So now that we have seen how to compute the expected shortfall using

the variance-covariance approach, let me focus on the historical approach.

So this approach will be very similar to what we saw for the value at risk.

So, what I will do is simply look at the history of return, and so

I will collect data, for example, on the S&P 500.

So return on the S&P 500, and I will take a minus.

Then what I will do is only focus on the loss return which

are above the value at risk, and I will compute the empirical average.

This is what you see in the formula.

You see that I have a sum.

Okay? I have a sum on the return of the S&P

500 multiplied by the indicator function.

So which takes value 1 if you are above the value at risk, and

zero if you are below.

I have a sum divided by T, so this correspond to an empirical average,

but only for the return which are above the value at risk.

Now, of course, I would also have to divide by 1 minus alpha,

which correspond to the 1% if I use alpha is equal to 99%.

So what are the learning outcomes of this session?

So, again, we have three learning outcomes.

The first one is that the expected shortfall is indeed a simple synthetic and

quantitative risk measure.

It correspond to the average loss return when my loss return is above the value

at risk, and the value at risk correspond to the quantile of a loss distribution.

And the third outcome that we saw is that it's very, very simple to compute.

Under the variance-covariance approach, you compute an empirical mean.

You compute an empirical standard deviation to the volatility, and

you replace in the formula.

And for the historical approach, the only study that you have to do is to compute

an empirical average of the loss return which are above the value at risk.

So this is very simple to implement.

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