 dollars.
Well we just evaluate, this that you want
to know what is This is [inaudible] what's
the area under the normal curve to the
left of $9,000 where the normal curve has
this mean and this standard deviation,
okay?
So, we would, draw this.
Here's our normal distribution and our
mean, 10,500, and we wanna know, here is
$9,000.
What is this probability?
This is the probability that end of period
wealth is less than $9,000.
'Kay?
And so we can use say, the Excel NORMDIST
function so we put in $9000, we put in the
mean, the standard deviation, turns out
that this is 6.7%.
So this area right here is 0.067.
So, there's a 6.7 percent chance that our
end of period wealth is less than $9000.
What is the rate of return that produces a
$9,000 loss?
Okay?
The rate of return that produces a $9,000
loss.
Well, if end of period wealth is $9,000,
this is our initial wealth is $10,000
divided by $10,000.
So if we lose ten percent of our
investment, right?
We start with $1000, we lose ten%.
That means we've lost $1000, and what
we're left with is $9000.
So, the rate of return now produces this
end of period wealth is -ten%.
Kay.
What is the likelihood, you know, that
we'll loose more than $9,000?
We already calculated that.
It was 6.7%, right?
So another way of saying that is, the 6.7
percent quantile of the return
distribution is -ten%.
So, if we think of the return
distribution.
Here.
This has mean 0.05 and if we have, -1.0,
then the area here is -0.067.
So this is the.067 quantile of the return
distribution.
So that means the probability that the
return is ten, is ten%, -ten percent or
less is 6.7%.
Alright, now in a valiant risk context
what we do is we specify our probability.
We want to know with five percent
probability, you know, how much could we
lose?
Alright, so.
With five percent probability, what is the
five percent quantile of the return
distribution, right?
So the five percent quantile return
distribution, we use the quantile
function, we take the inverse CDF of the
normal at the five percent probability
level with the mean and standard deviation
of a return distribution, and we see that
the five percent quantile turns out to be
11.4%.
So if we go to our distribution up over
here.
If we want area, this blue area.
To be five percent, then this quantile
here is minus zero point one, one four.
So what this means is, with five percent
probability or five percent of the returns
are less than 11.4%.
'Kay.
Now, if the return is equal to this
quantile, so if the return is -11.4%, how
much money do we lose?
Right.
We lose our initial investment, times the
rate of return.
We lose $1,144, 'kay.
So this is known as the five percent value
at risk.
We calculate the five percent value at
risk by looking at the five percent
quantile of our return distribution, which
is -11.4%.
So we know that the return can be -11.4
percent or smaller, with five percent
probability.
And then we say our dollar loss is our
initial investment times that negative
return.
Okay.
Now, notice when we calculate [inaudible]
risk, this is a, a conservative figure.
Because we could lose $1,144 or more with
five percent probability.
Because with five percent, percent
probability, we're here, but we can also
have returns less than that, right?
So this is kind of a conservative number.
When we say five percent probability, it's
not exactly five percent probability.
We're giving ourselves this, this, upper
bound [inaudible].
Or, sorry, sorry.
Lower bound on the loss.
>> Okay.
>> So in general when you calculate value
at risk on an investment the, alpha times
100 percent value at risk for an initial
investment of W dollars and we use this
notation capital V, little a, capital R
and the value at risk is often just called
the VaR.
It's equal to your initial dollar
investment times the alpha quartile of
your simple return distribution.
So this is why quantiles are very
important in finance.
Because when we're calculating risk
measures, for example value at risk, the
value at risk depends on the quantile of
your distribution.
The example that we did here, we computed
this quantile by assuming that the rate of
return followed the normal distribution,
right?
Remember, the normal distribution has a
certain tail shape.
If we use a student's t distribution, then
we would get a different quantile.
We would get a quantile that's generally
larger in, in absolute value.
Okay.
So we calculate Bar.
It's our initial investment times
quantaltive distribution.
And whatever distribution we assume then
determines what this, this quantal number
is.
Very often, and, and this is true in
practice in the real world.
People just assume that rates of return
are normally distributed, and that gives
them an easy way to compute that quantile.
And what we'll find when we look at data.
The normal distribution often isn't a very
good assumption for the data.
And so if we use a normal quantile, we
will often underestimate the loss.
