So now, the whole point of this
bootstrapping section was to show how we
can use a very simple method for
computing, say, the standard error on a
more complicated statistic like value at
risk.
So if we estimate our five percent value
of risk, our estimate is, is e to the five
percent quantile minus one times my
initial wealth, okay?
Now, we are going to use the bootstrap to
compute a standard error for this value of
risk estimate, and then we will see how
accurate is our estimate value of risk,
alright?
So, we have to write a function that will
evaluate value at risk on our observed
data.
So, this function is slightly more
complicated than the other ones.
It has two lines, three lines of code
instead of two lines of code.
This function takes in data or randomly
commuted indices and there are two
additional arguments, the probability
level for your value at risk and your
initial wealth, right?
So, how do you compute value at risk?
This is my return data.
So, the first thing I do is I compute the
quantile for the continuously compounded
return, which is the mean evaluated at my
randomly permuted data plus the standard
deviation on the randomly permuted data
times the quantile of a standard normal
based on the probability.
Then, my value at risk is e to the
quantile minus one times my initial
wealth.
And then, I return the result, okay?
So, I run the boot function and here's the
output.
So, my estimate for variant risk is
-13,769, right?
The bootstrap estimate of bias is 210.
So, this is saying that, I over-predict my
value at risk by $210 on average, okay?
So, there's you know, again, the, is this
a large bias?
Well, again, this is an estimate of loss
of $13,000 and I have a bias of $200.
So, the bias from a, from a economic
perspective is not very big.
What about the standard error of my
estimate?
The standard error of the estimate is
$1,800.
So, you think of the help, so the standard
error is about, you know, a little less
than one, a little more than one-tenth of
the size of the value at risk number.
So, there's a little bit of uncertainty.
And then, you would like to compute a 95
percent confidence interval, then we can
either do estimate plus or minus two times
the standard error or, or look at the
quantiles.
So, here's the bootstrap distribution of
value at risk and we can see that it's a
little asymmetric, right?
So, it's dipping down a little bit in the
q, q plot but, so this isn't quite normal.
So, the percentile confidence interval is
probably more accurate than the estimate
plus or minus two times the standard
error.
So, we're going to run the boot CI
function, we'll compute the two confidence
intervals like we did before and so here's
the normal confidence intervals.
So, this is estimate plus or minus two
times the bootstrap standard error.
So, on the low end, we loose $17,000 and
on the upper of the confidence interval,
we loose $10,000.
So, with 95 percent confidence, we think
our true loss is somewhere between 17,000
and 10,000.
Now, that's a difference of $7,000.
That's, that's actually pretty big.
The percentile method is similar on the
low end, we lose $17,000 and on the high
end, we lose $10,000.
You know, again, the range of values here
is about $7,000 and again, that's a pretty
wide confidence interval.
So how, how well do we estimate value at
risk at the five percent level?
Okay, I mean the confidence interval is
reasonably wide.
Now, one of the things that you will learn
as, as you go out in the real world, is
nobody when they, when you see s value at
risk report and practice, you never see
confidence intervals on these reports.
And you know, and that could be quiet
dangerous particularly, because, you know,
we see there is a, a fair bit of
uncertainty here.
So, your risk manager is saying, you know,
my best guess is, you know, in the middle
of the confidence interval, right?
But they are not saying, well, this guess
could be off by a certain amount.
And, and that's important to know.
So anyway, its just a, it's just a fact of
life that you often, don't often see these
things.
And one of the things that you should take
away from this lecture is there's no
reason why you shouldn't report a
confidence interval, right?
It was three lines of code to compute the
standard error on value at risk.
It was extremely simple to do.
And because it's so easy to do, it should
be done routinely.
And, and, and you, and you do get much
more information that way.
When you read about the bootstrap, most,
you often see this quotation that the
bootstrap gets its name from this phrase,
picking yourself up from your bootstraps.
And, and so literally, a bootstrap is a
strap on a boot, right?
And so, the idea is, you know, maybe it's
the idea is you're, you're kind of
starting from the ground up.
I don't know, that's sort of my
interpretation.
So, I mean, that's the origin of the
phrase bootstrapping.
The origin of this word for the technique
comes from picking yourself up from your
bootstraps.
And you know, how you're supposed to
interpret that.
I sort of, kind of, interpret that as you
know, kind of you know, you're doing a
calculation that, you know, starting from
the original sample.
And then, you know, you know, coming up
from there or something like that I guess.
It's a little bit of an unusual word.
Sometimes you will hear the bootstrap
referred to as a re-sampling algorithm.
And that is very descriptive because
that's exactly what you are doing.
When you are bootstrapping, you are just
re-sampling from your observed data and,
and doing a computation.
You want to re-sample from the data object
that you are computing your statistics
from.
So you know, I'm computing a mean of the
continuously compounded returns.
Well, returns are computed from prices.
So, and the prices are and, and the
returns are the first difference in the
prices.
So, if you re-sample just from the prices,
then when you're computing returns it's a
little bit weird because returns depend
upon two adjacent observations.
If you bootstrap from prices you know,
you've destroyed this relationship between
prices that you've seen in the day, you
know, the prices are sort of going up.
And if you ran, if you just bootstrap from
prices then, then you kind of destroy the,
the price distribution.
I, I guess the answer is the return data
is closer, is generally close to being
covariant stationary.
It's not trending, it's uncorrelated over
time and so on.
But the price data is nonstationary, and
it's not uncorrelated over time.
The price today is correlated with the
price yesterday and, and so on and so
forth.
So, so, for the purpose of what we're
doing here, it's not the same thing to
boot strap from prices and then compute
returns and then compute your statistics.
You would not get the same distribution of
returns as you would get from this
bootstrapping return.