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So now, the whole point of this
bootstrapping section was to show how we

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can use a very simple method for
computing, say, the standard error on a

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more complicated statistic like value at
risk.

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So if we estimate our five percent value
of risk, our estimate is, is e to the five

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percent quantile minus one times my
initial wealth, okay?

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Now, we are going to use the bootstrap to
compute a standard error for this value of

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risk estimate, and then we will see how
accurate is our estimate value of risk,

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alright?
So, we have to write a function that will

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evaluate value at risk on our observed
data.

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So, this function is slightly more
complicated than the other ones.

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It has two lines, three lines of code
instead of two lines of code.

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This function takes in data or randomly
commuted indices and there are two

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additional arguments, the probability
level for your value at risk and your

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initial wealth, right?
So, how do you compute value at risk?

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This is my return data.
So, the first thing I do is I compute the

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quantile for the continuously compounded
return, which is the mean evaluated at my

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randomly permuted data plus the standard
deviation on the randomly permuted data

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times the quantile of a standard normal
based on the probability.

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Then, my value at risk is e to the
quantile minus one times my initial

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wealth.
And then, I return the result, okay?

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So, I run the boot function and here's the
output.

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So, my estimate for variant risk is
-13,769, right?

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The bootstrap estimate of bias is 210.
So, this is saying that, I over-predict my

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value at risk by $210 on average, okay?
So, there's you know, again, the, is this

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a large bias?
Well, again, this is an estimate of loss

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of $13,000 and I have a bias of $200.
So, the bias from a, from a economic

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perspective is not very big.
What about the standard error of my

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estimate?
The standard error of the estimate is

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$1,800.
So, you think of the help, so the standard

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error is about, you know, a little less
than one, a little more than one-tenth of

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the size of the value at risk number.
So, there's a little bit of uncertainty.

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And then, you would like to compute a 95
percent confidence interval, then we can

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either do estimate plus or minus two times
the standard error or, or look at the

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quantiles.
So, here's the bootstrap distribution of

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value at risk and we can see that it's a
little asymmetric, right?

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So, it's dipping down a little bit in the
q, q plot but, so this isn't quite normal.

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So, the percentile confidence interval is
probably more accurate than the estimate

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plus or minus two times the standard
error.

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So, we're going to run the boot CI
function, we'll compute the two confidence

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intervals like we did before and so here's
the normal confidence intervals.

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So, this is estimate plus or minus two
times the bootstrap standard error.

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So, on the low end, we loose $17,000 and
on the upper of the confidence interval,

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we loose $10,000.
So, with 95 percent confidence, we think

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our true loss is somewhere between 17,000
and 10,000.

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Now, that's a difference of $7,000.
That's, that's actually pretty big.

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The percentile method is similar on the
low end, we lose $17,000 and on the high

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end, we lose $10,000.
You know, again, the range of values here

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is about $7,000 and again, that's a pretty
wide confidence interval.

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So how, how well do we estimate value at
risk at the five percent level?

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Okay, I mean the confidence interval is
reasonably wide.

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Now, one of the things that you will learn
as, as you go out in the real world, is

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nobody when they, when you see s value at
risk report and practice, you never see

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confidence intervals on these reports.
And you know, and that could be quiet

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dangerous particularly, because, you know,
we see there is a, a fair bit of

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uncertainty here.
So, your risk manager is saying, you know,

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my best guess is, you know, in the middle
of the confidence interval, right?

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But they are not saying, well, this guess
could be off by a certain amount.

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And, and that's important to know.
So anyway, its just a, it's just a fact of

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life that you often, don't often see these
things.

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And one of the things that you should take
away from this lecture is there's no

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reason why you shouldn't report a
confidence interval, right?

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It was three lines of code to compute the
standard error on value at risk.

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It was extremely simple to do.
And because it's so easy to do, it should

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be done routinely.
And, and, and you, and you do get much

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more information that way.
When you read about the bootstrap, most,

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you often see this quotation that the
bootstrap gets its name from this phrase,

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picking yourself up from your bootstraps.
And, and so literally, a bootstrap is a

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strap on a boot, right?
And so, the idea is, you know, maybe it's

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the idea is you're, you're kind of
starting from the ground up.

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I don't know, that's sort of my
interpretation.

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So, I mean, that's the origin of the
phrase bootstrapping.

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The origin of this word for the technique
comes from picking yourself up from your

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bootstraps.
And you know, how you're supposed to

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interpret that.
I sort of, kind of, interpret that as you

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know, kind of you know, you're doing a
calculation that, you know, starting from

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the original sample.
And then, you know, you know, coming up

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from there or something like that I guess.
It's a little bit of an unusual word.

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Sometimes you will hear the bootstrap
referred to as a re-sampling algorithm.

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And that is very descriptive because
that's exactly what you are doing.

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When you are bootstrapping, you are just
re-sampling from your observed data and,

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and doing a computation.
You want to re-sample from the data object

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that you are computing your statistics
from.

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So you know, I'm computing a mean of the
continuously compounded returns.

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Well, returns are computed from prices.
So, and the prices are and, and the

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returns are the first difference in the
prices.

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So, if you re-sample just from the prices,
then when you're computing returns it's a

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little bit weird because returns depend
upon two adjacent observations.

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If you bootstrap from prices you know,
you've destroyed this relationship between

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prices that you've seen in the day, you
know, the prices are sort of going up.

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And if you ran, if you just bootstrap from
prices then, then you kind of destroy the,

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the price distribution.
I, I guess the answer is the return data

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is closer, is generally close to being
covariant stationary.

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It's not trending, it's uncorrelated over
time and so on.

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But the price data is nonstationary, and
it's not uncorrelated over time.

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The price today is correlated with the
price yesterday and, and so on and so

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forth.
So, so, for the purpose of what we're

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doing here, it's not the same thing to
boot strap from prices and then compute

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returns and then compute your statistics.
You would not get the same distribution of

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returns as you would get from this
bootstrapping return.