Right.
Because if the, if the true distribution
has fatter tails than the normal, then the
five percent quantile of the true
distribution is gonna be a smaller
negative number, than the quantile for a
normal.
And then just to illustrate that point.
So, if this is the normal distribution.
Kay, here's the five percent quantile of
the normal.
Say this is equal to, you know, minus
11.4%.
If you have a distribution with fatter
tails than normal, then you're gonna get a
distribution that's gonna look like this.
The five percent quantile of this
distribution.
You know it might be something like minus
20.6%.
So when you have a fatter tail
distribution from the normal the left tail
quantiles are bigger negative numbers.
So Valiant risk is based upon taking a
quantile from a distribution.
And the previous example, we, assumed that
the simple rate of return was normally
distributed.
And then that gave us the computation for,
the quantile and the valiant risk number.
Now, at the beginning of lecture, I gave
you an example about why the normal
distribution may not be good for the
simple return.
Okay?
So.
And, and I gave an example to show that
the normal distribution may be more
appropriate for the continuously
compounded return.
So let's go back and do the valued risk
computation, where we assume that the
continuously compounded return follows a
normal distribution, and then deduce what
is the value at risk under this
assumption.
Because when we, when the continuously
compounded return is Normally distributed,
one plus the simple return has a log
normal distribution, and so that impacts
the quantile that we use in the
computation.
So let's, see what happens.
So we have so we start with the assumption
that the continuously confounded return is
normal with some mean and standard
deviation.
Then the process that we go through for
computing the value at risk is the
following: we compute the quantile based
on the continuously compounded
distribution.
Okay?
So the we know that, well, I, I stated as
a result previously, that if a variable
has a normal distribution, then the alpha
percent quantile of that distribution is
equal to the mean plus the standard
deviation, times the quantile.
You actually have a homework ex, example,
where you, you, you show this result.
But you can deduce this from just a,
standardization process of, of a random
variable.
So anyway, so the, quantile, we compute
for the continuously compounded
distribution.
Computed this way.
Now.
This is a quantile for the continuously
compounded distribution.
When we compute value at risk, we actually
need to use the simple return.
So we need to convert the continuously
compounded quantile into the quantile for
the simple return.
And it turns out that the quantile for the
simple return is E to the quantile for the
continuously compounded return -one.
Okay and this is because you know this the
function that we use to, to convert the
continuously compounded return to the
simple return.
Is monetonically increasing and so the
quantile is preserved under this, this
transformation.
So the con-, the simple quantile is E to
the continuously compounded quantile -one.
So now that we have this simple return,
quantile.
Then we compute our value at risk, in the
usual way.
We take our initial investment, and you
multiply it by the simple quantile that
we've computed.
Okay, so let's rawk, walk through an
example to, to see this in, in action.
So take our previous example here, the
continuously compounded return has the
mean of five percent and a standard
deviation of ten%.
Our initial investment is $10,000.'Kay.
So we're gonna compute five percent value
at risk.
That is, we want to know with five percent
probability, how much could we lose or
more.
So we compute the continuously compounded
return quantile.
We take the mean plus the standard
deviation times the five percent quantile
of a standard normal.
So that's equal to.05 plus.1 times the
five percent quantile of a standard normal
is minus 1.645.
Now we compute the simple return quantile.
We take e to the continuously compounded
return quantile minus one, so we compute e
to the minus.114 minus one.
That's minus.108.
So the simple quantile is actually less
than the continuously compounded quantile,
right?'Coz we always know, remember, 'coz
the continuously compounded return is
always smaller than the simple return.
So then we calculate our five percent bar.
We calc, , take our initial investment,
and you multiply it by the simple return
quantile.
And that's, -1077.
So if we start with the assumption that
contingency compatible returns are
normally distributed, and then we do our
bar calculation.
You know, we get a different quantile.
And our bar number tends to be a little
bit smaller.
That is, it, it looks like we don't lose
as much in that context.
And one of the reasons why, is, is
because, if the continuously compared to
return is normal we know that the simple
return is log normal, and the simple
return has a right skewness to it, right?
And so the left tail quantile is not as
big for the log normal return as it is for
the continuously compounded return.
So, that's another sort of illustration of
the point about how, asymmetry impacts the
quantiles of the distribution.